Hindawi Publishing CorporationScience and Technology of Nuclear InstallationsVolume 2013 Article ID 641863 26 pageshttpdxdoiorg1011552013641863
Research ArticleUnstructured Grids and the Multigroup NeutronDiffusion Equation
German Theler
TECNA Estudios y Proyectos de Ingenierıa SA Encarnacion Ezcurra 365 C1107CLA Buenos Aires Argentina
Correspondence should be addressed to GermanTheler gthelertecnacom
Received 22 May 2013 Revised 20 July 2013 Accepted 20 July 2013
Academic Editor Arkady Serikov
Copyright copy 2013 GermanTheler This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The neutron diffusion equation is often used to perform core-level neutronic calculations It consists of a set of second-orderpartial differential equations over the spatial coordinates that are both in the academia and in the industry usually solved bydiscretizing the neutron leakage term using a structured grid This work introduces the alternatives that unstructured grids canprovide to aid the engineers to solve the neutron diffusion problem and gives a brief overview of the variety of possibilities theyoffer It is by understanding the basic mathematics that lie beneath the equations that model real physical systems better technicaldecisions can be made It is in this spirit that this paper is written giving a first introduction to the basic concepts which can beincorporated into core-level neutron flux computations A simple two-dimensional homogeneous circular reactor is solved usinga coarse unstructured grid in order to illustrate some basic differences between the finite volumes and the finite elements methodAlso the classic 2D IAEA PWR benchmark problem is solved for eighty combinations of symmetries meshing algorithms basicgeometric entities discretization schemes and characteristic grid lengths giving even more insight into the peculiarities that arisewhen solving the neutron diffusion equation using unstructured grids
1 Introduction
The better we engineers are able to solve the equationsthat model the real physical plants we design and buildthe better services we can provide to our customers andthus general people can be benefited with better nuclearfacilities and installations The Boltzmann neutron transportequation describes how neutrons move and interact withmatter It involves continuous energy and space-dependentmacroscopic cross-sections that should be knownbeforehandand gives an integrodifferential equation for the vectorial fluxas a function of seven independent scalar variables namelythree spatial coordinates two angular directions energy andtime It represents a balance that holds at every point in spaceand at every instant in time Such an equationmay be tackledusing a variety of approaches one of them is a simplificationthat leads to the so-called neutron diffusion approxima-tion that states that the neutron current is proportional tothe gradient of the neutron flux by means of a diffusion
coefficient which is a function of the macroscopic transportcross-sectionWhen this approximationmdashwhich is analogousto Fickrsquos law in species diffusion and to the Fourier expressionof the heat fluxmdashis replaced into the transport equation apartial differential equation of second order on the spatialcoordinates is obtained Formally the neutron diffusionequation may be derived from the transport equation byexpanding the angular dependance of the vectorial neutronflux in a spherical harmonics series and retaining both thezero and one-moment terms neglecting the contributions ofhigher moments [1 2] As crude as it may seem this diffusionequation gives fairly accurate results when applied under theconditions in which thermal nuclear reactors usually operateIndeed it can be shown that in a homogeneous bare criticalreactor the neutron current is proportional to the neutrongradient for every neutron energy [3]
The energy domain is usually divided into a finite numberof groups thus transforming one partial differential equationover space and energy into several coupled equationsmdashone
2 Science and Technology of Nuclear Installations
for each groupmdashcontaining differential operators appliedonly over the spatial coordinatesThis resulting set of second-order PDEs is known as the multigroup neutron diffusionequation and is usually used to model design and analyzenuclear reactor cores by the so-called core-level calculationcodesThese programs take homogenizedmacroscopic cross-sections (which may depend on the spatial coordinatesthrough changes of fuel burnup materials temperature orother properties) computed by lattice-level codes as an inputand solve the diffusion equation to obtain the flux (and itsrelated quantities such as power xenon etc) distributionwithin the core
Given a certain spatial distribution of materials and itsproperties inside a reactor core chances are that the resultingreactor will not be critical That is to say in general therate of absorptions and leakages will not exactly overcomethe neutrons born by fissions sources and some kind offeedbackmdasheither through an external control system or bymeans of an inherent stability mechanism of the core [4]mdashis needed to keep the reactor power within a certain intervalMathematically this means that the transportmdashand thus thediffusionmdashequation does not have a steady-state solution Inpractice given a certain reactor configuration its steady-stateflux distribution is computed by setting all the time deriva-tives to zero as usual but also by dividing the fission sources bya positive value 119896eff called the effective multiplication factorwhich becomes also one unknown and turns the formulationinto a eigenvalue problemThe largest (or smallest dependingon the formulation) eigenvalue is therefore the effectivemultiplication factor If 119896eff lt 1 the original configurationwas subcritical and conversely As successive configurationsmake 119896eff rarr 1 the associated eigenfunctions approach thesteady-state critical flux distribution [5]
Core-level codes traditionally use regular grids to dis-cretize the differential operators over the space Dependingon the characteristics and symmetry of the reactor coreeither squares or hexagons are used as the basic shapeof the mesh Usual discretization schemes involve cell-centered finite differences or two-step coarse-meshcouplingcoefficient methods [6ndash8] which are accurate and efficientenough formost applicationsThere are nevertheless certainlimitations that are not inherently related to structured gridsbut to the way their coarseness is utilized and how they arecomputed and applied to the reactor geometry These issuescan be overcome by discretizing the spatial operators usinga scheme which could be applied to nonstructured gridsnamely finite volumes or finite elements
This way unstructured grids may be used to studyanalyze and understand the numerical errors introduced bythe discretization of the leakage operator with a difference-based scheme over a coarse structured grid by successivelyrefining the mesh whilst comparing the solutions with thestructured one as depicted in Figure 1 Another example ofapplication may be the improvement of the response matrixof boundary conditions over cylindrical surfaces which isthe case for most of the nuclear power plants cores Whena coarse structured mesh is applied to a curved geometrythere appears a geometric condition known as the staircaseeffect Even though arbitrary shapes cannot be perfectly
tessellated with unstructured grids for the same number ofunknowns they better represent the geometry than struc-tured meshes as Figure 2 illustrates Again by refining themesh and comparing solutions the matrix responses usedto set the boundary conditions on the coarse meshes canbe optimized Finally other complex geometries such asslanted control rods (Figure 3) or irradiation chambers can bedirectly taken into account by unstructured grids possibly byrefining the mesh in the locations around said complexities
2 The Steady-State Multigroup NeutronDiffusion Equation
In the present work we take the steady-state multigroupneutron diffusion equation for granted That is to say wefocus on the mathematical aspects of the eigenvalue problemand make no further reference to its derivation from thetransport equation nor to the validity of its applicationto reactor problems as these subjects that are extensivelydiscussed in the classic literature [1 5 11]
The differential formulation of the steady-state multi-group neutron diffusion equation over an 119898-dimensionaldomain 119880 with 119866 groups of energy is the set of 119866 coupleddifferential equations
div [119863119892 (x) sdot grad 120601119892 (x)] minus Σ119886119892 (x) sdot 120601119892 (x)
+
119866
sum
1198921015840 = 119892
Σ1199041198921015840rarr119892 (x) sdot 1206011198921015840 (x)
+ 120594119892 sdot
119866
sum
1198921015840=1
]Σ1198911198921015840 (x)119896eff
sdot 1206011198921015840 (x) = 0
(1)
where 120601119892 are the 119892 = 1 119866 unknown flux distributionsand the Σs and the 119863s are the macroscopic cross-sectionsand diffusion coefficients which ought to be computed by alattice-level code and taken as an input to core-level codesHowever for the purposes of fulfilling the objectives of thiswork we take the macroscopic cross-sections as functions ofthe spatial coordinate x isin R119898 which is known beforehandThe coefficient 120594119892 represents the fission spectrum and is thefraction of the fission neutrons that are born into group 119892 Asstated above the ]-fissions are divided by a positive effectivemultiplication factor 119896eff and all the time derivatives that isthe right hand of (1) is set to zero We note that in order todefine a consistent nomenclature we use the absorption crosssection Σ119886119892 of the group 119892 which is equal to the total crosssection Σ119905119892 minus the self-scattering cross section Σ119904119892rarr119892Therefore the sum over the scattering sources excludes theterm corresponding to 119892
1015840= 119892 If we had used the total
cross section in the second term of (1) we would have hadto include the self-scattering term as a source
If the cross sections depend only in an explicit wayon the spatial coordinate x then (1) is linear If as is thegeneral case the cross sections depend on x through the flux120601119892(x) itselfmdashsuch as by means of the xenon concentrationor by local temperature distributionsmdashthen (1) is nonlinearNevertheless this general case can be solved by successive
Science and Technology of Nuclear Installations 3
(a) Coarse structured grid commonly used in diffu-sion codes such as in [9] Each lattice cell is divided intoa 2 times 2 grid for solving the neutron diffusion equation
(b) Discretization of the lattice cell using a fineunstructured grid as proposed in this work
(c) Further refinement of the unstructured grid
Figure 1 Discretization of a homogenized PHWR array of fuel channels for a core-level diffusion computation Each square is a lattice-levelcell comprising one fuel channel and the surrounding moderator
linear iterations so the basic problem can be regarded as beingpurely linear
It should be noted that if at least one of the diffu-sion coefficients 119863119892(x) is discontinuous over space thenthe divergence operator is not defined at the discontinuitypoints Therefore the differential formulationmdashalso knownas the strong formulationmdashis not complete when there existmaterial discontinuities that involve the diffusion coefficientsAt thesematerial interfaces the differential equation has to bereplaced by a neutron current conservation condition
119863+
119892(x) sdot grad 120601
+
119892(x) = 119863
minus
119892(x) sdot grad 120601
minus
119892(x) (2)
where the plus and minus sign denote both sides of the inter-face As the diffusion coefficients are different the resultingflux distribution120601119892(x) ought to have a discontinuous gradientat the interface in order to conserve the current
When transforming the strong formulation into a weakformulationmdashnot just into an integral formulationmdashboth (1)and (2) can be taken into account by a single expression Ineffect let 120593119892(x) be arbitrary functions of x for 119892 = 1 119866Multiplying each of the 119866 equations (1) by 120601119892(x) integrating
over the domain 119880 isin R119898 and applying Greenrsquos formula [12]to the leakage term we obtain
int119880
119863119892 (x) sdot [grad 120593119892 (x) sdot grad 120601119892 (x)] 119889119898x
+ int120597119880
120593119892 (x) sdot 119863119892 (x) sdot [grad 120601119892 (x) sdot n] 119889119878
+ int119880
120593119892 (x) sdot Σ119886119892 (x) sdot 120601119892 (x) 119889119898x
+ int119880
120593119892 (x) sdot119866
sum
1198921015840 = 119892
Σ1199041198921015840rarr119892 (x) sdot 1206011198921015840 (x) 119889119898x
+ 120594119892 sdot int119880
120593119892 (x) sdot119866
sum
1198921015840=1
]Σ1198911198921015840 (x)119896eff
sdot 1206011198921015840 (x) 119889119898x
(3)
These 119866 coupled equations should hold for any arbitraryset of functions120593119892(x)Making an analogy between (3) and theprinciple of virtual work for structural problems we call thesefunctions virtual fluxes This formulation does not involveany differential operator over the diffusion coefficient and yet
4 Science and Technology of Nuclear Installations
(a) Continuous two-dimensional domain (b) Structured grid
(c) Unstructured grid
Figure 2 When a continuous domain (a) is meshed with an unstructured grid there appears a geometric condition known as the staircaseeffect (b) For the same number of nodes unstructured grids reproduce the original geometry better (c)
Figure 3 Cross section of a hypothetical reactor in which thecontrol rods enter into the core from above with a certain attackangle with respect to the vertical direction
at the same time can be shown to be equivalent to the strongformulation given by (1)
In any case both formulations involve the computationof 119866 unknown functions of x and one unknown real value119896eff Except for homogeneous cross sections in canonicaldomains 119880 the multigroup diffusion equation needs to besolved numerically As discussed below any nodal-baseddiscretization scheme replaces a continuous unknown func-tion 120601119892(x) by 119873 discrete values 120601119892(119894) for 119894 = 1 119873
Figure 4 The finite volumes method computes the unknown fluxin the cell centers (squares) whilst the finite elements methodcomputes the fluxes at the nodes (circles)
If we arrange these unknowns into a vector 120601 isin R119873119866
such as
120601 =
[[[[[[[[[[[[[[[[
[
1206011 (1)
1206012 (1)
120601119866 (1)
1206011 (2)
120601119892 (119894)
120601119866 (119873)
]]]]]]]]]]]]]]]]
]
(4)
Science and Technology of Nuclear Installations 5
(a) Fast flux distribution
minus100minus50 0 50 100minus100
minus500
50100
0
06
xy
(b) Fast flux unknowns
(c) Matrix 119877 structure (d) Matrix 119865 structure
Figure 5 A bare homogeneous circle solved with finite volumes (184 unknowns)
then the continuous eigenvalue problemmdashin either for-mulationmdashcan be transformed into a generalized matrixeigenvalueeigenvector problem casted in either of the follow-ing forms
119877 sdot 120601 =1
119896effsdot 119865 sdot 120601
119865 sdot 120601 = 119896eff sdot 119877 sdot 120601
119877minus1sdot 119865 sdot 120601 = 119896eff sdot 120601
(5)
where 119877 and 119865 are square 119873119866 times 119873119866 matrices We call119865 the fission matrix as it contains all the ]-fission termswhich are the ones that we artificially divided by 119896eff Therest of the terms are grouped into the removal or transportmatrix 119877 which includes the rest of the neutron-matterinteraction mechanisms It can be shown [11] that for anyreal set of cross sections 119877minus1 exists and that the119873119866 pairs ofeigenvalueeigenvector solutions of (5) satisfy that
(1) there is a unique real positive eigenvalue greater inmagnitude than any other eigenvalue
(2) all the elements of the eigenvector corresponding tothat eigenvalue are real and positive
(3) all other eigenvectors either have some elements thatare zero or have elements that differ in sign from eachother
21 Boundary Conditions Being a differential equation overspace the neutron diffusion equation needs proper boundaryconditions to conform a properly definedmathematical prob-lem These can be imposed flux (Dirichlet) imposed current(Neumann) or a linear combination (Robin) However dueto the fact that in the linear problem in absence of externalsourcesmdashsuch as (1) or (3)mdashthe problem is homogeneousif 120601119892(x) is a solution then any multiple 120572120601119892(x) is also asolution That is to say the eigenvectors are defined up to amultiplicative constant whose value is usually chosen as toobtain a certain total thermal power Thus the prescribedboundary conditions should also be homogeneous and bedefined up to amultiplicative constantTherefore the allowedDirichlet conditions are zero flux at the boundary theNeumann conditions should prescribe that the derivative in
6 Science and Technology of Nuclear Installations
(a) Fast flux distribution
minus100minus50 0 50 100minus100
minus500
50100
0
06
xy
(b) Fast flux unknowns
(c) Matrix 119877 structure (d) Matrix 119865 structure
Figure 6 A bare homogeneous circle solved with finite elements (218 unknowns)
the normal direction should be zero (ie symmetry condi-tion) and the Robin conditions are restricted to the followingform
grad 120601119892 (x) sdot n + 119886119892 (x) sdot 120601119892 (x) = 0 (6)
n being the outward unit normal to the boundary 120597119880 of thedomain 119880
22 Grids and Schemes One way of solving the neutrondiffusion equationmdashand in general any partial differentialequation over spacemdashis by discretizing the differential oper-ators with some kind of scheme that is applied over a certainspatial grid Given an 119898-dimensional domain a grid definesa partition composed of a finite number of simple geometricentities In structured grids these elementary entities arearranged following a well-defined structure in such a waythat each entity can be identified without needing furtherinformation than the one provided by the intrinsic structureOn the other hand the geometric entities that composean unstructured grid are arranged in an irregular patternand the identification of the elementary entities needs to
be separately specified for example by means of a sortedlist For instance Figure 2(b) shows a structured grid thatapproximates the continuous domain of Figure 2(a) Eachsquare may be uniquely identified by means of two integerindexes that indicate its relative position in each of thehorizontal and vertical directions In the case of Figure 2(c)that shows an unstructured grid there is no way to system-atically make a reference to a particular quadrangle withoutany further information
Almost all of the grid-based schemesmdashwhich are knownas nodal schemes which are to be differentiated from modalschemes based on series expansionsmdashare based in either thefinite differences volumes or elements method Finite differ-ences schemes provide themost simple and basic approach toreplace differential (ie continuous) operators by difference(ie discrete) approximations However they are not suitablefor unstructured meshes and may introduce convergenceproblems with parameters that are discontinuous in spacewhich is the case for any reactor core composed of atleast two different materials Moreover boundary conditionsare hard to incorporate and usually give rise to incorrectresults
Science and Technology of Nuclear Installations 7
20 40 600995
09975
1
Analytical solutionFinite volumesFinite elements
keff
a985747c
Figure 7 The effective multiplication factor of a two-group barecircular reactor of radius 119886 as a function of the quotient 119886ℓ
119888between
the radius and the characteristic length of the cellelement
20 40 60
Finite volumesFinite elements
Erro
r ink
eff
10minus3
10minus4
10minus5
a985747c
Figure 8 Absolute value of the error committed in the computationof 119896eff versus 119886ℓ119888
Methods of the finite volumes family involve the inte-gration of the differential equation over each elementaryentity applying the divergence theorem to transform volumeintegrals into surface integrals and providing a mean to esti-mate the fluxes through the entityrsquos surface using informationcontained in its neighbors In this context each elementaryentity is called a cell and finite volumes methods give themean value of each of the group fluxes 120601119892(119894) at the 119894th cellwhich may be associated to the value of 120601119892(x119894) where x119894 isthe location of the cell barycenter Being an integral-basedmethod spatial-discontinuous parameters are handled moreefficiently than in finite differences and boundary conditionscan be easily incorporated as forced cell fluxes Nonethelesseven though these methods may be applied to unstructuredgrids the estimation of the fluxes on the cellsrsquo surfaces isusually performed by using some geometric approximationsthat may lose validity as the quality of the grid is worsened
10
70
90
130
150
170
10 30 50 70 90 110 130 150 170
4
1 38
36 37353
3 32 33 34312
26 27 28 29 3025
19 20 21 22 23 2418
10 11 12 13 14 15 16 17
2 331 4 5 6 7 8 9
x (cm)
y (cm
)
Figure 9 The 2D IAEA PWR benchmark geometry
Moreover the simple integral approach cannot take intoaccount discontinuous diffusion coefficients so when twoneighbors pertain to different materials the flux has to becomputed in a certain particular way to conserve the neutroncurrent according to (2)
Finally finite elements methods rely on a weak formula-tion of the differential problem similar to (3) that maintainsall the mathematical characteristics of the original strongformulation plus its boundary conditions The method isbased on shape functions that are local to each elementaryentitymdashnow called element defined by nodes as cornersmdashand on finding a set of nodal values such that a certaincondition is met which is usually that the residual of thesolution has to be orthogonal to each of the shape functionsThis condition is known as the Galerkin method and impliesthat the error committed in the approximate solution ofthe continuous problem is confined into a small subset ofthe original vector space of the continuous functions Thesemathematical properties make finite elements schemes veryattractive However these features depend on a large numberof integrations that ought to be performed numerically soa computational effortdesired accuracy tradeoff has to beconsidered Not only does the finite elements method givethe values of the flux 120601119892(x119894) at the 119894th node but also theshape functions provide explicit expressions to interpolateand to evaluate the unknown functions at any location x ofthe domain119880 Boundary conditions are divided into essentialand naturalThefirst group comprises theDirichlet boundaryconditions which are satisfied exactlymdashwithin the precisionof the eigenvalue problem solvermdashby the obtained solutionThe latter include the Neumann and the Robin conditionsthat are satisfied only approximately by the derivatives of theinterpolated solution through the shape functions with finermeshes giving better agreement with the prescribed values
The same unstructured grid may be used either for thefinite volumes or for the finite elements method In the first
8 Science and Technology of Nuclear Installations
Largest eigenvalue 1029690 (288336 pcm)1135 (3066 3079)1306 (5171 13076)
Number of unknowns 15740Outer iterations 3Linear iterations 32Inner iterations 1967Residual normRelative errorError estimateMemory used 49824 kBSoft page faults 13401Hard page faults 0Total CPU time 074 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
2483 times 10minus8
1223 times 10minus8
403 times 10minus9
keff
quarter-symmetry core meshed using Delaunay (triangles 985747c = 3) solved with finite volumesMilongarsquos 2D LWR IAEA benchmark problem case no 002
(a)
123456789
1011121314151617181920212223242526272829303132333435363738
k Pk 1206011k 1206012k075 3282 555133 4242 984147 4645 1090123 3921 911061 2677 453094 2995 694093 2927 689074 2011 549mdash 322 756146 4597 1078150 4730 1110133 4199 985107 3405 789104 3271 767095 2971 700072 1956 534mdash 306 733149 4695 1102136 4289 1007118 3733 876107 3367 792097 2892 718066 1628 491mdash 234 549120 3797 891097 3096 718090 2842 669084 2221 619mdash 568 1215mdash 067 257047 2042 348068 2055 504058 1414 430mdash 222 562057 1374 425mdash 388 820mdash 053 203mdash 060 229
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 10 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
case the unknowns are themean value of the fluxes over eachcell whilst in the latter the unknowns are the fluxes evaluatedat each node Therefore the number of unknowns 119873119866 isdifferent for each method even when using the same gridFigure 4 shows this situation and in Section 31 we further
illustrate these differences A fully detailed mathematicaldescription of the actual algorithms for both volumes andelements-based proposed discretizations can be found in anacademic monograph written by the author of this paper[13]
Science and Technology of Nuclear Installations 9
Largest eigenvalue 1029927 (290571 pcm)1131 (3175 2999)1220 (13097 5098)
Number of unknowns 15576Outer iterations 3Linear iterations 32Inner iterations 1947Residual normRelative errorError estimateMemory used 56256 kBSoft page faults 15645Hard page faults 0Total CPU time 09201 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
1394 times 10minus8
6865 times 10minus9
3425 times 10minus9
keff
quarter-symmetry core meshed using Delaunay (quads 985747c = 2) solved with finite volumesMilongarsquos 2D LWR IAEA benchmark problem case no 013
(a)
123456789
1011121314151617181920212223242526272829303132333435363738
k Pk 1206011k 1206012k074 3244 549133 4230 982147 4651 1092124 3946 917060 2645 443094 2997 696093 2938 691074 2005 545mdash 301 746145 4588 1076150 4720 1108133 4198 984109 3465 805104 3280 769095 2986 704071 1962 528mdash 293 715148 4680 1098136 4288 1007119 3755 881107 3385 796097 2905 721067 1653 493mdash 220 556121 3808 893098 3115 724092 2889 680083 2247 615mdash 543 1241mdash 067 260045 1977 330069 2076 509058 1460 428mdash 226 574056 1364 413mdash 371 821mdash 052 200mdash 060 238
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 11 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
3 Results
We now proceed to show two illustrative results that are tobe taken as an overview of the possibilities that unstructuredgrids can provide in order to tackle the multigroup neutron
diffusion problem The examples are two-dimensional prob-lems as they contain some of the complexities a real three-dimensional reactor posse yet the reported results are notso complicated as to be easily understood and analyzed Inparticular we state some basic differences between the finite
10 Science and Technology of Nuclear Installations
Largest eigenvalue 1029695(288388 pcm)1112 (3076 3032)838 (5000 13000)
Number of unknowns 15922Outer iterations 3Linear iterations 32Inner iterations 1990Residual normRelative errorError estimateMemory used 109964 kBSoft page faults 30591Hard page faults 0Total CPU time 158 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
1056 times 10minus8
5202 times 10minus9
5122 times 10minus9
keff
quarter-symmetry core meshed using Delaunay (quads 985747c = 2) solved with finite volumesMilongarsquos 2D LWR IAEA benchmark problem case no 018
(a)
123456789
1011121314151617181920212223242526272829303132333435363738
k Pk 1206011k 1206012k074 3220 549130 4152 961144 4651 1068120 3843 890061 2648 452094 2996 693094 2959 696072 2061 569mdash 358 862142 4495 1054147 4634 1088131 4127 968107 3409 789104 3275 768096 3003 708071 2014 554mdash 342 820146 4606 1081134 4222 991118 3711 871107 3375 794098 2929 727062 1692 522mdash 258 632119 3752 880096 3084 715091 2858 673080 2280 633mdash 611 1277mdash 080 318047 2043 350069 2088 510054 1463 450mdash 255 648051 1418 439mdash 410 854mdash 064 252mdash 071 285
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 12 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
volumes and the finite elements methods by solving a two-group homogeneous bare reactor with the Robin boundaryconditions over the very same grid although a rather coarseone so the differences can be observed directly into theresulting figuresWe then solve the classical two-dimensionalLWR problem also known as the 2D IAEA PWR benchmark
Not only do we show again the differences between the finitevolumes and elements formulation but also we solve theproblemusing different combinations ofmeshing algorithmsbasic shapes and characteristic lengths of the mesh
To solve the two examples shown below we employed themilonga code which was written from scratch by the author
Science and Technology of Nuclear Installations 11
Largest eigenvalue 1029159 (283326 pcm)1174 (2851 3224)1560 (5200 13200)
Number of unknowns 3132Outer iterations 3Linear iterations 32Inner iterations 391Residual normRelative errorError estimateMemory used 26888 kBSoft page faults 7322Hard page faults 0Total CPU time 0308 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
5408 times 10minus8
2665 times 10minus8
8424 times 10minus9
keff
Milongarsquos 2D LWR IAEA benchmark problem case no 031quarter-symmetry core meshed using Delquad (quads 985747c = 4) solved with finite volumes
(a)
123456789
1011121314151617181920212223242526272829303132333435363738
k Pk 1206011k 1206012k077 3422 570140 4468 1040153 4836 1135128 4070 948060 2704 448095 3017 702091 2861 673069 1918 508mdash 264 689151 4785 1122155 4897 1149137 4334 1017111 3535 823104 3272 768092 2908 685067 1876 496mdash 254 658154 4854 1139139 4392 1031120 3792 890107 3360 790095 2831 702061 1574 451mdash 197 515123 3868 907099 3132 730090 2818 663078 2168 579mdash 481 1191mdash 055 217044 1984 328067 2027 498052 1360 388mdash 190 520052 1329 383mdash 321 792mdash 043 168mdash 048 195
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 13 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
of this paper and is currently still under development withinhis ongoing PhD thesis There exists a first public release[14] under the terms of the GNU General Public Licensemdashthat is it is a free softwaremdashthat can only handle structuredgrids There is a second release being prepared whose
main relevance is that it can work with nonstructured gridsas well which is the main feature of the code It uses ageneral mathematical frameworkmdashcoded from scratch aswellmdashwhich provides input file parsing algebraic expres-sions evaluation one- and multidimensional interpolation of
12 Science and Technology of Nuclear Installations
Largest eigenvalue 1029788 (289260 pcm)1120 (3149 3011)1091 (13038 5114)
Number of unknowns 15776Outer iterations 3Linear iterations 32Inner iterations 1972Residual normRelative errorError estimateMemory used 56996 kBSoft page faults 15436Hard page faults 0Total CPU time 09121 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
2296 times 10minus12
1131 times 10minus12
6877 times 10minus13
keff
Milongarsquos 2D LWR IAEA benchmark problem case no 043eighth-symmetry core meshed using Delaunaya (triangs985747c = 2) solved with finite volumes
(a)
1 074 3241 5482 131 4188 9713 145 4582 10764 122 3877 9005 060 2644 4456 093 2986 6927 093 2941 6928 075 2030 5529 mdash 322 779
10 143 4523 106111 148 4667 109512 131 4145 97213 107 3423 79414 103 3269 76715 095 2993 70616 073 1988 53917 mdash 312 75118 147 4637 108819 134 4243 99620 118 3721 87321 107 3370 79322 098 2914 72323 067 1651 49924 mdash 235 57125 119 3763 88226 096 3077 71527 091 2847 67128 083 2255 61729 mdash 576 122530 mdash 068 26631 046 2016 34132 068 2058 50533 059 1443 43534 mdash 233 58935 057 1381 42036 mdash 381 82337 mdash 054 20838 mdash 062 244
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 14 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
scattered data shared-memory access numerical integrationfacilities and so forth The milonga code was designed withfour design basis vectors in mindmdashwhich are thoroughlydiscussed in the documentation [15]mdashwhich includes thetype of problems it can handle the code scalability which
features are expected and what to do with the obtainedresults
It works by first reading an input file that using plain-text English keywords and arguments defines the number119898 of spatial dimensions the number 119866 of group energies
Science and Technology of Nuclear Installations 13
Largest eigenvalue 1029726 (288675 pcm)1104 (3069 3069)864 (13000 5000)
Number of unknowns 4040Outer iterations 2Linear iterations 24Inner iterations 505Residual normRelative errorError estimateMemory used 42940 kBSoft page faults 11437Hard page faults 0Total CPU time 1064 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
2019 times 10minus8
9946 times 10minus9
7138 times 10minus9
keff
Milongarsquos 2D LWR IAEA benchmark problem case no 047eighth-symmetry core meshed using Delaunay (triangles 985747c = 3) solved with finite volumes
(a)
1 074 3194 5452 129 4117 9533 143 4515 10604 119 3814 8835 061 2634 4506 093 2984 6907 094 2954 6958 071 2064 5719 mdash 362 865
10 141 4458 104511 146 4599 107912 130 4097 96113 106 3388 78414 103 3263 76515 095 2998 70716 068 2015 55717 mdash 346 82318 145 4572 107319 133 4194 98420 117 3691 86621 107 3363 79222 098 2925 72623 059 1696 52824 mdash 263 63525 118 3728 87426 096 3068 71127 091 2850 67228 079 2280 63529 mdash 619 127630 mdash 081 32131 047 2036 34932 068 2085 50933 051 1465 45434 mdash 258 65035 049 1419 44336 mdash 416 85437 mdash 065 25438 mdash 071 288
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 15 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
and optionally a mesh file Currently only grids generatedwith the code gmsh [16] are only supported mainly becauseit is also a free software and it suits perfectly well milongarsquosdesign basis in the sense that the continuous geometry can bedefined as a function of a number of parameters Afterwardthe physical entities defined in the grid are mapped into
materials with macroscopic cross sections which maydepend on the spatial coordinates x bymeans of intermediatefunctions such as burn-up or temperatures distributionswhich in turn may be defined by algebraic expressions byinterpolating data located in files by reading shared-memoryobjects or by a combination of them In the same sense
14 Science and Technology of Nuclear Installations
Largest eigenvalue 1029695 (288387 pcm)1108 (3045 3045)833 (13000 5000)
Number of unknowns 8234Outer iterations 3Linear iterations 32Inner iterations 1029Residual normRelative error
Max 1206012(x y) coreMax 1206012(x y) reflector
2028 times 10minus12
9993 times 10minus13
1012 times 10minus12
keff
Relative errorError estimateMemory used 75468 kBSoft page faults 20094Hard page faults 0Total CPU time 1256 seconds
9993 times 10
1012 times 10minus12
Milongarsquos 2D LWR IAEA benchmark problem case no 058eighth-symmetry core meshed using Delaunay (quads 985747c = 2) solved with finite volumes
(a)
1 074 3207 5472 129 4136 9573 144 4534 10644 120 3828 8875 061 2638 4506 093 2985 6907 094 2948 6948 072 2053 5669 mdash 357 859
10 142 4478 105011 146 4617 108412 130 4111 96413 106 3396 78614 103 3263 76515 095 2991 70616 071 2005 55217 mdash 340 81618 145 4588 107719 133 4206 98720 117 3698 86821 107 3362 79122 098 2918 72523 062 1686 52024 mdash 258 62925 118 3737 87626 096 3073 71227 091 2848 67128 080 2272 63129 mdash 609 127230 mdash 080 31731 047 2035 34832 069 2081 50933 054 1458 44834 mdash 255 64635 052 1413 43736 mdash 409 85137 mdash 064 25138 mdash 071 284
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 16 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
boundary conditions are applied to appropriate physicalentities of dimension119898minus1The problemmatrices119877 and119865 arethen built and stored in an appropriate sparse format usingthe free PETSc [17 18] library and the eigenvalue problemis solved using the free SLEPc [19] library The results arestored into milongarsquos variables and functions which may be
evaluated integratedmdashusually using the free GNU ScientificLibrary [20] routinesmdashand of course written into appropriateoutputs Milonga can also solve problems parametrically andbe used to solve optimization problems As stated above thecode is a free software so corrections and contributions aremore than welcome
Science and Technology of Nuclear Installations 15
Largest eigenvalue 1029462 (286185 pcm)1136 (3093 2953)1233 (13100 5100)
Number of unknowns 6228Outer iterations 2Linear iterations 24Inner iterations 778Residual normR l ti
Max 1206012(x y) coreMax 1206012(x y) reflector
26 times 10minus8
1281 times 10minus8
4996 times 10minus9
keff
Relative errorError estimateMemory used 41460 kBSoft page faults 10930Hard page faults 0Total CPU time 0604 seconds
1281 times 10 8
4996 times 10minus9
Milongarsquos 2D LWR IAEA benchmark problem case no 073eighth-symmetry core meshed using Delaunay (quads 985747c = 2) solved with finite volumes
(a)
1 075 3297 5552 134 4268 9923 148 4660 10944 123 3926 9145 059 2645 4406 094 2997 6977 093 2914 6858 072 2002 5349 mdash 301 773
10 146 4606 108011 150 4735 111112 133 4212 98813 108 3448 80214 104 3271 76715 094 2954 69616 070 1954 52117 mdash 288 73518 149 4691 110119 136 4289 100720 119 3748 88021 107 3359 79022 096 2880 71423 064 1636 47524 mdash 217 56325 121 3810 89426 098 3110 72427 090 2829 66628 081 2227 59929 mdash 536 125730 mdash 062 25031 045 2002 33632 068 2040 50033 055 1414 40934 mdash 215 57935 055 1359 40836 mdash 359 83437 mdash 050 19938 mdash 061 236
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 17 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
31 A Coarse Bare Homogeneous Circle Figures 5 and 6 showthe results of solving a bare two-dimensional circular homo-geneous reactor of radius 119886 over an unstructured grid whichis deliberately coarse using the finite volumes method andthe finite elements method respectively Two group energies
were used and a null-flux boundary conditionwas fixed at theexternal surface Figures 5(a) and 6(a) compare the obtainedfast flux distributions In the first case the numerical solutionprovidesmean values for each neutron flux group in each cellwhile in the latter the solution is computed at the nodes and
16 Science and Technology of Nuclear Installations
Largest eigenvalue 1029851 (289858 pcm)1090 (3182 3182)851 (13000 5000)
Number of unknowns 1752Outer iterations 3Linear iterations 32Inner iterations 219Residual norm
Max 1206012(x y) coreMax 1206012(x y) reflector
566 times 10minus12
2789 times 10minus12
1595 times 10minus12
keff
Residual normRelative errorError estimateMemory used 42564 kBSoft page faults 11213Hard page faults 0Total CPU time 09921 seconds
566 times 10
2789 times 10minus12
1595 times 10minus12
Milongarsquos 2D LWR IAEA benchmark problem case no 076eighth-symmetry core meshed using Delaunay (quads 985747c = 4) solved with finite volumes
(a)
1 073 3151 5402 127 4059 9393 141 4457 10464 118 3768 8715 061 2615 4506 093 2979 6887 094 2971 6998 067 2092 5829 mdash 375 885
10 139 4397 103111 144 4541 106612 128 4052 95013 105 3365 77814 103 3259 76415 096 3016 71216 066 2045 56817 mdash 359 84118 143 4520 106119 132 4155 97520 116 3672 86221 107 3366 79222 098 2951 73323 054 1728 54324 mdash 273 65125 117 3700 86826 095 3057 70827 091 2860 67428 074 2311 64829 mdash 642 130030 mdash 084 33231 048 2041 35332 069 2103 51333 046 1493 46734 mdash 269 66735 046 1444 45536 mdash 432 87137 mdash 067 26338 mdash 074 297
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 18 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
continuous functions are evaluated by means of the shapefunctions used in the formulation Figures 5(b) and 6(b)illustrate the fast fluxes unknowns and its relative position inspace It can be noted that the mesh coarseness gives resultsthat differ substantially in both cases Finally the structure ofthe sparse eigenvalue problemmatrices is shown for each case
with blue red and cyan representing positive negative andexplicitly inserted zero values In the finite volume case thereare 184 unknowns (92 cells times 2 groups) while in the secondcase there are 218 unknowns (109 nodes times 2 groups) Thevolumesrsquo fissionmatrix is almost diagonal it has a bandwidthequal to the number of energy groups which in this case is
Science and Technology of Nuclear Installations 17
500 1000 2000 5000 10000 20000 50000Number of unknowns
2840
2860
2880
2900
2920
ORNLRisoslashKWUOntarioQuarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumes
Quarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elementsEighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumes
Eighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
120588(p
cm)
105 2 times 105 5 times 105
Figure 19 Static reactivity versus number of unknowns The four original solutions as published in 1977 [10] are included as reference
twoThe off-diagonal values appear right next to the diagonalelements because of the chosen ordering of the unknownsin the flux vector 120601 isin R184 Other orderings may be usedbut the rate of convergence of the eigenvalue problem can bedeteriorated On the other hand the elementsrsquo fission matrixhas a nontrivial structure because the grid is unstructuredand the net fission rate at each element depends on the fluxesat the nodes whose location inside thematrix depends in turnon how the gridwas generatedThis effect is similar to the onethat appears in structural analysis where mass matrices needto be lumped in order to simplify transient computations [21]The diagonal block of elementrsquos 119877 matrix and the null blockin 119865 correspond to the discrete equations that set the null-flux boundary conditions on the nodes located at the externalsurface Other types of boundary conditions lead to differentkinds of structures within the problem matrices
In case a part of the domain contained a nonmultiplicativematerial such as a reflector then there would appear sectionsof the fission matrix with null values in both methodsrendering 119865 singular in both methods Care should be takenwhen dealing with the numerical schemes for the eigenvalueproblem solution It can be seen in the elementsrsquo matricesa particular structure that implements the boundary condi-tions at the external surfaces This structure does not appearin the volumesrsquo matrices because the boundary conditions
appear as flux terms which are summed up over all thesurfaces of each cell so they aremasked inside the volumetricdiscretization of the divergence and gradient operators
As the two-group neutron diffusion equation with uni-form cross sections over a circle subject to null-flux bound-ary conditions has an analytical solution it is adequate tocompare how the two proposed numerical schemes relate toit In the studied problem we ignored upscattering and fastfissions Then the analytical effective multiplication factor is
119896eff =]Σ1198912 sdot Σ1199041rarr2
[Σ1198861 + Σ1199041rarr2 + 1198631(]0119886)2] [Σ1198862 + 1198632(]0119886)
2]
(7)
being ]0 = 24048 the smallest root of Besselrsquos first-kindfunction of order zero 1198690(119903)
Figure 7 shows how the two numerical effective multi-plication factors compare to the analytical solution givenby (7) as a function of the mesh refinement indicated bythe quotient 119886ℓ119888 between the radius of the circle and thecharacteristic length of the cellelements We can see thatthe 119896eff computed by the finite volumes (elements) methodis always greater (less) than the analytical solution Indeedit can be proven that for a bare one-dimensional slab thisis always the case [13] However this result does not hold
18 Science and Technology of Nuclear Installations
2000 3000 6000 10000 20000 30000 60000
01
003
001
03
1
3
10
30
100
300To
tal w
all t
ime (
seco
nds)
Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
Figure 20 Total wall time versus number of unknowns
for problems with nonuniform cross sections even in simpleone-dimensional reflected reactors
We may draw two other conclusions from Figure 7 Firstthat the finite element method seems to provide a bettersolution than the volumes-based scheme and second thateven though the error committed tends to decrease with finergrids its behavior is not monotonic for finite volumes Infact Figure 8 shows the difference between the numericaland analytical solutions using a logarithmic vertical scalewhere both conclusions are even more evident We deferthe discussion of such differences between the discretizationscheme until the next section It is worth to note howeverthat the fact that a finite-element-based scheme throws betterresults for the particular bare homogeneous circular reactorunder study than the finite volumes does not imply thatfinite volumes ought to be discarded as a valid tool forsolving the neutron diffusion equation in general cases Thecombination of lattice and core-level computations is usuallyperformedusing cell-based resultswhichwhen fed into node-based methods to solve the few-group neutron diffusionequation may introduce errors which potentially can leadto unacceptable solutions Nevertheless this analysis is farbeyond the scope of this paper which focuses on solving amathematical equation over unstructured grids
32 The 2D IAEA PWR Benchmark Problem This is aclassical two-group neutron diffusion problem first designedin the early 1970s and taken as a reference benchmark forcomputational codes A number of codes were used to solveeither this problem or its three-dimensional formulation[22 23] including milonga using a structured grid [24] Theoriginal formulation can be found in the reference [10] andthere is a reproduction that may be easily found online inreference [24] The geometry consists of a one-quarter ofa PWR core depicted in Figure 9 and the homogeneousmacroscopic cross sections are listed in Table 1 An axialbuckling term 119861
2
119911119892= 08times10
minus4 should be taken into accountThe external surface should be subject to a zero incomingcurrent condition which may be written as
120597120601119892
120597119899= minus
04692
119863119892
sdot 120601119892 (8)
The expected results are as follows
(1) Maximum eigenvalue(2) Fundamental flux distributions
(a) Radial flux traverses 120601119892(119909 0) and 120601119892(119909 119909)
Science and Technology of Nuclear Installations 19
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
1
3
10
30
100
300Ti
me c
onsu
med
to m
esh
the g
eom
etry
(sec
onds
)
Figure 21 Time needed to mesh the geometry versus number of unknowns
Table 1 Macroscopic cross-sections (units are not stated in the original reference but they are assumed to be in cmminus1 or cm as appropriate)
1198631
1198632
Σ1rarr2
Σ1198861
Σ1198862
]Σ1198912
Material1 15 04 002 001 0080 0135 Fuel 12 15 04 002 001 0085 0135 Fuel 23 15 04 002 001 0130 0135 Fuel 2 + rod4 20 03 004 0 0010 0 Reflector
Note the fluxes shall be normalized such that
1
119881coreint119881core
sum
119892
]Σ119891119892 sdot 120601119892119889119881 = 1 (9)
(b) Value and location of maximum power densityThis corresponds to maximum of 1206012 in thecore It is recommended that the maximumvalues of 1206012 both in the inner core and at thecorereflector interface be given
(3) Average subassembly powers 119875119896
119875119896 =1
119881119896
int119881119896
sum
119892
]Σ119891119892 sdot 120601119892119889119881 (10)
where 119881119896 is the volume of the 119896th subassembly and119896 designates the fuel subassemblies as shown inFigure 9
(4) Number of unknowns in the problem number ofiterations and total and outer
(5) Total computing time iteration time IO-time andcomputer used
(6) Type and numerical values of convergence criteria(7) Table of average group fluxes for a square mesh grid
of 20 times 20 cm(8) Dependence of results on mesh spacing
Even though the original problem is based on a quarter-core situation the problem has an eighth-core symmetry
20 Science and Technology of Nuclear Installations
Quarter Delaunay triangles elements
Tim
e con
sum
ed to
read
the m
esh
(sec
onds
)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumes
Quarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
1
3
10
30
100
300
Figure 22 Time needed to read the mesh versus number of unknowns
which cannot be taken into account by structured gridswhich are the main target of the benchmark Howevernonstructured grids can take into consideration any kind ofsymmetry almost without loss of accuracy and at the sametime reducing roughly the number of unknowns by a halfand the associated computational effort needed to solve theproblem by a factor of four Answers to items (1)ndash(7) canbe given completely by milonga using a single input file(see Supplementary data in SupplementaryMaterial availableonline at httpdxdoiorg1011552013641863) As asked initem (8) how results depend not only on the mesh spacingbut also on the meshing algorithm on the gridrsquos elementarygeometric shape and on the discretization scheme may shedlights on the subject whichmay be evenmore interesting thanthe numerical results themselves
Taking advantage of milongarsquos capability of reading andparsing command-line arguments the selection of the coregeometry (quarter or eighth) the meshing algorithm (delau-nay [16] or delquad [25]) the shape of the elementaryentities (triangles or quadrangles) the discretization scheme(volumes or elements) and the characteristic length ℓ119888 ofthe mesh can be provided at run time Fixing five values forℓ119888 = 4 3 2 1 05 gives 2 times 2 times 2 times 2 times 5 = 80 possiblecombinations which we solve with successive invocations to
milonga with the same input file but with different argumentsfrom a simple script Figures 10 11 12 13 14 15 16 17 and18 show the results corresponding to items (1)ndash(7) for someillustrative cases The complete set of figures and the codeused to generate themmay be provided upon request Table 2compiles the answers to the problem for every case studied
As the milonga code is still under development itsnumerical routines are not yet fully optimized nor designedfor parallel computation Therefore the reported times areonly rough estimates and should be taken with care Thesolution comprises five steps
(1) generate the grid with the requested geometry mesh-ing algorithm basic shape and characteristic lengthby calling to gmsh
(2) read the generated mesh(3) build the matrices(4) solve the eigenvalue problem(5) compute the requested results
The CPU time reported in Figures 10ndash18 is thus thesum of all of these five steps but not the time needed togenerate the figuresmdashwhich in some caseswith coarsemeshes
Science and Technology of Nuclear Installations 21
Table2
Resultsofthe2
DIAEA
PWRBe
nchm
arkp
roblem
obtained
with
them
ilong
acod
efor
thee
ightyp
ropo
sedcombinatio
nsofsymmetrym
eshing
algorithm
basicshapediscretization
schemeandcharacteris
ticelem
entc
elllengthTh
ereference
solutio
nis120588=28749times10minus5w
hich
correspo
ndsto119896eff=10296
Case
no
Symm-
etry
Mesh
algorithm
Basic
shape
Solutio
nmetho
dℓ119888
(cm)
Δ120588
(pcm
)max1206012
(mdash)
Total
unkn
owns
Outer
iter
Linear
iter
Inner
iter
Resid
ual
norm
Relativ
eerror
Error
estim
ate
Mem
ory
(Mb)
Page
faults
114
Delaunay
998779Vo
lumes
40
minus120
1145
8264
332
1033
6119890minus09
3119890minus09
1119890minus09
349470
214
Delaunay
998779Vo
lumes
30
85
1135
15740
332
1967
2119890minus08
1119890minus08
4119890minus09
4813401
314
Delaunay
998779Vo
lumes
20
197
1124
31188
332
3898
1119890minus08
6119890minus09
3119890minus09
7621932
414
Delaunay
998779Vo
lumes
10153
1123
127476
332
15934
7119890minus09
3119890minus09
3119890minus09
273
81886
514
Delaunay
998779Vo
lumes
05
187
1119
510676
332
63834
2119890minus09
9119890minus10
1119890minus09
1148
352261
614
Delaunay
998779Elem
ents
40
173
1100
4308
332
538
2119890minus08
8119890minus09
4119890minus09
318794
714
Delaunay
998779Elem
ents
30
117
1107
8108
332
1013
5119890minus09
2119890minus09
2119890minus09
4712946
814
Delaunay
998779Elem
ents
20
84
1112
15936
332
1992
6119890minus09
3119890minus09
2119890minus09
7321347
914
Delaunay
998779Elem
ents
1062
1116
64478
332
8059
6119890minus09
3119890minus09
4119890minus09
262
77105
1014
Delaunay
998779Elem
ents
05
58
1117
256700
332
32087
1119890minus09
5119890minus10
1119890minus09
1091
331969
1114
Delaunay
◻Vo
lumes
40
53
1149
5042
332
630
2119890minus08
1119890minus08
5119890minus09
288007
1214
Delaunay
◻Vo
lumes
30
346
1133
8560
332
1070
7119890minus09
3119890minus09
2119890minus09
3910895
1314
Delaunay
◻Vo
lumes
20
308
1131
15576
332
1947
1119890minus08
7119890minus09
3119890minus09
541564
514
14Delaunay
◻Vo
lumes
10177
1122
60774
332
7596
8119890minus09
4119890minus09
3119890minus09
167
50115
1514
Delaunay
◻Vo
lumes
05
199
1120
244936
332
30617
4119890minus09
2119890minus09
2119890minus09
698
215678
1614
Delaunay
◻Elem
ents
40
196
1100
5222
332
652
3119890minus08
1119890minus08
8119890minus09
4713098
1714
Delaunay
◻Elem
ents
30
123
1107
8810
332
1101
2119890minus08
9119890minus09
7119890minus09
6918598
1814
Delaunay
◻Elem
ents
20
901112
15922
332
1990
1119890minus08
5119890minus09
5119890minus09
107
30591
1914
Delaunay
◻Elem
ents
1063
1116
61456
332
7682
5119890minus09
2119890minus09
4119890minus09
389
113237
2014
Delaunay
◻Elem
ents
05
58
1117
246332
332
30791
8119890minus10
4119890minus10
1119890minus09
1676
498192
2114
Delq
uad
998779Vo
lumes
40
minus45
1144
6228
332
778
4119890minus08
2119890minus08
7119890minus09
308495
2214
Delq
uad
998779Vo
lumes
30
570
1153
11820
332
1477
3119890minus08
1119890minus08
4119890minus09
4010879
2314
Delq
uad
998779Vo
lumes
20
459
1103
24100
332
3012
2119890minus08
1119890minus08
5119890minus09
6217715
2414
Delq
uad
998779Vo
lumes
10512
1104
9640
03
3212050
2119890minus08
1119890minus08
9119890minus09
207
61714
2514
Delq
uad
998779Vo
lumes
05
528
1106
385600
332
48200
2119890minus09
1119890minus09
2119890minus09
838
255735
2614
Delqu
ad998779
Elem
ents
40
220
1093
3288
332
411
3119890minus08
2119890minus08
6119890minus09
308495
2714
Delq
uad
998779Elem
ents
30
140
1104
6148
332
768
4119890minus08
2119890minus08
8119890minus09
39110
0028
14Delq
uad
998779Elem
ents
20
82
1110
12392
332
1549
2119890minus08
9119890minus09
6119890minus09
6117067
2914
Delqu
ad998779
Elem
ents
1063
1115
48882
332
6110
2119890minus09
1119890minus09
1119890minus09
197
5804
630
14Delq
uad
998779Elem
ents
05
58
1116
194162
332
24270
2119890minus09
9119890minus10
2119890minus09
790
238769
3114
Delq
uad
◻Vo
lumes
40
minus416
1174
3132
332
391
5119890minus08
3119890minus08
8119890minus09
267322
3214
Delq
uad
◻Vo
lumes
30
minus248
1151
5922
332
740
9119890minus09
4119890minus09
2119890minus09
318779
3314
Delq
uad
◻Vo
lumes
20
minus132
1139
12050
332
1506
1119890minus08
7119890minus09
4119890minus09
4412239
3414
Delq
uad
◻Vo
lumes
10minus49
1126
48200
332
6025
6119890minus09
3119890minus09
2119890minus09
122
36094
3514
Delq
uad
◻Vo
lumes
05
minus23
1123
192800
332
24100
5119890minus09
2119890minus09
3119890minus09
464
140228
3614
Delq
uad
◻Elem
ents
40
224
1093
3294
332
411
3119890minus08
1119890minus08
7119890minus09
359852
3714
Delq
uad
◻Elem
ents
30
141
1104
6144
332
768
3119890minus08
2119890minus08
9119890minus09
5114018
3814
Delq
uad
◻Elem
ents
20
921111
12392
332
1549
1119890minus08
6119890minus09
5119890minus09
8122526
3914
Delq
uad
◻Elem
ents
1065
1115
48882
332
6110
5119890minus09
3119890minus09
4119890minus09
276
78431
22 Science and Technology of Nuclear Installations
Table2Con
tinued
Case
no
Symm-
etry
Mesh
algorithm
Basic
shape
Solutio
nmetho
dℓ119888
(cm)
Δ120588
(pcm
)max1206012
(mdash)
Total
unkn
owns
Outer
iter
Linear
iter
Inner
iter
Resid
ual
norm
Relativ
eerror
Error
estim
ate
Mem
ory
(Mb)
Page
faults
4014
Delq
uad
◻Elem
ents
05
59
1117
194162
332
24270
1119890minus09
5119890minus10
1119890minus09
1104
318967
4118
Delaunay
998779Vo
lumes
40
minus99
1139
4228
332
528
4119890minus12
2119890minus12
1119890minus12
359630
4218
Delaunay
998779Vo
lumes
30
50
1132
7712
332
964
1119890minus11
6119890minus12
2119890minus12
4211578
4318
Delaunay
998779Vo
lumes
20
177
1120
15776
332
1972
2119890minus12
1119890minus12
7119890minus13
5515436
4418
Delaunay
998779Vo
lumes
10142
1118
63902
332
7987
3119890minus12
1119890minus12
1119890minus12
160
46691
4518
Delaunay
998779Vo
lumes
05
176
1115
254612
332
31826
2119890minus12
8119890minus13
1119890minus12
555
168526
4618
Delaunay
998779Elem
ents
40
183
1096
2252
332
281
6119890minus12
3119890minus12
2119890minus12
359589
4718
Delaunay
998779Elem
ents
30
119
1104
4040
224
505
2119890minus08
1119890minus08
7119890minus09
41114
3748
18Delaunay
998779Elem
ents
20
85
1108
8156
332
1019
2119890minus12
8119890minus13
6119890minus13
5515053
4918
Delaunay
998779Elem
ents
1063
1112
32480
332
4060
1119890minus12
6119890minus13
8119890minus13
151
43671
5018
Delaunay
998779Elem
ents
05
58
1113
128358
332
1604
47119890minus13
3119890minus13
7119890minus13
527
157581
5118
Delaunay
◻Vo
lumes
40
37
1144
2380
224
297
5119890minus08
2119890minus08
1119890minus08
5314173
5218
Delaunay
◻Vo
lumes
30
218
1120
4194
332
524
7119890minus12
4119890minus12
2119890minus12
3610034
5318
Delaunay
◻Vo
lumes
20
345
1110
7960
332
995
5119890minus12
3119890minus12
1119890minus12
4712879
5418
Delaunay
◻Vo
lumes
10165
1117
30652
332
3831
2119890minus12
1119890minus12
7119890minus13
100
28851
5518
Delaunay
◻Vo
lumes
05
191
1117
122066
224
15258
2119890minus08
8119890minus09
9119890minus09
343
103825
5618
Delaunay
◻Elem
ents
40
175
1097
2526
332
315
4119890minus12
2119890minus12
1119890minus12
6016159
5718
Delaunay
◻Elem
ents
30
114
1104
4386
332
548
4119890minus12
2119890minus12
2119890minus12
5013768
5818
Delaunay
◻Elem
ents
20
901108
8234
332
1029
2119890minus12
1119890minus12
1119890minus12
7320094
5918
Delaunay
◻Elem
ents
1062
1112
31186
224
3898
2119890minus08
7119890minus09
9119890minus09
208
59589
6018
Delaunay
◻Elem
ents
05
58
1113
123122
224
15390
9119890minus09
5119890minus09
1119890minus08
807
2366
7061
18Delq
uad
998779Vo
lumes
40
minus210
1168
3224
332
403
1119890minus11
5119890minus12
2119890minus12
43117
8462
18Delq
uad
998779Vo
lumes
30
369
1150
6000
332
750
1119890minus11
6119890minus12
3119890minus12
5013502
6318
Delq
uad
998779Vo
lumes
20
312
1104
12316
332
1539
2119890minus11
8119890minus12
5119890minus12
6016395
6418
Delq
uad
998779Vo
lumes
1044
01094
48714
332
6089
9119890minus12
4119890minus12
4119890minus12
1183444
165
18Delq
uad
998779Vo
lumes
05
534
1081
193980
332
24247
5119890minus12
2119890minus12
4119890minus12
423
126768
6618
Delq
uad
998779Elem
ents
40
240
1090
1750
332
218
9119890minus12
4119890minus12
2119890minus12
43117
8667
18Delqu
ad998779
Elem
ents
30
150
1100
3184
332
398
1119890minus11
5119890minus12
2119890minus12
5013509
6818
Delq
uad
998779Elem
ents
20
89
1106
6426
332
803
5119890minus12
3119890minus12
2119890minus12
5916047
6918
Delq
uad
998779Elem
ents
1064
1111
24886
332
3110
2119890minus12
1119890minus12
1119890minus12
113
32788
7018
Delq
uad
998779Elem
ents
05
59
1113
98052
224
12256
1119890minus08
5119890minus09
8119890minus09
398
118153
7118
Delq
uad
◻Vo
lumes
40
minus398
1172
1650
332
206
1119890minus11
5119890minus12
2119890minus12
369907
7218
Delq
uad
◻Vo
lumes
30
minus249
1146
3058
332
382
4119890minus12
2119890minus12
1119890minus12
349349
7318
Delq
uad
◻Vo
lumes
20
minus131
1136
6228
224
778
3119890minus08
1119890minus08
5119890minus09
4010930
7418
Delq
uad
◻Vo
lumes
10minus51
1123
24536
224
3067
3119890minus08
2119890minus08
1119890minus08
7822716
7618
Delq
uad
◻Elem
ents
40
237
1090
1752
332
219
6119890minus12
3119890minus12
2119890minus12
4111213
7718
Delq
uad
◻Elem
ents
30
145
1101
3184
332
398
6119890minus12
3119890minus12
2119890minus12
43118
1178
18Delq
uad
◻Elem
ents
20
941107
6426
332
803
3119890minus12
1119890minus12
1119890minus12
6016220
7918
Delqu
ad◻
Elem
ents
1065
1111
24874
332
3109
1119890minus12
6119890minus13
8119890minus13
156
4400
080
18Delq
uad
◻Elem
ents
05
59
1113
9804
22
2412255
8119890minus09
4119890minus09
8119890minus09
567
162168
Science and Technology of Nuclear Installations 23
Tim
e con
sum
ed to
bui
ld th
e mat
rices
(sec
onds
)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay trianglesvolumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
10
30
100
300
1
3
Figure 23 Time needed to build the matrices versus number of unknowns
was considerably larger than the solution time itself Theeigenvalue problem was solved using a multilayer iterativeKrylov-Schur method [26] with a tolerance relative to thematrices norm
1003817100381710038171003817119877 sdot 120601 minus (1119896eff) sdot 119865 sdot 1206011003817100381710038171003817
119877 + (1119896eff) sdot 119865lt 10minus8 (11)
The reported residual norm and relative error are10038171003817100381710038171003817100381710038171003817119877 sdot 120601 minus
1
119896effsdot 119865 sdot 120601
10038171003817100381710038171003817100381710038171003817
1003817100381710038171003817119877 sdot 120601 minus (1119896eff) sdot 119865 sdot 1206011003817100381710038171003817
1003817100381710038171003817(1119896eff) sdot 1206011003817100381710038171003817
(12)
respectively The fields marked as outer linear and inneriterations refer to the number of steps needed to attainthe requested tolerance in each layer of the Krylov-Schuralgorithm The computer used to solve the problem has anIntel i7 920 267GHz processor with 4Gb of RAM runningDebian GNULinux Wheezy
When using a finite volumes-based scheme over anunstructured mesh the solver has to gather information
about which cells are neighbors and which are not Currentlygmsh does not write this kind of lists in its output filesso milonga has to explicitly solve the neighbors problemPerforming a linear search is an 119874(119873
2) task which is
unacceptable for problem sizes 119873 of interest The code usesa search based on a 119896-dimensional tree [27] which can inprinciple be performed in119874(119873) steps Still for large values of119873 the time needed to read and parse the mesh (number twopreviously mentioned) is the bottleneck of the solution Thisstep is not needed in finite elements although the construc-tion of the elementary matrices involves the computationof the multidimensional Jacobians and integrals which thenhave to be assembled Again for large 119873 this step (numberthree) takes up most of the time needed to solve the problem
The Delaunay algorithm is a standard method for gen-erating two-dimensional grids [16] by tessellating a planewith triangles If the mesh needs to be based on quadranglesinstead a recombination algorithm can be used to transformtwo adjacent triangles in one quadrangle whenever is possi-ble However for geometries which are based on rectangularshapes there exist other algorithms [25] both for meshingand for recombining the triangles that give rise to elementaryentities with right angles almost everywhere which may be a
24 Science and Technology of Nuclear Installations
Tim
e con
sum
ed to
solv
e the
eige
nval
ue p
robl
em (s
econ
ds)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
10
30
100
300
1
3
Figure 24 Time needed to solve the eigenvalue problem versus number of unknowns
desired property of the resulting grid For finite elements thesteps needed to build the matrices depend on the selectionof triangles or quadrangles as the basic elementary geometrybecause the shape functions change However the number ofunknowns is the same as the number of nodes that does notchange after a recombination procedure On the other handthe number of unknowns in the finite volumes schemes withtriangles is roughly twice as the number of unknowns withquadrangles for the same grid
Figure 19 shows how the computed static reactivity (ie1 minus 1119896eff) depends on the number of unknowns for eachof the sixteen combinations of geometry algorithm shapescheme The four accepted results published in the originalreference [10] are included for reference although it shouldbe taken into account that said reactivities were computedalmost forty years ago Figure 20 shows the total wall timeneeded to solve the problem as a function of 119873119866 whileFigures 21 22 23 and 24 show the times needed for eachof the first four steps involved in the solution maintainingthe same logarithmic scale for both the abscissas and theordinates Green data represent finite volumes whilst bluebullets correspond to finite elements Solid lines indicatequarter-core and dashed lines eighth-symmetry Fillet bullets
are results obtained by theDelaunay triangulation and emptybullets were computed with the delquad algorithm Finallytriangle-shaped data correspond to triangles and squares anddiamonds denote quadrangles as the basic geometry of thegrid
It can be seen that finite elements produce amuch smallerdispersion of eigenvalues 119896eff than finite volumes with therefinement of the mesh This can be explained because finitevolumes methods rely on a geometric condition of the meshwhich may change abruptly if the meshing algorithm decidesto allocate the cells in a rather different form for small changesin ℓ119888 Finite elements methods are less influenced by thesediscontinuous lay-out changes of the elements As expectedeighth-core symmetries give almost the same results as thequarter-core geometries with roughly half the unknownsFor small problems finite volumes run faster that finiteelements because the time needed to solve the neighborproblem is negligible When the problem size grows thistime increases and exceeds the overhead implied in the con-struction and assembly of the finite elements matrices Alsoat least for this configuration it is seen that the eigenvalueproblem is solved faster for finite volumes than for finiteelements
Science and Technology of Nuclear Installations 25
4 Conclusions
Unstructured grids provide the cognizant engineer witha wide variety of possibilities to deal with the design oranalysis of nuclear reactor cores These kinds of grids cansuccessfully reproduce continuous geometries commonlyfound in reactor cores such as cylinders and thereforenot only can the diffusion equation be better approximatedinside the domain of definition but also the fulfillment ofboundary conditions is improved A free computer codewas written from scratch that is able to completely solvethe 2D IAEA PWR Benchmark using unstructured gridsfor sixteen combinations of geometry meshing algorithmbasic shape and discretization scheme plus any value ofthe gridrsquos characteristic length by using a single input fileThe complete set of input files and codemdashexecutable andsourcemdashis available either online or upon request withcomments experiences suggestions and corrections beingmore than welcome Further development should includetackling full three-dimensional geometries with completethermal hydraulic feedback in order to analyze how thesolutions of the coupled neutronic-thermal problem dependon the spatial discretization scheme of the neutron leakageterm Parallelization of the computation and assembly of thematrices and of the solution of the eigenvalue problem andits implementation using GPUs are also desired features toimplement A problemwith direct applications that the futureversions of milonga ought to solve is the analysis of how thegeometry of the absorbing materials should be taken intoaccount in structured coarse grids in order to mitigate effectssuch as the rod-cusp problem
Suitable schemes for approximating the continuous dif-ferential operators by discrete matrix expressions includefinite volumes and finite elements families Finite volumesmethods compute cell mean values whilst finite elementsgive functional values at the gridrsquos nodes In general finiteelements are less sensitive to changes in the mesh so theresults they provide do not change significantly for differentmeshing algorithms or elementary shapes Small problemsare best solved by finite volumes as the neighbor-findingproblem is faster than the process of building and assemblingthe eigenvalue-problem matrices For a large number ofunknowns the process of finding which cell is neighbor ofwhichmdasheven based on a 119896-dimensional treemdashoverwhelmsthe computation and assembly of elementary matrices andfinite-element methods perform better
References
[1] G Glasstone and S Bell Nuclear Reactor Theory KriegerPublishing Company 1970
[2] J J Duderstadt and L J Hamilton Nuclear Reactor AnalysisJohn Wiley amp Sons New York NY USA 1976
[3] C Gho Reactor Physics Undergraduate Course Lecture NotesInstituto Balseiro 2006
[4] G G Theler and F J Bonetto ldquoOn the stability of the pointreactor kinetics equationsrdquoNuclear Engineering andDesign vol240 no 6 pp 1443ndash1449 2010
[5] E E Lewis andW F Miller Computational Methods of NeutronTransport John Wiley amp Sons 1984
[6] E Masiello ldquoAnalytical stability analysis of Coarse-Mesh finitedifference methodrdquo in Proceedings of the International Confer-ence on the Physics of Reactors (PHYSOR rsquo08) Nuclear Power ASustainable Resource pp 456ndash464 September 2008
[7] M L Zerkle Development of a polynomial nodal methodwith flux and current discontinuity factors [PhD thesis] Mas-sachusets Institute of Technology 1992
[8] J I Yoon andHG Joo ldquoTwo-level coarsemesh finite differenceformulation with multigroup source expansion nodal kernelsrdquoJournal of Nuclear Science andTechnology vol 45 no 7 pp 668ndash682 2008
[9] O Mazzantini M Schivo J Di Cesare R Garbero M Riveroand G Theler ldquoA coupled calculation suite for Atucha II oper-ational transients analysisrdquo Science and Technology of NuclearInstallations vol 2011 Article ID 785304 12 pages 2011
[10] Computational Benchmark Problem Comitee for the Mathe-matics and Computation Division of the American NuclearSociety ldquoArgonne Code Center Benchmark problem bookrdquoTech Rep ANL-7416 Supplement 2 Argonne National Labo-ratory 1977
[11] A F Henry Nuclear Reactor Analysis MIT Cambridge UK1975
[12] O C Zienkiewicz R L Taylor and J Z ZhuTheFinite ElementMethod Its Basis and Fundamentals vol 1 Elsevier 6th edition2005
[13] G Theler Difusion de neutrones en mallas no estructuradascomparacion entre volumenes y elementos finitos AcademicMonograph Universidad de Buenos Aires 2013
[14] G Theler ldquoMilonga a free nuclear reactor core analysis coderdquoAvailable online 2011
[15] G Theler F J Bonetto and A Clausse ldquoOptimizacion deparametros en reactores de potencia base de diseno delcodigo neutronico milongardquo in Reunion Anual de la AsociacionArgentina de Tecnologia Nuclear XXXVII 2010
[16] C Geuzaine and J-F Remacle ldquoGmsh a 3-D finite elementmesh generator with built-in pre- and post-processing facili-tiesrdquo International Journal for Numerical Methods in Engineer-ing vol 79 no 11 pp 1309ndash1331 2009
[17] S Balay J Brown K Buschelman et al ldquoPETSc users manualrdquoTech Rep ANL-9511-Revision 34 Argonne National Labora-tory 2013
[18] S Balay W D Gropp L Curfman McInnes and B F SmithldquoEfficient management of parallelism in object oriented numer-ical software librariesrdquo in Modern Software Tools in ScientificComputing E Arge A M Bruaset and H P Langtangen Edspp 163ndash202 Birkhauser 1997
[19] V Hernandez J E Roman and V Vidal ldquoSLEPC a scalable andflexible toolkit for the solution of eigenvalue problemsrdquo ACMTransactions on Mathematical Software vol 31 no 3 pp 351ndash362 2005
[20] M Galassi J Davies J Theiler et al GNU Scientific LibraryReference Manual 3rd edition 2009
[21] O C Zienkiewicz and R L Taylor The Finite Element MethodFor Solid and Structural Mechanics vol 3 Elsevier 6th edition2005
[22] A Imelda and K Doddy Evaluation of the IAEA 3-D PWRbenchmark problem using NESTLE code 2004
[23] R Mosteller ldquoStatic benchmarking of the NESTLE advancednodal coderdquo in Proceedings of the Joint International Conference
26 Science and Technology of Nuclear Installations
on Mathematical Methods and Supercomputing for NuclearApplications vol 2 pp 1596ndash1605 1997
[24] GTheler F J Bonetto andAClausse ldquoSolution of the 2D IAEAPWR Benchmark with the neutronic code milongardquo in Actasde la Reunion Anual de la Asociacion Argentina de TecnologıaNuclear XXXVIII 2011
[25] J F Remacle F Henrotte T Carrier-Baudouin et al ldquoAfrontal delaunay quad mesh generator using the 119871
infin normrdquoInternational Journal For Numerical Methods in Engineeringvol 94 no 5 pp 494ndash512 2013
[26] G W Stewart ldquoA Krylov-Schur algorithm for large eigenprob-lemsrdquo SIAM Journal on Matrix Analysis and Applications vol23 no 3 pp 601ndash614 2002
[27] J L Bentley ldquoMultidimensional binary search trees used forassociative searchingrdquo Communications of the ACM vol 18 no9 pp 509ndash517 1975
TribologyAdvances in
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High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
2 Science and Technology of Nuclear Installations
for each groupmdashcontaining differential operators appliedonly over the spatial coordinatesThis resulting set of second-order PDEs is known as the multigroup neutron diffusionequation and is usually used to model design and analyzenuclear reactor cores by the so-called core-level calculationcodesThese programs take homogenizedmacroscopic cross-sections (which may depend on the spatial coordinatesthrough changes of fuel burnup materials temperature orother properties) computed by lattice-level codes as an inputand solve the diffusion equation to obtain the flux (and itsrelated quantities such as power xenon etc) distributionwithin the core
Given a certain spatial distribution of materials and itsproperties inside a reactor core chances are that the resultingreactor will not be critical That is to say in general therate of absorptions and leakages will not exactly overcomethe neutrons born by fissions sources and some kind offeedbackmdasheither through an external control system or bymeans of an inherent stability mechanism of the core [4]mdashis needed to keep the reactor power within a certain intervalMathematically this means that the transportmdashand thus thediffusionmdashequation does not have a steady-state solution Inpractice given a certain reactor configuration its steady-stateflux distribution is computed by setting all the time deriva-tives to zero as usual but also by dividing the fission sources bya positive value 119896eff called the effective multiplication factorwhich becomes also one unknown and turns the formulationinto a eigenvalue problemThe largest (or smallest dependingon the formulation) eigenvalue is therefore the effectivemultiplication factor If 119896eff lt 1 the original configurationwas subcritical and conversely As successive configurationsmake 119896eff rarr 1 the associated eigenfunctions approach thesteady-state critical flux distribution [5]
Core-level codes traditionally use regular grids to dis-cretize the differential operators over the space Dependingon the characteristics and symmetry of the reactor coreeither squares or hexagons are used as the basic shapeof the mesh Usual discretization schemes involve cell-centered finite differences or two-step coarse-meshcouplingcoefficient methods [6ndash8] which are accurate and efficientenough formost applicationsThere are nevertheless certainlimitations that are not inherently related to structured gridsbut to the way their coarseness is utilized and how they arecomputed and applied to the reactor geometry These issuescan be overcome by discretizing the spatial operators usinga scheme which could be applied to nonstructured gridsnamely finite volumes or finite elements
This way unstructured grids may be used to studyanalyze and understand the numerical errors introduced bythe discretization of the leakage operator with a difference-based scheme over a coarse structured grid by successivelyrefining the mesh whilst comparing the solutions with thestructured one as depicted in Figure 1 Another example ofapplication may be the improvement of the response matrixof boundary conditions over cylindrical surfaces which isthe case for most of the nuclear power plants cores Whena coarse structured mesh is applied to a curved geometrythere appears a geometric condition known as the staircaseeffect Even though arbitrary shapes cannot be perfectly
tessellated with unstructured grids for the same number ofunknowns they better represent the geometry than struc-tured meshes as Figure 2 illustrates Again by refining themesh and comparing solutions the matrix responses usedto set the boundary conditions on the coarse meshes canbe optimized Finally other complex geometries such asslanted control rods (Figure 3) or irradiation chambers can bedirectly taken into account by unstructured grids possibly byrefining the mesh in the locations around said complexities
2 The Steady-State Multigroup NeutronDiffusion Equation
In the present work we take the steady-state multigroupneutron diffusion equation for granted That is to say wefocus on the mathematical aspects of the eigenvalue problemand make no further reference to its derivation from thetransport equation nor to the validity of its applicationto reactor problems as these subjects that are extensivelydiscussed in the classic literature [1 5 11]
The differential formulation of the steady-state multi-group neutron diffusion equation over an 119898-dimensionaldomain 119880 with 119866 groups of energy is the set of 119866 coupleddifferential equations
div [119863119892 (x) sdot grad 120601119892 (x)] minus Σ119886119892 (x) sdot 120601119892 (x)
+
119866
sum
1198921015840 = 119892
Σ1199041198921015840rarr119892 (x) sdot 1206011198921015840 (x)
+ 120594119892 sdot
119866
sum
1198921015840=1
]Σ1198911198921015840 (x)119896eff
sdot 1206011198921015840 (x) = 0
(1)
where 120601119892 are the 119892 = 1 119866 unknown flux distributionsand the Σs and the 119863s are the macroscopic cross-sectionsand diffusion coefficients which ought to be computed by alattice-level code and taken as an input to core-level codesHowever for the purposes of fulfilling the objectives of thiswork we take the macroscopic cross-sections as functions ofthe spatial coordinate x isin R119898 which is known beforehandThe coefficient 120594119892 represents the fission spectrum and is thefraction of the fission neutrons that are born into group 119892 Asstated above the ]-fissions are divided by a positive effectivemultiplication factor 119896eff and all the time derivatives that isthe right hand of (1) is set to zero We note that in order todefine a consistent nomenclature we use the absorption crosssection Σ119886119892 of the group 119892 which is equal to the total crosssection Σ119905119892 minus the self-scattering cross section Σ119904119892rarr119892Therefore the sum over the scattering sources excludes theterm corresponding to 119892
1015840= 119892 If we had used the total
cross section in the second term of (1) we would have hadto include the self-scattering term as a source
If the cross sections depend only in an explicit wayon the spatial coordinate x then (1) is linear If as is thegeneral case the cross sections depend on x through the flux120601119892(x) itselfmdashsuch as by means of the xenon concentrationor by local temperature distributionsmdashthen (1) is nonlinearNevertheless this general case can be solved by successive
Science and Technology of Nuclear Installations 3
(a) Coarse structured grid commonly used in diffu-sion codes such as in [9] Each lattice cell is divided intoa 2 times 2 grid for solving the neutron diffusion equation
(b) Discretization of the lattice cell using a fineunstructured grid as proposed in this work
(c) Further refinement of the unstructured grid
Figure 1 Discretization of a homogenized PHWR array of fuel channels for a core-level diffusion computation Each square is a lattice-levelcell comprising one fuel channel and the surrounding moderator
linear iterations so the basic problem can be regarded as beingpurely linear
It should be noted that if at least one of the diffu-sion coefficients 119863119892(x) is discontinuous over space thenthe divergence operator is not defined at the discontinuitypoints Therefore the differential formulationmdashalso knownas the strong formulationmdashis not complete when there existmaterial discontinuities that involve the diffusion coefficientsAt thesematerial interfaces the differential equation has to bereplaced by a neutron current conservation condition
119863+
119892(x) sdot grad 120601
+
119892(x) = 119863
minus
119892(x) sdot grad 120601
minus
119892(x) (2)
where the plus and minus sign denote both sides of the inter-face As the diffusion coefficients are different the resultingflux distribution120601119892(x) ought to have a discontinuous gradientat the interface in order to conserve the current
When transforming the strong formulation into a weakformulationmdashnot just into an integral formulationmdashboth (1)and (2) can be taken into account by a single expression Ineffect let 120593119892(x) be arbitrary functions of x for 119892 = 1 119866Multiplying each of the 119866 equations (1) by 120601119892(x) integrating
over the domain 119880 isin R119898 and applying Greenrsquos formula [12]to the leakage term we obtain
int119880
119863119892 (x) sdot [grad 120593119892 (x) sdot grad 120601119892 (x)] 119889119898x
+ int120597119880
120593119892 (x) sdot 119863119892 (x) sdot [grad 120601119892 (x) sdot n] 119889119878
+ int119880
120593119892 (x) sdot Σ119886119892 (x) sdot 120601119892 (x) 119889119898x
+ int119880
120593119892 (x) sdot119866
sum
1198921015840 = 119892
Σ1199041198921015840rarr119892 (x) sdot 1206011198921015840 (x) 119889119898x
+ 120594119892 sdot int119880
120593119892 (x) sdot119866
sum
1198921015840=1
]Σ1198911198921015840 (x)119896eff
sdot 1206011198921015840 (x) 119889119898x
(3)
These 119866 coupled equations should hold for any arbitraryset of functions120593119892(x)Making an analogy between (3) and theprinciple of virtual work for structural problems we call thesefunctions virtual fluxes This formulation does not involveany differential operator over the diffusion coefficient and yet
4 Science and Technology of Nuclear Installations
(a) Continuous two-dimensional domain (b) Structured grid
(c) Unstructured grid
Figure 2 When a continuous domain (a) is meshed with an unstructured grid there appears a geometric condition known as the staircaseeffect (b) For the same number of nodes unstructured grids reproduce the original geometry better (c)
Figure 3 Cross section of a hypothetical reactor in which thecontrol rods enter into the core from above with a certain attackangle with respect to the vertical direction
at the same time can be shown to be equivalent to the strongformulation given by (1)
In any case both formulations involve the computationof 119866 unknown functions of x and one unknown real value119896eff Except for homogeneous cross sections in canonicaldomains 119880 the multigroup diffusion equation needs to besolved numerically As discussed below any nodal-baseddiscretization scheme replaces a continuous unknown func-tion 120601119892(x) by 119873 discrete values 120601119892(119894) for 119894 = 1 119873
Figure 4 The finite volumes method computes the unknown fluxin the cell centers (squares) whilst the finite elements methodcomputes the fluxes at the nodes (circles)
If we arrange these unknowns into a vector 120601 isin R119873119866
such as
120601 =
[[[[[[[[[[[[[[[[
[
1206011 (1)
1206012 (1)
120601119866 (1)
1206011 (2)
120601119892 (119894)
120601119866 (119873)
]]]]]]]]]]]]]]]]
]
(4)
Science and Technology of Nuclear Installations 5
(a) Fast flux distribution
minus100minus50 0 50 100minus100
minus500
50100
0
06
xy
(b) Fast flux unknowns
(c) Matrix 119877 structure (d) Matrix 119865 structure
Figure 5 A bare homogeneous circle solved with finite volumes (184 unknowns)
then the continuous eigenvalue problemmdashin either for-mulationmdashcan be transformed into a generalized matrixeigenvalueeigenvector problem casted in either of the follow-ing forms
119877 sdot 120601 =1
119896effsdot 119865 sdot 120601
119865 sdot 120601 = 119896eff sdot 119877 sdot 120601
119877minus1sdot 119865 sdot 120601 = 119896eff sdot 120601
(5)
where 119877 and 119865 are square 119873119866 times 119873119866 matrices We call119865 the fission matrix as it contains all the ]-fission termswhich are the ones that we artificially divided by 119896eff Therest of the terms are grouped into the removal or transportmatrix 119877 which includes the rest of the neutron-matterinteraction mechanisms It can be shown [11] that for anyreal set of cross sections 119877minus1 exists and that the119873119866 pairs ofeigenvalueeigenvector solutions of (5) satisfy that
(1) there is a unique real positive eigenvalue greater inmagnitude than any other eigenvalue
(2) all the elements of the eigenvector corresponding tothat eigenvalue are real and positive
(3) all other eigenvectors either have some elements thatare zero or have elements that differ in sign from eachother
21 Boundary Conditions Being a differential equation overspace the neutron diffusion equation needs proper boundaryconditions to conform a properly definedmathematical prob-lem These can be imposed flux (Dirichlet) imposed current(Neumann) or a linear combination (Robin) However dueto the fact that in the linear problem in absence of externalsourcesmdashsuch as (1) or (3)mdashthe problem is homogeneousif 120601119892(x) is a solution then any multiple 120572120601119892(x) is also asolution That is to say the eigenvectors are defined up to amultiplicative constant whose value is usually chosen as toobtain a certain total thermal power Thus the prescribedboundary conditions should also be homogeneous and bedefined up to amultiplicative constantTherefore the allowedDirichlet conditions are zero flux at the boundary theNeumann conditions should prescribe that the derivative in
6 Science and Technology of Nuclear Installations
(a) Fast flux distribution
minus100minus50 0 50 100minus100
minus500
50100
0
06
xy
(b) Fast flux unknowns
(c) Matrix 119877 structure (d) Matrix 119865 structure
Figure 6 A bare homogeneous circle solved with finite elements (218 unknowns)
the normal direction should be zero (ie symmetry condi-tion) and the Robin conditions are restricted to the followingform
grad 120601119892 (x) sdot n + 119886119892 (x) sdot 120601119892 (x) = 0 (6)
n being the outward unit normal to the boundary 120597119880 of thedomain 119880
22 Grids and Schemes One way of solving the neutrondiffusion equationmdashand in general any partial differentialequation over spacemdashis by discretizing the differential oper-ators with some kind of scheme that is applied over a certainspatial grid Given an 119898-dimensional domain a grid definesa partition composed of a finite number of simple geometricentities In structured grids these elementary entities arearranged following a well-defined structure in such a waythat each entity can be identified without needing furtherinformation than the one provided by the intrinsic structureOn the other hand the geometric entities that composean unstructured grid are arranged in an irregular patternand the identification of the elementary entities needs to
be separately specified for example by means of a sortedlist For instance Figure 2(b) shows a structured grid thatapproximates the continuous domain of Figure 2(a) Eachsquare may be uniquely identified by means of two integerindexes that indicate its relative position in each of thehorizontal and vertical directions In the case of Figure 2(c)that shows an unstructured grid there is no way to system-atically make a reference to a particular quadrangle withoutany further information
Almost all of the grid-based schemesmdashwhich are knownas nodal schemes which are to be differentiated from modalschemes based on series expansionsmdashare based in either thefinite differences volumes or elements method Finite differ-ences schemes provide themost simple and basic approach toreplace differential (ie continuous) operators by difference(ie discrete) approximations However they are not suitablefor unstructured meshes and may introduce convergenceproblems with parameters that are discontinuous in spacewhich is the case for any reactor core composed of atleast two different materials Moreover boundary conditionsare hard to incorporate and usually give rise to incorrectresults
Science and Technology of Nuclear Installations 7
20 40 600995
09975
1
Analytical solutionFinite volumesFinite elements
keff
a985747c
Figure 7 The effective multiplication factor of a two-group barecircular reactor of radius 119886 as a function of the quotient 119886ℓ
119888between
the radius and the characteristic length of the cellelement
20 40 60
Finite volumesFinite elements
Erro
r ink
eff
10minus3
10minus4
10minus5
a985747c
Figure 8 Absolute value of the error committed in the computationof 119896eff versus 119886ℓ119888
Methods of the finite volumes family involve the inte-gration of the differential equation over each elementaryentity applying the divergence theorem to transform volumeintegrals into surface integrals and providing a mean to esti-mate the fluxes through the entityrsquos surface using informationcontained in its neighbors In this context each elementaryentity is called a cell and finite volumes methods give themean value of each of the group fluxes 120601119892(119894) at the 119894th cellwhich may be associated to the value of 120601119892(x119894) where x119894 isthe location of the cell barycenter Being an integral-basedmethod spatial-discontinuous parameters are handled moreefficiently than in finite differences and boundary conditionscan be easily incorporated as forced cell fluxes Nonethelesseven though these methods may be applied to unstructuredgrids the estimation of the fluxes on the cellsrsquo surfaces isusually performed by using some geometric approximationsthat may lose validity as the quality of the grid is worsened
10
70
90
130
150
170
10 30 50 70 90 110 130 150 170
4
1 38
36 37353
3 32 33 34312
26 27 28 29 3025
19 20 21 22 23 2418
10 11 12 13 14 15 16 17
2 331 4 5 6 7 8 9
x (cm)
y (cm
)
Figure 9 The 2D IAEA PWR benchmark geometry
Moreover the simple integral approach cannot take intoaccount discontinuous diffusion coefficients so when twoneighbors pertain to different materials the flux has to becomputed in a certain particular way to conserve the neutroncurrent according to (2)
Finally finite elements methods rely on a weak formula-tion of the differential problem similar to (3) that maintainsall the mathematical characteristics of the original strongformulation plus its boundary conditions The method isbased on shape functions that are local to each elementaryentitymdashnow called element defined by nodes as cornersmdashand on finding a set of nodal values such that a certaincondition is met which is usually that the residual of thesolution has to be orthogonal to each of the shape functionsThis condition is known as the Galerkin method and impliesthat the error committed in the approximate solution ofthe continuous problem is confined into a small subset ofthe original vector space of the continuous functions Thesemathematical properties make finite elements schemes veryattractive However these features depend on a large numberof integrations that ought to be performed numerically soa computational effortdesired accuracy tradeoff has to beconsidered Not only does the finite elements method givethe values of the flux 120601119892(x119894) at the 119894th node but also theshape functions provide explicit expressions to interpolateand to evaluate the unknown functions at any location x ofthe domain119880 Boundary conditions are divided into essentialand naturalThefirst group comprises theDirichlet boundaryconditions which are satisfied exactlymdashwithin the precisionof the eigenvalue problem solvermdashby the obtained solutionThe latter include the Neumann and the Robin conditionsthat are satisfied only approximately by the derivatives of theinterpolated solution through the shape functions with finermeshes giving better agreement with the prescribed values
The same unstructured grid may be used either for thefinite volumes or for the finite elements method In the first
8 Science and Technology of Nuclear Installations
Largest eigenvalue 1029690 (288336 pcm)1135 (3066 3079)1306 (5171 13076)
Number of unknowns 15740Outer iterations 3Linear iterations 32Inner iterations 1967Residual normRelative errorError estimateMemory used 49824 kBSoft page faults 13401Hard page faults 0Total CPU time 074 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
2483 times 10minus8
1223 times 10minus8
403 times 10minus9
keff
quarter-symmetry core meshed using Delaunay (triangles 985747c = 3) solved with finite volumesMilongarsquos 2D LWR IAEA benchmark problem case no 002
(a)
123456789
1011121314151617181920212223242526272829303132333435363738
k Pk 1206011k 1206012k075 3282 555133 4242 984147 4645 1090123 3921 911061 2677 453094 2995 694093 2927 689074 2011 549mdash 322 756146 4597 1078150 4730 1110133 4199 985107 3405 789104 3271 767095 2971 700072 1956 534mdash 306 733149 4695 1102136 4289 1007118 3733 876107 3367 792097 2892 718066 1628 491mdash 234 549120 3797 891097 3096 718090 2842 669084 2221 619mdash 568 1215mdash 067 257047 2042 348068 2055 504058 1414 430mdash 222 562057 1374 425mdash 388 820mdash 053 203mdash 060 229
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 10 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
case the unknowns are themean value of the fluxes over eachcell whilst in the latter the unknowns are the fluxes evaluatedat each node Therefore the number of unknowns 119873119866 isdifferent for each method even when using the same gridFigure 4 shows this situation and in Section 31 we further
illustrate these differences A fully detailed mathematicaldescription of the actual algorithms for both volumes andelements-based proposed discretizations can be found in anacademic monograph written by the author of this paper[13]
Science and Technology of Nuclear Installations 9
Largest eigenvalue 1029927 (290571 pcm)1131 (3175 2999)1220 (13097 5098)
Number of unknowns 15576Outer iterations 3Linear iterations 32Inner iterations 1947Residual normRelative errorError estimateMemory used 56256 kBSoft page faults 15645Hard page faults 0Total CPU time 09201 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
1394 times 10minus8
6865 times 10minus9
3425 times 10minus9
keff
quarter-symmetry core meshed using Delaunay (quads 985747c = 2) solved with finite volumesMilongarsquos 2D LWR IAEA benchmark problem case no 013
(a)
123456789
1011121314151617181920212223242526272829303132333435363738
k Pk 1206011k 1206012k074 3244 549133 4230 982147 4651 1092124 3946 917060 2645 443094 2997 696093 2938 691074 2005 545mdash 301 746145 4588 1076150 4720 1108133 4198 984109 3465 805104 3280 769095 2986 704071 1962 528mdash 293 715148 4680 1098136 4288 1007119 3755 881107 3385 796097 2905 721067 1653 493mdash 220 556121 3808 893098 3115 724092 2889 680083 2247 615mdash 543 1241mdash 067 260045 1977 330069 2076 509058 1460 428mdash 226 574056 1364 413mdash 371 821mdash 052 200mdash 060 238
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 11 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
3 Results
We now proceed to show two illustrative results that are tobe taken as an overview of the possibilities that unstructuredgrids can provide in order to tackle the multigroup neutron
diffusion problem The examples are two-dimensional prob-lems as they contain some of the complexities a real three-dimensional reactor posse yet the reported results are notso complicated as to be easily understood and analyzed Inparticular we state some basic differences between the finite
10 Science and Technology of Nuclear Installations
Largest eigenvalue 1029695(288388 pcm)1112 (3076 3032)838 (5000 13000)
Number of unknowns 15922Outer iterations 3Linear iterations 32Inner iterations 1990Residual normRelative errorError estimateMemory used 109964 kBSoft page faults 30591Hard page faults 0Total CPU time 158 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
1056 times 10minus8
5202 times 10minus9
5122 times 10minus9
keff
quarter-symmetry core meshed using Delaunay (quads 985747c = 2) solved with finite volumesMilongarsquos 2D LWR IAEA benchmark problem case no 018
(a)
123456789
1011121314151617181920212223242526272829303132333435363738
k Pk 1206011k 1206012k074 3220 549130 4152 961144 4651 1068120 3843 890061 2648 452094 2996 693094 2959 696072 2061 569mdash 358 862142 4495 1054147 4634 1088131 4127 968107 3409 789104 3275 768096 3003 708071 2014 554mdash 342 820146 4606 1081134 4222 991118 3711 871107 3375 794098 2929 727062 1692 522mdash 258 632119 3752 880096 3084 715091 2858 673080 2280 633mdash 611 1277mdash 080 318047 2043 350069 2088 510054 1463 450mdash 255 648051 1418 439mdash 410 854mdash 064 252mdash 071 285
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 12 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
volumes and the finite elements methods by solving a two-group homogeneous bare reactor with the Robin boundaryconditions over the very same grid although a rather coarseone so the differences can be observed directly into theresulting figuresWe then solve the classical two-dimensionalLWR problem also known as the 2D IAEA PWR benchmark
Not only do we show again the differences between the finitevolumes and elements formulation but also we solve theproblemusing different combinations ofmeshing algorithmsbasic shapes and characteristic lengths of the mesh
To solve the two examples shown below we employed themilonga code which was written from scratch by the author
Science and Technology of Nuclear Installations 11
Largest eigenvalue 1029159 (283326 pcm)1174 (2851 3224)1560 (5200 13200)
Number of unknowns 3132Outer iterations 3Linear iterations 32Inner iterations 391Residual normRelative errorError estimateMemory used 26888 kBSoft page faults 7322Hard page faults 0Total CPU time 0308 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
5408 times 10minus8
2665 times 10minus8
8424 times 10minus9
keff
Milongarsquos 2D LWR IAEA benchmark problem case no 031quarter-symmetry core meshed using Delquad (quads 985747c = 4) solved with finite volumes
(a)
123456789
1011121314151617181920212223242526272829303132333435363738
k Pk 1206011k 1206012k077 3422 570140 4468 1040153 4836 1135128 4070 948060 2704 448095 3017 702091 2861 673069 1918 508mdash 264 689151 4785 1122155 4897 1149137 4334 1017111 3535 823104 3272 768092 2908 685067 1876 496mdash 254 658154 4854 1139139 4392 1031120 3792 890107 3360 790095 2831 702061 1574 451mdash 197 515123 3868 907099 3132 730090 2818 663078 2168 579mdash 481 1191mdash 055 217044 1984 328067 2027 498052 1360 388mdash 190 520052 1329 383mdash 321 792mdash 043 168mdash 048 195
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 13 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
of this paper and is currently still under development withinhis ongoing PhD thesis There exists a first public release[14] under the terms of the GNU General Public Licensemdashthat is it is a free softwaremdashthat can only handle structuredgrids There is a second release being prepared whose
main relevance is that it can work with nonstructured gridsas well which is the main feature of the code It uses ageneral mathematical frameworkmdashcoded from scratch aswellmdashwhich provides input file parsing algebraic expres-sions evaluation one- and multidimensional interpolation of
12 Science and Technology of Nuclear Installations
Largest eigenvalue 1029788 (289260 pcm)1120 (3149 3011)1091 (13038 5114)
Number of unknowns 15776Outer iterations 3Linear iterations 32Inner iterations 1972Residual normRelative errorError estimateMemory used 56996 kBSoft page faults 15436Hard page faults 0Total CPU time 09121 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
2296 times 10minus12
1131 times 10minus12
6877 times 10minus13
keff
Milongarsquos 2D LWR IAEA benchmark problem case no 043eighth-symmetry core meshed using Delaunaya (triangs985747c = 2) solved with finite volumes
(a)
1 074 3241 5482 131 4188 9713 145 4582 10764 122 3877 9005 060 2644 4456 093 2986 6927 093 2941 6928 075 2030 5529 mdash 322 779
10 143 4523 106111 148 4667 109512 131 4145 97213 107 3423 79414 103 3269 76715 095 2993 70616 073 1988 53917 mdash 312 75118 147 4637 108819 134 4243 99620 118 3721 87321 107 3370 79322 098 2914 72323 067 1651 49924 mdash 235 57125 119 3763 88226 096 3077 71527 091 2847 67128 083 2255 61729 mdash 576 122530 mdash 068 26631 046 2016 34132 068 2058 50533 059 1443 43534 mdash 233 58935 057 1381 42036 mdash 381 82337 mdash 054 20838 mdash 062 244
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 14 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
scattered data shared-memory access numerical integrationfacilities and so forth The milonga code was designed withfour design basis vectors in mindmdashwhich are thoroughlydiscussed in the documentation [15]mdashwhich includes thetype of problems it can handle the code scalability which
features are expected and what to do with the obtainedresults
It works by first reading an input file that using plain-text English keywords and arguments defines the number119898 of spatial dimensions the number 119866 of group energies
Science and Technology of Nuclear Installations 13
Largest eigenvalue 1029726 (288675 pcm)1104 (3069 3069)864 (13000 5000)
Number of unknowns 4040Outer iterations 2Linear iterations 24Inner iterations 505Residual normRelative errorError estimateMemory used 42940 kBSoft page faults 11437Hard page faults 0Total CPU time 1064 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
2019 times 10minus8
9946 times 10minus9
7138 times 10minus9
keff
Milongarsquos 2D LWR IAEA benchmark problem case no 047eighth-symmetry core meshed using Delaunay (triangles 985747c = 3) solved with finite volumes
(a)
1 074 3194 5452 129 4117 9533 143 4515 10604 119 3814 8835 061 2634 4506 093 2984 6907 094 2954 6958 071 2064 5719 mdash 362 865
10 141 4458 104511 146 4599 107912 130 4097 96113 106 3388 78414 103 3263 76515 095 2998 70716 068 2015 55717 mdash 346 82318 145 4572 107319 133 4194 98420 117 3691 86621 107 3363 79222 098 2925 72623 059 1696 52824 mdash 263 63525 118 3728 87426 096 3068 71127 091 2850 67228 079 2280 63529 mdash 619 127630 mdash 081 32131 047 2036 34932 068 2085 50933 051 1465 45434 mdash 258 65035 049 1419 44336 mdash 416 85437 mdash 065 25438 mdash 071 288
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 15 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
and optionally a mesh file Currently only grids generatedwith the code gmsh [16] are only supported mainly becauseit is also a free software and it suits perfectly well milongarsquosdesign basis in the sense that the continuous geometry can bedefined as a function of a number of parameters Afterwardthe physical entities defined in the grid are mapped into
materials with macroscopic cross sections which maydepend on the spatial coordinates x bymeans of intermediatefunctions such as burn-up or temperatures distributionswhich in turn may be defined by algebraic expressions byinterpolating data located in files by reading shared-memoryobjects or by a combination of them In the same sense
14 Science and Technology of Nuclear Installations
Largest eigenvalue 1029695 (288387 pcm)1108 (3045 3045)833 (13000 5000)
Number of unknowns 8234Outer iterations 3Linear iterations 32Inner iterations 1029Residual normRelative error
Max 1206012(x y) coreMax 1206012(x y) reflector
2028 times 10minus12
9993 times 10minus13
1012 times 10minus12
keff
Relative errorError estimateMemory used 75468 kBSoft page faults 20094Hard page faults 0Total CPU time 1256 seconds
9993 times 10
1012 times 10minus12
Milongarsquos 2D LWR IAEA benchmark problem case no 058eighth-symmetry core meshed using Delaunay (quads 985747c = 2) solved with finite volumes
(a)
1 074 3207 5472 129 4136 9573 144 4534 10644 120 3828 8875 061 2638 4506 093 2985 6907 094 2948 6948 072 2053 5669 mdash 357 859
10 142 4478 105011 146 4617 108412 130 4111 96413 106 3396 78614 103 3263 76515 095 2991 70616 071 2005 55217 mdash 340 81618 145 4588 107719 133 4206 98720 117 3698 86821 107 3362 79122 098 2918 72523 062 1686 52024 mdash 258 62925 118 3737 87626 096 3073 71227 091 2848 67128 080 2272 63129 mdash 609 127230 mdash 080 31731 047 2035 34832 069 2081 50933 054 1458 44834 mdash 255 64635 052 1413 43736 mdash 409 85137 mdash 064 25138 mdash 071 284
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 16 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
boundary conditions are applied to appropriate physicalentities of dimension119898minus1The problemmatrices119877 and119865 arethen built and stored in an appropriate sparse format usingthe free PETSc [17 18] library and the eigenvalue problemis solved using the free SLEPc [19] library The results arestored into milongarsquos variables and functions which may be
evaluated integratedmdashusually using the free GNU ScientificLibrary [20] routinesmdashand of course written into appropriateoutputs Milonga can also solve problems parametrically andbe used to solve optimization problems As stated above thecode is a free software so corrections and contributions aremore than welcome
Science and Technology of Nuclear Installations 15
Largest eigenvalue 1029462 (286185 pcm)1136 (3093 2953)1233 (13100 5100)
Number of unknowns 6228Outer iterations 2Linear iterations 24Inner iterations 778Residual normR l ti
Max 1206012(x y) coreMax 1206012(x y) reflector
26 times 10minus8
1281 times 10minus8
4996 times 10minus9
keff
Relative errorError estimateMemory used 41460 kBSoft page faults 10930Hard page faults 0Total CPU time 0604 seconds
1281 times 10 8
4996 times 10minus9
Milongarsquos 2D LWR IAEA benchmark problem case no 073eighth-symmetry core meshed using Delaunay (quads 985747c = 2) solved with finite volumes
(a)
1 075 3297 5552 134 4268 9923 148 4660 10944 123 3926 9145 059 2645 4406 094 2997 6977 093 2914 6858 072 2002 5349 mdash 301 773
10 146 4606 108011 150 4735 111112 133 4212 98813 108 3448 80214 104 3271 76715 094 2954 69616 070 1954 52117 mdash 288 73518 149 4691 110119 136 4289 100720 119 3748 88021 107 3359 79022 096 2880 71423 064 1636 47524 mdash 217 56325 121 3810 89426 098 3110 72427 090 2829 66628 081 2227 59929 mdash 536 125730 mdash 062 25031 045 2002 33632 068 2040 50033 055 1414 40934 mdash 215 57935 055 1359 40836 mdash 359 83437 mdash 050 19938 mdash 061 236
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 17 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
31 A Coarse Bare Homogeneous Circle Figures 5 and 6 showthe results of solving a bare two-dimensional circular homo-geneous reactor of radius 119886 over an unstructured grid whichis deliberately coarse using the finite volumes method andthe finite elements method respectively Two group energies
were used and a null-flux boundary conditionwas fixed at theexternal surface Figures 5(a) and 6(a) compare the obtainedfast flux distributions In the first case the numerical solutionprovidesmean values for each neutron flux group in each cellwhile in the latter the solution is computed at the nodes and
16 Science and Technology of Nuclear Installations
Largest eigenvalue 1029851 (289858 pcm)1090 (3182 3182)851 (13000 5000)
Number of unknowns 1752Outer iterations 3Linear iterations 32Inner iterations 219Residual norm
Max 1206012(x y) coreMax 1206012(x y) reflector
566 times 10minus12
2789 times 10minus12
1595 times 10minus12
keff
Residual normRelative errorError estimateMemory used 42564 kBSoft page faults 11213Hard page faults 0Total CPU time 09921 seconds
566 times 10
2789 times 10minus12
1595 times 10minus12
Milongarsquos 2D LWR IAEA benchmark problem case no 076eighth-symmetry core meshed using Delaunay (quads 985747c = 4) solved with finite volumes
(a)
1 073 3151 5402 127 4059 9393 141 4457 10464 118 3768 8715 061 2615 4506 093 2979 6887 094 2971 6998 067 2092 5829 mdash 375 885
10 139 4397 103111 144 4541 106612 128 4052 95013 105 3365 77814 103 3259 76415 096 3016 71216 066 2045 56817 mdash 359 84118 143 4520 106119 132 4155 97520 116 3672 86221 107 3366 79222 098 2951 73323 054 1728 54324 mdash 273 65125 117 3700 86826 095 3057 70827 091 2860 67428 074 2311 64829 mdash 642 130030 mdash 084 33231 048 2041 35332 069 2103 51333 046 1493 46734 mdash 269 66735 046 1444 45536 mdash 432 87137 mdash 067 26338 mdash 074 297
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 18 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
continuous functions are evaluated by means of the shapefunctions used in the formulation Figures 5(b) and 6(b)illustrate the fast fluxes unknowns and its relative position inspace It can be noted that the mesh coarseness gives resultsthat differ substantially in both cases Finally the structure ofthe sparse eigenvalue problemmatrices is shown for each case
with blue red and cyan representing positive negative andexplicitly inserted zero values In the finite volume case thereare 184 unknowns (92 cells times 2 groups) while in the secondcase there are 218 unknowns (109 nodes times 2 groups) Thevolumesrsquo fissionmatrix is almost diagonal it has a bandwidthequal to the number of energy groups which in this case is
Science and Technology of Nuclear Installations 17
500 1000 2000 5000 10000 20000 50000Number of unknowns
2840
2860
2880
2900
2920
ORNLRisoslashKWUOntarioQuarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumes
Quarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elementsEighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumes
Eighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
120588(p
cm)
105 2 times 105 5 times 105
Figure 19 Static reactivity versus number of unknowns The four original solutions as published in 1977 [10] are included as reference
twoThe off-diagonal values appear right next to the diagonalelements because of the chosen ordering of the unknownsin the flux vector 120601 isin R184 Other orderings may be usedbut the rate of convergence of the eigenvalue problem can bedeteriorated On the other hand the elementsrsquo fission matrixhas a nontrivial structure because the grid is unstructuredand the net fission rate at each element depends on the fluxesat the nodes whose location inside thematrix depends in turnon how the gridwas generatedThis effect is similar to the onethat appears in structural analysis where mass matrices needto be lumped in order to simplify transient computations [21]The diagonal block of elementrsquos 119877 matrix and the null blockin 119865 correspond to the discrete equations that set the null-flux boundary conditions on the nodes located at the externalsurface Other types of boundary conditions lead to differentkinds of structures within the problem matrices
In case a part of the domain contained a nonmultiplicativematerial such as a reflector then there would appear sectionsof the fission matrix with null values in both methodsrendering 119865 singular in both methods Care should be takenwhen dealing with the numerical schemes for the eigenvalueproblem solution It can be seen in the elementsrsquo matricesa particular structure that implements the boundary condi-tions at the external surfaces This structure does not appearin the volumesrsquo matrices because the boundary conditions
appear as flux terms which are summed up over all thesurfaces of each cell so they aremasked inside the volumetricdiscretization of the divergence and gradient operators
As the two-group neutron diffusion equation with uni-form cross sections over a circle subject to null-flux bound-ary conditions has an analytical solution it is adequate tocompare how the two proposed numerical schemes relate toit In the studied problem we ignored upscattering and fastfissions Then the analytical effective multiplication factor is
119896eff =]Σ1198912 sdot Σ1199041rarr2
[Σ1198861 + Σ1199041rarr2 + 1198631(]0119886)2] [Σ1198862 + 1198632(]0119886)
2]
(7)
being ]0 = 24048 the smallest root of Besselrsquos first-kindfunction of order zero 1198690(119903)
Figure 7 shows how the two numerical effective multi-plication factors compare to the analytical solution givenby (7) as a function of the mesh refinement indicated bythe quotient 119886ℓ119888 between the radius of the circle and thecharacteristic length of the cellelements We can see thatthe 119896eff computed by the finite volumes (elements) methodis always greater (less) than the analytical solution Indeedit can be proven that for a bare one-dimensional slab thisis always the case [13] However this result does not hold
18 Science and Technology of Nuclear Installations
2000 3000 6000 10000 20000 30000 60000
01
003
001
03
1
3
10
30
100
300To
tal w
all t
ime (
seco
nds)
Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
Figure 20 Total wall time versus number of unknowns
for problems with nonuniform cross sections even in simpleone-dimensional reflected reactors
We may draw two other conclusions from Figure 7 Firstthat the finite element method seems to provide a bettersolution than the volumes-based scheme and second thateven though the error committed tends to decrease with finergrids its behavior is not monotonic for finite volumes Infact Figure 8 shows the difference between the numericaland analytical solutions using a logarithmic vertical scalewhere both conclusions are even more evident We deferthe discussion of such differences between the discretizationscheme until the next section It is worth to note howeverthat the fact that a finite-element-based scheme throws betterresults for the particular bare homogeneous circular reactorunder study than the finite volumes does not imply thatfinite volumes ought to be discarded as a valid tool forsolving the neutron diffusion equation in general cases Thecombination of lattice and core-level computations is usuallyperformedusing cell-based resultswhichwhen fed into node-based methods to solve the few-group neutron diffusionequation may introduce errors which potentially can leadto unacceptable solutions Nevertheless this analysis is farbeyond the scope of this paper which focuses on solving amathematical equation over unstructured grids
32 The 2D IAEA PWR Benchmark Problem This is aclassical two-group neutron diffusion problem first designedin the early 1970s and taken as a reference benchmark forcomputational codes A number of codes were used to solveeither this problem or its three-dimensional formulation[22 23] including milonga using a structured grid [24] Theoriginal formulation can be found in the reference [10] andthere is a reproduction that may be easily found online inreference [24] The geometry consists of a one-quarter ofa PWR core depicted in Figure 9 and the homogeneousmacroscopic cross sections are listed in Table 1 An axialbuckling term 119861
2
119911119892= 08times10
minus4 should be taken into accountThe external surface should be subject to a zero incomingcurrent condition which may be written as
120597120601119892
120597119899= minus
04692
119863119892
sdot 120601119892 (8)
The expected results are as follows
(1) Maximum eigenvalue(2) Fundamental flux distributions
(a) Radial flux traverses 120601119892(119909 0) and 120601119892(119909 119909)
Science and Technology of Nuclear Installations 19
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
1
3
10
30
100
300Ti
me c
onsu
med
to m
esh
the g
eom
etry
(sec
onds
)
Figure 21 Time needed to mesh the geometry versus number of unknowns
Table 1 Macroscopic cross-sections (units are not stated in the original reference but they are assumed to be in cmminus1 or cm as appropriate)
1198631
1198632
Σ1rarr2
Σ1198861
Σ1198862
]Σ1198912
Material1 15 04 002 001 0080 0135 Fuel 12 15 04 002 001 0085 0135 Fuel 23 15 04 002 001 0130 0135 Fuel 2 + rod4 20 03 004 0 0010 0 Reflector
Note the fluxes shall be normalized such that
1
119881coreint119881core
sum
119892
]Σ119891119892 sdot 120601119892119889119881 = 1 (9)
(b) Value and location of maximum power densityThis corresponds to maximum of 1206012 in thecore It is recommended that the maximumvalues of 1206012 both in the inner core and at thecorereflector interface be given
(3) Average subassembly powers 119875119896
119875119896 =1
119881119896
int119881119896
sum
119892
]Σ119891119892 sdot 120601119892119889119881 (10)
where 119881119896 is the volume of the 119896th subassembly and119896 designates the fuel subassemblies as shown inFigure 9
(4) Number of unknowns in the problem number ofiterations and total and outer
(5) Total computing time iteration time IO-time andcomputer used
(6) Type and numerical values of convergence criteria(7) Table of average group fluxes for a square mesh grid
of 20 times 20 cm(8) Dependence of results on mesh spacing
Even though the original problem is based on a quarter-core situation the problem has an eighth-core symmetry
20 Science and Technology of Nuclear Installations
Quarter Delaunay triangles elements
Tim
e con
sum
ed to
read
the m
esh
(sec
onds
)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumes
Quarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
1
3
10
30
100
300
Figure 22 Time needed to read the mesh versus number of unknowns
which cannot be taken into account by structured gridswhich are the main target of the benchmark Howevernonstructured grids can take into consideration any kind ofsymmetry almost without loss of accuracy and at the sametime reducing roughly the number of unknowns by a halfand the associated computational effort needed to solve theproblem by a factor of four Answers to items (1)ndash(7) canbe given completely by milonga using a single input file(see Supplementary data in SupplementaryMaterial availableonline at httpdxdoiorg1011552013641863) As asked initem (8) how results depend not only on the mesh spacingbut also on the meshing algorithm on the gridrsquos elementarygeometric shape and on the discretization scheme may shedlights on the subject whichmay be evenmore interesting thanthe numerical results themselves
Taking advantage of milongarsquos capability of reading andparsing command-line arguments the selection of the coregeometry (quarter or eighth) the meshing algorithm (delau-nay [16] or delquad [25]) the shape of the elementaryentities (triangles or quadrangles) the discretization scheme(volumes or elements) and the characteristic length ℓ119888 ofthe mesh can be provided at run time Fixing five values forℓ119888 = 4 3 2 1 05 gives 2 times 2 times 2 times 2 times 5 = 80 possiblecombinations which we solve with successive invocations to
milonga with the same input file but with different argumentsfrom a simple script Figures 10 11 12 13 14 15 16 17 and18 show the results corresponding to items (1)ndash(7) for someillustrative cases The complete set of figures and the codeused to generate themmay be provided upon request Table 2compiles the answers to the problem for every case studied
As the milonga code is still under development itsnumerical routines are not yet fully optimized nor designedfor parallel computation Therefore the reported times areonly rough estimates and should be taken with care Thesolution comprises five steps
(1) generate the grid with the requested geometry mesh-ing algorithm basic shape and characteristic lengthby calling to gmsh
(2) read the generated mesh(3) build the matrices(4) solve the eigenvalue problem(5) compute the requested results
The CPU time reported in Figures 10ndash18 is thus thesum of all of these five steps but not the time needed togenerate the figuresmdashwhich in some caseswith coarsemeshes
Science and Technology of Nuclear Installations 21
Table2
Resultsofthe2
DIAEA
PWRBe
nchm
arkp
roblem
obtained
with
them
ilong
acod
efor
thee
ightyp
ropo
sedcombinatio
nsofsymmetrym
eshing
algorithm
basicshapediscretization
schemeandcharacteris
ticelem
entc
elllengthTh
ereference
solutio
nis120588=28749times10minus5w
hich
correspo
ndsto119896eff=10296
Case
no
Symm-
etry
Mesh
algorithm
Basic
shape
Solutio
nmetho
dℓ119888
(cm)
Δ120588
(pcm
)max1206012
(mdash)
Total
unkn
owns
Outer
iter
Linear
iter
Inner
iter
Resid
ual
norm
Relativ
eerror
Error
estim
ate
Mem
ory
(Mb)
Page
faults
114
Delaunay
998779Vo
lumes
40
minus120
1145
8264
332
1033
6119890minus09
3119890minus09
1119890minus09
349470
214
Delaunay
998779Vo
lumes
30
85
1135
15740
332
1967
2119890minus08
1119890minus08
4119890minus09
4813401
314
Delaunay
998779Vo
lumes
20
197
1124
31188
332
3898
1119890minus08
6119890minus09
3119890minus09
7621932
414
Delaunay
998779Vo
lumes
10153
1123
127476
332
15934
7119890minus09
3119890minus09
3119890minus09
273
81886
514
Delaunay
998779Vo
lumes
05
187
1119
510676
332
63834
2119890minus09
9119890minus10
1119890minus09
1148
352261
614
Delaunay
998779Elem
ents
40
173
1100
4308
332
538
2119890minus08
8119890minus09
4119890minus09
318794
714
Delaunay
998779Elem
ents
30
117
1107
8108
332
1013
5119890minus09
2119890minus09
2119890minus09
4712946
814
Delaunay
998779Elem
ents
20
84
1112
15936
332
1992
6119890minus09
3119890minus09
2119890minus09
7321347
914
Delaunay
998779Elem
ents
1062
1116
64478
332
8059
6119890minus09
3119890minus09
4119890minus09
262
77105
1014
Delaunay
998779Elem
ents
05
58
1117
256700
332
32087
1119890minus09
5119890minus10
1119890minus09
1091
331969
1114
Delaunay
◻Vo
lumes
40
53
1149
5042
332
630
2119890minus08
1119890minus08
5119890minus09
288007
1214
Delaunay
◻Vo
lumes
30
346
1133
8560
332
1070
7119890minus09
3119890minus09
2119890minus09
3910895
1314
Delaunay
◻Vo
lumes
20
308
1131
15576
332
1947
1119890minus08
7119890minus09
3119890minus09
541564
514
14Delaunay
◻Vo
lumes
10177
1122
60774
332
7596
8119890minus09
4119890minus09
3119890minus09
167
50115
1514
Delaunay
◻Vo
lumes
05
199
1120
244936
332
30617
4119890minus09
2119890minus09
2119890minus09
698
215678
1614
Delaunay
◻Elem
ents
40
196
1100
5222
332
652
3119890minus08
1119890minus08
8119890minus09
4713098
1714
Delaunay
◻Elem
ents
30
123
1107
8810
332
1101
2119890minus08
9119890minus09
7119890minus09
6918598
1814
Delaunay
◻Elem
ents
20
901112
15922
332
1990
1119890minus08
5119890minus09
5119890minus09
107
30591
1914
Delaunay
◻Elem
ents
1063
1116
61456
332
7682
5119890minus09
2119890minus09
4119890minus09
389
113237
2014
Delaunay
◻Elem
ents
05
58
1117
246332
332
30791
8119890minus10
4119890minus10
1119890minus09
1676
498192
2114
Delq
uad
998779Vo
lumes
40
minus45
1144
6228
332
778
4119890minus08
2119890minus08
7119890minus09
308495
2214
Delq
uad
998779Vo
lumes
30
570
1153
11820
332
1477
3119890minus08
1119890minus08
4119890minus09
4010879
2314
Delq
uad
998779Vo
lumes
20
459
1103
24100
332
3012
2119890minus08
1119890minus08
5119890minus09
6217715
2414
Delq
uad
998779Vo
lumes
10512
1104
9640
03
3212050
2119890minus08
1119890minus08
9119890minus09
207
61714
2514
Delq
uad
998779Vo
lumes
05
528
1106
385600
332
48200
2119890minus09
1119890minus09
2119890minus09
838
255735
2614
Delqu
ad998779
Elem
ents
40
220
1093
3288
332
411
3119890minus08
2119890minus08
6119890minus09
308495
2714
Delq
uad
998779Elem
ents
30
140
1104
6148
332
768
4119890minus08
2119890minus08
8119890minus09
39110
0028
14Delq
uad
998779Elem
ents
20
82
1110
12392
332
1549
2119890minus08
9119890minus09
6119890minus09
6117067
2914
Delqu
ad998779
Elem
ents
1063
1115
48882
332
6110
2119890minus09
1119890minus09
1119890minus09
197
5804
630
14Delq
uad
998779Elem
ents
05
58
1116
194162
332
24270
2119890minus09
9119890minus10
2119890minus09
790
238769
3114
Delq
uad
◻Vo
lumes
40
minus416
1174
3132
332
391
5119890minus08
3119890minus08
8119890minus09
267322
3214
Delq
uad
◻Vo
lumes
30
minus248
1151
5922
332
740
9119890minus09
4119890minus09
2119890minus09
318779
3314
Delq
uad
◻Vo
lumes
20
minus132
1139
12050
332
1506
1119890minus08
7119890minus09
4119890minus09
4412239
3414
Delq
uad
◻Vo
lumes
10minus49
1126
48200
332
6025
6119890minus09
3119890minus09
2119890minus09
122
36094
3514
Delq
uad
◻Vo
lumes
05
minus23
1123
192800
332
24100
5119890minus09
2119890minus09
3119890minus09
464
140228
3614
Delq
uad
◻Elem
ents
40
224
1093
3294
332
411
3119890minus08
1119890minus08
7119890minus09
359852
3714
Delq
uad
◻Elem
ents
30
141
1104
6144
332
768
3119890minus08
2119890minus08
9119890minus09
5114018
3814
Delq
uad
◻Elem
ents
20
921111
12392
332
1549
1119890minus08
6119890minus09
5119890minus09
8122526
3914
Delq
uad
◻Elem
ents
1065
1115
48882
332
6110
5119890minus09
3119890minus09
4119890minus09
276
78431
22 Science and Technology of Nuclear Installations
Table2Con
tinued
Case
no
Symm-
etry
Mesh
algorithm
Basic
shape
Solutio
nmetho
dℓ119888
(cm)
Δ120588
(pcm
)max1206012
(mdash)
Total
unkn
owns
Outer
iter
Linear
iter
Inner
iter
Resid
ual
norm
Relativ
eerror
Error
estim
ate
Mem
ory
(Mb)
Page
faults
4014
Delq
uad
◻Elem
ents
05
59
1117
194162
332
24270
1119890minus09
5119890minus10
1119890minus09
1104
318967
4118
Delaunay
998779Vo
lumes
40
minus99
1139
4228
332
528
4119890minus12
2119890minus12
1119890minus12
359630
4218
Delaunay
998779Vo
lumes
30
50
1132
7712
332
964
1119890minus11
6119890minus12
2119890minus12
4211578
4318
Delaunay
998779Vo
lumes
20
177
1120
15776
332
1972
2119890minus12
1119890minus12
7119890minus13
5515436
4418
Delaunay
998779Vo
lumes
10142
1118
63902
332
7987
3119890minus12
1119890minus12
1119890minus12
160
46691
4518
Delaunay
998779Vo
lumes
05
176
1115
254612
332
31826
2119890minus12
8119890minus13
1119890minus12
555
168526
4618
Delaunay
998779Elem
ents
40
183
1096
2252
332
281
6119890minus12
3119890minus12
2119890minus12
359589
4718
Delaunay
998779Elem
ents
30
119
1104
4040
224
505
2119890minus08
1119890minus08
7119890minus09
41114
3748
18Delaunay
998779Elem
ents
20
85
1108
8156
332
1019
2119890minus12
8119890minus13
6119890minus13
5515053
4918
Delaunay
998779Elem
ents
1063
1112
32480
332
4060
1119890minus12
6119890minus13
8119890minus13
151
43671
5018
Delaunay
998779Elem
ents
05
58
1113
128358
332
1604
47119890minus13
3119890minus13
7119890minus13
527
157581
5118
Delaunay
◻Vo
lumes
40
37
1144
2380
224
297
5119890minus08
2119890minus08
1119890minus08
5314173
5218
Delaunay
◻Vo
lumes
30
218
1120
4194
332
524
7119890minus12
4119890minus12
2119890minus12
3610034
5318
Delaunay
◻Vo
lumes
20
345
1110
7960
332
995
5119890minus12
3119890minus12
1119890minus12
4712879
5418
Delaunay
◻Vo
lumes
10165
1117
30652
332
3831
2119890minus12
1119890minus12
7119890minus13
100
28851
5518
Delaunay
◻Vo
lumes
05
191
1117
122066
224
15258
2119890minus08
8119890minus09
9119890minus09
343
103825
5618
Delaunay
◻Elem
ents
40
175
1097
2526
332
315
4119890minus12
2119890minus12
1119890minus12
6016159
5718
Delaunay
◻Elem
ents
30
114
1104
4386
332
548
4119890minus12
2119890minus12
2119890minus12
5013768
5818
Delaunay
◻Elem
ents
20
901108
8234
332
1029
2119890minus12
1119890minus12
1119890minus12
7320094
5918
Delaunay
◻Elem
ents
1062
1112
31186
224
3898
2119890minus08
7119890minus09
9119890minus09
208
59589
6018
Delaunay
◻Elem
ents
05
58
1113
123122
224
15390
9119890minus09
5119890minus09
1119890minus08
807
2366
7061
18Delq
uad
998779Vo
lumes
40
minus210
1168
3224
332
403
1119890minus11
5119890minus12
2119890minus12
43117
8462
18Delq
uad
998779Vo
lumes
30
369
1150
6000
332
750
1119890minus11
6119890minus12
3119890minus12
5013502
6318
Delq
uad
998779Vo
lumes
20
312
1104
12316
332
1539
2119890minus11
8119890minus12
5119890minus12
6016395
6418
Delq
uad
998779Vo
lumes
1044
01094
48714
332
6089
9119890minus12
4119890minus12
4119890minus12
1183444
165
18Delq
uad
998779Vo
lumes
05
534
1081
193980
332
24247
5119890minus12
2119890minus12
4119890minus12
423
126768
6618
Delq
uad
998779Elem
ents
40
240
1090
1750
332
218
9119890minus12
4119890minus12
2119890minus12
43117
8667
18Delqu
ad998779
Elem
ents
30
150
1100
3184
332
398
1119890minus11
5119890minus12
2119890minus12
5013509
6818
Delq
uad
998779Elem
ents
20
89
1106
6426
332
803
5119890minus12
3119890minus12
2119890minus12
5916047
6918
Delq
uad
998779Elem
ents
1064
1111
24886
332
3110
2119890minus12
1119890minus12
1119890minus12
113
32788
7018
Delq
uad
998779Elem
ents
05
59
1113
98052
224
12256
1119890minus08
5119890minus09
8119890minus09
398
118153
7118
Delq
uad
◻Vo
lumes
40
minus398
1172
1650
332
206
1119890minus11
5119890minus12
2119890minus12
369907
7218
Delq
uad
◻Vo
lumes
30
minus249
1146
3058
332
382
4119890minus12
2119890minus12
1119890minus12
349349
7318
Delq
uad
◻Vo
lumes
20
minus131
1136
6228
224
778
3119890minus08
1119890minus08
5119890minus09
4010930
7418
Delq
uad
◻Vo
lumes
10minus51
1123
24536
224
3067
3119890minus08
2119890minus08
1119890minus08
7822716
7618
Delq
uad
◻Elem
ents
40
237
1090
1752
332
219
6119890minus12
3119890minus12
2119890minus12
4111213
7718
Delq
uad
◻Elem
ents
30
145
1101
3184
332
398
6119890minus12
3119890minus12
2119890minus12
43118
1178
18Delq
uad
◻Elem
ents
20
941107
6426
332
803
3119890minus12
1119890minus12
1119890minus12
6016220
7918
Delqu
ad◻
Elem
ents
1065
1111
24874
332
3109
1119890minus12
6119890minus13
8119890minus13
156
4400
080
18Delq
uad
◻Elem
ents
05
59
1113
9804
22
2412255
8119890minus09
4119890minus09
8119890minus09
567
162168
Science and Technology of Nuclear Installations 23
Tim
e con
sum
ed to
bui
ld th
e mat
rices
(sec
onds
)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay trianglesvolumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
10
30
100
300
1
3
Figure 23 Time needed to build the matrices versus number of unknowns
was considerably larger than the solution time itself Theeigenvalue problem was solved using a multilayer iterativeKrylov-Schur method [26] with a tolerance relative to thematrices norm
1003817100381710038171003817119877 sdot 120601 minus (1119896eff) sdot 119865 sdot 1206011003817100381710038171003817
119877 + (1119896eff) sdot 119865lt 10minus8 (11)
The reported residual norm and relative error are10038171003817100381710038171003817100381710038171003817119877 sdot 120601 minus
1
119896effsdot 119865 sdot 120601
10038171003817100381710038171003817100381710038171003817
1003817100381710038171003817119877 sdot 120601 minus (1119896eff) sdot 119865 sdot 1206011003817100381710038171003817
1003817100381710038171003817(1119896eff) sdot 1206011003817100381710038171003817
(12)
respectively The fields marked as outer linear and inneriterations refer to the number of steps needed to attainthe requested tolerance in each layer of the Krylov-Schuralgorithm The computer used to solve the problem has anIntel i7 920 267GHz processor with 4Gb of RAM runningDebian GNULinux Wheezy
When using a finite volumes-based scheme over anunstructured mesh the solver has to gather information
about which cells are neighbors and which are not Currentlygmsh does not write this kind of lists in its output filesso milonga has to explicitly solve the neighbors problemPerforming a linear search is an 119874(119873
2) task which is
unacceptable for problem sizes 119873 of interest The code usesa search based on a 119896-dimensional tree [27] which can inprinciple be performed in119874(119873) steps Still for large values of119873 the time needed to read and parse the mesh (number twopreviously mentioned) is the bottleneck of the solution Thisstep is not needed in finite elements although the construc-tion of the elementary matrices involves the computationof the multidimensional Jacobians and integrals which thenhave to be assembled Again for large 119873 this step (numberthree) takes up most of the time needed to solve the problem
The Delaunay algorithm is a standard method for gen-erating two-dimensional grids [16] by tessellating a planewith triangles If the mesh needs to be based on quadranglesinstead a recombination algorithm can be used to transformtwo adjacent triangles in one quadrangle whenever is possi-ble However for geometries which are based on rectangularshapes there exist other algorithms [25] both for meshingand for recombining the triangles that give rise to elementaryentities with right angles almost everywhere which may be a
24 Science and Technology of Nuclear Installations
Tim
e con
sum
ed to
solv
e the
eige
nval
ue p
robl
em (s
econ
ds)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
10
30
100
300
1
3
Figure 24 Time needed to solve the eigenvalue problem versus number of unknowns
desired property of the resulting grid For finite elements thesteps needed to build the matrices depend on the selectionof triangles or quadrangles as the basic elementary geometrybecause the shape functions change However the number ofunknowns is the same as the number of nodes that does notchange after a recombination procedure On the other handthe number of unknowns in the finite volumes schemes withtriangles is roughly twice as the number of unknowns withquadrangles for the same grid
Figure 19 shows how the computed static reactivity (ie1 minus 1119896eff) depends on the number of unknowns for eachof the sixteen combinations of geometry algorithm shapescheme The four accepted results published in the originalreference [10] are included for reference although it shouldbe taken into account that said reactivities were computedalmost forty years ago Figure 20 shows the total wall timeneeded to solve the problem as a function of 119873119866 whileFigures 21 22 23 and 24 show the times needed for eachof the first four steps involved in the solution maintainingthe same logarithmic scale for both the abscissas and theordinates Green data represent finite volumes whilst bluebullets correspond to finite elements Solid lines indicatequarter-core and dashed lines eighth-symmetry Fillet bullets
are results obtained by theDelaunay triangulation and emptybullets were computed with the delquad algorithm Finallytriangle-shaped data correspond to triangles and squares anddiamonds denote quadrangles as the basic geometry of thegrid
It can be seen that finite elements produce amuch smallerdispersion of eigenvalues 119896eff than finite volumes with therefinement of the mesh This can be explained because finitevolumes methods rely on a geometric condition of the meshwhich may change abruptly if the meshing algorithm decidesto allocate the cells in a rather different form for small changesin ℓ119888 Finite elements methods are less influenced by thesediscontinuous lay-out changes of the elements As expectedeighth-core symmetries give almost the same results as thequarter-core geometries with roughly half the unknownsFor small problems finite volumes run faster that finiteelements because the time needed to solve the neighborproblem is negligible When the problem size grows thistime increases and exceeds the overhead implied in the con-struction and assembly of the finite elements matrices Alsoat least for this configuration it is seen that the eigenvalueproblem is solved faster for finite volumes than for finiteelements
Science and Technology of Nuclear Installations 25
4 Conclusions
Unstructured grids provide the cognizant engineer witha wide variety of possibilities to deal with the design oranalysis of nuclear reactor cores These kinds of grids cansuccessfully reproduce continuous geometries commonlyfound in reactor cores such as cylinders and thereforenot only can the diffusion equation be better approximatedinside the domain of definition but also the fulfillment ofboundary conditions is improved A free computer codewas written from scratch that is able to completely solvethe 2D IAEA PWR Benchmark using unstructured gridsfor sixteen combinations of geometry meshing algorithmbasic shape and discretization scheme plus any value ofthe gridrsquos characteristic length by using a single input fileThe complete set of input files and codemdashexecutable andsourcemdashis available either online or upon request withcomments experiences suggestions and corrections beingmore than welcome Further development should includetackling full three-dimensional geometries with completethermal hydraulic feedback in order to analyze how thesolutions of the coupled neutronic-thermal problem dependon the spatial discretization scheme of the neutron leakageterm Parallelization of the computation and assembly of thematrices and of the solution of the eigenvalue problem andits implementation using GPUs are also desired features toimplement A problemwith direct applications that the futureversions of milonga ought to solve is the analysis of how thegeometry of the absorbing materials should be taken intoaccount in structured coarse grids in order to mitigate effectssuch as the rod-cusp problem
Suitable schemes for approximating the continuous dif-ferential operators by discrete matrix expressions includefinite volumes and finite elements families Finite volumesmethods compute cell mean values whilst finite elementsgive functional values at the gridrsquos nodes In general finiteelements are less sensitive to changes in the mesh so theresults they provide do not change significantly for differentmeshing algorithms or elementary shapes Small problemsare best solved by finite volumes as the neighbor-findingproblem is faster than the process of building and assemblingthe eigenvalue-problem matrices For a large number ofunknowns the process of finding which cell is neighbor ofwhichmdasheven based on a 119896-dimensional treemdashoverwhelmsthe computation and assembly of elementary matrices andfinite-element methods perform better
References
[1] G Glasstone and S Bell Nuclear Reactor Theory KriegerPublishing Company 1970
[2] J J Duderstadt and L J Hamilton Nuclear Reactor AnalysisJohn Wiley amp Sons New York NY USA 1976
[3] C Gho Reactor Physics Undergraduate Course Lecture NotesInstituto Balseiro 2006
[4] G G Theler and F J Bonetto ldquoOn the stability of the pointreactor kinetics equationsrdquoNuclear Engineering andDesign vol240 no 6 pp 1443ndash1449 2010
[5] E E Lewis andW F Miller Computational Methods of NeutronTransport John Wiley amp Sons 1984
[6] E Masiello ldquoAnalytical stability analysis of Coarse-Mesh finitedifference methodrdquo in Proceedings of the International Confer-ence on the Physics of Reactors (PHYSOR rsquo08) Nuclear Power ASustainable Resource pp 456ndash464 September 2008
[7] M L Zerkle Development of a polynomial nodal methodwith flux and current discontinuity factors [PhD thesis] Mas-sachusets Institute of Technology 1992
[8] J I Yoon andHG Joo ldquoTwo-level coarsemesh finite differenceformulation with multigroup source expansion nodal kernelsrdquoJournal of Nuclear Science andTechnology vol 45 no 7 pp 668ndash682 2008
[9] O Mazzantini M Schivo J Di Cesare R Garbero M Riveroand G Theler ldquoA coupled calculation suite for Atucha II oper-ational transients analysisrdquo Science and Technology of NuclearInstallations vol 2011 Article ID 785304 12 pages 2011
[10] Computational Benchmark Problem Comitee for the Mathe-matics and Computation Division of the American NuclearSociety ldquoArgonne Code Center Benchmark problem bookrdquoTech Rep ANL-7416 Supplement 2 Argonne National Labo-ratory 1977
[11] A F Henry Nuclear Reactor Analysis MIT Cambridge UK1975
[12] O C Zienkiewicz R L Taylor and J Z ZhuTheFinite ElementMethod Its Basis and Fundamentals vol 1 Elsevier 6th edition2005
[13] G Theler Difusion de neutrones en mallas no estructuradascomparacion entre volumenes y elementos finitos AcademicMonograph Universidad de Buenos Aires 2013
[14] G Theler ldquoMilonga a free nuclear reactor core analysis coderdquoAvailable online 2011
[15] G Theler F J Bonetto and A Clausse ldquoOptimizacion deparametros en reactores de potencia base de diseno delcodigo neutronico milongardquo in Reunion Anual de la AsociacionArgentina de Tecnologia Nuclear XXXVII 2010
[16] C Geuzaine and J-F Remacle ldquoGmsh a 3-D finite elementmesh generator with built-in pre- and post-processing facili-tiesrdquo International Journal for Numerical Methods in Engineer-ing vol 79 no 11 pp 1309ndash1331 2009
[17] S Balay J Brown K Buschelman et al ldquoPETSc users manualrdquoTech Rep ANL-9511-Revision 34 Argonne National Labora-tory 2013
[18] S Balay W D Gropp L Curfman McInnes and B F SmithldquoEfficient management of parallelism in object oriented numer-ical software librariesrdquo in Modern Software Tools in ScientificComputing E Arge A M Bruaset and H P Langtangen Edspp 163ndash202 Birkhauser 1997
[19] V Hernandez J E Roman and V Vidal ldquoSLEPC a scalable andflexible toolkit for the solution of eigenvalue problemsrdquo ACMTransactions on Mathematical Software vol 31 no 3 pp 351ndash362 2005
[20] M Galassi J Davies J Theiler et al GNU Scientific LibraryReference Manual 3rd edition 2009
[21] O C Zienkiewicz and R L Taylor The Finite Element MethodFor Solid and Structural Mechanics vol 3 Elsevier 6th edition2005
[22] A Imelda and K Doddy Evaluation of the IAEA 3-D PWRbenchmark problem using NESTLE code 2004
[23] R Mosteller ldquoStatic benchmarking of the NESTLE advancednodal coderdquo in Proceedings of the Joint International Conference
26 Science and Technology of Nuclear Installations
on Mathematical Methods and Supercomputing for NuclearApplications vol 2 pp 1596ndash1605 1997
[24] GTheler F J Bonetto andAClausse ldquoSolution of the 2D IAEAPWR Benchmark with the neutronic code milongardquo in Actasde la Reunion Anual de la Asociacion Argentina de TecnologıaNuclear XXXVIII 2011
[25] J F Remacle F Henrotte T Carrier-Baudouin et al ldquoAfrontal delaunay quad mesh generator using the 119871
infin normrdquoInternational Journal For Numerical Methods in Engineeringvol 94 no 5 pp 494ndash512 2013
[26] G W Stewart ldquoA Krylov-Schur algorithm for large eigenprob-lemsrdquo SIAM Journal on Matrix Analysis and Applications vol23 no 3 pp 601ndash614 2002
[27] J L Bentley ldquoMultidimensional binary search trees used forassociative searchingrdquo Communications of the ACM vol 18 no9 pp 509ndash517 1975
TribologyAdvances in
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International Journal of
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FuelsJournal of
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Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Journal of
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Renewable Energy
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
StructuresJournal of
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
EnergyJournal of
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Journal ofEngineeringVolume 2014
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Wind EnergyJournal of
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Nuclear EnergyInternational Journal of
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High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Science and Technology of Nuclear Installations 3
(a) Coarse structured grid commonly used in diffu-sion codes such as in [9] Each lattice cell is divided intoa 2 times 2 grid for solving the neutron diffusion equation
(b) Discretization of the lattice cell using a fineunstructured grid as proposed in this work
(c) Further refinement of the unstructured grid
Figure 1 Discretization of a homogenized PHWR array of fuel channels for a core-level diffusion computation Each square is a lattice-levelcell comprising one fuel channel and the surrounding moderator
linear iterations so the basic problem can be regarded as beingpurely linear
It should be noted that if at least one of the diffu-sion coefficients 119863119892(x) is discontinuous over space thenthe divergence operator is not defined at the discontinuitypoints Therefore the differential formulationmdashalso knownas the strong formulationmdashis not complete when there existmaterial discontinuities that involve the diffusion coefficientsAt thesematerial interfaces the differential equation has to bereplaced by a neutron current conservation condition
119863+
119892(x) sdot grad 120601
+
119892(x) = 119863
minus
119892(x) sdot grad 120601
minus
119892(x) (2)
where the plus and minus sign denote both sides of the inter-face As the diffusion coefficients are different the resultingflux distribution120601119892(x) ought to have a discontinuous gradientat the interface in order to conserve the current
When transforming the strong formulation into a weakformulationmdashnot just into an integral formulationmdashboth (1)and (2) can be taken into account by a single expression Ineffect let 120593119892(x) be arbitrary functions of x for 119892 = 1 119866Multiplying each of the 119866 equations (1) by 120601119892(x) integrating
over the domain 119880 isin R119898 and applying Greenrsquos formula [12]to the leakage term we obtain
int119880
119863119892 (x) sdot [grad 120593119892 (x) sdot grad 120601119892 (x)] 119889119898x
+ int120597119880
120593119892 (x) sdot 119863119892 (x) sdot [grad 120601119892 (x) sdot n] 119889119878
+ int119880
120593119892 (x) sdot Σ119886119892 (x) sdot 120601119892 (x) 119889119898x
+ int119880
120593119892 (x) sdot119866
sum
1198921015840 = 119892
Σ1199041198921015840rarr119892 (x) sdot 1206011198921015840 (x) 119889119898x
+ 120594119892 sdot int119880
120593119892 (x) sdot119866
sum
1198921015840=1
]Σ1198911198921015840 (x)119896eff
sdot 1206011198921015840 (x) 119889119898x
(3)
These 119866 coupled equations should hold for any arbitraryset of functions120593119892(x)Making an analogy between (3) and theprinciple of virtual work for structural problems we call thesefunctions virtual fluxes This formulation does not involveany differential operator over the diffusion coefficient and yet
4 Science and Technology of Nuclear Installations
(a) Continuous two-dimensional domain (b) Structured grid
(c) Unstructured grid
Figure 2 When a continuous domain (a) is meshed with an unstructured grid there appears a geometric condition known as the staircaseeffect (b) For the same number of nodes unstructured grids reproduce the original geometry better (c)
Figure 3 Cross section of a hypothetical reactor in which thecontrol rods enter into the core from above with a certain attackangle with respect to the vertical direction
at the same time can be shown to be equivalent to the strongformulation given by (1)
In any case both formulations involve the computationof 119866 unknown functions of x and one unknown real value119896eff Except for homogeneous cross sections in canonicaldomains 119880 the multigroup diffusion equation needs to besolved numerically As discussed below any nodal-baseddiscretization scheme replaces a continuous unknown func-tion 120601119892(x) by 119873 discrete values 120601119892(119894) for 119894 = 1 119873
Figure 4 The finite volumes method computes the unknown fluxin the cell centers (squares) whilst the finite elements methodcomputes the fluxes at the nodes (circles)
If we arrange these unknowns into a vector 120601 isin R119873119866
such as
120601 =
[[[[[[[[[[[[[[[[
[
1206011 (1)
1206012 (1)
120601119866 (1)
1206011 (2)
120601119892 (119894)
120601119866 (119873)
]]]]]]]]]]]]]]]]
]
(4)
Science and Technology of Nuclear Installations 5
(a) Fast flux distribution
minus100minus50 0 50 100minus100
minus500
50100
0
06
xy
(b) Fast flux unknowns
(c) Matrix 119877 structure (d) Matrix 119865 structure
Figure 5 A bare homogeneous circle solved with finite volumes (184 unknowns)
then the continuous eigenvalue problemmdashin either for-mulationmdashcan be transformed into a generalized matrixeigenvalueeigenvector problem casted in either of the follow-ing forms
119877 sdot 120601 =1
119896effsdot 119865 sdot 120601
119865 sdot 120601 = 119896eff sdot 119877 sdot 120601
119877minus1sdot 119865 sdot 120601 = 119896eff sdot 120601
(5)
where 119877 and 119865 are square 119873119866 times 119873119866 matrices We call119865 the fission matrix as it contains all the ]-fission termswhich are the ones that we artificially divided by 119896eff Therest of the terms are grouped into the removal or transportmatrix 119877 which includes the rest of the neutron-matterinteraction mechanisms It can be shown [11] that for anyreal set of cross sections 119877minus1 exists and that the119873119866 pairs ofeigenvalueeigenvector solutions of (5) satisfy that
(1) there is a unique real positive eigenvalue greater inmagnitude than any other eigenvalue
(2) all the elements of the eigenvector corresponding tothat eigenvalue are real and positive
(3) all other eigenvectors either have some elements thatare zero or have elements that differ in sign from eachother
21 Boundary Conditions Being a differential equation overspace the neutron diffusion equation needs proper boundaryconditions to conform a properly definedmathematical prob-lem These can be imposed flux (Dirichlet) imposed current(Neumann) or a linear combination (Robin) However dueto the fact that in the linear problem in absence of externalsourcesmdashsuch as (1) or (3)mdashthe problem is homogeneousif 120601119892(x) is a solution then any multiple 120572120601119892(x) is also asolution That is to say the eigenvectors are defined up to amultiplicative constant whose value is usually chosen as toobtain a certain total thermal power Thus the prescribedboundary conditions should also be homogeneous and bedefined up to amultiplicative constantTherefore the allowedDirichlet conditions are zero flux at the boundary theNeumann conditions should prescribe that the derivative in
6 Science and Technology of Nuclear Installations
(a) Fast flux distribution
minus100minus50 0 50 100minus100
minus500
50100
0
06
xy
(b) Fast flux unknowns
(c) Matrix 119877 structure (d) Matrix 119865 structure
Figure 6 A bare homogeneous circle solved with finite elements (218 unknowns)
the normal direction should be zero (ie symmetry condi-tion) and the Robin conditions are restricted to the followingform
grad 120601119892 (x) sdot n + 119886119892 (x) sdot 120601119892 (x) = 0 (6)
n being the outward unit normal to the boundary 120597119880 of thedomain 119880
22 Grids and Schemes One way of solving the neutrondiffusion equationmdashand in general any partial differentialequation over spacemdashis by discretizing the differential oper-ators with some kind of scheme that is applied over a certainspatial grid Given an 119898-dimensional domain a grid definesa partition composed of a finite number of simple geometricentities In structured grids these elementary entities arearranged following a well-defined structure in such a waythat each entity can be identified without needing furtherinformation than the one provided by the intrinsic structureOn the other hand the geometric entities that composean unstructured grid are arranged in an irregular patternand the identification of the elementary entities needs to
be separately specified for example by means of a sortedlist For instance Figure 2(b) shows a structured grid thatapproximates the continuous domain of Figure 2(a) Eachsquare may be uniquely identified by means of two integerindexes that indicate its relative position in each of thehorizontal and vertical directions In the case of Figure 2(c)that shows an unstructured grid there is no way to system-atically make a reference to a particular quadrangle withoutany further information
Almost all of the grid-based schemesmdashwhich are knownas nodal schemes which are to be differentiated from modalschemes based on series expansionsmdashare based in either thefinite differences volumes or elements method Finite differ-ences schemes provide themost simple and basic approach toreplace differential (ie continuous) operators by difference(ie discrete) approximations However they are not suitablefor unstructured meshes and may introduce convergenceproblems with parameters that are discontinuous in spacewhich is the case for any reactor core composed of atleast two different materials Moreover boundary conditionsare hard to incorporate and usually give rise to incorrectresults
Science and Technology of Nuclear Installations 7
20 40 600995
09975
1
Analytical solutionFinite volumesFinite elements
keff
a985747c
Figure 7 The effective multiplication factor of a two-group barecircular reactor of radius 119886 as a function of the quotient 119886ℓ
119888between
the radius and the characteristic length of the cellelement
20 40 60
Finite volumesFinite elements
Erro
r ink
eff
10minus3
10minus4
10minus5
a985747c
Figure 8 Absolute value of the error committed in the computationof 119896eff versus 119886ℓ119888
Methods of the finite volumes family involve the inte-gration of the differential equation over each elementaryentity applying the divergence theorem to transform volumeintegrals into surface integrals and providing a mean to esti-mate the fluxes through the entityrsquos surface using informationcontained in its neighbors In this context each elementaryentity is called a cell and finite volumes methods give themean value of each of the group fluxes 120601119892(119894) at the 119894th cellwhich may be associated to the value of 120601119892(x119894) where x119894 isthe location of the cell barycenter Being an integral-basedmethod spatial-discontinuous parameters are handled moreefficiently than in finite differences and boundary conditionscan be easily incorporated as forced cell fluxes Nonethelesseven though these methods may be applied to unstructuredgrids the estimation of the fluxes on the cellsrsquo surfaces isusually performed by using some geometric approximationsthat may lose validity as the quality of the grid is worsened
10
70
90
130
150
170
10 30 50 70 90 110 130 150 170
4
1 38
36 37353
3 32 33 34312
26 27 28 29 3025
19 20 21 22 23 2418
10 11 12 13 14 15 16 17
2 331 4 5 6 7 8 9
x (cm)
y (cm
)
Figure 9 The 2D IAEA PWR benchmark geometry
Moreover the simple integral approach cannot take intoaccount discontinuous diffusion coefficients so when twoneighbors pertain to different materials the flux has to becomputed in a certain particular way to conserve the neutroncurrent according to (2)
Finally finite elements methods rely on a weak formula-tion of the differential problem similar to (3) that maintainsall the mathematical characteristics of the original strongformulation plus its boundary conditions The method isbased on shape functions that are local to each elementaryentitymdashnow called element defined by nodes as cornersmdashand on finding a set of nodal values such that a certaincondition is met which is usually that the residual of thesolution has to be orthogonal to each of the shape functionsThis condition is known as the Galerkin method and impliesthat the error committed in the approximate solution ofthe continuous problem is confined into a small subset ofthe original vector space of the continuous functions Thesemathematical properties make finite elements schemes veryattractive However these features depend on a large numberof integrations that ought to be performed numerically soa computational effortdesired accuracy tradeoff has to beconsidered Not only does the finite elements method givethe values of the flux 120601119892(x119894) at the 119894th node but also theshape functions provide explicit expressions to interpolateand to evaluate the unknown functions at any location x ofthe domain119880 Boundary conditions are divided into essentialand naturalThefirst group comprises theDirichlet boundaryconditions which are satisfied exactlymdashwithin the precisionof the eigenvalue problem solvermdashby the obtained solutionThe latter include the Neumann and the Robin conditionsthat are satisfied only approximately by the derivatives of theinterpolated solution through the shape functions with finermeshes giving better agreement with the prescribed values
The same unstructured grid may be used either for thefinite volumes or for the finite elements method In the first
8 Science and Technology of Nuclear Installations
Largest eigenvalue 1029690 (288336 pcm)1135 (3066 3079)1306 (5171 13076)
Number of unknowns 15740Outer iterations 3Linear iterations 32Inner iterations 1967Residual normRelative errorError estimateMemory used 49824 kBSoft page faults 13401Hard page faults 0Total CPU time 074 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
2483 times 10minus8
1223 times 10minus8
403 times 10minus9
keff
quarter-symmetry core meshed using Delaunay (triangles 985747c = 3) solved with finite volumesMilongarsquos 2D LWR IAEA benchmark problem case no 002
(a)
123456789
1011121314151617181920212223242526272829303132333435363738
k Pk 1206011k 1206012k075 3282 555133 4242 984147 4645 1090123 3921 911061 2677 453094 2995 694093 2927 689074 2011 549mdash 322 756146 4597 1078150 4730 1110133 4199 985107 3405 789104 3271 767095 2971 700072 1956 534mdash 306 733149 4695 1102136 4289 1007118 3733 876107 3367 792097 2892 718066 1628 491mdash 234 549120 3797 891097 3096 718090 2842 669084 2221 619mdash 568 1215mdash 067 257047 2042 348068 2055 504058 1414 430mdash 222 562057 1374 425mdash 388 820mdash 053 203mdash 060 229
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 10 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
case the unknowns are themean value of the fluxes over eachcell whilst in the latter the unknowns are the fluxes evaluatedat each node Therefore the number of unknowns 119873119866 isdifferent for each method even when using the same gridFigure 4 shows this situation and in Section 31 we further
illustrate these differences A fully detailed mathematicaldescription of the actual algorithms for both volumes andelements-based proposed discretizations can be found in anacademic monograph written by the author of this paper[13]
Science and Technology of Nuclear Installations 9
Largest eigenvalue 1029927 (290571 pcm)1131 (3175 2999)1220 (13097 5098)
Number of unknowns 15576Outer iterations 3Linear iterations 32Inner iterations 1947Residual normRelative errorError estimateMemory used 56256 kBSoft page faults 15645Hard page faults 0Total CPU time 09201 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
1394 times 10minus8
6865 times 10minus9
3425 times 10minus9
keff
quarter-symmetry core meshed using Delaunay (quads 985747c = 2) solved with finite volumesMilongarsquos 2D LWR IAEA benchmark problem case no 013
(a)
123456789
1011121314151617181920212223242526272829303132333435363738
k Pk 1206011k 1206012k074 3244 549133 4230 982147 4651 1092124 3946 917060 2645 443094 2997 696093 2938 691074 2005 545mdash 301 746145 4588 1076150 4720 1108133 4198 984109 3465 805104 3280 769095 2986 704071 1962 528mdash 293 715148 4680 1098136 4288 1007119 3755 881107 3385 796097 2905 721067 1653 493mdash 220 556121 3808 893098 3115 724092 2889 680083 2247 615mdash 543 1241mdash 067 260045 1977 330069 2076 509058 1460 428mdash 226 574056 1364 413mdash 371 821mdash 052 200mdash 060 238
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 11 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
3 Results
We now proceed to show two illustrative results that are tobe taken as an overview of the possibilities that unstructuredgrids can provide in order to tackle the multigroup neutron
diffusion problem The examples are two-dimensional prob-lems as they contain some of the complexities a real three-dimensional reactor posse yet the reported results are notso complicated as to be easily understood and analyzed Inparticular we state some basic differences between the finite
10 Science and Technology of Nuclear Installations
Largest eigenvalue 1029695(288388 pcm)1112 (3076 3032)838 (5000 13000)
Number of unknowns 15922Outer iterations 3Linear iterations 32Inner iterations 1990Residual normRelative errorError estimateMemory used 109964 kBSoft page faults 30591Hard page faults 0Total CPU time 158 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
1056 times 10minus8
5202 times 10minus9
5122 times 10minus9
keff
quarter-symmetry core meshed using Delaunay (quads 985747c = 2) solved with finite volumesMilongarsquos 2D LWR IAEA benchmark problem case no 018
(a)
123456789
1011121314151617181920212223242526272829303132333435363738
k Pk 1206011k 1206012k074 3220 549130 4152 961144 4651 1068120 3843 890061 2648 452094 2996 693094 2959 696072 2061 569mdash 358 862142 4495 1054147 4634 1088131 4127 968107 3409 789104 3275 768096 3003 708071 2014 554mdash 342 820146 4606 1081134 4222 991118 3711 871107 3375 794098 2929 727062 1692 522mdash 258 632119 3752 880096 3084 715091 2858 673080 2280 633mdash 611 1277mdash 080 318047 2043 350069 2088 510054 1463 450mdash 255 648051 1418 439mdash 410 854mdash 064 252mdash 071 285
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 12 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
volumes and the finite elements methods by solving a two-group homogeneous bare reactor with the Robin boundaryconditions over the very same grid although a rather coarseone so the differences can be observed directly into theresulting figuresWe then solve the classical two-dimensionalLWR problem also known as the 2D IAEA PWR benchmark
Not only do we show again the differences between the finitevolumes and elements formulation but also we solve theproblemusing different combinations ofmeshing algorithmsbasic shapes and characteristic lengths of the mesh
To solve the two examples shown below we employed themilonga code which was written from scratch by the author
Science and Technology of Nuclear Installations 11
Largest eigenvalue 1029159 (283326 pcm)1174 (2851 3224)1560 (5200 13200)
Number of unknowns 3132Outer iterations 3Linear iterations 32Inner iterations 391Residual normRelative errorError estimateMemory used 26888 kBSoft page faults 7322Hard page faults 0Total CPU time 0308 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
5408 times 10minus8
2665 times 10minus8
8424 times 10minus9
keff
Milongarsquos 2D LWR IAEA benchmark problem case no 031quarter-symmetry core meshed using Delquad (quads 985747c = 4) solved with finite volumes
(a)
123456789
1011121314151617181920212223242526272829303132333435363738
k Pk 1206011k 1206012k077 3422 570140 4468 1040153 4836 1135128 4070 948060 2704 448095 3017 702091 2861 673069 1918 508mdash 264 689151 4785 1122155 4897 1149137 4334 1017111 3535 823104 3272 768092 2908 685067 1876 496mdash 254 658154 4854 1139139 4392 1031120 3792 890107 3360 790095 2831 702061 1574 451mdash 197 515123 3868 907099 3132 730090 2818 663078 2168 579mdash 481 1191mdash 055 217044 1984 328067 2027 498052 1360 388mdash 190 520052 1329 383mdash 321 792mdash 043 168mdash 048 195
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 13 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
of this paper and is currently still under development withinhis ongoing PhD thesis There exists a first public release[14] under the terms of the GNU General Public Licensemdashthat is it is a free softwaremdashthat can only handle structuredgrids There is a second release being prepared whose
main relevance is that it can work with nonstructured gridsas well which is the main feature of the code It uses ageneral mathematical frameworkmdashcoded from scratch aswellmdashwhich provides input file parsing algebraic expres-sions evaluation one- and multidimensional interpolation of
12 Science and Technology of Nuclear Installations
Largest eigenvalue 1029788 (289260 pcm)1120 (3149 3011)1091 (13038 5114)
Number of unknowns 15776Outer iterations 3Linear iterations 32Inner iterations 1972Residual normRelative errorError estimateMemory used 56996 kBSoft page faults 15436Hard page faults 0Total CPU time 09121 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
2296 times 10minus12
1131 times 10minus12
6877 times 10minus13
keff
Milongarsquos 2D LWR IAEA benchmark problem case no 043eighth-symmetry core meshed using Delaunaya (triangs985747c = 2) solved with finite volumes
(a)
1 074 3241 5482 131 4188 9713 145 4582 10764 122 3877 9005 060 2644 4456 093 2986 6927 093 2941 6928 075 2030 5529 mdash 322 779
10 143 4523 106111 148 4667 109512 131 4145 97213 107 3423 79414 103 3269 76715 095 2993 70616 073 1988 53917 mdash 312 75118 147 4637 108819 134 4243 99620 118 3721 87321 107 3370 79322 098 2914 72323 067 1651 49924 mdash 235 57125 119 3763 88226 096 3077 71527 091 2847 67128 083 2255 61729 mdash 576 122530 mdash 068 26631 046 2016 34132 068 2058 50533 059 1443 43534 mdash 233 58935 057 1381 42036 mdash 381 82337 mdash 054 20838 mdash 062 244
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 14 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
scattered data shared-memory access numerical integrationfacilities and so forth The milonga code was designed withfour design basis vectors in mindmdashwhich are thoroughlydiscussed in the documentation [15]mdashwhich includes thetype of problems it can handle the code scalability which
features are expected and what to do with the obtainedresults
It works by first reading an input file that using plain-text English keywords and arguments defines the number119898 of spatial dimensions the number 119866 of group energies
Science and Technology of Nuclear Installations 13
Largest eigenvalue 1029726 (288675 pcm)1104 (3069 3069)864 (13000 5000)
Number of unknowns 4040Outer iterations 2Linear iterations 24Inner iterations 505Residual normRelative errorError estimateMemory used 42940 kBSoft page faults 11437Hard page faults 0Total CPU time 1064 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
2019 times 10minus8
9946 times 10minus9
7138 times 10minus9
keff
Milongarsquos 2D LWR IAEA benchmark problem case no 047eighth-symmetry core meshed using Delaunay (triangles 985747c = 3) solved with finite volumes
(a)
1 074 3194 5452 129 4117 9533 143 4515 10604 119 3814 8835 061 2634 4506 093 2984 6907 094 2954 6958 071 2064 5719 mdash 362 865
10 141 4458 104511 146 4599 107912 130 4097 96113 106 3388 78414 103 3263 76515 095 2998 70716 068 2015 55717 mdash 346 82318 145 4572 107319 133 4194 98420 117 3691 86621 107 3363 79222 098 2925 72623 059 1696 52824 mdash 263 63525 118 3728 87426 096 3068 71127 091 2850 67228 079 2280 63529 mdash 619 127630 mdash 081 32131 047 2036 34932 068 2085 50933 051 1465 45434 mdash 258 65035 049 1419 44336 mdash 416 85437 mdash 065 25438 mdash 071 288
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 15 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
and optionally a mesh file Currently only grids generatedwith the code gmsh [16] are only supported mainly becauseit is also a free software and it suits perfectly well milongarsquosdesign basis in the sense that the continuous geometry can bedefined as a function of a number of parameters Afterwardthe physical entities defined in the grid are mapped into
materials with macroscopic cross sections which maydepend on the spatial coordinates x bymeans of intermediatefunctions such as burn-up or temperatures distributionswhich in turn may be defined by algebraic expressions byinterpolating data located in files by reading shared-memoryobjects or by a combination of them In the same sense
14 Science and Technology of Nuclear Installations
Largest eigenvalue 1029695 (288387 pcm)1108 (3045 3045)833 (13000 5000)
Number of unknowns 8234Outer iterations 3Linear iterations 32Inner iterations 1029Residual normRelative error
Max 1206012(x y) coreMax 1206012(x y) reflector
2028 times 10minus12
9993 times 10minus13
1012 times 10minus12
keff
Relative errorError estimateMemory used 75468 kBSoft page faults 20094Hard page faults 0Total CPU time 1256 seconds
9993 times 10
1012 times 10minus12
Milongarsquos 2D LWR IAEA benchmark problem case no 058eighth-symmetry core meshed using Delaunay (quads 985747c = 2) solved with finite volumes
(a)
1 074 3207 5472 129 4136 9573 144 4534 10644 120 3828 8875 061 2638 4506 093 2985 6907 094 2948 6948 072 2053 5669 mdash 357 859
10 142 4478 105011 146 4617 108412 130 4111 96413 106 3396 78614 103 3263 76515 095 2991 70616 071 2005 55217 mdash 340 81618 145 4588 107719 133 4206 98720 117 3698 86821 107 3362 79122 098 2918 72523 062 1686 52024 mdash 258 62925 118 3737 87626 096 3073 71227 091 2848 67128 080 2272 63129 mdash 609 127230 mdash 080 31731 047 2035 34832 069 2081 50933 054 1458 44834 mdash 255 64635 052 1413 43736 mdash 409 85137 mdash 064 25138 mdash 071 284
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 16 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
boundary conditions are applied to appropriate physicalentities of dimension119898minus1The problemmatrices119877 and119865 arethen built and stored in an appropriate sparse format usingthe free PETSc [17 18] library and the eigenvalue problemis solved using the free SLEPc [19] library The results arestored into milongarsquos variables and functions which may be
evaluated integratedmdashusually using the free GNU ScientificLibrary [20] routinesmdashand of course written into appropriateoutputs Milonga can also solve problems parametrically andbe used to solve optimization problems As stated above thecode is a free software so corrections and contributions aremore than welcome
Science and Technology of Nuclear Installations 15
Largest eigenvalue 1029462 (286185 pcm)1136 (3093 2953)1233 (13100 5100)
Number of unknowns 6228Outer iterations 2Linear iterations 24Inner iterations 778Residual normR l ti
Max 1206012(x y) coreMax 1206012(x y) reflector
26 times 10minus8
1281 times 10minus8
4996 times 10minus9
keff
Relative errorError estimateMemory used 41460 kBSoft page faults 10930Hard page faults 0Total CPU time 0604 seconds
1281 times 10 8
4996 times 10minus9
Milongarsquos 2D LWR IAEA benchmark problem case no 073eighth-symmetry core meshed using Delaunay (quads 985747c = 2) solved with finite volumes
(a)
1 075 3297 5552 134 4268 9923 148 4660 10944 123 3926 9145 059 2645 4406 094 2997 6977 093 2914 6858 072 2002 5349 mdash 301 773
10 146 4606 108011 150 4735 111112 133 4212 98813 108 3448 80214 104 3271 76715 094 2954 69616 070 1954 52117 mdash 288 73518 149 4691 110119 136 4289 100720 119 3748 88021 107 3359 79022 096 2880 71423 064 1636 47524 mdash 217 56325 121 3810 89426 098 3110 72427 090 2829 66628 081 2227 59929 mdash 536 125730 mdash 062 25031 045 2002 33632 068 2040 50033 055 1414 40934 mdash 215 57935 055 1359 40836 mdash 359 83437 mdash 050 19938 mdash 061 236
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 17 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
31 A Coarse Bare Homogeneous Circle Figures 5 and 6 showthe results of solving a bare two-dimensional circular homo-geneous reactor of radius 119886 over an unstructured grid whichis deliberately coarse using the finite volumes method andthe finite elements method respectively Two group energies
were used and a null-flux boundary conditionwas fixed at theexternal surface Figures 5(a) and 6(a) compare the obtainedfast flux distributions In the first case the numerical solutionprovidesmean values for each neutron flux group in each cellwhile in the latter the solution is computed at the nodes and
16 Science and Technology of Nuclear Installations
Largest eigenvalue 1029851 (289858 pcm)1090 (3182 3182)851 (13000 5000)
Number of unknowns 1752Outer iterations 3Linear iterations 32Inner iterations 219Residual norm
Max 1206012(x y) coreMax 1206012(x y) reflector
566 times 10minus12
2789 times 10minus12
1595 times 10minus12
keff
Residual normRelative errorError estimateMemory used 42564 kBSoft page faults 11213Hard page faults 0Total CPU time 09921 seconds
566 times 10
2789 times 10minus12
1595 times 10minus12
Milongarsquos 2D LWR IAEA benchmark problem case no 076eighth-symmetry core meshed using Delaunay (quads 985747c = 4) solved with finite volumes
(a)
1 073 3151 5402 127 4059 9393 141 4457 10464 118 3768 8715 061 2615 4506 093 2979 6887 094 2971 6998 067 2092 5829 mdash 375 885
10 139 4397 103111 144 4541 106612 128 4052 95013 105 3365 77814 103 3259 76415 096 3016 71216 066 2045 56817 mdash 359 84118 143 4520 106119 132 4155 97520 116 3672 86221 107 3366 79222 098 2951 73323 054 1728 54324 mdash 273 65125 117 3700 86826 095 3057 70827 091 2860 67428 074 2311 64829 mdash 642 130030 mdash 084 33231 048 2041 35332 069 2103 51333 046 1493 46734 mdash 269 66735 046 1444 45536 mdash 432 87137 mdash 067 26338 mdash 074 297
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 18 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
continuous functions are evaluated by means of the shapefunctions used in the formulation Figures 5(b) and 6(b)illustrate the fast fluxes unknowns and its relative position inspace It can be noted that the mesh coarseness gives resultsthat differ substantially in both cases Finally the structure ofthe sparse eigenvalue problemmatrices is shown for each case
with blue red and cyan representing positive negative andexplicitly inserted zero values In the finite volume case thereare 184 unknowns (92 cells times 2 groups) while in the secondcase there are 218 unknowns (109 nodes times 2 groups) Thevolumesrsquo fissionmatrix is almost diagonal it has a bandwidthequal to the number of energy groups which in this case is
Science and Technology of Nuclear Installations 17
500 1000 2000 5000 10000 20000 50000Number of unknowns
2840
2860
2880
2900
2920
ORNLRisoslashKWUOntarioQuarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumes
Quarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elementsEighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumes
Eighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
120588(p
cm)
105 2 times 105 5 times 105
Figure 19 Static reactivity versus number of unknowns The four original solutions as published in 1977 [10] are included as reference
twoThe off-diagonal values appear right next to the diagonalelements because of the chosen ordering of the unknownsin the flux vector 120601 isin R184 Other orderings may be usedbut the rate of convergence of the eigenvalue problem can bedeteriorated On the other hand the elementsrsquo fission matrixhas a nontrivial structure because the grid is unstructuredand the net fission rate at each element depends on the fluxesat the nodes whose location inside thematrix depends in turnon how the gridwas generatedThis effect is similar to the onethat appears in structural analysis where mass matrices needto be lumped in order to simplify transient computations [21]The diagonal block of elementrsquos 119877 matrix and the null blockin 119865 correspond to the discrete equations that set the null-flux boundary conditions on the nodes located at the externalsurface Other types of boundary conditions lead to differentkinds of structures within the problem matrices
In case a part of the domain contained a nonmultiplicativematerial such as a reflector then there would appear sectionsof the fission matrix with null values in both methodsrendering 119865 singular in both methods Care should be takenwhen dealing with the numerical schemes for the eigenvalueproblem solution It can be seen in the elementsrsquo matricesa particular structure that implements the boundary condi-tions at the external surfaces This structure does not appearin the volumesrsquo matrices because the boundary conditions
appear as flux terms which are summed up over all thesurfaces of each cell so they aremasked inside the volumetricdiscretization of the divergence and gradient operators
As the two-group neutron diffusion equation with uni-form cross sections over a circle subject to null-flux bound-ary conditions has an analytical solution it is adequate tocompare how the two proposed numerical schemes relate toit In the studied problem we ignored upscattering and fastfissions Then the analytical effective multiplication factor is
119896eff =]Σ1198912 sdot Σ1199041rarr2
[Σ1198861 + Σ1199041rarr2 + 1198631(]0119886)2] [Σ1198862 + 1198632(]0119886)
2]
(7)
being ]0 = 24048 the smallest root of Besselrsquos first-kindfunction of order zero 1198690(119903)
Figure 7 shows how the two numerical effective multi-plication factors compare to the analytical solution givenby (7) as a function of the mesh refinement indicated bythe quotient 119886ℓ119888 between the radius of the circle and thecharacteristic length of the cellelements We can see thatthe 119896eff computed by the finite volumes (elements) methodis always greater (less) than the analytical solution Indeedit can be proven that for a bare one-dimensional slab thisis always the case [13] However this result does not hold
18 Science and Technology of Nuclear Installations
2000 3000 6000 10000 20000 30000 60000
01
003
001
03
1
3
10
30
100
300To
tal w
all t
ime (
seco
nds)
Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
Figure 20 Total wall time versus number of unknowns
for problems with nonuniform cross sections even in simpleone-dimensional reflected reactors
We may draw two other conclusions from Figure 7 Firstthat the finite element method seems to provide a bettersolution than the volumes-based scheme and second thateven though the error committed tends to decrease with finergrids its behavior is not monotonic for finite volumes Infact Figure 8 shows the difference between the numericaland analytical solutions using a logarithmic vertical scalewhere both conclusions are even more evident We deferthe discussion of such differences between the discretizationscheme until the next section It is worth to note howeverthat the fact that a finite-element-based scheme throws betterresults for the particular bare homogeneous circular reactorunder study than the finite volumes does not imply thatfinite volumes ought to be discarded as a valid tool forsolving the neutron diffusion equation in general cases Thecombination of lattice and core-level computations is usuallyperformedusing cell-based resultswhichwhen fed into node-based methods to solve the few-group neutron diffusionequation may introduce errors which potentially can leadto unacceptable solutions Nevertheless this analysis is farbeyond the scope of this paper which focuses on solving amathematical equation over unstructured grids
32 The 2D IAEA PWR Benchmark Problem This is aclassical two-group neutron diffusion problem first designedin the early 1970s and taken as a reference benchmark forcomputational codes A number of codes were used to solveeither this problem or its three-dimensional formulation[22 23] including milonga using a structured grid [24] Theoriginal formulation can be found in the reference [10] andthere is a reproduction that may be easily found online inreference [24] The geometry consists of a one-quarter ofa PWR core depicted in Figure 9 and the homogeneousmacroscopic cross sections are listed in Table 1 An axialbuckling term 119861
2
119911119892= 08times10
minus4 should be taken into accountThe external surface should be subject to a zero incomingcurrent condition which may be written as
120597120601119892
120597119899= minus
04692
119863119892
sdot 120601119892 (8)
The expected results are as follows
(1) Maximum eigenvalue(2) Fundamental flux distributions
(a) Radial flux traverses 120601119892(119909 0) and 120601119892(119909 119909)
Science and Technology of Nuclear Installations 19
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
1
3
10
30
100
300Ti
me c
onsu
med
to m
esh
the g
eom
etry
(sec
onds
)
Figure 21 Time needed to mesh the geometry versus number of unknowns
Table 1 Macroscopic cross-sections (units are not stated in the original reference but they are assumed to be in cmminus1 or cm as appropriate)
1198631
1198632
Σ1rarr2
Σ1198861
Σ1198862
]Σ1198912
Material1 15 04 002 001 0080 0135 Fuel 12 15 04 002 001 0085 0135 Fuel 23 15 04 002 001 0130 0135 Fuel 2 + rod4 20 03 004 0 0010 0 Reflector
Note the fluxes shall be normalized such that
1
119881coreint119881core
sum
119892
]Σ119891119892 sdot 120601119892119889119881 = 1 (9)
(b) Value and location of maximum power densityThis corresponds to maximum of 1206012 in thecore It is recommended that the maximumvalues of 1206012 both in the inner core and at thecorereflector interface be given
(3) Average subassembly powers 119875119896
119875119896 =1
119881119896
int119881119896
sum
119892
]Σ119891119892 sdot 120601119892119889119881 (10)
where 119881119896 is the volume of the 119896th subassembly and119896 designates the fuel subassemblies as shown inFigure 9
(4) Number of unknowns in the problem number ofiterations and total and outer
(5) Total computing time iteration time IO-time andcomputer used
(6) Type and numerical values of convergence criteria(7) Table of average group fluxes for a square mesh grid
of 20 times 20 cm(8) Dependence of results on mesh spacing
Even though the original problem is based on a quarter-core situation the problem has an eighth-core symmetry
20 Science and Technology of Nuclear Installations
Quarter Delaunay triangles elements
Tim
e con
sum
ed to
read
the m
esh
(sec
onds
)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumes
Quarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
1
3
10
30
100
300
Figure 22 Time needed to read the mesh versus number of unknowns
which cannot be taken into account by structured gridswhich are the main target of the benchmark Howevernonstructured grids can take into consideration any kind ofsymmetry almost without loss of accuracy and at the sametime reducing roughly the number of unknowns by a halfand the associated computational effort needed to solve theproblem by a factor of four Answers to items (1)ndash(7) canbe given completely by milonga using a single input file(see Supplementary data in SupplementaryMaterial availableonline at httpdxdoiorg1011552013641863) As asked initem (8) how results depend not only on the mesh spacingbut also on the meshing algorithm on the gridrsquos elementarygeometric shape and on the discretization scheme may shedlights on the subject whichmay be evenmore interesting thanthe numerical results themselves
Taking advantage of milongarsquos capability of reading andparsing command-line arguments the selection of the coregeometry (quarter or eighth) the meshing algorithm (delau-nay [16] or delquad [25]) the shape of the elementaryentities (triangles or quadrangles) the discretization scheme(volumes or elements) and the characteristic length ℓ119888 ofthe mesh can be provided at run time Fixing five values forℓ119888 = 4 3 2 1 05 gives 2 times 2 times 2 times 2 times 5 = 80 possiblecombinations which we solve with successive invocations to
milonga with the same input file but with different argumentsfrom a simple script Figures 10 11 12 13 14 15 16 17 and18 show the results corresponding to items (1)ndash(7) for someillustrative cases The complete set of figures and the codeused to generate themmay be provided upon request Table 2compiles the answers to the problem for every case studied
As the milonga code is still under development itsnumerical routines are not yet fully optimized nor designedfor parallel computation Therefore the reported times areonly rough estimates and should be taken with care Thesolution comprises five steps
(1) generate the grid with the requested geometry mesh-ing algorithm basic shape and characteristic lengthby calling to gmsh
(2) read the generated mesh(3) build the matrices(4) solve the eigenvalue problem(5) compute the requested results
The CPU time reported in Figures 10ndash18 is thus thesum of all of these five steps but not the time needed togenerate the figuresmdashwhich in some caseswith coarsemeshes
Science and Technology of Nuclear Installations 21
Table2
Resultsofthe2
DIAEA
PWRBe
nchm
arkp
roblem
obtained
with
them
ilong
acod
efor
thee
ightyp
ropo
sedcombinatio
nsofsymmetrym
eshing
algorithm
basicshapediscretization
schemeandcharacteris
ticelem
entc
elllengthTh
ereference
solutio
nis120588=28749times10minus5w
hich
correspo
ndsto119896eff=10296
Case
no
Symm-
etry
Mesh
algorithm
Basic
shape
Solutio
nmetho
dℓ119888
(cm)
Δ120588
(pcm
)max1206012
(mdash)
Total
unkn
owns
Outer
iter
Linear
iter
Inner
iter
Resid
ual
norm
Relativ
eerror
Error
estim
ate
Mem
ory
(Mb)
Page
faults
114
Delaunay
998779Vo
lumes
40
minus120
1145
8264
332
1033
6119890minus09
3119890minus09
1119890minus09
349470
214
Delaunay
998779Vo
lumes
30
85
1135
15740
332
1967
2119890minus08
1119890minus08
4119890minus09
4813401
314
Delaunay
998779Vo
lumes
20
197
1124
31188
332
3898
1119890minus08
6119890minus09
3119890minus09
7621932
414
Delaunay
998779Vo
lumes
10153
1123
127476
332
15934
7119890minus09
3119890minus09
3119890minus09
273
81886
514
Delaunay
998779Vo
lumes
05
187
1119
510676
332
63834
2119890minus09
9119890minus10
1119890minus09
1148
352261
614
Delaunay
998779Elem
ents
40
173
1100
4308
332
538
2119890minus08
8119890minus09
4119890minus09
318794
714
Delaunay
998779Elem
ents
30
117
1107
8108
332
1013
5119890minus09
2119890minus09
2119890minus09
4712946
814
Delaunay
998779Elem
ents
20
84
1112
15936
332
1992
6119890minus09
3119890minus09
2119890minus09
7321347
914
Delaunay
998779Elem
ents
1062
1116
64478
332
8059
6119890minus09
3119890minus09
4119890minus09
262
77105
1014
Delaunay
998779Elem
ents
05
58
1117
256700
332
32087
1119890minus09
5119890minus10
1119890minus09
1091
331969
1114
Delaunay
◻Vo
lumes
40
53
1149
5042
332
630
2119890minus08
1119890minus08
5119890minus09
288007
1214
Delaunay
◻Vo
lumes
30
346
1133
8560
332
1070
7119890minus09
3119890minus09
2119890minus09
3910895
1314
Delaunay
◻Vo
lumes
20
308
1131
15576
332
1947
1119890minus08
7119890minus09
3119890minus09
541564
514
14Delaunay
◻Vo
lumes
10177
1122
60774
332
7596
8119890minus09
4119890minus09
3119890minus09
167
50115
1514
Delaunay
◻Vo
lumes
05
199
1120
244936
332
30617
4119890minus09
2119890minus09
2119890minus09
698
215678
1614
Delaunay
◻Elem
ents
40
196
1100
5222
332
652
3119890minus08
1119890minus08
8119890minus09
4713098
1714
Delaunay
◻Elem
ents
30
123
1107
8810
332
1101
2119890minus08
9119890minus09
7119890minus09
6918598
1814
Delaunay
◻Elem
ents
20
901112
15922
332
1990
1119890minus08
5119890minus09
5119890minus09
107
30591
1914
Delaunay
◻Elem
ents
1063
1116
61456
332
7682
5119890minus09
2119890minus09
4119890minus09
389
113237
2014
Delaunay
◻Elem
ents
05
58
1117
246332
332
30791
8119890minus10
4119890minus10
1119890minus09
1676
498192
2114
Delq
uad
998779Vo
lumes
40
minus45
1144
6228
332
778
4119890minus08
2119890minus08
7119890minus09
308495
2214
Delq
uad
998779Vo
lumes
30
570
1153
11820
332
1477
3119890minus08
1119890minus08
4119890minus09
4010879
2314
Delq
uad
998779Vo
lumes
20
459
1103
24100
332
3012
2119890minus08
1119890minus08
5119890minus09
6217715
2414
Delq
uad
998779Vo
lumes
10512
1104
9640
03
3212050
2119890minus08
1119890minus08
9119890minus09
207
61714
2514
Delq
uad
998779Vo
lumes
05
528
1106
385600
332
48200
2119890minus09
1119890minus09
2119890minus09
838
255735
2614
Delqu
ad998779
Elem
ents
40
220
1093
3288
332
411
3119890minus08
2119890minus08
6119890minus09
308495
2714
Delq
uad
998779Elem
ents
30
140
1104
6148
332
768
4119890minus08
2119890minus08
8119890minus09
39110
0028
14Delq
uad
998779Elem
ents
20
82
1110
12392
332
1549
2119890minus08
9119890minus09
6119890minus09
6117067
2914
Delqu
ad998779
Elem
ents
1063
1115
48882
332
6110
2119890minus09
1119890minus09
1119890minus09
197
5804
630
14Delq
uad
998779Elem
ents
05
58
1116
194162
332
24270
2119890minus09
9119890minus10
2119890minus09
790
238769
3114
Delq
uad
◻Vo
lumes
40
minus416
1174
3132
332
391
5119890minus08
3119890minus08
8119890minus09
267322
3214
Delq
uad
◻Vo
lumes
30
minus248
1151
5922
332
740
9119890minus09
4119890minus09
2119890minus09
318779
3314
Delq
uad
◻Vo
lumes
20
minus132
1139
12050
332
1506
1119890minus08
7119890minus09
4119890minus09
4412239
3414
Delq
uad
◻Vo
lumes
10minus49
1126
48200
332
6025
6119890minus09
3119890minus09
2119890minus09
122
36094
3514
Delq
uad
◻Vo
lumes
05
minus23
1123
192800
332
24100
5119890minus09
2119890minus09
3119890minus09
464
140228
3614
Delq
uad
◻Elem
ents
40
224
1093
3294
332
411
3119890minus08
1119890minus08
7119890minus09
359852
3714
Delq
uad
◻Elem
ents
30
141
1104
6144
332
768
3119890minus08
2119890minus08
9119890minus09
5114018
3814
Delq
uad
◻Elem
ents
20
921111
12392
332
1549
1119890minus08
6119890minus09
5119890minus09
8122526
3914
Delq
uad
◻Elem
ents
1065
1115
48882
332
6110
5119890minus09
3119890minus09
4119890minus09
276
78431
22 Science and Technology of Nuclear Installations
Table2Con
tinued
Case
no
Symm-
etry
Mesh
algorithm
Basic
shape
Solutio
nmetho
dℓ119888
(cm)
Δ120588
(pcm
)max1206012
(mdash)
Total
unkn
owns
Outer
iter
Linear
iter
Inner
iter
Resid
ual
norm
Relativ
eerror
Error
estim
ate
Mem
ory
(Mb)
Page
faults
4014
Delq
uad
◻Elem
ents
05
59
1117
194162
332
24270
1119890minus09
5119890minus10
1119890minus09
1104
318967
4118
Delaunay
998779Vo
lumes
40
minus99
1139
4228
332
528
4119890minus12
2119890minus12
1119890minus12
359630
4218
Delaunay
998779Vo
lumes
30
50
1132
7712
332
964
1119890minus11
6119890minus12
2119890minus12
4211578
4318
Delaunay
998779Vo
lumes
20
177
1120
15776
332
1972
2119890minus12
1119890minus12
7119890minus13
5515436
4418
Delaunay
998779Vo
lumes
10142
1118
63902
332
7987
3119890minus12
1119890minus12
1119890minus12
160
46691
4518
Delaunay
998779Vo
lumes
05
176
1115
254612
332
31826
2119890minus12
8119890minus13
1119890minus12
555
168526
4618
Delaunay
998779Elem
ents
40
183
1096
2252
332
281
6119890minus12
3119890minus12
2119890minus12
359589
4718
Delaunay
998779Elem
ents
30
119
1104
4040
224
505
2119890minus08
1119890minus08
7119890minus09
41114
3748
18Delaunay
998779Elem
ents
20
85
1108
8156
332
1019
2119890minus12
8119890minus13
6119890minus13
5515053
4918
Delaunay
998779Elem
ents
1063
1112
32480
332
4060
1119890minus12
6119890minus13
8119890minus13
151
43671
5018
Delaunay
998779Elem
ents
05
58
1113
128358
332
1604
47119890minus13
3119890minus13
7119890minus13
527
157581
5118
Delaunay
◻Vo
lumes
40
37
1144
2380
224
297
5119890minus08
2119890minus08
1119890minus08
5314173
5218
Delaunay
◻Vo
lumes
30
218
1120
4194
332
524
7119890minus12
4119890minus12
2119890minus12
3610034
5318
Delaunay
◻Vo
lumes
20
345
1110
7960
332
995
5119890minus12
3119890minus12
1119890minus12
4712879
5418
Delaunay
◻Vo
lumes
10165
1117
30652
332
3831
2119890minus12
1119890minus12
7119890minus13
100
28851
5518
Delaunay
◻Vo
lumes
05
191
1117
122066
224
15258
2119890minus08
8119890minus09
9119890minus09
343
103825
5618
Delaunay
◻Elem
ents
40
175
1097
2526
332
315
4119890minus12
2119890minus12
1119890minus12
6016159
5718
Delaunay
◻Elem
ents
30
114
1104
4386
332
548
4119890minus12
2119890minus12
2119890minus12
5013768
5818
Delaunay
◻Elem
ents
20
901108
8234
332
1029
2119890minus12
1119890minus12
1119890minus12
7320094
5918
Delaunay
◻Elem
ents
1062
1112
31186
224
3898
2119890minus08
7119890minus09
9119890minus09
208
59589
6018
Delaunay
◻Elem
ents
05
58
1113
123122
224
15390
9119890minus09
5119890minus09
1119890minus08
807
2366
7061
18Delq
uad
998779Vo
lumes
40
minus210
1168
3224
332
403
1119890minus11
5119890minus12
2119890minus12
43117
8462
18Delq
uad
998779Vo
lumes
30
369
1150
6000
332
750
1119890minus11
6119890minus12
3119890minus12
5013502
6318
Delq
uad
998779Vo
lumes
20
312
1104
12316
332
1539
2119890minus11
8119890minus12
5119890minus12
6016395
6418
Delq
uad
998779Vo
lumes
1044
01094
48714
332
6089
9119890minus12
4119890minus12
4119890minus12
1183444
165
18Delq
uad
998779Vo
lumes
05
534
1081
193980
332
24247
5119890minus12
2119890minus12
4119890minus12
423
126768
6618
Delq
uad
998779Elem
ents
40
240
1090
1750
332
218
9119890minus12
4119890minus12
2119890minus12
43117
8667
18Delqu
ad998779
Elem
ents
30
150
1100
3184
332
398
1119890minus11
5119890minus12
2119890minus12
5013509
6818
Delq
uad
998779Elem
ents
20
89
1106
6426
332
803
5119890minus12
3119890minus12
2119890minus12
5916047
6918
Delq
uad
998779Elem
ents
1064
1111
24886
332
3110
2119890minus12
1119890minus12
1119890minus12
113
32788
7018
Delq
uad
998779Elem
ents
05
59
1113
98052
224
12256
1119890minus08
5119890minus09
8119890minus09
398
118153
7118
Delq
uad
◻Vo
lumes
40
minus398
1172
1650
332
206
1119890minus11
5119890minus12
2119890minus12
369907
7218
Delq
uad
◻Vo
lumes
30
minus249
1146
3058
332
382
4119890minus12
2119890minus12
1119890minus12
349349
7318
Delq
uad
◻Vo
lumes
20
minus131
1136
6228
224
778
3119890minus08
1119890minus08
5119890minus09
4010930
7418
Delq
uad
◻Vo
lumes
10minus51
1123
24536
224
3067
3119890minus08
2119890minus08
1119890minus08
7822716
7618
Delq
uad
◻Elem
ents
40
237
1090
1752
332
219
6119890minus12
3119890minus12
2119890minus12
4111213
7718
Delq
uad
◻Elem
ents
30
145
1101
3184
332
398
6119890minus12
3119890minus12
2119890minus12
43118
1178
18Delq
uad
◻Elem
ents
20
941107
6426
332
803
3119890minus12
1119890minus12
1119890minus12
6016220
7918
Delqu
ad◻
Elem
ents
1065
1111
24874
332
3109
1119890minus12
6119890minus13
8119890minus13
156
4400
080
18Delq
uad
◻Elem
ents
05
59
1113
9804
22
2412255
8119890minus09
4119890minus09
8119890minus09
567
162168
Science and Technology of Nuclear Installations 23
Tim
e con
sum
ed to
bui
ld th
e mat
rices
(sec
onds
)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay trianglesvolumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
10
30
100
300
1
3
Figure 23 Time needed to build the matrices versus number of unknowns
was considerably larger than the solution time itself Theeigenvalue problem was solved using a multilayer iterativeKrylov-Schur method [26] with a tolerance relative to thematrices norm
1003817100381710038171003817119877 sdot 120601 minus (1119896eff) sdot 119865 sdot 1206011003817100381710038171003817
119877 + (1119896eff) sdot 119865lt 10minus8 (11)
The reported residual norm and relative error are10038171003817100381710038171003817100381710038171003817119877 sdot 120601 minus
1
119896effsdot 119865 sdot 120601
10038171003817100381710038171003817100381710038171003817
1003817100381710038171003817119877 sdot 120601 minus (1119896eff) sdot 119865 sdot 1206011003817100381710038171003817
1003817100381710038171003817(1119896eff) sdot 1206011003817100381710038171003817
(12)
respectively The fields marked as outer linear and inneriterations refer to the number of steps needed to attainthe requested tolerance in each layer of the Krylov-Schuralgorithm The computer used to solve the problem has anIntel i7 920 267GHz processor with 4Gb of RAM runningDebian GNULinux Wheezy
When using a finite volumes-based scheme over anunstructured mesh the solver has to gather information
about which cells are neighbors and which are not Currentlygmsh does not write this kind of lists in its output filesso milonga has to explicitly solve the neighbors problemPerforming a linear search is an 119874(119873
2) task which is
unacceptable for problem sizes 119873 of interest The code usesa search based on a 119896-dimensional tree [27] which can inprinciple be performed in119874(119873) steps Still for large values of119873 the time needed to read and parse the mesh (number twopreviously mentioned) is the bottleneck of the solution Thisstep is not needed in finite elements although the construc-tion of the elementary matrices involves the computationof the multidimensional Jacobians and integrals which thenhave to be assembled Again for large 119873 this step (numberthree) takes up most of the time needed to solve the problem
The Delaunay algorithm is a standard method for gen-erating two-dimensional grids [16] by tessellating a planewith triangles If the mesh needs to be based on quadranglesinstead a recombination algorithm can be used to transformtwo adjacent triangles in one quadrangle whenever is possi-ble However for geometries which are based on rectangularshapes there exist other algorithms [25] both for meshingand for recombining the triangles that give rise to elementaryentities with right angles almost everywhere which may be a
24 Science and Technology of Nuclear Installations
Tim
e con
sum
ed to
solv
e the
eige
nval
ue p
robl
em (s
econ
ds)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
10
30
100
300
1
3
Figure 24 Time needed to solve the eigenvalue problem versus number of unknowns
desired property of the resulting grid For finite elements thesteps needed to build the matrices depend on the selectionof triangles or quadrangles as the basic elementary geometrybecause the shape functions change However the number ofunknowns is the same as the number of nodes that does notchange after a recombination procedure On the other handthe number of unknowns in the finite volumes schemes withtriangles is roughly twice as the number of unknowns withquadrangles for the same grid
Figure 19 shows how the computed static reactivity (ie1 minus 1119896eff) depends on the number of unknowns for eachof the sixteen combinations of geometry algorithm shapescheme The four accepted results published in the originalreference [10] are included for reference although it shouldbe taken into account that said reactivities were computedalmost forty years ago Figure 20 shows the total wall timeneeded to solve the problem as a function of 119873119866 whileFigures 21 22 23 and 24 show the times needed for eachof the first four steps involved in the solution maintainingthe same logarithmic scale for both the abscissas and theordinates Green data represent finite volumes whilst bluebullets correspond to finite elements Solid lines indicatequarter-core and dashed lines eighth-symmetry Fillet bullets
are results obtained by theDelaunay triangulation and emptybullets were computed with the delquad algorithm Finallytriangle-shaped data correspond to triangles and squares anddiamonds denote quadrangles as the basic geometry of thegrid
It can be seen that finite elements produce amuch smallerdispersion of eigenvalues 119896eff than finite volumes with therefinement of the mesh This can be explained because finitevolumes methods rely on a geometric condition of the meshwhich may change abruptly if the meshing algorithm decidesto allocate the cells in a rather different form for small changesin ℓ119888 Finite elements methods are less influenced by thesediscontinuous lay-out changes of the elements As expectedeighth-core symmetries give almost the same results as thequarter-core geometries with roughly half the unknownsFor small problems finite volumes run faster that finiteelements because the time needed to solve the neighborproblem is negligible When the problem size grows thistime increases and exceeds the overhead implied in the con-struction and assembly of the finite elements matrices Alsoat least for this configuration it is seen that the eigenvalueproblem is solved faster for finite volumes than for finiteelements
Science and Technology of Nuclear Installations 25
4 Conclusions
Unstructured grids provide the cognizant engineer witha wide variety of possibilities to deal with the design oranalysis of nuclear reactor cores These kinds of grids cansuccessfully reproduce continuous geometries commonlyfound in reactor cores such as cylinders and thereforenot only can the diffusion equation be better approximatedinside the domain of definition but also the fulfillment ofboundary conditions is improved A free computer codewas written from scratch that is able to completely solvethe 2D IAEA PWR Benchmark using unstructured gridsfor sixteen combinations of geometry meshing algorithmbasic shape and discretization scheme plus any value ofthe gridrsquos characteristic length by using a single input fileThe complete set of input files and codemdashexecutable andsourcemdashis available either online or upon request withcomments experiences suggestions and corrections beingmore than welcome Further development should includetackling full three-dimensional geometries with completethermal hydraulic feedback in order to analyze how thesolutions of the coupled neutronic-thermal problem dependon the spatial discretization scheme of the neutron leakageterm Parallelization of the computation and assembly of thematrices and of the solution of the eigenvalue problem andits implementation using GPUs are also desired features toimplement A problemwith direct applications that the futureversions of milonga ought to solve is the analysis of how thegeometry of the absorbing materials should be taken intoaccount in structured coarse grids in order to mitigate effectssuch as the rod-cusp problem
Suitable schemes for approximating the continuous dif-ferential operators by discrete matrix expressions includefinite volumes and finite elements families Finite volumesmethods compute cell mean values whilst finite elementsgive functional values at the gridrsquos nodes In general finiteelements are less sensitive to changes in the mesh so theresults they provide do not change significantly for differentmeshing algorithms or elementary shapes Small problemsare best solved by finite volumes as the neighbor-findingproblem is faster than the process of building and assemblingthe eigenvalue-problem matrices For a large number ofunknowns the process of finding which cell is neighbor ofwhichmdasheven based on a 119896-dimensional treemdashoverwhelmsthe computation and assembly of elementary matrices andfinite-element methods perform better
References
[1] G Glasstone and S Bell Nuclear Reactor Theory KriegerPublishing Company 1970
[2] J J Duderstadt and L J Hamilton Nuclear Reactor AnalysisJohn Wiley amp Sons New York NY USA 1976
[3] C Gho Reactor Physics Undergraduate Course Lecture NotesInstituto Balseiro 2006
[4] G G Theler and F J Bonetto ldquoOn the stability of the pointreactor kinetics equationsrdquoNuclear Engineering andDesign vol240 no 6 pp 1443ndash1449 2010
[5] E E Lewis andW F Miller Computational Methods of NeutronTransport John Wiley amp Sons 1984
[6] E Masiello ldquoAnalytical stability analysis of Coarse-Mesh finitedifference methodrdquo in Proceedings of the International Confer-ence on the Physics of Reactors (PHYSOR rsquo08) Nuclear Power ASustainable Resource pp 456ndash464 September 2008
[7] M L Zerkle Development of a polynomial nodal methodwith flux and current discontinuity factors [PhD thesis] Mas-sachusets Institute of Technology 1992
[8] J I Yoon andHG Joo ldquoTwo-level coarsemesh finite differenceformulation with multigroup source expansion nodal kernelsrdquoJournal of Nuclear Science andTechnology vol 45 no 7 pp 668ndash682 2008
[9] O Mazzantini M Schivo J Di Cesare R Garbero M Riveroand G Theler ldquoA coupled calculation suite for Atucha II oper-ational transients analysisrdquo Science and Technology of NuclearInstallations vol 2011 Article ID 785304 12 pages 2011
[10] Computational Benchmark Problem Comitee for the Mathe-matics and Computation Division of the American NuclearSociety ldquoArgonne Code Center Benchmark problem bookrdquoTech Rep ANL-7416 Supplement 2 Argonne National Labo-ratory 1977
[11] A F Henry Nuclear Reactor Analysis MIT Cambridge UK1975
[12] O C Zienkiewicz R L Taylor and J Z ZhuTheFinite ElementMethod Its Basis and Fundamentals vol 1 Elsevier 6th edition2005
[13] G Theler Difusion de neutrones en mallas no estructuradascomparacion entre volumenes y elementos finitos AcademicMonograph Universidad de Buenos Aires 2013
[14] G Theler ldquoMilonga a free nuclear reactor core analysis coderdquoAvailable online 2011
[15] G Theler F J Bonetto and A Clausse ldquoOptimizacion deparametros en reactores de potencia base de diseno delcodigo neutronico milongardquo in Reunion Anual de la AsociacionArgentina de Tecnologia Nuclear XXXVII 2010
[16] C Geuzaine and J-F Remacle ldquoGmsh a 3-D finite elementmesh generator with built-in pre- and post-processing facili-tiesrdquo International Journal for Numerical Methods in Engineer-ing vol 79 no 11 pp 1309ndash1331 2009
[17] S Balay J Brown K Buschelman et al ldquoPETSc users manualrdquoTech Rep ANL-9511-Revision 34 Argonne National Labora-tory 2013
[18] S Balay W D Gropp L Curfman McInnes and B F SmithldquoEfficient management of parallelism in object oriented numer-ical software librariesrdquo in Modern Software Tools in ScientificComputing E Arge A M Bruaset and H P Langtangen Edspp 163ndash202 Birkhauser 1997
[19] V Hernandez J E Roman and V Vidal ldquoSLEPC a scalable andflexible toolkit for the solution of eigenvalue problemsrdquo ACMTransactions on Mathematical Software vol 31 no 3 pp 351ndash362 2005
[20] M Galassi J Davies J Theiler et al GNU Scientific LibraryReference Manual 3rd edition 2009
[21] O C Zienkiewicz and R L Taylor The Finite Element MethodFor Solid and Structural Mechanics vol 3 Elsevier 6th edition2005
[22] A Imelda and K Doddy Evaluation of the IAEA 3-D PWRbenchmark problem using NESTLE code 2004
[23] R Mosteller ldquoStatic benchmarking of the NESTLE advancednodal coderdquo in Proceedings of the Joint International Conference
26 Science and Technology of Nuclear Installations
on Mathematical Methods and Supercomputing for NuclearApplications vol 2 pp 1596ndash1605 1997
[24] GTheler F J Bonetto andAClausse ldquoSolution of the 2D IAEAPWR Benchmark with the neutronic code milongardquo in Actasde la Reunion Anual de la Asociacion Argentina de TecnologıaNuclear XXXVIII 2011
[25] J F Remacle F Henrotte T Carrier-Baudouin et al ldquoAfrontal delaunay quad mesh generator using the 119871
infin normrdquoInternational Journal For Numerical Methods in Engineeringvol 94 no 5 pp 494ndash512 2013
[26] G W Stewart ldquoA Krylov-Schur algorithm for large eigenprob-lemsrdquo SIAM Journal on Matrix Analysis and Applications vol23 no 3 pp 601ndash614 2002
[27] J L Bentley ldquoMultidimensional binary search trees used forassociative searchingrdquo Communications of the ACM vol 18 no9 pp 509ndash517 1975
TribologyAdvances in
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FuelsJournal of
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Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Renewable Energy
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
StructuresJournal of
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear InstallationsScience and Technology of
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Wind EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear EnergyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
4 Science and Technology of Nuclear Installations
(a) Continuous two-dimensional domain (b) Structured grid
(c) Unstructured grid
Figure 2 When a continuous domain (a) is meshed with an unstructured grid there appears a geometric condition known as the staircaseeffect (b) For the same number of nodes unstructured grids reproduce the original geometry better (c)
Figure 3 Cross section of a hypothetical reactor in which thecontrol rods enter into the core from above with a certain attackangle with respect to the vertical direction
at the same time can be shown to be equivalent to the strongformulation given by (1)
In any case both formulations involve the computationof 119866 unknown functions of x and one unknown real value119896eff Except for homogeneous cross sections in canonicaldomains 119880 the multigroup diffusion equation needs to besolved numerically As discussed below any nodal-baseddiscretization scheme replaces a continuous unknown func-tion 120601119892(x) by 119873 discrete values 120601119892(119894) for 119894 = 1 119873
Figure 4 The finite volumes method computes the unknown fluxin the cell centers (squares) whilst the finite elements methodcomputes the fluxes at the nodes (circles)
If we arrange these unknowns into a vector 120601 isin R119873119866
such as
120601 =
[[[[[[[[[[[[[[[[
[
1206011 (1)
1206012 (1)
120601119866 (1)
1206011 (2)
120601119892 (119894)
120601119866 (119873)
]]]]]]]]]]]]]]]]
]
(4)
Science and Technology of Nuclear Installations 5
(a) Fast flux distribution
minus100minus50 0 50 100minus100
minus500
50100
0
06
xy
(b) Fast flux unknowns
(c) Matrix 119877 structure (d) Matrix 119865 structure
Figure 5 A bare homogeneous circle solved with finite volumes (184 unknowns)
then the continuous eigenvalue problemmdashin either for-mulationmdashcan be transformed into a generalized matrixeigenvalueeigenvector problem casted in either of the follow-ing forms
119877 sdot 120601 =1
119896effsdot 119865 sdot 120601
119865 sdot 120601 = 119896eff sdot 119877 sdot 120601
119877minus1sdot 119865 sdot 120601 = 119896eff sdot 120601
(5)
where 119877 and 119865 are square 119873119866 times 119873119866 matrices We call119865 the fission matrix as it contains all the ]-fission termswhich are the ones that we artificially divided by 119896eff Therest of the terms are grouped into the removal or transportmatrix 119877 which includes the rest of the neutron-matterinteraction mechanisms It can be shown [11] that for anyreal set of cross sections 119877minus1 exists and that the119873119866 pairs ofeigenvalueeigenvector solutions of (5) satisfy that
(1) there is a unique real positive eigenvalue greater inmagnitude than any other eigenvalue
(2) all the elements of the eigenvector corresponding tothat eigenvalue are real and positive
(3) all other eigenvectors either have some elements thatare zero or have elements that differ in sign from eachother
21 Boundary Conditions Being a differential equation overspace the neutron diffusion equation needs proper boundaryconditions to conform a properly definedmathematical prob-lem These can be imposed flux (Dirichlet) imposed current(Neumann) or a linear combination (Robin) However dueto the fact that in the linear problem in absence of externalsourcesmdashsuch as (1) or (3)mdashthe problem is homogeneousif 120601119892(x) is a solution then any multiple 120572120601119892(x) is also asolution That is to say the eigenvectors are defined up to amultiplicative constant whose value is usually chosen as toobtain a certain total thermal power Thus the prescribedboundary conditions should also be homogeneous and bedefined up to amultiplicative constantTherefore the allowedDirichlet conditions are zero flux at the boundary theNeumann conditions should prescribe that the derivative in
6 Science and Technology of Nuclear Installations
(a) Fast flux distribution
minus100minus50 0 50 100minus100
minus500
50100
0
06
xy
(b) Fast flux unknowns
(c) Matrix 119877 structure (d) Matrix 119865 structure
Figure 6 A bare homogeneous circle solved with finite elements (218 unknowns)
the normal direction should be zero (ie symmetry condi-tion) and the Robin conditions are restricted to the followingform
grad 120601119892 (x) sdot n + 119886119892 (x) sdot 120601119892 (x) = 0 (6)
n being the outward unit normal to the boundary 120597119880 of thedomain 119880
22 Grids and Schemes One way of solving the neutrondiffusion equationmdashand in general any partial differentialequation over spacemdashis by discretizing the differential oper-ators with some kind of scheme that is applied over a certainspatial grid Given an 119898-dimensional domain a grid definesa partition composed of a finite number of simple geometricentities In structured grids these elementary entities arearranged following a well-defined structure in such a waythat each entity can be identified without needing furtherinformation than the one provided by the intrinsic structureOn the other hand the geometric entities that composean unstructured grid are arranged in an irregular patternand the identification of the elementary entities needs to
be separately specified for example by means of a sortedlist For instance Figure 2(b) shows a structured grid thatapproximates the continuous domain of Figure 2(a) Eachsquare may be uniquely identified by means of two integerindexes that indicate its relative position in each of thehorizontal and vertical directions In the case of Figure 2(c)that shows an unstructured grid there is no way to system-atically make a reference to a particular quadrangle withoutany further information
Almost all of the grid-based schemesmdashwhich are knownas nodal schemes which are to be differentiated from modalschemes based on series expansionsmdashare based in either thefinite differences volumes or elements method Finite differ-ences schemes provide themost simple and basic approach toreplace differential (ie continuous) operators by difference(ie discrete) approximations However they are not suitablefor unstructured meshes and may introduce convergenceproblems with parameters that are discontinuous in spacewhich is the case for any reactor core composed of atleast two different materials Moreover boundary conditionsare hard to incorporate and usually give rise to incorrectresults
Science and Technology of Nuclear Installations 7
20 40 600995
09975
1
Analytical solutionFinite volumesFinite elements
keff
a985747c
Figure 7 The effective multiplication factor of a two-group barecircular reactor of radius 119886 as a function of the quotient 119886ℓ
119888between
the radius and the characteristic length of the cellelement
20 40 60
Finite volumesFinite elements
Erro
r ink
eff
10minus3
10minus4
10minus5
a985747c
Figure 8 Absolute value of the error committed in the computationof 119896eff versus 119886ℓ119888
Methods of the finite volumes family involve the inte-gration of the differential equation over each elementaryentity applying the divergence theorem to transform volumeintegrals into surface integrals and providing a mean to esti-mate the fluxes through the entityrsquos surface using informationcontained in its neighbors In this context each elementaryentity is called a cell and finite volumes methods give themean value of each of the group fluxes 120601119892(119894) at the 119894th cellwhich may be associated to the value of 120601119892(x119894) where x119894 isthe location of the cell barycenter Being an integral-basedmethod spatial-discontinuous parameters are handled moreefficiently than in finite differences and boundary conditionscan be easily incorporated as forced cell fluxes Nonethelesseven though these methods may be applied to unstructuredgrids the estimation of the fluxes on the cellsrsquo surfaces isusually performed by using some geometric approximationsthat may lose validity as the quality of the grid is worsened
10
70
90
130
150
170
10 30 50 70 90 110 130 150 170
4
1 38
36 37353
3 32 33 34312
26 27 28 29 3025
19 20 21 22 23 2418
10 11 12 13 14 15 16 17
2 331 4 5 6 7 8 9
x (cm)
y (cm
)
Figure 9 The 2D IAEA PWR benchmark geometry
Moreover the simple integral approach cannot take intoaccount discontinuous diffusion coefficients so when twoneighbors pertain to different materials the flux has to becomputed in a certain particular way to conserve the neutroncurrent according to (2)
Finally finite elements methods rely on a weak formula-tion of the differential problem similar to (3) that maintainsall the mathematical characteristics of the original strongformulation plus its boundary conditions The method isbased on shape functions that are local to each elementaryentitymdashnow called element defined by nodes as cornersmdashand on finding a set of nodal values such that a certaincondition is met which is usually that the residual of thesolution has to be orthogonal to each of the shape functionsThis condition is known as the Galerkin method and impliesthat the error committed in the approximate solution ofthe continuous problem is confined into a small subset ofthe original vector space of the continuous functions Thesemathematical properties make finite elements schemes veryattractive However these features depend on a large numberof integrations that ought to be performed numerically soa computational effortdesired accuracy tradeoff has to beconsidered Not only does the finite elements method givethe values of the flux 120601119892(x119894) at the 119894th node but also theshape functions provide explicit expressions to interpolateand to evaluate the unknown functions at any location x ofthe domain119880 Boundary conditions are divided into essentialand naturalThefirst group comprises theDirichlet boundaryconditions which are satisfied exactlymdashwithin the precisionof the eigenvalue problem solvermdashby the obtained solutionThe latter include the Neumann and the Robin conditionsthat are satisfied only approximately by the derivatives of theinterpolated solution through the shape functions with finermeshes giving better agreement with the prescribed values
The same unstructured grid may be used either for thefinite volumes or for the finite elements method In the first
8 Science and Technology of Nuclear Installations
Largest eigenvalue 1029690 (288336 pcm)1135 (3066 3079)1306 (5171 13076)
Number of unknowns 15740Outer iterations 3Linear iterations 32Inner iterations 1967Residual normRelative errorError estimateMemory used 49824 kBSoft page faults 13401Hard page faults 0Total CPU time 074 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
2483 times 10minus8
1223 times 10minus8
403 times 10minus9
keff
quarter-symmetry core meshed using Delaunay (triangles 985747c = 3) solved with finite volumesMilongarsquos 2D LWR IAEA benchmark problem case no 002
(a)
123456789
1011121314151617181920212223242526272829303132333435363738
k Pk 1206011k 1206012k075 3282 555133 4242 984147 4645 1090123 3921 911061 2677 453094 2995 694093 2927 689074 2011 549mdash 322 756146 4597 1078150 4730 1110133 4199 985107 3405 789104 3271 767095 2971 700072 1956 534mdash 306 733149 4695 1102136 4289 1007118 3733 876107 3367 792097 2892 718066 1628 491mdash 234 549120 3797 891097 3096 718090 2842 669084 2221 619mdash 568 1215mdash 067 257047 2042 348068 2055 504058 1414 430mdash 222 562057 1374 425mdash 388 820mdash 053 203mdash 060 229
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 10 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
case the unknowns are themean value of the fluxes over eachcell whilst in the latter the unknowns are the fluxes evaluatedat each node Therefore the number of unknowns 119873119866 isdifferent for each method even when using the same gridFigure 4 shows this situation and in Section 31 we further
illustrate these differences A fully detailed mathematicaldescription of the actual algorithms for both volumes andelements-based proposed discretizations can be found in anacademic monograph written by the author of this paper[13]
Science and Technology of Nuclear Installations 9
Largest eigenvalue 1029927 (290571 pcm)1131 (3175 2999)1220 (13097 5098)
Number of unknowns 15576Outer iterations 3Linear iterations 32Inner iterations 1947Residual normRelative errorError estimateMemory used 56256 kBSoft page faults 15645Hard page faults 0Total CPU time 09201 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
1394 times 10minus8
6865 times 10minus9
3425 times 10minus9
keff
quarter-symmetry core meshed using Delaunay (quads 985747c = 2) solved with finite volumesMilongarsquos 2D LWR IAEA benchmark problem case no 013
(a)
123456789
1011121314151617181920212223242526272829303132333435363738
k Pk 1206011k 1206012k074 3244 549133 4230 982147 4651 1092124 3946 917060 2645 443094 2997 696093 2938 691074 2005 545mdash 301 746145 4588 1076150 4720 1108133 4198 984109 3465 805104 3280 769095 2986 704071 1962 528mdash 293 715148 4680 1098136 4288 1007119 3755 881107 3385 796097 2905 721067 1653 493mdash 220 556121 3808 893098 3115 724092 2889 680083 2247 615mdash 543 1241mdash 067 260045 1977 330069 2076 509058 1460 428mdash 226 574056 1364 413mdash 371 821mdash 052 200mdash 060 238
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 11 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
3 Results
We now proceed to show two illustrative results that are tobe taken as an overview of the possibilities that unstructuredgrids can provide in order to tackle the multigroup neutron
diffusion problem The examples are two-dimensional prob-lems as they contain some of the complexities a real three-dimensional reactor posse yet the reported results are notso complicated as to be easily understood and analyzed Inparticular we state some basic differences between the finite
10 Science and Technology of Nuclear Installations
Largest eigenvalue 1029695(288388 pcm)1112 (3076 3032)838 (5000 13000)
Number of unknowns 15922Outer iterations 3Linear iterations 32Inner iterations 1990Residual normRelative errorError estimateMemory used 109964 kBSoft page faults 30591Hard page faults 0Total CPU time 158 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
1056 times 10minus8
5202 times 10minus9
5122 times 10minus9
keff
quarter-symmetry core meshed using Delaunay (quads 985747c = 2) solved with finite volumesMilongarsquos 2D LWR IAEA benchmark problem case no 018
(a)
123456789
1011121314151617181920212223242526272829303132333435363738
k Pk 1206011k 1206012k074 3220 549130 4152 961144 4651 1068120 3843 890061 2648 452094 2996 693094 2959 696072 2061 569mdash 358 862142 4495 1054147 4634 1088131 4127 968107 3409 789104 3275 768096 3003 708071 2014 554mdash 342 820146 4606 1081134 4222 991118 3711 871107 3375 794098 2929 727062 1692 522mdash 258 632119 3752 880096 3084 715091 2858 673080 2280 633mdash 611 1277mdash 080 318047 2043 350069 2088 510054 1463 450mdash 255 648051 1418 439mdash 410 854mdash 064 252mdash 071 285
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 12 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
volumes and the finite elements methods by solving a two-group homogeneous bare reactor with the Robin boundaryconditions over the very same grid although a rather coarseone so the differences can be observed directly into theresulting figuresWe then solve the classical two-dimensionalLWR problem also known as the 2D IAEA PWR benchmark
Not only do we show again the differences between the finitevolumes and elements formulation but also we solve theproblemusing different combinations ofmeshing algorithmsbasic shapes and characteristic lengths of the mesh
To solve the two examples shown below we employed themilonga code which was written from scratch by the author
Science and Technology of Nuclear Installations 11
Largest eigenvalue 1029159 (283326 pcm)1174 (2851 3224)1560 (5200 13200)
Number of unknowns 3132Outer iterations 3Linear iterations 32Inner iterations 391Residual normRelative errorError estimateMemory used 26888 kBSoft page faults 7322Hard page faults 0Total CPU time 0308 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
5408 times 10minus8
2665 times 10minus8
8424 times 10minus9
keff
Milongarsquos 2D LWR IAEA benchmark problem case no 031quarter-symmetry core meshed using Delquad (quads 985747c = 4) solved with finite volumes
(a)
123456789
1011121314151617181920212223242526272829303132333435363738
k Pk 1206011k 1206012k077 3422 570140 4468 1040153 4836 1135128 4070 948060 2704 448095 3017 702091 2861 673069 1918 508mdash 264 689151 4785 1122155 4897 1149137 4334 1017111 3535 823104 3272 768092 2908 685067 1876 496mdash 254 658154 4854 1139139 4392 1031120 3792 890107 3360 790095 2831 702061 1574 451mdash 197 515123 3868 907099 3132 730090 2818 663078 2168 579mdash 481 1191mdash 055 217044 1984 328067 2027 498052 1360 388mdash 190 520052 1329 383mdash 321 792mdash 043 168mdash 048 195
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 13 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
of this paper and is currently still under development withinhis ongoing PhD thesis There exists a first public release[14] under the terms of the GNU General Public Licensemdashthat is it is a free softwaremdashthat can only handle structuredgrids There is a second release being prepared whose
main relevance is that it can work with nonstructured gridsas well which is the main feature of the code It uses ageneral mathematical frameworkmdashcoded from scratch aswellmdashwhich provides input file parsing algebraic expres-sions evaluation one- and multidimensional interpolation of
12 Science and Technology of Nuclear Installations
Largest eigenvalue 1029788 (289260 pcm)1120 (3149 3011)1091 (13038 5114)
Number of unknowns 15776Outer iterations 3Linear iterations 32Inner iterations 1972Residual normRelative errorError estimateMemory used 56996 kBSoft page faults 15436Hard page faults 0Total CPU time 09121 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
2296 times 10minus12
1131 times 10minus12
6877 times 10minus13
keff
Milongarsquos 2D LWR IAEA benchmark problem case no 043eighth-symmetry core meshed using Delaunaya (triangs985747c = 2) solved with finite volumes
(a)
1 074 3241 5482 131 4188 9713 145 4582 10764 122 3877 9005 060 2644 4456 093 2986 6927 093 2941 6928 075 2030 5529 mdash 322 779
10 143 4523 106111 148 4667 109512 131 4145 97213 107 3423 79414 103 3269 76715 095 2993 70616 073 1988 53917 mdash 312 75118 147 4637 108819 134 4243 99620 118 3721 87321 107 3370 79322 098 2914 72323 067 1651 49924 mdash 235 57125 119 3763 88226 096 3077 71527 091 2847 67128 083 2255 61729 mdash 576 122530 mdash 068 26631 046 2016 34132 068 2058 50533 059 1443 43534 mdash 233 58935 057 1381 42036 mdash 381 82337 mdash 054 20838 mdash 062 244
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 14 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
scattered data shared-memory access numerical integrationfacilities and so forth The milonga code was designed withfour design basis vectors in mindmdashwhich are thoroughlydiscussed in the documentation [15]mdashwhich includes thetype of problems it can handle the code scalability which
features are expected and what to do with the obtainedresults
It works by first reading an input file that using plain-text English keywords and arguments defines the number119898 of spatial dimensions the number 119866 of group energies
Science and Technology of Nuclear Installations 13
Largest eigenvalue 1029726 (288675 pcm)1104 (3069 3069)864 (13000 5000)
Number of unknowns 4040Outer iterations 2Linear iterations 24Inner iterations 505Residual normRelative errorError estimateMemory used 42940 kBSoft page faults 11437Hard page faults 0Total CPU time 1064 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
2019 times 10minus8
9946 times 10minus9
7138 times 10minus9
keff
Milongarsquos 2D LWR IAEA benchmark problem case no 047eighth-symmetry core meshed using Delaunay (triangles 985747c = 3) solved with finite volumes
(a)
1 074 3194 5452 129 4117 9533 143 4515 10604 119 3814 8835 061 2634 4506 093 2984 6907 094 2954 6958 071 2064 5719 mdash 362 865
10 141 4458 104511 146 4599 107912 130 4097 96113 106 3388 78414 103 3263 76515 095 2998 70716 068 2015 55717 mdash 346 82318 145 4572 107319 133 4194 98420 117 3691 86621 107 3363 79222 098 2925 72623 059 1696 52824 mdash 263 63525 118 3728 87426 096 3068 71127 091 2850 67228 079 2280 63529 mdash 619 127630 mdash 081 32131 047 2036 34932 068 2085 50933 051 1465 45434 mdash 258 65035 049 1419 44336 mdash 416 85437 mdash 065 25438 mdash 071 288
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 15 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
and optionally a mesh file Currently only grids generatedwith the code gmsh [16] are only supported mainly becauseit is also a free software and it suits perfectly well milongarsquosdesign basis in the sense that the continuous geometry can bedefined as a function of a number of parameters Afterwardthe physical entities defined in the grid are mapped into
materials with macroscopic cross sections which maydepend on the spatial coordinates x bymeans of intermediatefunctions such as burn-up or temperatures distributionswhich in turn may be defined by algebraic expressions byinterpolating data located in files by reading shared-memoryobjects or by a combination of them In the same sense
14 Science and Technology of Nuclear Installations
Largest eigenvalue 1029695 (288387 pcm)1108 (3045 3045)833 (13000 5000)
Number of unknowns 8234Outer iterations 3Linear iterations 32Inner iterations 1029Residual normRelative error
Max 1206012(x y) coreMax 1206012(x y) reflector
2028 times 10minus12
9993 times 10minus13
1012 times 10minus12
keff
Relative errorError estimateMemory used 75468 kBSoft page faults 20094Hard page faults 0Total CPU time 1256 seconds
9993 times 10
1012 times 10minus12
Milongarsquos 2D LWR IAEA benchmark problem case no 058eighth-symmetry core meshed using Delaunay (quads 985747c = 2) solved with finite volumes
(a)
1 074 3207 5472 129 4136 9573 144 4534 10644 120 3828 8875 061 2638 4506 093 2985 6907 094 2948 6948 072 2053 5669 mdash 357 859
10 142 4478 105011 146 4617 108412 130 4111 96413 106 3396 78614 103 3263 76515 095 2991 70616 071 2005 55217 mdash 340 81618 145 4588 107719 133 4206 98720 117 3698 86821 107 3362 79122 098 2918 72523 062 1686 52024 mdash 258 62925 118 3737 87626 096 3073 71227 091 2848 67128 080 2272 63129 mdash 609 127230 mdash 080 31731 047 2035 34832 069 2081 50933 054 1458 44834 mdash 255 64635 052 1413 43736 mdash 409 85137 mdash 064 25138 mdash 071 284
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 16 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
boundary conditions are applied to appropriate physicalentities of dimension119898minus1The problemmatrices119877 and119865 arethen built and stored in an appropriate sparse format usingthe free PETSc [17 18] library and the eigenvalue problemis solved using the free SLEPc [19] library The results arestored into milongarsquos variables and functions which may be
evaluated integratedmdashusually using the free GNU ScientificLibrary [20] routinesmdashand of course written into appropriateoutputs Milonga can also solve problems parametrically andbe used to solve optimization problems As stated above thecode is a free software so corrections and contributions aremore than welcome
Science and Technology of Nuclear Installations 15
Largest eigenvalue 1029462 (286185 pcm)1136 (3093 2953)1233 (13100 5100)
Number of unknowns 6228Outer iterations 2Linear iterations 24Inner iterations 778Residual normR l ti
Max 1206012(x y) coreMax 1206012(x y) reflector
26 times 10minus8
1281 times 10minus8
4996 times 10minus9
keff
Relative errorError estimateMemory used 41460 kBSoft page faults 10930Hard page faults 0Total CPU time 0604 seconds
1281 times 10 8
4996 times 10minus9
Milongarsquos 2D LWR IAEA benchmark problem case no 073eighth-symmetry core meshed using Delaunay (quads 985747c = 2) solved with finite volumes
(a)
1 075 3297 5552 134 4268 9923 148 4660 10944 123 3926 9145 059 2645 4406 094 2997 6977 093 2914 6858 072 2002 5349 mdash 301 773
10 146 4606 108011 150 4735 111112 133 4212 98813 108 3448 80214 104 3271 76715 094 2954 69616 070 1954 52117 mdash 288 73518 149 4691 110119 136 4289 100720 119 3748 88021 107 3359 79022 096 2880 71423 064 1636 47524 mdash 217 56325 121 3810 89426 098 3110 72427 090 2829 66628 081 2227 59929 mdash 536 125730 mdash 062 25031 045 2002 33632 068 2040 50033 055 1414 40934 mdash 215 57935 055 1359 40836 mdash 359 83437 mdash 050 19938 mdash 061 236
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 17 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
31 A Coarse Bare Homogeneous Circle Figures 5 and 6 showthe results of solving a bare two-dimensional circular homo-geneous reactor of radius 119886 over an unstructured grid whichis deliberately coarse using the finite volumes method andthe finite elements method respectively Two group energies
were used and a null-flux boundary conditionwas fixed at theexternal surface Figures 5(a) and 6(a) compare the obtainedfast flux distributions In the first case the numerical solutionprovidesmean values for each neutron flux group in each cellwhile in the latter the solution is computed at the nodes and
16 Science and Technology of Nuclear Installations
Largest eigenvalue 1029851 (289858 pcm)1090 (3182 3182)851 (13000 5000)
Number of unknowns 1752Outer iterations 3Linear iterations 32Inner iterations 219Residual norm
Max 1206012(x y) coreMax 1206012(x y) reflector
566 times 10minus12
2789 times 10minus12
1595 times 10minus12
keff
Residual normRelative errorError estimateMemory used 42564 kBSoft page faults 11213Hard page faults 0Total CPU time 09921 seconds
566 times 10
2789 times 10minus12
1595 times 10minus12
Milongarsquos 2D LWR IAEA benchmark problem case no 076eighth-symmetry core meshed using Delaunay (quads 985747c = 4) solved with finite volumes
(a)
1 073 3151 5402 127 4059 9393 141 4457 10464 118 3768 8715 061 2615 4506 093 2979 6887 094 2971 6998 067 2092 5829 mdash 375 885
10 139 4397 103111 144 4541 106612 128 4052 95013 105 3365 77814 103 3259 76415 096 3016 71216 066 2045 56817 mdash 359 84118 143 4520 106119 132 4155 97520 116 3672 86221 107 3366 79222 098 2951 73323 054 1728 54324 mdash 273 65125 117 3700 86826 095 3057 70827 091 2860 67428 074 2311 64829 mdash 642 130030 mdash 084 33231 048 2041 35332 069 2103 51333 046 1493 46734 mdash 269 66735 046 1444 45536 mdash 432 87137 mdash 067 26338 mdash 074 297
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 18 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
continuous functions are evaluated by means of the shapefunctions used in the formulation Figures 5(b) and 6(b)illustrate the fast fluxes unknowns and its relative position inspace It can be noted that the mesh coarseness gives resultsthat differ substantially in both cases Finally the structure ofthe sparse eigenvalue problemmatrices is shown for each case
with blue red and cyan representing positive negative andexplicitly inserted zero values In the finite volume case thereare 184 unknowns (92 cells times 2 groups) while in the secondcase there are 218 unknowns (109 nodes times 2 groups) Thevolumesrsquo fissionmatrix is almost diagonal it has a bandwidthequal to the number of energy groups which in this case is
Science and Technology of Nuclear Installations 17
500 1000 2000 5000 10000 20000 50000Number of unknowns
2840
2860
2880
2900
2920
ORNLRisoslashKWUOntarioQuarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumes
Quarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elementsEighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumes
Eighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
120588(p
cm)
105 2 times 105 5 times 105
Figure 19 Static reactivity versus number of unknowns The four original solutions as published in 1977 [10] are included as reference
twoThe off-diagonal values appear right next to the diagonalelements because of the chosen ordering of the unknownsin the flux vector 120601 isin R184 Other orderings may be usedbut the rate of convergence of the eigenvalue problem can bedeteriorated On the other hand the elementsrsquo fission matrixhas a nontrivial structure because the grid is unstructuredand the net fission rate at each element depends on the fluxesat the nodes whose location inside thematrix depends in turnon how the gridwas generatedThis effect is similar to the onethat appears in structural analysis where mass matrices needto be lumped in order to simplify transient computations [21]The diagonal block of elementrsquos 119877 matrix and the null blockin 119865 correspond to the discrete equations that set the null-flux boundary conditions on the nodes located at the externalsurface Other types of boundary conditions lead to differentkinds of structures within the problem matrices
In case a part of the domain contained a nonmultiplicativematerial such as a reflector then there would appear sectionsof the fission matrix with null values in both methodsrendering 119865 singular in both methods Care should be takenwhen dealing with the numerical schemes for the eigenvalueproblem solution It can be seen in the elementsrsquo matricesa particular structure that implements the boundary condi-tions at the external surfaces This structure does not appearin the volumesrsquo matrices because the boundary conditions
appear as flux terms which are summed up over all thesurfaces of each cell so they aremasked inside the volumetricdiscretization of the divergence and gradient operators
As the two-group neutron diffusion equation with uni-form cross sections over a circle subject to null-flux bound-ary conditions has an analytical solution it is adequate tocompare how the two proposed numerical schemes relate toit In the studied problem we ignored upscattering and fastfissions Then the analytical effective multiplication factor is
119896eff =]Σ1198912 sdot Σ1199041rarr2
[Σ1198861 + Σ1199041rarr2 + 1198631(]0119886)2] [Σ1198862 + 1198632(]0119886)
2]
(7)
being ]0 = 24048 the smallest root of Besselrsquos first-kindfunction of order zero 1198690(119903)
Figure 7 shows how the two numerical effective multi-plication factors compare to the analytical solution givenby (7) as a function of the mesh refinement indicated bythe quotient 119886ℓ119888 between the radius of the circle and thecharacteristic length of the cellelements We can see thatthe 119896eff computed by the finite volumes (elements) methodis always greater (less) than the analytical solution Indeedit can be proven that for a bare one-dimensional slab thisis always the case [13] However this result does not hold
18 Science and Technology of Nuclear Installations
2000 3000 6000 10000 20000 30000 60000
01
003
001
03
1
3
10
30
100
300To
tal w
all t
ime (
seco
nds)
Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
Figure 20 Total wall time versus number of unknowns
for problems with nonuniform cross sections even in simpleone-dimensional reflected reactors
We may draw two other conclusions from Figure 7 Firstthat the finite element method seems to provide a bettersolution than the volumes-based scheme and second thateven though the error committed tends to decrease with finergrids its behavior is not monotonic for finite volumes Infact Figure 8 shows the difference between the numericaland analytical solutions using a logarithmic vertical scalewhere both conclusions are even more evident We deferthe discussion of such differences between the discretizationscheme until the next section It is worth to note howeverthat the fact that a finite-element-based scheme throws betterresults for the particular bare homogeneous circular reactorunder study than the finite volumes does not imply thatfinite volumes ought to be discarded as a valid tool forsolving the neutron diffusion equation in general cases Thecombination of lattice and core-level computations is usuallyperformedusing cell-based resultswhichwhen fed into node-based methods to solve the few-group neutron diffusionequation may introduce errors which potentially can leadto unacceptable solutions Nevertheless this analysis is farbeyond the scope of this paper which focuses on solving amathematical equation over unstructured grids
32 The 2D IAEA PWR Benchmark Problem This is aclassical two-group neutron diffusion problem first designedin the early 1970s and taken as a reference benchmark forcomputational codes A number of codes were used to solveeither this problem or its three-dimensional formulation[22 23] including milonga using a structured grid [24] Theoriginal formulation can be found in the reference [10] andthere is a reproduction that may be easily found online inreference [24] The geometry consists of a one-quarter ofa PWR core depicted in Figure 9 and the homogeneousmacroscopic cross sections are listed in Table 1 An axialbuckling term 119861
2
119911119892= 08times10
minus4 should be taken into accountThe external surface should be subject to a zero incomingcurrent condition which may be written as
120597120601119892
120597119899= minus
04692
119863119892
sdot 120601119892 (8)
The expected results are as follows
(1) Maximum eigenvalue(2) Fundamental flux distributions
(a) Radial flux traverses 120601119892(119909 0) and 120601119892(119909 119909)
Science and Technology of Nuclear Installations 19
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
1
3
10
30
100
300Ti
me c
onsu
med
to m
esh
the g
eom
etry
(sec
onds
)
Figure 21 Time needed to mesh the geometry versus number of unknowns
Table 1 Macroscopic cross-sections (units are not stated in the original reference but they are assumed to be in cmminus1 or cm as appropriate)
1198631
1198632
Σ1rarr2
Σ1198861
Σ1198862
]Σ1198912
Material1 15 04 002 001 0080 0135 Fuel 12 15 04 002 001 0085 0135 Fuel 23 15 04 002 001 0130 0135 Fuel 2 + rod4 20 03 004 0 0010 0 Reflector
Note the fluxes shall be normalized such that
1
119881coreint119881core
sum
119892
]Σ119891119892 sdot 120601119892119889119881 = 1 (9)
(b) Value and location of maximum power densityThis corresponds to maximum of 1206012 in thecore It is recommended that the maximumvalues of 1206012 both in the inner core and at thecorereflector interface be given
(3) Average subassembly powers 119875119896
119875119896 =1
119881119896
int119881119896
sum
119892
]Σ119891119892 sdot 120601119892119889119881 (10)
where 119881119896 is the volume of the 119896th subassembly and119896 designates the fuel subassemblies as shown inFigure 9
(4) Number of unknowns in the problem number ofiterations and total and outer
(5) Total computing time iteration time IO-time andcomputer used
(6) Type and numerical values of convergence criteria(7) Table of average group fluxes for a square mesh grid
of 20 times 20 cm(8) Dependence of results on mesh spacing
Even though the original problem is based on a quarter-core situation the problem has an eighth-core symmetry
20 Science and Technology of Nuclear Installations
Quarter Delaunay triangles elements
Tim
e con
sum
ed to
read
the m
esh
(sec
onds
)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumes
Quarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
1
3
10
30
100
300
Figure 22 Time needed to read the mesh versus number of unknowns
which cannot be taken into account by structured gridswhich are the main target of the benchmark Howevernonstructured grids can take into consideration any kind ofsymmetry almost without loss of accuracy and at the sametime reducing roughly the number of unknowns by a halfand the associated computational effort needed to solve theproblem by a factor of four Answers to items (1)ndash(7) canbe given completely by milonga using a single input file(see Supplementary data in SupplementaryMaterial availableonline at httpdxdoiorg1011552013641863) As asked initem (8) how results depend not only on the mesh spacingbut also on the meshing algorithm on the gridrsquos elementarygeometric shape and on the discretization scheme may shedlights on the subject whichmay be evenmore interesting thanthe numerical results themselves
Taking advantage of milongarsquos capability of reading andparsing command-line arguments the selection of the coregeometry (quarter or eighth) the meshing algorithm (delau-nay [16] or delquad [25]) the shape of the elementaryentities (triangles or quadrangles) the discretization scheme(volumes or elements) and the characteristic length ℓ119888 ofthe mesh can be provided at run time Fixing five values forℓ119888 = 4 3 2 1 05 gives 2 times 2 times 2 times 2 times 5 = 80 possiblecombinations which we solve with successive invocations to
milonga with the same input file but with different argumentsfrom a simple script Figures 10 11 12 13 14 15 16 17 and18 show the results corresponding to items (1)ndash(7) for someillustrative cases The complete set of figures and the codeused to generate themmay be provided upon request Table 2compiles the answers to the problem for every case studied
As the milonga code is still under development itsnumerical routines are not yet fully optimized nor designedfor parallel computation Therefore the reported times areonly rough estimates and should be taken with care Thesolution comprises five steps
(1) generate the grid with the requested geometry mesh-ing algorithm basic shape and characteristic lengthby calling to gmsh
(2) read the generated mesh(3) build the matrices(4) solve the eigenvalue problem(5) compute the requested results
The CPU time reported in Figures 10ndash18 is thus thesum of all of these five steps but not the time needed togenerate the figuresmdashwhich in some caseswith coarsemeshes
Science and Technology of Nuclear Installations 21
Table2
Resultsofthe2
DIAEA
PWRBe
nchm
arkp
roblem
obtained
with
them
ilong
acod
efor
thee
ightyp
ropo
sedcombinatio
nsofsymmetrym
eshing
algorithm
basicshapediscretization
schemeandcharacteris
ticelem
entc
elllengthTh
ereference
solutio
nis120588=28749times10minus5w
hich
correspo
ndsto119896eff=10296
Case
no
Symm-
etry
Mesh
algorithm
Basic
shape
Solutio
nmetho
dℓ119888
(cm)
Δ120588
(pcm
)max1206012
(mdash)
Total
unkn
owns
Outer
iter
Linear
iter
Inner
iter
Resid
ual
norm
Relativ
eerror
Error
estim
ate
Mem
ory
(Mb)
Page
faults
114
Delaunay
998779Vo
lumes
40
minus120
1145
8264
332
1033
6119890minus09
3119890minus09
1119890minus09
349470
214
Delaunay
998779Vo
lumes
30
85
1135
15740
332
1967
2119890minus08
1119890minus08
4119890minus09
4813401
314
Delaunay
998779Vo
lumes
20
197
1124
31188
332
3898
1119890minus08
6119890minus09
3119890minus09
7621932
414
Delaunay
998779Vo
lumes
10153
1123
127476
332
15934
7119890minus09
3119890minus09
3119890minus09
273
81886
514
Delaunay
998779Vo
lumes
05
187
1119
510676
332
63834
2119890minus09
9119890minus10
1119890minus09
1148
352261
614
Delaunay
998779Elem
ents
40
173
1100
4308
332
538
2119890minus08
8119890minus09
4119890minus09
318794
714
Delaunay
998779Elem
ents
30
117
1107
8108
332
1013
5119890minus09
2119890minus09
2119890minus09
4712946
814
Delaunay
998779Elem
ents
20
84
1112
15936
332
1992
6119890minus09
3119890minus09
2119890minus09
7321347
914
Delaunay
998779Elem
ents
1062
1116
64478
332
8059
6119890minus09
3119890minus09
4119890minus09
262
77105
1014
Delaunay
998779Elem
ents
05
58
1117
256700
332
32087
1119890minus09
5119890minus10
1119890minus09
1091
331969
1114
Delaunay
◻Vo
lumes
40
53
1149
5042
332
630
2119890minus08
1119890minus08
5119890minus09
288007
1214
Delaunay
◻Vo
lumes
30
346
1133
8560
332
1070
7119890minus09
3119890minus09
2119890minus09
3910895
1314
Delaunay
◻Vo
lumes
20
308
1131
15576
332
1947
1119890minus08
7119890minus09
3119890minus09
541564
514
14Delaunay
◻Vo
lumes
10177
1122
60774
332
7596
8119890minus09
4119890minus09
3119890minus09
167
50115
1514
Delaunay
◻Vo
lumes
05
199
1120
244936
332
30617
4119890minus09
2119890minus09
2119890minus09
698
215678
1614
Delaunay
◻Elem
ents
40
196
1100
5222
332
652
3119890minus08
1119890minus08
8119890minus09
4713098
1714
Delaunay
◻Elem
ents
30
123
1107
8810
332
1101
2119890minus08
9119890minus09
7119890minus09
6918598
1814
Delaunay
◻Elem
ents
20
901112
15922
332
1990
1119890minus08
5119890minus09
5119890minus09
107
30591
1914
Delaunay
◻Elem
ents
1063
1116
61456
332
7682
5119890minus09
2119890minus09
4119890minus09
389
113237
2014
Delaunay
◻Elem
ents
05
58
1117
246332
332
30791
8119890minus10
4119890minus10
1119890minus09
1676
498192
2114
Delq
uad
998779Vo
lumes
40
minus45
1144
6228
332
778
4119890minus08
2119890minus08
7119890minus09
308495
2214
Delq
uad
998779Vo
lumes
30
570
1153
11820
332
1477
3119890minus08
1119890minus08
4119890minus09
4010879
2314
Delq
uad
998779Vo
lumes
20
459
1103
24100
332
3012
2119890minus08
1119890minus08
5119890minus09
6217715
2414
Delq
uad
998779Vo
lumes
10512
1104
9640
03
3212050
2119890minus08
1119890minus08
9119890minus09
207
61714
2514
Delq
uad
998779Vo
lumes
05
528
1106
385600
332
48200
2119890minus09
1119890minus09
2119890minus09
838
255735
2614
Delqu
ad998779
Elem
ents
40
220
1093
3288
332
411
3119890minus08
2119890minus08
6119890minus09
308495
2714
Delq
uad
998779Elem
ents
30
140
1104
6148
332
768
4119890minus08
2119890minus08
8119890minus09
39110
0028
14Delq
uad
998779Elem
ents
20
82
1110
12392
332
1549
2119890minus08
9119890minus09
6119890minus09
6117067
2914
Delqu
ad998779
Elem
ents
1063
1115
48882
332
6110
2119890minus09
1119890minus09
1119890minus09
197
5804
630
14Delq
uad
998779Elem
ents
05
58
1116
194162
332
24270
2119890minus09
9119890minus10
2119890minus09
790
238769
3114
Delq
uad
◻Vo
lumes
40
minus416
1174
3132
332
391
5119890minus08
3119890minus08
8119890minus09
267322
3214
Delq
uad
◻Vo
lumes
30
minus248
1151
5922
332
740
9119890minus09
4119890minus09
2119890minus09
318779
3314
Delq
uad
◻Vo
lumes
20
minus132
1139
12050
332
1506
1119890minus08
7119890minus09
4119890minus09
4412239
3414
Delq
uad
◻Vo
lumes
10minus49
1126
48200
332
6025
6119890minus09
3119890minus09
2119890minus09
122
36094
3514
Delq
uad
◻Vo
lumes
05
minus23
1123
192800
332
24100
5119890minus09
2119890minus09
3119890minus09
464
140228
3614
Delq
uad
◻Elem
ents
40
224
1093
3294
332
411
3119890minus08
1119890minus08
7119890minus09
359852
3714
Delq
uad
◻Elem
ents
30
141
1104
6144
332
768
3119890minus08
2119890minus08
9119890minus09
5114018
3814
Delq
uad
◻Elem
ents
20
921111
12392
332
1549
1119890minus08
6119890minus09
5119890minus09
8122526
3914
Delq
uad
◻Elem
ents
1065
1115
48882
332
6110
5119890minus09
3119890minus09
4119890minus09
276
78431
22 Science and Technology of Nuclear Installations
Table2Con
tinued
Case
no
Symm-
etry
Mesh
algorithm
Basic
shape
Solutio
nmetho
dℓ119888
(cm)
Δ120588
(pcm
)max1206012
(mdash)
Total
unkn
owns
Outer
iter
Linear
iter
Inner
iter
Resid
ual
norm
Relativ
eerror
Error
estim
ate
Mem
ory
(Mb)
Page
faults
4014
Delq
uad
◻Elem
ents
05
59
1117
194162
332
24270
1119890minus09
5119890minus10
1119890minus09
1104
318967
4118
Delaunay
998779Vo
lumes
40
minus99
1139
4228
332
528
4119890minus12
2119890minus12
1119890minus12
359630
4218
Delaunay
998779Vo
lumes
30
50
1132
7712
332
964
1119890minus11
6119890minus12
2119890minus12
4211578
4318
Delaunay
998779Vo
lumes
20
177
1120
15776
332
1972
2119890minus12
1119890minus12
7119890minus13
5515436
4418
Delaunay
998779Vo
lumes
10142
1118
63902
332
7987
3119890minus12
1119890minus12
1119890minus12
160
46691
4518
Delaunay
998779Vo
lumes
05
176
1115
254612
332
31826
2119890minus12
8119890minus13
1119890minus12
555
168526
4618
Delaunay
998779Elem
ents
40
183
1096
2252
332
281
6119890minus12
3119890minus12
2119890minus12
359589
4718
Delaunay
998779Elem
ents
30
119
1104
4040
224
505
2119890minus08
1119890minus08
7119890minus09
41114
3748
18Delaunay
998779Elem
ents
20
85
1108
8156
332
1019
2119890minus12
8119890minus13
6119890minus13
5515053
4918
Delaunay
998779Elem
ents
1063
1112
32480
332
4060
1119890minus12
6119890minus13
8119890minus13
151
43671
5018
Delaunay
998779Elem
ents
05
58
1113
128358
332
1604
47119890minus13
3119890minus13
7119890minus13
527
157581
5118
Delaunay
◻Vo
lumes
40
37
1144
2380
224
297
5119890minus08
2119890minus08
1119890minus08
5314173
5218
Delaunay
◻Vo
lumes
30
218
1120
4194
332
524
7119890minus12
4119890minus12
2119890minus12
3610034
5318
Delaunay
◻Vo
lumes
20
345
1110
7960
332
995
5119890minus12
3119890minus12
1119890minus12
4712879
5418
Delaunay
◻Vo
lumes
10165
1117
30652
332
3831
2119890minus12
1119890minus12
7119890minus13
100
28851
5518
Delaunay
◻Vo
lumes
05
191
1117
122066
224
15258
2119890minus08
8119890minus09
9119890minus09
343
103825
5618
Delaunay
◻Elem
ents
40
175
1097
2526
332
315
4119890minus12
2119890minus12
1119890minus12
6016159
5718
Delaunay
◻Elem
ents
30
114
1104
4386
332
548
4119890minus12
2119890minus12
2119890minus12
5013768
5818
Delaunay
◻Elem
ents
20
901108
8234
332
1029
2119890minus12
1119890minus12
1119890minus12
7320094
5918
Delaunay
◻Elem
ents
1062
1112
31186
224
3898
2119890minus08
7119890minus09
9119890minus09
208
59589
6018
Delaunay
◻Elem
ents
05
58
1113
123122
224
15390
9119890minus09
5119890minus09
1119890minus08
807
2366
7061
18Delq
uad
998779Vo
lumes
40
minus210
1168
3224
332
403
1119890minus11
5119890minus12
2119890minus12
43117
8462
18Delq
uad
998779Vo
lumes
30
369
1150
6000
332
750
1119890minus11
6119890minus12
3119890minus12
5013502
6318
Delq
uad
998779Vo
lumes
20
312
1104
12316
332
1539
2119890minus11
8119890minus12
5119890minus12
6016395
6418
Delq
uad
998779Vo
lumes
1044
01094
48714
332
6089
9119890minus12
4119890minus12
4119890minus12
1183444
165
18Delq
uad
998779Vo
lumes
05
534
1081
193980
332
24247
5119890minus12
2119890minus12
4119890minus12
423
126768
6618
Delq
uad
998779Elem
ents
40
240
1090
1750
332
218
9119890minus12
4119890minus12
2119890minus12
43117
8667
18Delqu
ad998779
Elem
ents
30
150
1100
3184
332
398
1119890minus11
5119890minus12
2119890minus12
5013509
6818
Delq
uad
998779Elem
ents
20
89
1106
6426
332
803
5119890minus12
3119890minus12
2119890minus12
5916047
6918
Delq
uad
998779Elem
ents
1064
1111
24886
332
3110
2119890minus12
1119890minus12
1119890minus12
113
32788
7018
Delq
uad
998779Elem
ents
05
59
1113
98052
224
12256
1119890minus08
5119890minus09
8119890minus09
398
118153
7118
Delq
uad
◻Vo
lumes
40
minus398
1172
1650
332
206
1119890minus11
5119890minus12
2119890minus12
369907
7218
Delq
uad
◻Vo
lumes
30
minus249
1146
3058
332
382
4119890minus12
2119890minus12
1119890minus12
349349
7318
Delq
uad
◻Vo
lumes
20
minus131
1136
6228
224
778
3119890minus08
1119890minus08
5119890minus09
4010930
7418
Delq
uad
◻Vo
lumes
10minus51
1123
24536
224
3067
3119890minus08
2119890minus08
1119890minus08
7822716
7618
Delq
uad
◻Elem
ents
40
237
1090
1752
332
219
6119890minus12
3119890minus12
2119890minus12
4111213
7718
Delq
uad
◻Elem
ents
30
145
1101
3184
332
398
6119890minus12
3119890minus12
2119890minus12
43118
1178
18Delq
uad
◻Elem
ents
20
941107
6426
332
803
3119890minus12
1119890minus12
1119890minus12
6016220
7918
Delqu
ad◻
Elem
ents
1065
1111
24874
332
3109
1119890minus12
6119890minus13
8119890minus13
156
4400
080
18Delq
uad
◻Elem
ents
05
59
1113
9804
22
2412255
8119890minus09
4119890minus09
8119890minus09
567
162168
Science and Technology of Nuclear Installations 23
Tim
e con
sum
ed to
bui
ld th
e mat
rices
(sec
onds
)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay trianglesvolumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
10
30
100
300
1
3
Figure 23 Time needed to build the matrices versus number of unknowns
was considerably larger than the solution time itself Theeigenvalue problem was solved using a multilayer iterativeKrylov-Schur method [26] with a tolerance relative to thematrices norm
1003817100381710038171003817119877 sdot 120601 minus (1119896eff) sdot 119865 sdot 1206011003817100381710038171003817
119877 + (1119896eff) sdot 119865lt 10minus8 (11)
The reported residual norm and relative error are10038171003817100381710038171003817100381710038171003817119877 sdot 120601 minus
1
119896effsdot 119865 sdot 120601
10038171003817100381710038171003817100381710038171003817
1003817100381710038171003817119877 sdot 120601 minus (1119896eff) sdot 119865 sdot 1206011003817100381710038171003817
1003817100381710038171003817(1119896eff) sdot 1206011003817100381710038171003817
(12)
respectively The fields marked as outer linear and inneriterations refer to the number of steps needed to attainthe requested tolerance in each layer of the Krylov-Schuralgorithm The computer used to solve the problem has anIntel i7 920 267GHz processor with 4Gb of RAM runningDebian GNULinux Wheezy
When using a finite volumes-based scheme over anunstructured mesh the solver has to gather information
about which cells are neighbors and which are not Currentlygmsh does not write this kind of lists in its output filesso milonga has to explicitly solve the neighbors problemPerforming a linear search is an 119874(119873
2) task which is
unacceptable for problem sizes 119873 of interest The code usesa search based on a 119896-dimensional tree [27] which can inprinciple be performed in119874(119873) steps Still for large values of119873 the time needed to read and parse the mesh (number twopreviously mentioned) is the bottleneck of the solution Thisstep is not needed in finite elements although the construc-tion of the elementary matrices involves the computationof the multidimensional Jacobians and integrals which thenhave to be assembled Again for large 119873 this step (numberthree) takes up most of the time needed to solve the problem
The Delaunay algorithm is a standard method for gen-erating two-dimensional grids [16] by tessellating a planewith triangles If the mesh needs to be based on quadranglesinstead a recombination algorithm can be used to transformtwo adjacent triangles in one quadrangle whenever is possi-ble However for geometries which are based on rectangularshapes there exist other algorithms [25] both for meshingand for recombining the triangles that give rise to elementaryentities with right angles almost everywhere which may be a
24 Science and Technology of Nuclear Installations
Tim
e con
sum
ed to
solv
e the
eige
nval
ue p
robl
em (s
econ
ds)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
10
30
100
300
1
3
Figure 24 Time needed to solve the eigenvalue problem versus number of unknowns
desired property of the resulting grid For finite elements thesteps needed to build the matrices depend on the selectionof triangles or quadrangles as the basic elementary geometrybecause the shape functions change However the number ofunknowns is the same as the number of nodes that does notchange after a recombination procedure On the other handthe number of unknowns in the finite volumes schemes withtriangles is roughly twice as the number of unknowns withquadrangles for the same grid
Figure 19 shows how the computed static reactivity (ie1 minus 1119896eff) depends on the number of unknowns for eachof the sixteen combinations of geometry algorithm shapescheme The four accepted results published in the originalreference [10] are included for reference although it shouldbe taken into account that said reactivities were computedalmost forty years ago Figure 20 shows the total wall timeneeded to solve the problem as a function of 119873119866 whileFigures 21 22 23 and 24 show the times needed for eachof the first four steps involved in the solution maintainingthe same logarithmic scale for both the abscissas and theordinates Green data represent finite volumes whilst bluebullets correspond to finite elements Solid lines indicatequarter-core and dashed lines eighth-symmetry Fillet bullets
are results obtained by theDelaunay triangulation and emptybullets were computed with the delquad algorithm Finallytriangle-shaped data correspond to triangles and squares anddiamonds denote quadrangles as the basic geometry of thegrid
It can be seen that finite elements produce amuch smallerdispersion of eigenvalues 119896eff than finite volumes with therefinement of the mesh This can be explained because finitevolumes methods rely on a geometric condition of the meshwhich may change abruptly if the meshing algorithm decidesto allocate the cells in a rather different form for small changesin ℓ119888 Finite elements methods are less influenced by thesediscontinuous lay-out changes of the elements As expectedeighth-core symmetries give almost the same results as thequarter-core geometries with roughly half the unknownsFor small problems finite volumes run faster that finiteelements because the time needed to solve the neighborproblem is negligible When the problem size grows thistime increases and exceeds the overhead implied in the con-struction and assembly of the finite elements matrices Alsoat least for this configuration it is seen that the eigenvalueproblem is solved faster for finite volumes than for finiteelements
Science and Technology of Nuclear Installations 25
4 Conclusions
Unstructured grids provide the cognizant engineer witha wide variety of possibilities to deal with the design oranalysis of nuclear reactor cores These kinds of grids cansuccessfully reproduce continuous geometries commonlyfound in reactor cores such as cylinders and thereforenot only can the diffusion equation be better approximatedinside the domain of definition but also the fulfillment ofboundary conditions is improved A free computer codewas written from scratch that is able to completely solvethe 2D IAEA PWR Benchmark using unstructured gridsfor sixteen combinations of geometry meshing algorithmbasic shape and discretization scheme plus any value ofthe gridrsquos characteristic length by using a single input fileThe complete set of input files and codemdashexecutable andsourcemdashis available either online or upon request withcomments experiences suggestions and corrections beingmore than welcome Further development should includetackling full three-dimensional geometries with completethermal hydraulic feedback in order to analyze how thesolutions of the coupled neutronic-thermal problem dependon the spatial discretization scheme of the neutron leakageterm Parallelization of the computation and assembly of thematrices and of the solution of the eigenvalue problem andits implementation using GPUs are also desired features toimplement A problemwith direct applications that the futureversions of milonga ought to solve is the analysis of how thegeometry of the absorbing materials should be taken intoaccount in structured coarse grids in order to mitigate effectssuch as the rod-cusp problem
Suitable schemes for approximating the continuous dif-ferential operators by discrete matrix expressions includefinite volumes and finite elements families Finite volumesmethods compute cell mean values whilst finite elementsgive functional values at the gridrsquos nodes In general finiteelements are less sensitive to changes in the mesh so theresults they provide do not change significantly for differentmeshing algorithms or elementary shapes Small problemsare best solved by finite volumes as the neighbor-findingproblem is faster than the process of building and assemblingthe eigenvalue-problem matrices For a large number ofunknowns the process of finding which cell is neighbor ofwhichmdasheven based on a 119896-dimensional treemdashoverwhelmsthe computation and assembly of elementary matrices andfinite-element methods perform better
References
[1] G Glasstone and S Bell Nuclear Reactor Theory KriegerPublishing Company 1970
[2] J J Duderstadt and L J Hamilton Nuclear Reactor AnalysisJohn Wiley amp Sons New York NY USA 1976
[3] C Gho Reactor Physics Undergraduate Course Lecture NotesInstituto Balseiro 2006
[4] G G Theler and F J Bonetto ldquoOn the stability of the pointreactor kinetics equationsrdquoNuclear Engineering andDesign vol240 no 6 pp 1443ndash1449 2010
[5] E E Lewis andW F Miller Computational Methods of NeutronTransport John Wiley amp Sons 1984
[6] E Masiello ldquoAnalytical stability analysis of Coarse-Mesh finitedifference methodrdquo in Proceedings of the International Confer-ence on the Physics of Reactors (PHYSOR rsquo08) Nuclear Power ASustainable Resource pp 456ndash464 September 2008
[7] M L Zerkle Development of a polynomial nodal methodwith flux and current discontinuity factors [PhD thesis] Mas-sachusets Institute of Technology 1992
[8] J I Yoon andHG Joo ldquoTwo-level coarsemesh finite differenceformulation with multigroup source expansion nodal kernelsrdquoJournal of Nuclear Science andTechnology vol 45 no 7 pp 668ndash682 2008
[9] O Mazzantini M Schivo J Di Cesare R Garbero M Riveroand G Theler ldquoA coupled calculation suite for Atucha II oper-ational transients analysisrdquo Science and Technology of NuclearInstallations vol 2011 Article ID 785304 12 pages 2011
[10] Computational Benchmark Problem Comitee for the Mathe-matics and Computation Division of the American NuclearSociety ldquoArgonne Code Center Benchmark problem bookrdquoTech Rep ANL-7416 Supplement 2 Argonne National Labo-ratory 1977
[11] A F Henry Nuclear Reactor Analysis MIT Cambridge UK1975
[12] O C Zienkiewicz R L Taylor and J Z ZhuTheFinite ElementMethod Its Basis and Fundamentals vol 1 Elsevier 6th edition2005
[13] G Theler Difusion de neutrones en mallas no estructuradascomparacion entre volumenes y elementos finitos AcademicMonograph Universidad de Buenos Aires 2013
[14] G Theler ldquoMilonga a free nuclear reactor core analysis coderdquoAvailable online 2011
[15] G Theler F J Bonetto and A Clausse ldquoOptimizacion deparametros en reactores de potencia base de diseno delcodigo neutronico milongardquo in Reunion Anual de la AsociacionArgentina de Tecnologia Nuclear XXXVII 2010
[16] C Geuzaine and J-F Remacle ldquoGmsh a 3-D finite elementmesh generator with built-in pre- and post-processing facili-tiesrdquo International Journal for Numerical Methods in Engineer-ing vol 79 no 11 pp 1309ndash1331 2009
[17] S Balay J Brown K Buschelman et al ldquoPETSc users manualrdquoTech Rep ANL-9511-Revision 34 Argonne National Labora-tory 2013
[18] S Balay W D Gropp L Curfman McInnes and B F SmithldquoEfficient management of parallelism in object oriented numer-ical software librariesrdquo in Modern Software Tools in ScientificComputing E Arge A M Bruaset and H P Langtangen Edspp 163ndash202 Birkhauser 1997
[19] V Hernandez J E Roman and V Vidal ldquoSLEPC a scalable andflexible toolkit for the solution of eigenvalue problemsrdquo ACMTransactions on Mathematical Software vol 31 no 3 pp 351ndash362 2005
[20] M Galassi J Davies J Theiler et al GNU Scientific LibraryReference Manual 3rd edition 2009
[21] O C Zienkiewicz and R L Taylor The Finite Element MethodFor Solid and Structural Mechanics vol 3 Elsevier 6th edition2005
[22] A Imelda and K Doddy Evaluation of the IAEA 3-D PWRbenchmark problem using NESTLE code 2004
[23] R Mosteller ldquoStatic benchmarking of the NESTLE advancednodal coderdquo in Proceedings of the Joint International Conference
26 Science and Technology of Nuclear Installations
on Mathematical Methods and Supercomputing for NuclearApplications vol 2 pp 1596ndash1605 1997
[24] GTheler F J Bonetto andAClausse ldquoSolution of the 2D IAEAPWR Benchmark with the neutronic code milongardquo in Actasde la Reunion Anual de la Asociacion Argentina de TecnologıaNuclear XXXVIII 2011
[25] J F Remacle F Henrotte T Carrier-Baudouin et al ldquoAfrontal delaunay quad mesh generator using the 119871
infin normrdquoInternational Journal For Numerical Methods in Engineeringvol 94 no 5 pp 494ndash512 2013
[26] G W Stewart ldquoA Krylov-Schur algorithm for large eigenprob-lemsrdquo SIAM Journal on Matrix Analysis and Applications vol23 no 3 pp 601ndash614 2002
[27] J L Bentley ldquoMultidimensional binary search trees used forassociative searchingrdquo Communications of the ACM vol 18 no9 pp 509ndash517 1975
TribologyAdvances in
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Renewable Energy
Submit your manuscripts athttpwwwhindawicom
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High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Science and Technology of Nuclear Installations 5
(a) Fast flux distribution
minus100minus50 0 50 100minus100
minus500
50100
0
06
xy
(b) Fast flux unknowns
(c) Matrix 119877 structure (d) Matrix 119865 structure
Figure 5 A bare homogeneous circle solved with finite volumes (184 unknowns)
then the continuous eigenvalue problemmdashin either for-mulationmdashcan be transformed into a generalized matrixeigenvalueeigenvector problem casted in either of the follow-ing forms
119877 sdot 120601 =1
119896effsdot 119865 sdot 120601
119865 sdot 120601 = 119896eff sdot 119877 sdot 120601
119877minus1sdot 119865 sdot 120601 = 119896eff sdot 120601
(5)
where 119877 and 119865 are square 119873119866 times 119873119866 matrices We call119865 the fission matrix as it contains all the ]-fission termswhich are the ones that we artificially divided by 119896eff Therest of the terms are grouped into the removal or transportmatrix 119877 which includes the rest of the neutron-matterinteraction mechanisms It can be shown [11] that for anyreal set of cross sections 119877minus1 exists and that the119873119866 pairs ofeigenvalueeigenvector solutions of (5) satisfy that
(1) there is a unique real positive eigenvalue greater inmagnitude than any other eigenvalue
(2) all the elements of the eigenvector corresponding tothat eigenvalue are real and positive
(3) all other eigenvectors either have some elements thatare zero or have elements that differ in sign from eachother
21 Boundary Conditions Being a differential equation overspace the neutron diffusion equation needs proper boundaryconditions to conform a properly definedmathematical prob-lem These can be imposed flux (Dirichlet) imposed current(Neumann) or a linear combination (Robin) However dueto the fact that in the linear problem in absence of externalsourcesmdashsuch as (1) or (3)mdashthe problem is homogeneousif 120601119892(x) is a solution then any multiple 120572120601119892(x) is also asolution That is to say the eigenvectors are defined up to amultiplicative constant whose value is usually chosen as toobtain a certain total thermal power Thus the prescribedboundary conditions should also be homogeneous and bedefined up to amultiplicative constantTherefore the allowedDirichlet conditions are zero flux at the boundary theNeumann conditions should prescribe that the derivative in
6 Science and Technology of Nuclear Installations
(a) Fast flux distribution
minus100minus50 0 50 100minus100
minus500
50100
0
06
xy
(b) Fast flux unknowns
(c) Matrix 119877 structure (d) Matrix 119865 structure
Figure 6 A bare homogeneous circle solved with finite elements (218 unknowns)
the normal direction should be zero (ie symmetry condi-tion) and the Robin conditions are restricted to the followingform
grad 120601119892 (x) sdot n + 119886119892 (x) sdot 120601119892 (x) = 0 (6)
n being the outward unit normal to the boundary 120597119880 of thedomain 119880
22 Grids and Schemes One way of solving the neutrondiffusion equationmdashand in general any partial differentialequation over spacemdashis by discretizing the differential oper-ators with some kind of scheme that is applied over a certainspatial grid Given an 119898-dimensional domain a grid definesa partition composed of a finite number of simple geometricentities In structured grids these elementary entities arearranged following a well-defined structure in such a waythat each entity can be identified without needing furtherinformation than the one provided by the intrinsic structureOn the other hand the geometric entities that composean unstructured grid are arranged in an irregular patternand the identification of the elementary entities needs to
be separately specified for example by means of a sortedlist For instance Figure 2(b) shows a structured grid thatapproximates the continuous domain of Figure 2(a) Eachsquare may be uniquely identified by means of two integerindexes that indicate its relative position in each of thehorizontal and vertical directions In the case of Figure 2(c)that shows an unstructured grid there is no way to system-atically make a reference to a particular quadrangle withoutany further information
Almost all of the grid-based schemesmdashwhich are knownas nodal schemes which are to be differentiated from modalschemes based on series expansionsmdashare based in either thefinite differences volumes or elements method Finite differ-ences schemes provide themost simple and basic approach toreplace differential (ie continuous) operators by difference(ie discrete) approximations However they are not suitablefor unstructured meshes and may introduce convergenceproblems with parameters that are discontinuous in spacewhich is the case for any reactor core composed of atleast two different materials Moreover boundary conditionsare hard to incorporate and usually give rise to incorrectresults
Science and Technology of Nuclear Installations 7
20 40 600995
09975
1
Analytical solutionFinite volumesFinite elements
keff
a985747c
Figure 7 The effective multiplication factor of a two-group barecircular reactor of radius 119886 as a function of the quotient 119886ℓ
119888between
the radius and the characteristic length of the cellelement
20 40 60
Finite volumesFinite elements
Erro
r ink
eff
10minus3
10minus4
10minus5
a985747c
Figure 8 Absolute value of the error committed in the computationof 119896eff versus 119886ℓ119888
Methods of the finite volumes family involve the inte-gration of the differential equation over each elementaryentity applying the divergence theorem to transform volumeintegrals into surface integrals and providing a mean to esti-mate the fluxes through the entityrsquos surface using informationcontained in its neighbors In this context each elementaryentity is called a cell and finite volumes methods give themean value of each of the group fluxes 120601119892(119894) at the 119894th cellwhich may be associated to the value of 120601119892(x119894) where x119894 isthe location of the cell barycenter Being an integral-basedmethod spatial-discontinuous parameters are handled moreefficiently than in finite differences and boundary conditionscan be easily incorporated as forced cell fluxes Nonethelesseven though these methods may be applied to unstructuredgrids the estimation of the fluxes on the cellsrsquo surfaces isusually performed by using some geometric approximationsthat may lose validity as the quality of the grid is worsened
10
70
90
130
150
170
10 30 50 70 90 110 130 150 170
4
1 38
36 37353
3 32 33 34312
26 27 28 29 3025
19 20 21 22 23 2418
10 11 12 13 14 15 16 17
2 331 4 5 6 7 8 9
x (cm)
y (cm
)
Figure 9 The 2D IAEA PWR benchmark geometry
Moreover the simple integral approach cannot take intoaccount discontinuous diffusion coefficients so when twoneighbors pertain to different materials the flux has to becomputed in a certain particular way to conserve the neutroncurrent according to (2)
Finally finite elements methods rely on a weak formula-tion of the differential problem similar to (3) that maintainsall the mathematical characteristics of the original strongformulation plus its boundary conditions The method isbased on shape functions that are local to each elementaryentitymdashnow called element defined by nodes as cornersmdashand on finding a set of nodal values such that a certaincondition is met which is usually that the residual of thesolution has to be orthogonal to each of the shape functionsThis condition is known as the Galerkin method and impliesthat the error committed in the approximate solution ofthe continuous problem is confined into a small subset ofthe original vector space of the continuous functions Thesemathematical properties make finite elements schemes veryattractive However these features depend on a large numberof integrations that ought to be performed numerically soa computational effortdesired accuracy tradeoff has to beconsidered Not only does the finite elements method givethe values of the flux 120601119892(x119894) at the 119894th node but also theshape functions provide explicit expressions to interpolateand to evaluate the unknown functions at any location x ofthe domain119880 Boundary conditions are divided into essentialand naturalThefirst group comprises theDirichlet boundaryconditions which are satisfied exactlymdashwithin the precisionof the eigenvalue problem solvermdashby the obtained solutionThe latter include the Neumann and the Robin conditionsthat are satisfied only approximately by the derivatives of theinterpolated solution through the shape functions with finermeshes giving better agreement with the prescribed values
The same unstructured grid may be used either for thefinite volumes or for the finite elements method In the first
8 Science and Technology of Nuclear Installations
Largest eigenvalue 1029690 (288336 pcm)1135 (3066 3079)1306 (5171 13076)
Number of unknowns 15740Outer iterations 3Linear iterations 32Inner iterations 1967Residual normRelative errorError estimateMemory used 49824 kBSoft page faults 13401Hard page faults 0Total CPU time 074 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
2483 times 10minus8
1223 times 10minus8
403 times 10minus9
keff
quarter-symmetry core meshed using Delaunay (triangles 985747c = 3) solved with finite volumesMilongarsquos 2D LWR IAEA benchmark problem case no 002
(a)
123456789
1011121314151617181920212223242526272829303132333435363738
k Pk 1206011k 1206012k075 3282 555133 4242 984147 4645 1090123 3921 911061 2677 453094 2995 694093 2927 689074 2011 549mdash 322 756146 4597 1078150 4730 1110133 4199 985107 3405 789104 3271 767095 2971 700072 1956 534mdash 306 733149 4695 1102136 4289 1007118 3733 876107 3367 792097 2892 718066 1628 491mdash 234 549120 3797 891097 3096 718090 2842 669084 2221 619mdash 568 1215mdash 067 257047 2042 348068 2055 504058 1414 430mdash 222 562057 1374 425mdash 388 820mdash 053 203mdash 060 229
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 10 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
case the unknowns are themean value of the fluxes over eachcell whilst in the latter the unknowns are the fluxes evaluatedat each node Therefore the number of unknowns 119873119866 isdifferent for each method even when using the same gridFigure 4 shows this situation and in Section 31 we further
illustrate these differences A fully detailed mathematicaldescription of the actual algorithms for both volumes andelements-based proposed discretizations can be found in anacademic monograph written by the author of this paper[13]
Science and Technology of Nuclear Installations 9
Largest eigenvalue 1029927 (290571 pcm)1131 (3175 2999)1220 (13097 5098)
Number of unknowns 15576Outer iterations 3Linear iterations 32Inner iterations 1947Residual normRelative errorError estimateMemory used 56256 kBSoft page faults 15645Hard page faults 0Total CPU time 09201 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
1394 times 10minus8
6865 times 10minus9
3425 times 10minus9
keff
quarter-symmetry core meshed using Delaunay (quads 985747c = 2) solved with finite volumesMilongarsquos 2D LWR IAEA benchmark problem case no 013
(a)
123456789
1011121314151617181920212223242526272829303132333435363738
k Pk 1206011k 1206012k074 3244 549133 4230 982147 4651 1092124 3946 917060 2645 443094 2997 696093 2938 691074 2005 545mdash 301 746145 4588 1076150 4720 1108133 4198 984109 3465 805104 3280 769095 2986 704071 1962 528mdash 293 715148 4680 1098136 4288 1007119 3755 881107 3385 796097 2905 721067 1653 493mdash 220 556121 3808 893098 3115 724092 2889 680083 2247 615mdash 543 1241mdash 067 260045 1977 330069 2076 509058 1460 428mdash 226 574056 1364 413mdash 371 821mdash 052 200mdash 060 238
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 11 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
3 Results
We now proceed to show two illustrative results that are tobe taken as an overview of the possibilities that unstructuredgrids can provide in order to tackle the multigroup neutron
diffusion problem The examples are two-dimensional prob-lems as they contain some of the complexities a real three-dimensional reactor posse yet the reported results are notso complicated as to be easily understood and analyzed Inparticular we state some basic differences between the finite
10 Science and Technology of Nuclear Installations
Largest eigenvalue 1029695(288388 pcm)1112 (3076 3032)838 (5000 13000)
Number of unknowns 15922Outer iterations 3Linear iterations 32Inner iterations 1990Residual normRelative errorError estimateMemory used 109964 kBSoft page faults 30591Hard page faults 0Total CPU time 158 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
1056 times 10minus8
5202 times 10minus9
5122 times 10minus9
keff
quarter-symmetry core meshed using Delaunay (quads 985747c = 2) solved with finite volumesMilongarsquos 2D LWR IAEA benchmark problem case no 018
(a)
123456789
1011121314151617181920212223242526272829303132333435363738
k Pk 1206011k 1206012k074 3220 549130 4152 961144 4651 1068120 3843 890061 2648 452094 2996 693094 2959 696072 2061 569mdash 358 862142 4495 1054147 4634 1088131 4127 968107 3409 789104 3275 768096 3003 708071 2014 554mdash 342 820146 4606 1081134 4222 991118 3711 871107 3375 794098 2929 727062 1692 522mdash 258 632119 3752 880096 3084 715091 2858 673080 2280 633mdash 611 1277mdash 080 318047 2043 350069 2088 510054 1463 450mdash 255 648051 1418 439mdash 410 854mdash 064 252mdash 071 285
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 12 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
volumes and the finite elements methods by solving a two-group homogeneous bare reactor with the Robin boundaryconditions over the very same grid although a rather coarseone so the differences can be observed directly into theresulting figuresWe then solve the classical two-dimensionalLWR problem also known as the 2D IAEA PWR benchmark
Not only do we show again the differences between the finitevolumes and elements formulation but also we solve theproblemusing different combinations ofmeshing algorithmsbasic shapes and characteristic lengths of the mesh
To solve the two examples shown below we employed themilonga code which was written from scratch by the author
Science and Technology of Nuclear Installations 11
Largest eigenvalue 1029159 (283326 pcm)1174 (2851 3224)1560 (5200 13200)
Number of unknowns 3132Outer iterations 3Linear iterations 32Inner iterations 391Residual normRelative errorError estimateMemory used 26888 kBSoft page faults 7322Hard page faults 0Total CPU time 0308 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
5408 times 10minus8
2665 times 10minus8
8424 times 10minus9
keff
Milongarsquos 2D LWR IAEA benchmark problem case no 031quarter-symmetry core meshed using Delquad (quads 985747c = 4) solved with finite volumes
(a)
123456789
1011121314151617181920212223242526272829303132333435363738
k Pk 1206011k 1206012k077 3422 570140 4468 1040153 4836 1135128 4070 948060 2704 448095 3017 702091 2861 673069 1918 508mdash 264 689151 4785 1122155 4897 1149137 4334 1017111 3535 823104 3272 768092 2908 685067 1876 496mdash 254 658154 4854 1139139 4392 1031120 3792 890107 3360 790095 2831 702061 1574 451mdash 197 515123 3868 907099 3132 730090 2818 663078 2168 579mdash 481 1191mdash 055 217044 1984 328067 2027 498052 1360 388mdash 190 520052 1329 383mdash 321 792mdash 043 168mdash 048 195
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 13 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
of this paper and is currently still under development withinhis ongoing PhD thesis There exists a first public release[14] under the terms of the GNU General Public Licensemdashthat is it is a free softwaremdashthat can only handle structuredgrids There is a second release being prepared whose
main relevance is that it can work with nonstructured gridsas well which is the main feature of the code It uses ageneral mathematical frameworkmdashcoded from scratch aswellmdashwhich provides input file parsing algebraic expres-sions evaluation one- and multidimensional interpolation of
12 Science and Technology of Nuclear Installations
Largest eigenvalue 1029788 (289260 pcm)1120 (3149 3011)1091 (13038 5114)
Number of unknowns 15776Outer iterations 3Linear iterations 32Inner iterations 1972Residual normRelative errorError estimateMemory used 56996 kBSoft page faults 15436Hard page faults 0Total CPU time 09121 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
2296 times 10minus12
1131 times 10minus12
6877 times 10minus13
keff
Milongarsquos 2D LWR IAEA benchmark problem case no 043eighth-symmetry core meshed using Delaunaya (triangs985747c = 2) solved with finite volumes
(a)
1 074 3241 5482 131 4188 9713 145 4582 10764 122 3877 9005 060 2644 4456 093 2986 6927 093 2941 6928 075 2030 5529 mdash 322 779
10 143 4523 106111 148 4667 109512 131 4145 97213 107 3423 79414 103 3269 76715 095 2993 70616 073 1988 53917 mdash 312 75118 147 4637 108819 134 4243 99620 118 3721 87321 107 3370 79322 098 2914 72323 067 1651 49924 mdash 235 57125 119 3763 88226 096 3077 71527 091 2847 67128 083 2255 61729 mdash 576 122530 mdash 068 26631 046 2016 34132 068 2058 50533 059 1443 43534 mdash 233 58935 057 1381 42036 mdash 381 82337 mdash 054 20838 mdash 062 244
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 14 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
scattered data shared-memory access numerical integrationfacilities and so forth The milonga code was designed withfour design basis vectors in mindmdashwhich are thoroughlydiscussed in the documentation [15]mdashwhich includes thetype of problems it can handle the code scalability which
features are expected and what to do with the obtainedresults
It works by first reading an input file that using plain-text English keywords and arguments defines the number119898 of spatial dimensions the number 119866 of group energies
Science and Technology of Nuclear Installations 13
Largest eigenvalue 1029726 (288675 pcm)1104 (3069 3069)864 (13000 5000)
Number of unknowns 4040Outer iterations 2Linear iterations 24Inner iterations 505Residual normRelative errorError estimateMemory used 42940 kBSoft page faults 11437Hard page faults 0Total CPU time 1064 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
2019 times 10minus8
9946 times 10minus9
7138 times 10minus9
keff
Milongarsquos 2D LWR IAEA benchmark problem case no 047eighth-symmetry core meshed using Delaunay (triangles 985747c = 3) solved with finite volumes
(a)
1 074 3194 5452 129 4117 9533 143 4515 10604 119 3814 8835 061 2634 4506 093 2984 6907 094 2954 6958 071 2064 5719 mdash 362 865
10 141 4458 104511 146 4599 107912 130 4097 96113 106 3388 78414 103 3263 76515 095 2998 70716 068 2015 55717 mdash 346 82318 145 4572 107319 133 4194 98420 117 3691 86621 107 3363 79222 098 2925 72623 059 1696 52824 mdash 263 63525 118 3728 87426 096 3068 71127 091 2850 67228 079 2280 63529 mdash 619 127630 mdash 081 32131 047 2036 34932 068 2085 50933 051 1465 45434 mdash 258 65035 049 1419 44336 mdash 416 85437 mdash 065 25438 mdash 071 288
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 15 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
and optionally a mesh file Currently only grids generatedwith the code gmsh [16] are only supported mainly becauseit is also a free software and it suits perfectly well milongarsquosdesign basis in the sense that the continuous geometry can bedefined as a function of a number of parameters Afterwardthe physical entities defined in the grid are mapped into
materials with macroscopic cross sections which maydepend on the spatial coordinates x bymeans of intermediatefunctions such as burn-up or temperatures distributionswhich in turn may be defined by algebraic expressions byinterpolating data located in files by reading shared-memoryobjects or by a combination of them In the same sense
14 Science and Technology of Nuclear Installations
Largest eigenvalue 1029695 (288387 pcm)1108 (3045 3045)833 (13000 5000)
Number of unknowns 8234Outer iterations 3Linear iterations 32Inner iterations 1029Residual normRelative error
Max 1206012(x y) coreMax 1206012(x y) reflector
2028 times 10minus12
9993 times 10minus13
1012 times 10minus12
keff
Relative errorError estimateMemory used 75468 kBSoft page faults 20094Hard page faults 0Total CPU time 1256 seconds
9993 times 10
1012 times 10minus12
Milongarsquos 2D LWR IAEA benchmark problem case no 058eighth-symmetry core meshed using Delaunay (quads 985747c = 2) solved with finite volumes
(a)
1 074 3207 5472 129 4136 9573 144 4534 10644 120 3828 8875 061 2638 4506 093 2985 6907 094 2948 6948 072 2053 5669 mdash 357 859
10 142 4478 105011 146 4617 108412 130 4111 96413 106 3396 78614 103 3263 76515 095 2991 70616 071 2005 55217 mdash 340 81618 145 4588 107719 133 4206 98720 117 3698 86821 107 3362 79122 098 2918 72523 062 1686 52024 mdash 258 62925 118 3737 87626 096 3073 71227 091 2848 67128 080 2272 63129 mdash 609 127230 mdash 080 31731 047 2035 34832 069 2081 50933 054 1458 44834 mdash 255 64635 052 1413 43736 mdash 409 85137 mdash 064 25138 mdash 071 284
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 16 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
boundary conditions are applied to appropriate physicalentities of dimension119898minus1The problemmatrices119877 and119865 arethen built and stored in an appropriate sparse format usingthe free PETSc [17 18] library and the eigenvalue problemis solved using the free SLEPc [19] library The results arestored into milongarsquos variables and functions which may be
evaluated integratedmdashusually using the free GNU ScientificLibrary [20] routinesmdashand of course written into appropriateoutputs Milonga can also solve problems parametrically andbe used to solve optimization problems As stated above thecode is a free software so corrections and contributions aremore than welcome
Science and Technology of Nuclear Installations 15
Largest eigenvalue 1029462 (286185 pcm)1136 (3093 2953)1233 (13100 5100)
Number of unknowns 6228Outer iterations 2Linear iterations 24Inner iterations 778Residual normR l ti
Max 1206012(x y) coreMax 1206012(x y) reflector
26 times 10minus8
1281 times 10minus8
4996 times 10minus9
keff
Relative errorError estimateMemory used 41460 kBSoft page faults 10930Hard page faults 0Total CPU time 0604 seconds
1281 times 10 8
4996 times 10minus9
Milongarsquos 2D LWR IAEA benchmark problem case no 073eighth-symmetry core meshed using Delaunay (quads 985747c = 2) solved with finite volumes
(a)
1 075 3297 5552 134 4268 9923 148 4660 10944 123 3926 9145 059 2645 4406 094 2997 6977 093 2914 6858 072 2002 5349 mdash 301 773
10 146 4606 108011 150 4735 111112 133 4212 98813 108 3448 80214 104 3271 76715 094 2954 69616 070 1954 52117 mdash 288 73518 149 4691 110119 136 4289 100720 119 3748 88021 107 3359 79022 096 2880 71423 064 1636 47524 mdash 217 56325 121 3810 89426 098 3110 72427 090 2829 66628 081 2227 59929 mdash 536 125730 mdash 062 25031 045 2002 33632 068 2040 50033 055 1414 40934 mdash 215 57935 055 1359 40836 mdash 359 83437 mdash 050 19938 mdash 061 236
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 17 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
31 A Coarse Bare Homogeneous Circle Figures 5 and 6 showthe results of solving a bare two-dimensional circular homo-geneous reactor of radius 119886 over an unstructured grid whichis deliberately coarse using the finite volumes method andthe finite elements method respectively Two group energies
were used and a null-flux boundary conditionwas fixed at theexternal surface Figures 5(a) and 6(a) compare the obtainedfast flux distributions In the first case the numerical solutionprovidesmean values for each neutron flux group in each cellwhile in the latter the solution is computed at the nodes and
16 Science and Technology of Nuclear Installations
Largest eigenvalue 1029851 (289858 pcm)1090 (3182 3182)851 (13000 5000)
Number of unknowns 1752Outer iterations 3Linear iterations 32Inner iterations 219Residual norm
Max 1206012(x y) coreMax 1206012(x y) reflector
566 times 10minus12
2789 times 10minus12
1595 times 10minus12
keff
Residual normRelative errorError estimateMemory used 42564 kBSoft page faults 11213Hard page faults 0Total CPU time 09921 seconds
566 times 10
2789 times 10minus12
1595 times 10minus12
Milongarsquos 2D LWR IAEA benchmark problem case no 076eighth-symmetry core meshed using Delaunay (quads 985747c = 4) solved with finite volumes
(a)
1 073 3151 5402 127 4059 9393 141 4457 10464 118 3768 8715 061 2615 4506 093 2979 6887 094 2971 6998 067 2092 5829 mdash 375 885
10 139 4397 103111 144 4541 106612 128 4052 95013 105 3365 77814 103 3259 76415 096 3016 71216 066 2045 56817 mdash 359 84118 143 4520 106119 132 4155 97520 116 3672 86221 107 3366 79222 098 2951 73323 054 1728 54324 mdash 273 65125 117 3700 86826 095 3057 70827 091 2860 67428 074 2311 64829 mdash 642 130030 mdash 084 33231 048 2041 35332 069 2103 51333 046 1493 46734 mdash 269 66735 046 1444 45536 mdash 432 87137 mdash 067 26338 mdash 074 297
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 18 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
continuous functions are evaluated by means of the shapefunctions used in the formulation Figures 5(b) and 6(b)illustrate the fast fluxes unknowns and its relative position inspace It can be noted that the mesh coarseness gives resultsthat differ substantially in both cases Finally the structure ofthe sparse eigenvalue problemmatrices is shown for each case
with blue red and cyan representing positive negative andexplicitly inserted zero values In the finite volume case thereare 184 unknowns (92 cells times 2 groups) while in the secondcase there are 218 unknowns (109 nodes times 2 groups) Thevolumesrsquo fissionmatrix is almost diagonal it has a bandwidthequal to the number of energy groups which in this case is
Science and Technology of Nuclear Installations 17
500 1000 2000 5000 10000 20000 50000Number of unknowns
2840
2860
2880
2900
2920
ORNLRisoslashKWUOntarioQuarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumes
Quarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elementsEighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumes
Eighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
120588(p
cm)
105 2 times 105 5 times 105
Figure 19 Static reactivity versus number of unknowns The four original solutions as published in 1977 [10] are included as reference
twoThe off-diagonal values appear right next to the diagonalelements because of the chosen ordering of the unknownsin the flux vector 120601 isin R184 Other orderings may be usedbut the rate of convergence of the eigenvalue problem can bedeteriorated On the other hand the elementsrsquo fission matrixhas a nontrivial structure because the grid is unstructuredand the net fission rate at each element depends on the fluxesat the nodes whose location inside thematrix depends in turnon how the gridwas generatedThis effect is similar to the onethat appears in structural analysis where mass matrices needto be lumped in order to simplify transient computations [21]The diagonal block of elementrsquos 119877 matrix and the null blockin 119865 correspond to the discrete equations that set the null-flux boundary conditions on the nodes located at the externalsurface Other types of boundary conditions lead to differentkinds of structures within the problem matrices
In case a part of the domain contained a nonmultiplicativematerial such as a reflector then there would appear sectionsof the fission matrix with null values in both methodsrendering 119865 singular in both methods Care should be takenwhen dealing with the numerical schemes for the eigenvalueproblem solution It can be seen in the elementsrsquo matricesa particular structure that implements the boundary condi-tions at the external surfaces This structure does not appearin the volumesrsquo matrices because the boundary conditions
appear as flux terms which are summed up over all thesurfaces of each cell so they aremasked inside the volumetricdiscretization of the divergence and gradient operators
As the two-group neutron diffusion equation with uni-form cross sections over a circle subject to null-flux bound-ary conditions has an analytical solution it is adequate tocompare how the two proposed numerical schemes relate toit In the studied problem we ignored upscattering and fastfissions Then the analytical effective multiplication factor is
119896eff =]Σ1198912 sdot Σ1199041rarr2
[Σ1198861 + Σ1199041rarr2 + 1198631(]0119886)2] [Σ1198862 + 1198632(]0119886)
2]
(7)
being ]0 = 24048 the smallest root of Besselrsquos first-kindfunction of order zero 1198690(119903)
Figure 7 shows how the two numerical effective multi-plication factors compare to the analytical solution givenby (7) as a function of the mesh refinement indicated bythe quotient 119886ℓ119888 between the radius of the circle and thecharacteristic length of the cellelements We can see thatthe 119896eff computed by the finite volumes (elements) methodis always greater (less) than the analytical solution Indeedit can be proven that for a bare one-dimensional slab thisis always the case [13] However this result does not hold
18 Science and Technology of Nuclear Installations
2000 3000 6000 10000 20000 30000 60000
01
003
001
03
1
3
10
30
100
300To
tal w
all t
ime (
seco
nds)
Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
Figure 20 Total wall time versus number of unknowns
for problems with nonuniform cross sections even in simpleone-dimensional reflected reactors
We may draw two other conclusions from Figure 7 Firstthat the finite element method seems to provide a bettersolution than the volumes-based scheme and second thateven though the error committed tends to decrease with finergrids its behavior is not monotonic for finite volumes Infact Figure 8 shows the difference between the numericaland analytical solutions using a logarithmic vertical scalewhere both conclusions are even more evident We deferthe discussion of such differences between the discretizationscheme until the next section It is worth to note howeverthat the fact that a finite-element-based scheme throws betterresults for the particular bare homogeneous circular reactorunder study than the finite volumes does not imply thatfinite volumes ought to be discarded as a valid tool forsolving the neutron diffusion equation in general cases Thecombination of lattice and core-level computations is usuallyperformedusing cell-based resultswhichwhen fed into node-based methods to solve the few-group neutron diffusionequation may introduce errors which potentially can leadto unacceptable solutions Nevertheless this analysis is farbeyond the scope of this paper which focuses on solving amathematical equation over unstructured grids
32 The 2D IAEA PWR Benchmark Problem This is aclassical two-group neutron diffusion problem first designedin the early 1970s and taken as a reference benchmark forcomputational codes A number of codes were used to solveeither this problem or its three-dimensional formulation[22 23] including milonga using a structured grid [24] Theoriginal formulation can be found in the reference [10] andthere is a reproduction that may be easily found online inreference [24] The geometry consists of a one-quarter ofa PWR core depicted in Figure 9 and the homogeneousmacroscopic cross sections are listed in Table 1 An axialbuckling term 119861
2
119911119892= 08times10
minus4 should be taken into accountThe external surface should be subject to a zero incomingcurrent condition which may be written as
120597120601119892
120597119899= minus
04692
119863119892
sdot 120601119892 (8)
The expected results are as follows
(1) Maximum eigenvalue(2) Fundamental flux distributions
(a) Radial flux traverses 120601119892(119909 0) and 120601119892(119909 119909)
Science and Technology of Nuclear Installations 19
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
1
3
10
30
100
300Ti
me c
onsu
med
to m
esh
the g
eom
etry
(sec
onds
)
Figure 21 Time needed to mesh the geometry versus number of unknowns
Table 1 Macroscopic cross-sections (units are not stated in the original reference but they are assumed to be in cmminus1 or cm as appropriate)
1198631
1198632
Σ1rarr2
Σ1198861
Σ1198862
]Σ1198912
Material1 15 04 002 001 0080 0135 Fuel 12 15 04 002 001 0085 0135 Fuel 23 15 04 002 001 0130 0135 Fuel 2 + rod4 20 03 004 0 0010 0 Reflector
Note the fluxes shall be normalized such that
1
119881coreint119881core
sum
119892
]Σ119891119892 sdot 120601119892119889119881 = 1 (9)
(b) Value and location of maximum power densityThis corresponds to maximum of 1206012 in thecore It is recommended that the maximumvalues of 1206012 both in the inner core and at thecorereflector interface be given
(3) Average subassembly powers 119875119896
119875119896 =1
119881119896
int119881119896
sum
119892
]Σ119891119892 sdot 120601119892119889119881 (10)
where 119881119896 is the volume of the 119896th subassembly and119896 designates the fuel subassemblies as shown inFigure 9
(4) Number of unknowns in the problem number ofiterations and total and outer
(5) Total computing time iteration time IO-time andcomputer used
(6) Type and numerical values of convergence criteria(7) Table of average group fluxes for a square mesh grid
of 20 times 20 cm(8) Dependence of results on mesh spacing
Even though the original problem is based on a quarter-core situation the problem has an eighth-core symmetry
20 Science and Technology of Nuclear Installations
Quarter Delaunay triangles elements
Tim
e con
sum
ed to
read
the m
esh
(sec
onds
)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumes
Quarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
1
3
10
30
100
300
Figure 22 Time needed to read the mesh versus number of unknowns
which cannot be taken into account by structured gridswhich are the main target of the benchmark Howevernonstructured grids can take into consideration any kind ofsymmetry almost without loss of accuracy and at the sametime reducing roughly the number of unknowns by a halfand the associated computational effort needed to solve theproblem by a factor of four Answers to items (1)ndash(7) canbe given completely by milonga using a single input file(see Supplementary data in SupplementaryMaterial availableonline at httpdxdoiorg1011552013641863) As asked initem (8) how results depend not only on the mesh spacingbut also on the meshing algorithm on the gridrsquos elementarygeometric shape and on the discretization scheme may shedlights on the subject whichmay be evenmore interesting thanthe numerical results themselves
Taking advantage of milongarsquos capability of reading andparsing command-line arguments the selection of the coregeometry (quarter or eighth) the meshing algorithm (delau-nay [16] or delquad [25]) the shape of the elementaryentities (triangles or quadrangles) the discretization scheme(volumes or elements) and the characteristic length ℓ119888 ofthe mesh can be provided at run time Fixing five values forℓ119888 = 4 3 2 1 05 gives 2 times 2 times 2 times 2 times 5 = 80 possiblecombinations which we solve with successive invocations to
milonga with the same input file but with different argumentsfrom a simple script Figures 10 11 12 13 14 15 16 17 and18 show the results corresponding to items (1)ndash(7) for someillustrative cases The complete set of figures and the codeused to generate themmay be provided upon request Table 2compiles the answers to the problem for every case studied
As the milonga code is still under development itsnumerical routines are not yet fully optimized nor designedfor parallel computation Therefore the reported times areonly rough estimates and should be taken with care Thesolution comprises five steps
(1) generate the grid with the requested geometry mesh-ing algorithm basic shape and characteristic lengthby calling to gmsh
(2) read the generated mesh(3) build the matrices(4) solve the eigenvalue problem(5) compute the requested results
The CPU time reported in Figures 10ndash18 is thus thesum of all of these five steps but not the time needed togenerate the figuresmdashwhich in some caseswith coarsemeshes
Science and Technology of Nuclear Installations 21
Table2
Resultsofthe2
DIAEA
PWRBe
nchm
arkp
roblem
obtained
with
them
ilong
acod
efor
thee
ightyp
ropo
sedcombinatio
nsofsymmetrym
eshing
algorithm
basicshapediscretization
schemeandcharacteris
ticelem
entc
elllengthTh
ereference
solutio
nis120588=28749times10minus5w
hich
correspo
ndsto119896eff=10296
Case
no
Symm-
etry
Mesh
algorithm
Basic
shape
Solutio
nmetho
dℓ119888
(cm)
Δ120588
(pcm
)max1206012
(mdash)
Total
unkn
owns
Outer
iter
Linear
iter
Inner
iter
Resid
ual
norm
Relativ
eerror
Error
estim
ate
Mem
ory
(Mb)
Page
faults
114
Delaunay
998779Vo
lumes
40
minus120
1145
8264
332
1033
6119890minus09
3119890minus09
1119890minus09
349470
214
Delaunay
998779Vo
lumes
30
85
1135
15740
332
1967
2119890minus08
1119890minus08
4119890minus09
4813401
314
Delaunay
998779Vo
lumes
20
197
1124
31188
332
3898
1119890minus08
6119890minus09
3119890minus09
7621932
414
Delaunay
998779Vo
lumes
10153
1123
127476
332
15934
7119890minus09
3119890minus09
3119890minus09
273
81886
514
Delaunay
998779Vo
lumes
05
187
1119
510676
332
63834
2119890minus09
9119890minus10
1119890minus09
1148
352261
614
Delaunay
998779Elem
ents
40
173
1100
4308
332
538
2119890minus08
8119890minus09
4119890minus09
318794
714
Delaunay
998779Elem
ents
30
117
1107
8108
332
1013
5119890minus09
2119890minus09
2119890minus09
4712946
814
Delaunay
998779Elem
ents
20
84
1112
15936
332
1992
6119890minus09
3119890minus09
2119890minus09
7321347
914
Delaunay
998779Elem
ents
1062
1116
64478
332
8059
6119890minus09
3119890minus09
4119890minus09
262
77105
1014
Delaunay
998779Elem
ents
05
58
1117
256700
332
32087
1119890minus09
5119890minus10
1119890minus09
1091
331969
1114
Delaunay
◻Vo
lumes
40
53
1149
5042
332
630
2119890minus08
1119890minus08
5119890minus09
288007
1214
Delaunay
◻Vo
lumes
30
346
1133
8560
332
1070
7119890minus09
3119890minus09
2119890minus09
3910895
1314
Delaunay
◻Vo
lumes
20
308
1131
15576
332
1947
1119890minus08
7119890minus09
3119890minus09
541564
514
14Delaunay
◻Vo
lumes
10177
1122
60774
332
7596
8119890minus09
4119890minus09
3119890minus09
167
50115
1514
Delaunay
◻Vo
lumes
05
199
1120
244936
332
30617
4119890minus09
2119890minus09
2119890minus09
698
215678
1614
Delaunay
◻Elem
ents
40
196
1100
5222
332
652
3119890minus08
1119890minus08
8119890minus09
4713098
1714
Delaunay
◻Elem
ents
30
123
1107
8810
332
1101
2119890minus08
9119890minus09
7119890minus09
6918598
1814
Delaunay
◻Elem
ents
20
901112
15922
332
1990
1119890minus08
5119890minus09
5119890minus09
107
30591
1914
Delaunay
◻Elem
ents
1063
1116
61456
332
7682
5119890minus09
2119890minus09
4119890minus09
389
113237
2014
Delaunay
◻Elem
ents
05
58
1117
246332
332
30791
8119890minus10
4119890minus10
1119890minus09
1676
498192
2114
Delq
uad
998779Vo
lumes
40
minus45
1144
6228
332
778
4119890minus08
2119890minus08
7119890minus09
308495
2214
Delq
uad
998779Vo
lumes
30
570
1153
11820
332
1477
3119890minus08
1119890minus08
4119890minus09
4010879
2314
Delq
uad
998779Vo
lumes
20
459
1103
24100
332
3012
2119890minus08
1119890minus08
5119890minus09
6217715
2414
Delq
uad
998779Vo
lumes
10512
1104
9640
03
3212050
2119890minus08
1119890minus08
9119890minus09
207
61714
2514
Delq
uad
998779Vo
lumes
05
528
1106
385600
332
48200
2119890minus09
1119890minus09
2119890minus09
838
255735
2614
Delqu
ad998779
Elem
ents
40
220
1093
3288
332
411
3119890minus08
2119890minus08
6119890minus09
308495
2714
Delq
uad
998779Elem
ents
30
140
1104
6148
332
768
4119890minus08
2119890minus08
8119890minus09
39110
0028
14Delq
uad
998779Elem
ents
20
82
1110
12392
332
1549
2119890minus08
9119890minus09
6119890minus09
6117067
2914
Delqu
ad998779
Elem
ents
1063
1115
48882
332
6110
2119890minus09
1119890minus09
1119890minus09
197
5804
630
14Delq
uad
998779Elem
ents
05
58
1116
194162
332
24270
2119890minus09
9119890minus10
2119890minus09
790
238769
3114
Delq
uad
◻Vo
lumes
40
minus416
1174
3132
332
391
5119890minus08
3119890minus08
8119890minus09
267322
3214
Delq
uad
◻Vo
lumes
30
minus248
1151
5922
332
740
9119890minus09
4119890minus09
2119890minus09
318779
3314
Delq
uad
◻Vo
lumes
20
minus132
1139
12050
332
1506
1119890minus08
7119890minus09
4119890minus09
4412239
3414
Delq
uad
◻Vo
lumes
10minus49
1126
48200
332
6025
6119890minus09
3119890minus09
2119890minus09
122
36094
3514
Delq
uad
◻Vo
lumes
05
minus23
1123
192800
332
24100
5119890minus09
2119890minus09
3119890minus09
464
140228
3614
Delq
uad
◻Elem
ents
40
224
1093
3294
332
411
3119890minus08
1119890minus08
7119890minus09
359852
3714
Delq
uad
◻Elem
ents
30
141
1104
6144
332
768
3119890minus08
2119890minus08
9119890minus09
5114018
3814
Delq
uad
◻Elem
ents
20
921111
12392
332
1549
1119890minus08
6119890minus09
5119890minus09
8122526
3914
Delq
uad
◻Elem
ents
1065
1115
48882
332
6110
5119890minus09
3119890minus09
4119890minus09
276
78431
22 Science and Technology of Nuclear Installations
Table2Con
tinued
Case
no
Symm-
etry
Mesh
algorithm
Basic
shape
Solutio
nmetho
dℓ119888
(cm)
Δ120588
(pcm
)max1206012
(mdash)
Total
unkn
owns
Outer
iter
Linear
iter
Inner
iter
Resid
ual
norm
Relativ
eerror
Error
estim
ate
Mem
ory
(Mb)
Page
faults
4014
Delq
uad
◻Elem
ents
05
59
1117
194162
332
24270
1119890minus09
5119890minus10
1119890minus09
1104
318967
4118
Delaunay
998779Vo
lumes
40
minus99
1139
4228
332
528
4119890minus12
2119890minus12
1119890minus12
359630
4218
Delaunay
998779Vo
lumes
30
50
1132
7712
332
964
1119890minus11
6119890minus12
2119890minus12
4211578
4318
Delaunay
998779Vo
lumes
20
177
1120
15776
332
1972
2119890minus12
1119890minus12
7119890minus13
5515436
4418
Delaunay
998779Vo
lumes
10142
1118
63902
332
7987
3119890minus12
1119890minus12
1119890minus12
160
46691
4518
Delaunay
998779Vo
lumes
05
176
1115
254612
332
31826
2119890minus12
8119890minus13
1119890minus12
555
168526
4618
Delaunay
998779Elem
ents
40
183
1096
2252
332
281
6119890minus12
3119890minus12
2119890minus12
359589
4718
Delaunay
998779Elem
ents
30
119
1104
4040
224
505
2119890minus08
1119890minus08
7119890minus09
41114
3748
18Delaunay
998779Elem
ents
20
85
1108
8156
332
1019
2119890minus12
8119890minus13
6119890minus13
5515053
4918
Delaunay
998779Elem
ents
1063
1112
32480
332
4060
1119890minus12
6119890minus13
8119890minus13
151
43671
5018
Delaunay
998779Elem
ents
05
58
1113
128358
332
1604
47119890minus13
3119890minus13
7119890minus13
527
157581
5118
Delaunay
◻Vo
lumes
40
37
1144
2380
224
297
5119890minus08
2119890minus08
1119890minus08
5314173
5218
Delaunay
◻Vo
lumes
30
218
1120
4194
332
524
7119890minus12
4119890minus12
2119890minus12
3610034
5318
Delaunay
◻Vo
lumes
20
345
1110
7960
332
995
5119890minus12
3119890minus12
1119890minus12
4712879
5418
Delaunay
◻Vo
lumes
10165
1117
30652
332
3831
2119890minus12
1119890minus12
7119890minus13
100
28851
5518
Delaunay
◻Vo
lumes
05
191
1117
122066
224
15258
2119890minus08
8119890minus09
9119890minus09
343
103825
5618
Delaunay
◻Elem
ents
40
175
1097
2526
332
315
4119890minus12
2119890minus12
1119890minus12
6016159
5718
Delaunay
◻Elem
ents
30
114
1104
4386
332
548
4119890minus12
2119890minus12
2119890minus12
5013768
5818
Delaunay
◻Elem
ents
20
901108
8234
332
1029
2119890minus12
1119890minus12
1119890minus12
7320094
5918
Delaunay
◻Elem
ents
1062
1112
31186
224
3898
2119890minus08
7119890minus09
9119890minus09
208
59589
6018
Delaunay
◻Elem
ents
05
58
1113
123122
224
15390
9119890minus09
5119890minus09
1119890minus08
807
2366
7061
18Delq
uad
998779Vo
lumes
40
minus210
1168
3224
332
403
1119890minus11
5119890minus12
2119890minus12
43117
8462
18Delq
uad
998779Vo
lumes
30
369
1150
6000
332
750
1119890minus11
6119890minus12
3119890minus12
5013502
6318
Delq
uad
998779Vo
lumes
20
312
1104
12316
332
1539
2119890minus11
8119890minus12
5119890minus12
6016395
6418
Delq
uad
998779Vo
lumes
1044
01094
48714
332
6089
9119890minus12
4119890minus12
4119890minus12
1183444
165
18Delq
uad
998779Vo
lumes
05
534
1081
193980
332
24247
5119890minus12
2119890minus12
4119890minus12
423
126768
6618
Delq
uad
998779Elem
ents
40
240
1090
1750
332
218
9119890minus12
4119890minus12
2119890minus12
43117
8667
18Delqu
ad998779
Elem
ents
30
150
1100
3184
332
398
1119890minus11
5119890minus12
2119890minus12
5013509
6818
Delq
uad
998779Elem
ents
20
89
1106
6426
332
803
5119890minus12
3119890minus12
2119890minus12
5916047
6918
Delq
uad
998779Elem
ents
1064
1111
24886
332
3110
2119890minus12
1119890minus12
1119890minus12
113
32788
7018
Delq
uad
998779Elem
ents
05
59
1113
98052
224
12256
1119890minus08
5119890minus09
8119890minus09
398
118153
7118
Delq
uad
◻Vo
lumes
40
minus398
1172
1650
332
206
1119890minus11
5119890minus12
2119890minus12
369907
7218
Delq
uad
◻Vo
lumes
30
minus249
1146
3058
332
382
4119890minus12
2119890minus12
1119890minus12
349349
7318
Delq
uad
◻Vo
lumes
20
minus131
1136
6228
224
778
3119890minus08
1119890minus08
5119890minus09
4010930
7418
Delq
uad
◻Vo
lumes
10minus51
1123
24536
224
3067
3119890minus08
2119890minus08
1119890minus08
7822716
7618
Delq
uad
◻Elem
ents
40
237
1090
1752
332
219
6119890minus12
3119890minus12
2119890minus12
4111213
7718
Delq
uad
◻Elem
ents
30
145
1101
3184
332
398
6119890minus12
3119890minus12
2119890minus12
43118
1178
18Delq
uad
◻Elem
ents
20
941107
6426
332
803
3119890minus12
1119890minus12
1119890minus12
6016220
7918
Delqu
ad◻
Elem
ents
1065
1111
24874
332
3109
1119890minus12
6119890minus13
8119890minus13
156
4400
080
18Delq
uad
◻Elem
ents
05
59
1113
9804
22
2412255
8119890minus09
4119890minus09
8119890minus09
567
162168
Science and Technology of Nuclear Installations 23
Tim
e con
sum
ed to
bui
ld th
e mat
rices
(sec
onds
)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay trianglesvolumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
10
30
100
300
1
3
Figure 23 Time needed to build the matrices versus number of unknowns
was considerably larger than the solution time itself Theeigenvalue problem was solved using a multilayer iterativeKrylov-Schur method [26] with a tolerance relative to thematrices norm
1003817100381710038171003817119877 sdot 120601 minus (1119896eff) sdot 119865 sdot 1206011003817100381710038171003817
119877 + (1119896eff) sdot 119865lt 10minus8 (11)
The reported residual norm and relative error are10038171003817100381710038171003817100381710038171003817119877 sdot 120601 minus
1
119896effsdot 119865 sdot 120601
10038171003817100381710038171003817100381710038171003817
1003817100381710038171003817119877 sdot 120601 minus (1119896eff) sdot 119865 sdot 1206011003817100381710038171003817
1003817100381710038171003817(1119896eff) sdot 1206011003817100381710038171003817
(12)
respectively The fields marked as outer linear and inneriterations refer to the number of steps needed to attainthe requested tolerance in each layer of the Krylov-Schuralgorithm The computer used to solve the problem has anIntel i7 920 267GHz processor with 4Gb of RAM runningDebian GNULinux Wheezy
When using a finite volumes-based scheme over anunstructured mesh the solver has to gather information
about which cells are neighbors and which are not Currentlygmsh does not write this kind of lists in its output filesso milonga has to explicitly solve the neighbors problemPerforming a linear search is an 119874(119873
2) task which is
unacceptable for problem sizes 119873 of interest The code usesa search based on a 119896-dimensional tree [27] which can inprinciple be performed in119874(119873) steps Still for large values of119873 the time needed to read and parse the mesh (number twopreviously mentioned) is the bottleneck of the solution Thisstep is not needed in finite elements although the construc-tion of the elementary matrices involves the computationof the multidimensional Jacobians and integrals which thenhave to be assembled Again for large 119873 this step (numberthree) takes up most of the time needed to solve the problem
The Delaunay algorithm is a standard method for gen-erating two-dimensional grids [16] by tessellating a planewith triangles If the mesh needs to be based on quadranglesinstead a recombination algorithm can be used to transformtwo adjacent triangles in one quadrangle whenever is possi-ble However for geometries which are based on rectangularshapes there exist other algorithms [25] both for meshingand for recombining the triangles that give rise to elementaryentities with right angles almost everywhere which may be a
24 Science and Technology of Nuclear Installations
Tim
e con
sum
ed to
solv
e the
eige
nval
ue p
robl
em (s
econ
ds)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
10
30
100
300
1
3
Figure 24 Time needed to solve the eigenvalue problem versus number of unknowns
desired property of the resulting grid For finite elements thesteps needed to build the matrices depend on the selectionof triangles or quadrangles as the basic elementary geometrybecause the shape functions change However the number ofunknowns is the same as the number of nodes that does notchange after a recombination procedure On the other handthe number of unknowns in the finite volumes schemes withtriangles is roughly twice as the number of unknowns withquadrangles for the same grid
Figure 19 shows how the computed static reactivity (ie1 minus 1119896eff) depends on the number of unknowns for eachof the sixteen combinations of geometry algorithm shapescheme The four accepted results published in the originalreference [10] are included for reference although it shouldbe taken into account that said reactivities were computedalmost forty years ago Figure 20 shows the total wall timeneeded to solve the problem as a function of 119873119866 whileFigures 21 22 23 and 24 show the times needed for eachof the first four steps involved in the solution maintainingthe same logarithmic scale for both the abscissas and theordinates Green data represent finite volumes whilst bluebullets correspond to finite elements Solid lines indicatequarter-core and dashed lines eighth-symmetry Fillet bullets
are results obtained by theDelaunay triangulation and emptybullets were computed with the delquad algorithm Finallytriangle-shaped data correspond to triangles and squares anddiamonds denote quadrangles as the basic geometry of thegrid
It can be seen that finite elements produce amuch smallerdispersion of eigenvalues 119896eff than finite volumes with therefinement of the mesh This can be explained because finitevolumes methods rely on a geometric condition of the meshwhich may change abruptly if the meshing algorithm decidesto allocate the cells in a rather different form for small changesin ℓ119888 Finite elements methods are less influenced by thesediscontinuous lay-out changes of the elements As expectedeighth-core symmetries give almost the same results as thequarter-core geometries with roughly half the unknownsFor small problems finite volumes run faster that finiteelements because the time needed to solve the neighborproblem is negligible When the problem size grows thistime increases and exceeds the overhead implied in the con-struction and assembly of the finite elements matrices Alsoat least for this configuration it is seen that the eigenvalueproblem is solved faster for finite volumes than for finiteelements
Science and Technology of Nuclear Installations 25
4 Conclusions
Unstructured grids provide the cognizant engineer witha wide variety of possibilities to deal with the design oranalysis of nuclear reactor cores These kinds of grids cansuccessfully reproduce continuous geometries commonlyfound in reactor cores such as cylinders and thereforenot only can the diffusion equation be better approximatedinside the domain of definition but also the fulfillment ofboundary conditions is improved A free computer codewas written from scratch that is able to completely solvethe 2D IAEA PWR Benchmark using unstructured gridsfor sixteen combinations of geometry meshing algorithmbasic shape and discretization scheme plus any value ofthe gridrsquos characteristic length by using a single input fileThe complete set of input files and codemdashexecutable andsourcemdashis available either online or upon request withcomments experiences suggestions and corrections beingmore than welcome Further development should includetackling full three-dimensional geometries with completethermal hydraulic feedback in order to analyze how thesolutions of the coupled neutronic-thermal problem dependon the spatial discretization scheme of the neutron leakageterm Parallelization of the computation and assembly of thematrices and of the solution of the eigenvalue problem andits implementation using GPUs are also desired features toimplement A problemwith direct applications that the futureversions of milonga ought to solve is the analysis of how thegeometry of the absorbing materials should be taken intoaccount in structured coarse grids in order to mitigate effectssuch as the rod-cusp problem
Suitable schemes for approximating the continuous dif-ferential operators by discrete matrix expressions includefinite volumes and finite elements families Finite volumesmethods compute cell mean values whilst finite elementsgive functional values at the gridrsquos nodes In general finiteelements are less sensitive to changes in the mesh so theresults they provide do not change significantly for differentmeshing algorithms or elementary shapes Small problemsare best solved by finite volumes as the neighbor-findingproblem is faster than the process of building and assemblingthe eigenvalue-problem matrices For a large number ofunknowns the process of finding which cell is neighbor ofwhichmdasheven based on a 119896-dimensional treemdashoverwhelmsthe computation and assembly of elementary matrices andfinite-element methods perform better
References
[1] G Glasstone and S Bell Nuclear Reactor Theory KriegerPublishing Company 1970
[2] J J Duderstadt and L J Hamilton Nuclear Reactor AnalysisJohn Wiley amp Sons New York NY USA 1976
[3] C Gho Reactor Physics Undergraduate Course Lecture NotesInstituto Balseiro 2006
[4] G G Theler and F J Bonetto ldquoOn the stability of the pointreactor kinetics equationsrdquoNuclear Engineering andDesign vol240 no 6 pp 1443ndash1449 2010
[5] E E Lewis andW F Miller Computational Methods of NeutronTransport John Wiley amp Sons 1984
[6] E Masiello ldquoAnalytical stability analysis of Coarse-Mesh finitedifference methodrdquo in Proceedings of the International Confer-ence on the Physics of Reactors (PHYSOR rsquo08) Nuclear Power ASustainable Resource pp 456ndash464 September 2008
[7] M L Zerkle Development of a polynomial nodal methodwith flux and current discontinuity factors [PhD thesis] Mas-sachusets Institute of Technology 1992
[8] J I Yoon andHG Joo ldquoTwo-level coarsemesh finite differenceformulation with multigroup source expansion nodal kernelsrdquoJournal of Nuclear Science andTechnology vol 45 no 7 pp 668ndash682 2008
[9] O Mazzantini M Schivo J Di Cesare R Garbero M Riveroand G Theler ldquoA coupled calculation suite for Atucha II oper-ational transients analysisrdquo Science and Technology of NuclearInstallations vol 2011 Article ID 785304 12 pages 2011
[10] Computational Benchmark Problem Comitee for the Mathe-matics and Computation Division of the American NuclearSociety ldquoArgonne Code Center Benchmark problem bookrdquoTech Rep ANL-7416 Supplement 2 Argonne National Labo-ratory 1977
[11] A F Henry Nuclear Reactor Analysis MIT Cambridge UK1975
[12] O C Zienkiewicz R L Taylor and J Z ZhuTheFinite ElementMethod Its Basis and Fundamentals vol 1 Elsevier 6th edition2005
[13] G Theler Difusion de neutrones en mallas no estructuradascomparacion entre volumenes y elementos finitos AcademicMonograph Universidad de Buenos Aires 2013
[14] G Theler ldquoMilonga a free nuclear reactor core analysis coderdquoAvailable online 2011
[15] G Theler F J Bonetto and A Clausse ldquoOptimizacion deparametros en reactores de potencia base de diseno delcodigo neutronico milongardquo in Reunion Anual de la AsociacionArgentina de Tecnologia Nuclear XXXVII 2010
[16] C Geuzaine and J-F Remacle ldquoGmsh a 3-D finite elementmesh generator with built-in pre- and post-processing facili-tiesrdquo International Journal for Numerical Methods in Engineer-ing vol 79 no 11 pp 1309ndash1331 2009
[17] S Balay J Brown K Buschelman et al ldquoPETSc users manualrdquoTech Rep ANL-9511-Revision 34 Argonne National Labora-tory 2013
[18] S Balay W D Gropp L Curfman McInnes and B F SmithldquoEfficient management of parallelism in object oriented numer-ical software librariesrdquo in Modern Software Tools in ScientificComputing E Arge A M Bruaset and H P Langtangen Edspp 163ndash202 Birkhauser 1997
[19] V Hernandez J E Roman and V Vidal ldquoSLEPC a scalable andflexible toolkit for the solution of eigenvalue problemsrdquo ACMTransactions on Mathematical Software vol 31 no 3 pp 351ndash362 2005
[20] M Galassi J Davies J Theiler et al GNU Scientific LibraryReference Manual 3rd edition 2009
[21] O C Zienkiewicz and R L Taylor The Finite Element MethodFor Solid and Structural Mechanics vol 3 Elsevier 6th edition2005
[22] A Imelda and K Doddy Evaluation of the IAEA 3-D PWRbenchmark problem using NESTLE code 2004
[23] R Mosteller ldquoStatic benchmarking of the NESTLE advancednodal coderdquo in Proceedings of the Joint International Conference
26 Science and Technology of Nuclear Installations
on Mathematical Methods and Supercomputing for NuclearApplications vol 2 pp 1596ndash1605 1997
[24] GTheler F J Bonetto andAClausse ldquoSolution of the 2D IAEAPWR Benchmark with the neutronic code milongardquo in Actasde la Reunion Anual de la Asociacion Argentina de TecnologıaNuclear XXXVIII 2011
[25] J F Remacle F Henrotte T Carrier-Baudouin et al ldquoAfrontal delaunay quad mesh generator using the 119871
infin normrdquoInternational Journal For Numerical Methods in Engineeringvol 94 no 5 pp 494ndash512 2013
[26] G W Stewart ldquoA Krylov-Schur algorithm for large eigenprob-lemsrdquo SIAM Journal on Matrix Analysis and Applications vol23 no 3 pp 601ndash614 2002
[27] J L Bentley ldquoMultidimensional binary search trees used forassociative searchingrdquo Communications of the ACM vol 18 no9 pp 509ndash517 1975
TribologyAdvances in
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International Journal of
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FuelsJournal of
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Renewable Energy
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
StructuresJournal of
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear InstallationsScience and Technology of
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Solar EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Wind EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear EnergyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
6 Science and Technology of Nuclear Installations
(a) Fast flux distribution
minus100minus50 0 50 100minus100
minus500
50100
0
06
xy
(b) Fast flux unknowns
(c) Matrix 119877 structure (d) Matrix 119865 structure
Figure 6 A bare homogeneous circle solved with finite elements (218 unknowns)
the normal direction should be zero (ie symmetry condi-tion) and the Robin conditions are restricted to the followingform
grad 120601119892 (x) sdot n + 119886119892 (x) sdot 120601119892 (x) = 0 (6)
n being the outward unit normal to the boundary 120597119880 of thedomain 119880
22 Grids and Schemes One way of solving the neutrondiffusion equationmdashand in general any partial differentialequation over spacemdashis by discretizing the differential oper-ators with some kind of scheme that is applied over a certainspatial grid Given an 119898-dimensional domain a grid definesa partition composed of a finite number of simple geometricentities In structured grids these elementary entities arearranged following a well-defined structure in such a waythat each entity can be identified without needing furtherinformation than the one provided by the intrinsic structureOn the other hand the geometric entities that composean unstructured grid are arranged in an irregular patternand the identification of the elementary entities needs to
be separately specified for example by means of a sortedlist For instance Figure 2(b) shows a structured grid thatapproximates the continuous domain of Figure 2(a) Eachsquare may be uniquely identified by means of two integerindexes that indicate its relative position in each of thehorizontal and vertical directions In the case of Figure 2(c)that shows an unstructured grid there is no way to system-atically make a reference to a particular quadrangle withoutany further information
Almost all of the grid-based schemesmdashwhich are knownas nodal schemes which are to be differentiated from modalschemes based on series expansionsmdashare based in either thefinite differences volumes or elements method Finite differ-ences schemes provide themost simple and basic approach toreplace differential (ie continuous) operators by difference(ie discrete) approximations However they are not suitablefor unstructured meshes and may introduce convergenceproblems with parameters that are discontinuous in spacewhich is the case for any reactor core composed of atleast two different materials Moreover boundary conditionsare hard to incorporate and usually give rise to incorrectresults
Science and Technology of Nuclear Installations 7
20 40 600995
09975
1
Analytical solutionFinite volumesFinite elements
keff
a985747c
Figure 7 The effective multiplication factor of a two-group barecircular reactor of radius 119886 as a function of the quotient 119886ℓ
119888between
the radius and the characteristic length of the cellelement
20 40 60
Finite volumesFinite elements
Erro
r ink
eff
10minus3
10minus4
10minus5
a985747c
Figure 8 Absolute value of the error committed in the computationof 119896eff versus 119886ℓ119888
Methods of the finite volumes family involve the inte-gration of the differential equation over each elementaryentity applying the divergence theorem to transform volumeintegrals into surface integrals and providing a mean to esti-mate the fluxes through the entityrsquos surface using informationcontained in its neighbors In this context each elementaryentity is called a cell and finite volumes methods give themean value of each of the group fluxes 120601119892(119894) at the 119894th cellwhich may be associated to the value of 120601119892(x119894) where x119894 isthe location of the cell barycenter Being an integral-basedmethod spatial-discontinuous parameters are handled moreefficiently than in finite differences and boundary conditionscan be easily incorporated as forced cell fluxes Nonethelesseven though these methods may be applied to unstructuredgrids the estimation of the fluxes on the cellsrsquo surfaces isusually performed by using some geometric approximationsthat may lose validity as the quality of the grid is worsened
10
70
90
130
150
170
10 30 50 70 90 110 130 150 170
4
1 38
36 37353
3 32 33 34312
26 27 28 29 3025
19 20 21 22 23 2418
10 11 12 13 14 15 16 17
2 331 4 5 6 7 8 9
x (cm)
y (cm
)
Figure 9 The 2D IAEA PWR benchmark geometry
Moreover the simple integral approach cannot take intoaccount discontinuous diffusion coefficients so when twoneighbors pertain to different materials the flux has to becomputed in a certain particular way to conserve the neutroncurrent according to (2)
Finally finite elements methods rely on a weak formula-tion of the differential problem similar to (3) that maintainsall the mathematical characteristics of the original strongformulation plus its boundary conditions The method isbased on shape functions that are local to each elementaryentitymdashnow called element defined by nodes as cornersmdashand on finding a set of nodal values such that a certaincondition is met which is usually that the residual of thesolution has to be orthogonal to each of the shape functionsThis condition is known as the Galerkin method and impliesthat the error committed in the approximate solution ofthe continuous problem is confined into a small subset ofthe original vector space of the continuous functions Thesemathematical properties make finite elements schemes veryattractive However these features depend on a large numberof integrations that ought to be performed numerically soa computational effortdesired accuracy tradeoff has to beconsidered Not only does the finite elements method givethe values of the flux 120601119892(x119894) at the 119894th node but also theshape functions provide explicit expressions to interpolateand to evaluate the unknown functions at any location x ofthe domain119880 Boundary conditions are divided into essentialand naturalThefirst group comprises theDirichlet boundaryconditions which are satisfied exactlymdashwithin the precisionof the eigenvalue problem solvermdashby the obtained solutionThe latter include the Neumann and the Robin conditionsthat are satisfied only approximately by the derivatives of theinterpolated solution through the shape functions with finermeshes giving better agreement with the prescribed values
The same unstructured grid may be used either for thefinite volumes or for the finite elements method In the first
8 Science and Technology of Nuclear Installations
Largest eigenvalue 1029690 (288336 pcm)1135 (3066 3079)1306 (5171 13076)
Number of unknowns 15740Outer iterations 3Linear iterations 32Inner iterations 1967Residual normRelative errorError estimateMemory used 49824 kBSoft page faults 13401Hard page faults 0Total CPU time 074 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
2483 times 10minus8
1223 times 10minus8
403 times 10minus9
keff
quarter-symmetry core meshed using Delaunay (triangles 985747c = 3) solved with finite volumesMilongarsquos 2D LWR IAEA benchmark problem case no 002
(a)
123456789
1011121314151617181920212223242526272829303132333435363738
k Pk 1206011k 1206012k075 3282 555133 4242 984147 4645 1090123 3921 911061 2677 453094 2995 694093 2927 689074 2011 549mdash 322 756146 4597 1078150 4730 1110133 4199 985107 3405 789104 3271 767095 2971 700072 1956 534mdash 306 733149 4695 1102136 4289 1007118 3733 876107 3367 792097 2892 718066 1628 491mdash 234 549120 3797 891097 3096 718090 2842 669084 2221 619mdash 568 1215mdash 067 257047 2042 348068 2055 504058 1414 430mdash 222 562057 1374 425mdash 388 820mdash 053 203mdash 060 229
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 10 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
case the unknowns are themean value of the fluxes over eachcell whilst in the latter the unknowns are the fluxes evaluatedat each node Therefore the number of unknowns 119873119866 isdifferent for each method even when using the same gridFigure 4 shows this situation and in Section 31 we further
illustrate these differences A fully detailed mathematicaldescription of the actual algorithms for both volumes andelements-based proposed discretizations can be found in anacademic monograph written by the author of this paper[13]
Science and Technology of Nuclear Installations 9
Largest eigenvalue 1029927 (290571 pcm)1131 (3175 2999)1220 (13097 5098)
Number of unknowns 15576Outer iterations 3Linear iterations 32Inner iterations 1947Residual normRelative errorError estimateMemory used 56256 kBSoft page faults 15645Hard page faults 0Total CPU time 09201 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
1394 times 10minus8
6865 times 10minus9
3425 times 10minus9
keff
quarter-symmetry core meshed using Delaunay (quads 985747c = 2) solved with finite volumesMilongarsquos 2D LWR IAEA benchmark problem case no 013
(a)
123456789
1011121314151617181920212223242526272829303132333435363738
k Pk 1206011k 1206012k074 3244 549133 4230 982147 4651 1092124 3946 917060 2645 443094 2997 696093 2938 691074 2005 545mdash 301 746145 4588 1076150 4720 1108133 4198 984109 3465 805104 3280 769095 2986 704071 1962 528mdash 293 715148 4680 1098136 4288 1007119 3755 881107 3385 796097 2905 721067 1653 493mdash 220 556121 3808 893098 3115 724092 2889 680083 2247 615mdash 543 1241mdash 067 260045 1977 330069 2076 509058 1460 428mdash 226 574056 1364 413mdash 371 821mdash 052 200mdash 060 238
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 11 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
3 Results
We now proceed to show two illustrative results that are tobe taken as an overview of the possibilities that unstructuredgrids can provide in order to tackle the multigroup neutron
diffusion problem The examples are two-dimensional prob-lems as they contain some of the complexities a real three-dimensional reactor posse yet the reported results are notso complicated as to be easily understood and analyzed Inparticular we state some basic differences between the finite
10 Science and Technology of Nuclear Installations
Largest eigenvalue 1029695(288388 pcm)1112 (3076 3032)838 (5000 13000)
Number of unknowns 15922Outer iterations 3Linear iterations 32Inner iterations 1990Residual normRelative errorError estimateMemory used 109964 kBSoft page faults 30591Hard page faults 0Total CPU time 158 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
1056 times 10minus8
5202 times 10minus9
5122 times 10minus9
keff
quarter-symmetry core meshed using Delaunay (quads 985747c = 2) solved with finite volumesMilongarsquos 2D LWR IAEA benchmark problem case no 018
(a)
123456789
1011121314151617181920212223242526272829303132333435363738
k Pk 1206011k 1206012k074 3220 549130 4152 961144 4651 1068120 3843 890061 2648 452094 2996 693094 2959 696072 2061 569mdash 358 862142 4495 1054147 4634 1088131 4127 968107 3409 789104 3275 768096 3003 708071 2014 554mdash 342 820146 4606 1081134 4222 991118 3711 871107 3375 794098 2929 727062 1692 522mdash 258 632119 3752 880096 3084 715091 2858 673080 2280 633mdash 611 1277mdash 080 318047 2043 350069 2088 510054 1463 450mdash 255 648051 1418 439mdash 410 854mdash 064 252mdash 071 285
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 12 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
volumes and the finite elements methods by solving a two-group homogeneous bare reactor with the Robin boundaryconditions over the very same grid although a rather coarseone so the differences can be observed directly into theresulting figuresWe then solve the classical two-dimensionalLWR problem also known as the 2D IAEA PWR benchmark
Not only do we show again the differences between the finitevolumes and elements formulation but also we solve theproblemusing different combinations ofmeshing algorithmsbasic shapes and characteristic lengths of the mesh
To solve the two examples shown below we employed themilonga code which was written from scratch by the author
Science and Technology of Nuclear Installations 11
Largest eigenvalue 1029159 (283326 pcm)1174 (2851 3224)1560 (5200 13200)
Number of unknowns 3132Outer iterations 3Linear iterations 32Inner iterations 391Residual normRelative errorError estimateMemory used 26888 kBSoft page faults 7322Hard page faults 0Total CPU time 0308 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
5408 times 10minus8
2665 times 10minus8
8424 times 10minus9
keff
Milongarsquos 2D LWR IAEA benchmark problem case no 031quarter-symmetry core meshed using Delquad (quads 985747c = 4) solved with finite volumes
(a)
123456789
1011121314151617181920212223242526272829303132333435363738
k Pk 1206011k 1206012k077 3422 570140 4468 1040153 4836 1135128 4070 948060 2704 448095 3017 702091 2861 673069 1918 508mdash 264 689151 4785 1122155 4897 1149137 4334 1017111 3535 823104 3272 768092 2908 685067 1876 496mdash 254 658154 4854 1139139 4392 1031120 3792 890107 3360 790095 2831 702061 1574 451mdash 197 515123 3868 907099 3132 730090 2818 663078 2168 579mdash 481 1191mdash 055 217044 1984 328067 2027 498052 1360 388mdash 190 520052 1329 383mdash 321 792mdash 043 168mdash 048 195
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 13 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
of this paper and is currently still under development withinhis ongoing PhD thesis There exists a first public release[14] under the terms of the GNU General Public Licensemdashthat is it is a free softwaremdashthat can only handle structuredgrids There is a second release being prepared whose
main relevance is that it can work with nonstructured gridsas well which is the main feature of the code It uses ageneral mathematical frameworkmdashcoded from scratch aswellmdashwhich provides input file parsing algebraic expres-sions evaluation one- and multidimensional interpolation of
12 Science and Technology of Nuclear Installations
Largest eigenvalue 1029788 (289260 pcm)1120 (3149 3011)1091 (13038 5114)
Number of unknowns 15776Outer iterations 3Linear iterations 32Inner iterations 1972Residual normRelative errorError estimateMemory used 56996 kBSoft page faults 15436Hard page faults 0Total CPU time 09121 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
2296 times 10minus12
1131 times 10minus12
6877 times 10minus13
keff
Milongarsquos 2D LWR IAEA benchmark problem case no 043eighth-symmetry core meshed using Delaunaya (triangs985747c = 2) solved with finite volumes
(a)
1 074 3241 5482 131 4188 9713 145 4582 10764 122 3877 9005 060 2644 4456 093 2986 6927 093 2941 6928 075 2030 5529 mdash 322 779
10 143 4523 106111 148 4667 109512 131 4145 97213 107 3423 79414 103 3269 76715 095 2993 70616 073 1988 53917 mdash 312 75118 147 4637 108819 134 4243 99620 118 3721 87321 107 3370 79322 098 2914 72323 067 1651 49924 mdash 235 57125 119 3763 88226 096 3077 71527 091 2847 67128 083 2255 61729 mdash 576 122530 mdash 068 26631 046 2016 34132 068 2058 50533 059 1443 43534 mdash 233 58935 057 1381 42036 mdash 381 82337 mdash 054 20838 mdash 062 244
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 14 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
scattered data shared-memory access numerical integrationfacilities and so forth The milonga code was designed withfour design basis vectors in mindmdashwhich are thoroughlydiscussed in the documentation [15]mdashwhich includes thetype of problems it can handle the code scalability which
features are expected and what to do with the obtainedresults
It works by first reading an input file that using plain-text English keywords and arguments defines the number119898 of spatial dimensions the number 119866 of group energies
Science and Technology of Nuclear Installations 13
Largest eigenvalue 1029726 (288675 pcm)1104 (3069 3069)864 (13000 5000)
Number of unknowns 4040Outer iterations 2Linear iterations 24Inner iterations 505Residual normRelative errorError estimateMemory used 42940 kBSoft page faults 11437Hard page faults 0Total CPU time 1064 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
2019 times 10minus8
9946 times 10minus9
7138 times 10minus9
keff
Milongarsquos 2D LWR IAEA benchmark problem case no 047eighth-symmetry core meshed using Delaunay (triangles 985747c = 3) solved with finite volumes
(a)
1 074 3194 5452 129 4117 9533 143 4515 10604 119 3814 8835 061 2634 4506 093 2984 6907 094 2954 6958 071 2064 5719 mdash 362 865
10 141 4458 104511 146 4599 107912 130 4097 96113 106 3388 78414 103 3263 76515 095 2998 70716 068 2015 55717 mdash 346 82318 145 4572 107319 133 4194 98420 117 3691 86621 107 3363 79222 098 2925 72623 059 1696 52824 mdash 263 63525 118 3728 87426 096 3068 71127 091 2850 67228 079 2280 63529 mdash 619 127630 mdash 081 32131 047 2036 34932 068 2085 50933 051 1465 45434 mdash 258 65035 049 1419 44336 mdash 416 85437 mdash 065 25438 mdash 071 288
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 15 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
and optionally a mesh file Currently only grids generatedwith the code gmsh [16] are only supported mainly becauseit is also a free software and it suits perfectly well milongarsquosdesign basis in the sense that the continuous geometry can bedefined as a function of a number of parameters Afterwardthe physical entities defined in the grid are mapped into
materials with macroscopic cross sections which maydepend on the spatial coordinates x bymeans of intermediatefunctions such as burn-up or temperatures distributionswhich in turn may be defined by algebraic expressions byinterpolating data located in files by reading shared-memoryobjects or by a combination of them In the same sense
14 Science and Technology of Nuclear Installations
Largest eigenvalue 1029695 (288387 pcm)1108 (3045 3045)833 (13000 5000)
Number of unknowns 8234Outer iterations 3Linear iterations 32Inner iterations 1029Residual normRelative error
Max 1206012(x y) coreMax 1206012(x y) reflector
2028 times 10minus12
9993 times 10minus13
1012 times 10minus12
keff
Relative errorError estimateMemory used 75468 kBSoft page faults 20094Hard page faults 0Total CPU time 1256 seconds
9993 times 10
1012 times 10minus12
Milongarsquos 2D LWR IAEA benchmark problem case no 058eighth-symmetry core meshed using Delaunay (quads 985747c = 2) solved with finite volumes
(a)
1 074 3207 5472 129 4136 9573 144 4534 10644 120 3828 8875 061 2638 4506 093 2985 6907 094 2948 6948 072 2053 5669 mdash 357 859
10 142 4478 105011 146 4617 108412 130 4111 96413 106 3396 78614 103 3263 76515 095 2991 70616 071 2005 55217 mdash 340 81618 145 4588 107719 133 4206 98720 117 3698 86821 107 3362 79122 098 2918 72523 062 1686 52024 mdash 258 62925 118 3737 87626 096 3073 71227 091 2848 67128 080 2272 63129 mdash 609 127230 mdash 080 31731 047 2035 34832 069 2081 50933 054 1458 44834 mdash 255 64635 052 1413 43736 mdash 409 85137 mdash 064 25138 mdash 071 284
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 16 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
boundary conditions are applied to appropriate physicalentities of dimension119898minus1The problemmatrices119877 and119865 arethen built and stored in an appropriate sparse format usingthe free PETSc [17 18] library and the eigenvalue problemis solved using the free SLEPc [19] library The results arestored into milongarsquos variables and functions which may be
evaluated integratedmdashusually using the free GNU ScientificLibrary [20] routinesmdashand of course written into appropriateoutputs Milonga can also solve problems parametrically andbe used to solve optimization problems As stated above thecode is a free software so corrections and contributions aremore than welcome
Science and Technology of Nuclear Installations 15
Largest eigenvalue 1029462 (286185 pcm)1136 (3093 2953)1233 (13100 5100)
Number of unknowns 6228Outer iterations 2Linear iterations 24Inner iterations 778Residual normR l ti
Max 1206012(x y) coreMax 1206012(x y) reflector
26 times 10minus8
1281 times 10minus8
4996 times 10minus9
keff
Relative errorError estimateMemory used 41460 kBSoft page faults 10930Hard page faults 0Total CPU time 0604 seconds
1281 times 10 8
4996 times 10minus9
Milongarsquos 2D LWR IAEA benchmark problem case no 073eighth-symmetry core meshed using Delaunay (quads 985747c = 2) solved with finite volumes
(a)
1 075 3297 5552 134 4268 9923 148 4660 10944 123 3926 9145 059 2645 4406 094 2997 6977 093 2914 6858 072 2002 5349 mdash 301 773
10 146 4606 108011 150 4735 111112 133 4212 98813 108 3448 80214 104 3271 76715 094 2954 69616 070 1954 52117 mdash 288 73518 149 4691 110119 136 4289 100720 119 3748 88021 107 3359 79022 096 2880 71423 064 1636 47524 mdash 217 56325 121 3810 89426 098 3110 72427 090 2829 66628 081 2227 59929 mdash 536 125730 mdash 062 25031 045 2002 33632 068 2040 50033 055 1414 40934 mdash 215 57935 055 1359 40836 mdash 359 83437 mdash 050 19938 mdash 061 236
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 17 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
31 A Coarse Bare Homogeneous Circle Figures 5 and 6 showthe results of solving a bare two-dimensional circular homo-geneous reactor of radius 119886 over an unstructured grid whichis deliberately coarse using the finite volumes method andthe finite elements method respectively Two group energies
were used and a null-flux boundary conditionwas fixed at theexternal surface Figures 5(a) and 6(a) compare the obtainedfast flux distributions In the first case the numerical solutionprovidesmean values for each neutron flux group in each cellwhile in the latter the solution is computed at the nodes and
16 Science and Technology of Nuclear Installations
Largest eigenvalue 1029851 (289858 pcm)1090 (3182 3182)851 (13000 5000)
Number of unknowns 1752Outer iterations 3Linear iterations 32Inner iterations 219Residual norm
Max 1206012(x y) coreMax 1206012(x y) reflector
566 times 10minus12
2789 times 10minus12
1595 times 10minus12
keff
Residual normRelative errorError estimateMemory used 42564 kBSoft page faults 11213Hard page faults 0Total CPU time 09921 seconds
566 times 10
2789 times 10minus12
1595 times 10minus12
Milongarsquos 2D LWR IAEA benchmark problem case no 076eighth-symmetry core meshed using Delaunay (quads 985747c = 4) solved with finite volumes
(a)
1 073 3151 5402 127 4059 9393 141 4457 10464 118 3768 8715 061 2615 4506 093 2979 6887 094 2971 6998 067 2092 5829 mdash 375 885
10 139 4397 103111 144 4541 106612 128 4052 95013 105 3365 77814 103 3259 76415 096 3016 71216 066 2045 56817 mdash 359 84118 143 4520 106119 132 4155 97520 116 3672 86221 107 3366 79222 098 2951 73323 054 1728 54324 mdash 273 65125 117 3700 86826 095 3057 70827 091 2860 67428 074 2311 64829 mdash 642 130030 mdash 084 33231 048 2041 35332 069 2103 51333 046 1493 46734 mdash 269 66735 046 1444 45536 mdash 432 87137 mdash 067 26338 mdash 074 297
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 18 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
continuous functions are evaluated by means of the shapefunctions used in the formulation Figures 5(b) and 6(b)illustrate the fast fluxes unknowns and its relative position inspace It can be noted that the mesh coarseness gives resultsthat differ substantially in both cases Finally the structure ofthe sparse eigenvalue problemmatrices is shown for each case
with blue red and cyan representing positive negative andexplicitly inserted zero values In the finite volume case thereare 184 unknowns (92 cells times 2 groups) while in the secondcase there are 218 unknowns (109 nodes times 2 groups) Thevolumesrsquo fissionmatrix is almost diagonal it has a bandwidthequal to the number of energy groups which in this case is
Science and Technology of Nuclear Installations 17
500 1000 2000 5000 10000 20000 50000Number of unknowns
2840
2860
2880
2900
2920
ORNLRisoslashKWUOntarioQuarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumes
Quarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elementsEighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumes
Eighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
120588(p
cm)
105 2 times 105 5 times 105
Figure 19 Static reactivity versus number of unknowns The four original solutions as published in 1977 [10] are included as reference
twoThe off-diagonal values appear right next to the diagonalelements because of the chosen ordering of the unknownsin the flux vector 120601 isin R184 Other orderings may be usedbut the rate of convergence of the eigenvalue problem can bedeteriorated On the other hand the elementsrsquo fission matrixhas a nontrivial structure because the grid is unstructuredand the net fission rate at each element depends on the fluxesat the nodes whose location inside thematrix depends in turnon how the gridwas generatedThis effect is similar to the onethat appears in structural analysis where mass matrices needto be lumped in order to simplify transient computations [21]The diagonal block of elementrsquos 119877 matrix and the null blockin 119865 correspond to the discrete equations that set the null-flux boundary conditions on the nodes located at the externalsurface Other types of boundary conditions lead to differentkinds of structures within the problem matrices
In case a part of the domain contained a nonmultiplicativematerial such as a reflector then there would appear sectionsof the fission matrix with null values in both methodsrendering 119865 singular in both methods Care should be takenwhen dealing with the numerical schemes for the eigenvalueproblem solution It can be seen in the elementsrsquo matricesa particular structure that implements the boundary condi-tions at the external surfaces This structure does not appearin the volumesrsquo matrices because the boundary conditions
appear as flux terms which are summed up over all thesurfaces of each cell so they aremasked inside the volumetricdiscretization of the divergence and gradient operators
As the two-group neutron diffusion equation with uni-form cross sections over a circle subject to null-flux bound-ary conditions has an analytical solution it is adequate tocompare how the two proposed numerical schemes relate toit In the studied problem we ignored upscattering and fastfissions Then the analytical effective multiplication factor is
119896eff =]Σ1198912 sdot Σ1199041rarr2
[Σ1198861 + Σ1199041rarr2 + 1198631(]0119886)2] [Σ1198862 + 1198632(]0119886)
2]
(7)
being ]0 = 24048 the smallest root of Besselrsquos first-kindfunction of order zero 1198690(119903)
Figure 7 shows how the two numerical effective multi-plication factors compare to the analytical solution givenby (7) as a function of the mesh refinement indicated bythe quotient 119886ℓ119888 between the radius of the circle and thecharacteristic length of the cellelements We can see thatthe 119896eff computed by the finite volumes (elements) methodis always greater (less) than the analytical solution Indeedit can be proven that for a bare one-dimensional slab thisis always the case [13] However this result does not hold
18 Science and Technology of Nuclear Installations
2000 3000 6000 10000 20000 30000 60000
01
003
001
03
1
3
10
30
100
300To
tal w
all t
ime (
seco
nds)
Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
Figure 20 Total wall time versus number of unknowns
for problems with nonuniform cross sections even in simpleone-dimensional reflected reactors
We may draw two other conclusions from Figure 7 Firstthat the finite element method seems to provide a bettersolution than the volumes-based scheme and second thateven though the error committed tends to decrease with finergrids its behavior is not monotonic for finite volumes Infact Figure 8 shows the difference between the numericaland analytical solutions using a logarithmic vertical scalewhere both conclusions are even more evident We deferthe discussion of such differences between the discretizationscheme until the next section It is worth to note howeverthat the fact that a finite-element-based scheme throws betterresults for the particular bare homogeneous circular reactorunder study than the finite volumes does not imply thatfinite volumes ought to be discarded as a valid tool forsolving the neutron diffusion equation in general cases Thecombination of lattice and core-level computations is usuallyperformedusing cell-based resultswhichwhen fed into node-based methods to solve the few-group neutron diffusionequation may introduce errors which potentially can leadto unacceptable solutions Nevertheless this analysis is farbeyond the scope of this paper which focuses on solving amathematical equation over unstructured grids
32 The 2D IAEA PWR Benchmark Problem This is aclassical two-group neutron diffusion problem first designedin the early 1970s and taken as a reference benchmark forcomputational codes A number of codes were used to solveeither this problem or its three-dimensional formulation[22 23] including milonga using a structured grid [24] Theoriginal formulation can be found in the reference [10] andthere is a reproduction that may be easily found online inreference [24] The geometry consists of a one-quarter ofa PWR core depicted in Figure 9 and the homogeneousmacroscopic cross sections are listed in Table 1 An axialbuckling term 119861
2
119911119892= 08times10
minus4 should be taken into accountThe external surface should be subject to a zero incomingcurrent condition which may be written as
120597120601119892
120597119899= minus
04692
119863119892
sdot 120601119892 (8)
The expected results are as follows
(1) Maximum eigenvalue(2) Fundamental flux distributions
(a) Radial flux traverses 120601119892(119909 0) and 120601119892(119909 119909)
Science and Technology of Nuclear Installations 19
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
1
3
10
30
100
300Ti
me c
onsu
med
to m
esh
the g
eom
etry
(sec
onds
)
Figure 21 Time needed to mesh the geometry versus number of unknowns
Table 1 Macroscopic cross-sections (units are not stated in the original reference but they are assumed to be in cmminus1 or cm as appropriate)
1198631
1198632
Σ1rarr2
Σ1198861
Σ1198862
]Σ1198912
Material1 15 04 002 001 0080 0135 Fuel 12 15 04 002 001 0085 0135 Fuel 23 15 04 002 001 0130 0135 Fuel 2 + rod4 20 03 004 0 0010 0 Reflector
Note the fluxes shall be normalized such that
1
119881coreint119881core
sum
119892
]Σ119891119892 sdot 120601119892119889119881 = 1 (9)
(b) Value and location of maximum power densityThis corresponds to maximum of 1206012 in thecore It is recommended that the maximumvalues of 1206012 both in the inner core and at thecorereflector interface be given
(3) Average subassembly powers 119875119896
119875119896 =1
119881119896
int119881119896
sum
119892
]Σ119891119892 sdot 120601119892119889119881 (10)
where 119881119896 is the volume of the 119896th subassembly and119896 designates the fuel subassemblies as shown inFigure 9
(4) Number of unknowns in the problem number ofiterations and total and outer
(5) Total computing time iteration time IO-time andcomputer used
(6) Type and numerical values of convergence criteria(7) Table of average group fluxes for a square mesh grid
of 20 times 20 cm(8) Dependence of results on mesh spacing
Even though the original problem is based on a quarter-core situation the problem has an eighth-core symmetry
20 Science and Technology of Nuclear Installations
Quarter Delaunay triangles elements
Tim
e con
sum
ed to
read
the m
esh
(sec
onds
)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumes
Quarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
1
3
10
30
100
300
Figure 22 Time needed to read the mesh versus number of unknowns
which cannot be taken into account by structured gridswhich are the main target of the benchmark Howevernonstructured grids can take into consideration any kind ofsymmetry almost without loss of accuracy and at the sametime reducing roughly the number of unknowns by a halfand the associated computational effort needed to solve theproblem by a factor of four Answers to items (1)ndash(7) canbe given completely by milonga using a single input file(see Supplementary data in SupplementaryMaterial availableonline at httpdxdoiorg1011552013641863) As asked initem (8) how results depend not only on the mesh spacingbut also on the meshing algorithm on the gridrsquos elementarygeometric shape and on the discretization scheme may shedlights on the subject whichmay be evenmore interesting thanthe numerical results themselves
Taking advantage of milongarsquos capability of reading andparsing command-line arguments the selection of the coregeometry (quarter or eighth) the meshing algorithm (delau-nay [16] or delquad [25]) the shape of the elementaryentities (triangles or quadrangles) the discretization scheme(volumes or elements) and the characteristic length ℓ119888 ofthe mesh can be provided at run time Fixing five values forℓ119888 = 4 3 2 1 05 gives 2 times 2 times 2 times 2 times 5 = 80 possiblecombinations which we solve with successive invocations to
milonga with the same input file but with different argumentsfrom a simple script Figures 10 11 12 13 14 15 16 17 and18 show the results corresponding to items (1)ndash(7) for someillustrative cases The complete set of figures and the codeused to generate themmay be provided upon request Table 2compiles the answers to the problem for every case studied
As the milonga code is still under development itsnumerical routines are not yet fully optimized nor designedfor parallel computation Therefore the reported times areonly rough estimates and should be taken with care Thesolution comprises five steps
(1) generate the grid with the requested geometry mesh-ing algorithm basic shape and characteristic lengthby calling to gmsh
(2) read the generated mesh(3) build the matrices(4) solve the eigenvalue problem(5) compute the requested results
The CPU time reported in Figures 10ndash18 is thus thesum of all of these five steps but not the time needed togenerate the figuresmdashwhich in some caseswith coarsemeshes
Science and Technology of Nuclear Installations 21
Table2
Resultsofthe2
DIAEA
PWRBe
nchm
arkp
roblem
obtained
with
them
ilong
acod
efor
thee
ightyp
ropo
sedcombinatio
nsofsymmetrym
eshing
algorithm
basicshapediscretization
schemeandcharacteris
ticelem
entc
elllengthTh
ereference
solutio
nis120588=28749times10minus5w
hich
correspo
ndsto119896eff=10296
Case
no
Symm-
etry
Mesh
algorithm
Basic
shape
Solutio
nmetho
dℓ119888
(cm)
Δ120588
(pcm
)max1206012
(mdash)
Total
unkn
owns
Outer
iter
Linear
iter
Inner
iter
Resid
ual
norm
Relativ
eerror
Error
estim
ate
Mem
ory
(Mb)
Page
faults
114
Delaunay
998779Vo
lumes
40
minus120
1145
8264
332
1033
6119890minus09
3119890minus09
1119890minus09
349470
214
Delaunay
998779Vo
lumes
30
85
1135
15740
332
1967
2119890minus08
1119890minus08
4119890minus09
4813401
314
Delaunay
998779Vo
lumes
20
197
1124
31188
332
3898
1119890minus08
6119890minus09
3119890minus09
7621932
414
Delaunay
998779Vo
lumes
10153
1123
127476
332
15934
7119890minus09
3119890minus09
3119890minus09
273
81886
514
Delaunay
998779Vo
lumes
05
187
1119
510676
332
63834
2119890minus09
9119890minus10
1119890minus09
1148
352261
614
Delaunay
998779Elem
ents
40
173
1100
4308
332
538
2119890minus08
8119890minus09
4119890minus09
318794
714
Delaunay
998779Elem
ents
30
117
1107
8108
332
1013
5119890minus09
2119890minus09
2119890minus09
4712946
814
Delaunay
998779Elem
ents
20
84
1112
15936
332
1992
6119890minus09
3119890minus09
2119890minus09
7321347
914
Delaunay
998779Elem
ents
1062
1116
64478
332
8059
6119890minus09
3119890minus09
4119890minus09
262
77105
1014
Delaunay
998779Elem
ents
05
58
1117
256700
332
32087
1119890minus09
5119890minus10
1119890minus09
1091
331969
1114
Delaunay
◻Vo
lumes
40
53
1149
5042
332
630
2119890minus08
1119890minus08
5119890minus09
288007
1214
Delaunay
◻Vo
lumes
30
346
1133
8560
332
1070
7119890minus09
3119890minus09
2119890minus09
3910895
1314
Delaunay
◻Vo
lumes
20
308
1131
15576
332
1947
1119890minus08
7119890minus09
3119890minus09
541564
514
14Delaunay
◻Vo
lumes
10177
1122
60774
332
7596
8119890minus09
4119890minus09
3119890minus09
167
50115
1514
Delaunay
◻Vo
lumes
05
199
1120
244936
332
30617
4119890minus09
2119890minus09
2119890minus09
698
215678
1614
Delaunay
◻Elem
ents
40
196
1100
5222
332
652
3119890minus08
1119890minus08
8119890minus09
4713098
1714
Delaunay
◻Elem
ents
30
123
1107
8810
332
1101
2119890minus08
9119890minus09
7119890minus09
6918598
1814
Delaunay
◻Elem
ents
20
901112
15922
332
1990
1119890minus08
5119890minus09
5119890minus09
107
30591
1914
Delaunay
◻Elem
ents
1063
1116
61456
332
7682
5119890minus09
2119890minus09
4119890minus09
389
113237
2014
Delaunay
◻Elem
ents
05
58
1117
246332
332
30791
8119890minus10
4119890minus10
1119890minus09
1676
498192
2114
Delq
uad
998779Vo
lumes
40
minus45
1144
6228
332
778
4119890minus08
2119890minus08
7119890minus09
308495
2214
Delq
uad
998779Vo
lumes
30
570
1153
11820
332
1477
3119890minus08
1119890minus08
4119890minus09
4010879
2314
Delq
uad
998779Vo
lumes
20
459
1103
24100
332
3012
2119890minus08
1119890minus08
5119890minus09
6217715
2414
Delq
uad
998779Vo
lumes
10512
1104
9640
03
3212050
2119890minus08
1119890minus08
9119890minus09
207
61714
2514
Delq
uad
998779Vo
lumes
05
528
1106
385600
332
48200
2119890minus09
1119890minus09
2119890minus09
838
255735
2614
Delqu
ad998779
Elem
ents
40
220
1093
3288
332
411
3119890minus08
2119890minus08
6119890minus09
308495
2714
Delq
uad
998779Elem
ents
30
140
1104
6148
332
768
4119890minus08
2119890minus08
8119890minus09
39110
0028
14Delq
uad
998779Elem
ents
20
82
1110
12392
332
1549
2119890minus08
9119890minus09
6119890minus09
6117067
2914
Delqu
ad998779
Elem
ents
1063
1115
48882
332
6110
2119890minus09
1119890minus09
1119890minus09
197
5804
630
14Delq
uad
998779Elem
ents
05
58
1116
194162
332
24270
2119890minus09
9119890minus10
2119890minus09
790
238769
3114
Delq
uad
◻Vo
lumes
40
minus416
1174
3132
332
391
5119890minus08
3119890minus08
8119890minus09
267322
3214
Delq
uad
◻Vo
lumes
30
minus248
1151
5922
332
740
9119890minus09
4119890minus09
2119890minus09
318779
3314
Delq
uad
◻Vo
lumes
20
minus132
1139
12050
332
1506
1119890minus08
7119890minus09
4119890minus09
4412239
3414
Delq
uad
◻Vo
lumes
10minus49
1126
48200
332
6025
6119890minus09
3119890minus09
2119890minus09
122
36094
3514
Delq
uad
◻Vo
lumes
05
minus23
1123
192800
332
24100
5119890minus09
2119890minus09
3119890minus09
464
140228
3614
Delq
uad
◻Elem
ents
40
224
1093
3294
332
411
3119890minus08
1119890minus08
7119890minus09
359852
3714
Delq
uad
◻Elem
ents
30
141
1104
6144
332
768
3119890minus08
2119890minus08
9119890minus09
5114018
3814
Delq
uad
◻Elem
ents
20
921111
12392
332
1549
1119890minus08
6119890minus09
5119890minus09
8122526
3914
Delq
uad
◻Elem
ents
1065
1115
48882
332
6110
5119890minus09
3119890minus09
4119890minus09
276
78431
22 Science and Technology of Nuclear Installations
Table2Con
tinued
Case
no
Symm-
etry
Mesh
algorithm
Basic
shape
Solutio
nmetho
dℓ119888
(cm)
Δ120588
(pcm
)max1206012
(mdash)
Total
unkn
owns
Outer
iter
Linear
iter
Inner
iter
Resid
ual
norm
Relativ
eerror
Error
estim
ate
Mem
ory
(Mb)
Page
faults
4014
Delq
uad
◻Elem
ents
05
59
1117
194162
332
24270
1119890minus09
5119890minus10
1119890minus09
1104
318967
4118
Delaunay
998779Vo
lumes
40
minus99
1139
4228
332
528
4119890minus12
2119890minus12
1119890minus12
359630
4218
Delaunay
998779Vo
lumes
30
50
1132
7712
332
964
1119890minus11
6119890minus12
2119890minus12
4211578
4318
Delaunay
998779Vo
lumes
20
177
1120
15776
332
1972
2119890minus12
1119890minus12
7119890minus13
5515436
4418
Delaunay
998779Vo
lumes
10142
1118
63902
332
7987
3119890minus12
1119890minus12
1119890minus12
160
46691
4518
Delaunay
998779Vo
lumes
05
176
1115
254612
332
31826
2119890minus12
8119890minus13
1119890minus12
555
168526
4618
Delaunay
998779Elem
ents
40
183
1096
2252
332
281
6119890minus12
3119890minus12
2119890minus12
359589
4718
Delaunay
998779Elem
ents
30
119
1104
4040
224
505
2119890minus08
1119890minus08
7119890minus09
41114
3748
18Delaunay
998779Elem
ents
20
85
1108
8156
332
1019
2119890minus12
8119890minus13
6119890minus13
5515053
4918
Delaunay
998779Elem
ents
1063
1112
32480
332
4060
1119890minus12
6119890minus13
8119890minus13
151
43671
5018
Delaunay
998779Elem
ents
05
58
1113
128358
332
1604
47119890minus13
3119890minus13
7119890minus13
527
157581
5118
Delaunay
◻Vo
lumes
40
37
1144
2380
224
297
5119890minus08
2119890minus08
1119890minus08
5314173
5218
Delaunay
◻Vo
lumes
30
218
1120
4194
332
524
7119890minus12
4119890minus12
2119890minus12
3610034
5318
Delaunay
◻Vo
lumes
20
345
1110
7960
332
995
5119890minus12
3119890minus12
1119890minus12
4712879
5418
Delaunay
◻Vo
lumes
10165
1117
30652
332
3831
2119890minus12
1119890minus12
7119890minus13
100
28851
5518
Delaunay
◻Vo
lumes
05
191
1117
122066
224
15258
2119890minus08
8119890minus09
9119890minus09
343
103825
5618
Delaunay
◻Elem
ents
40
175
1097
2526
332
315
4119890minus12
2119890minus12
1119890minus12
6016159
5718
Delaunay
◻Elem
ents
30
114
1104
4386
332
548
4119890minus12
2119890minus12
2119890minus12
5013768
5818
Delaunay
◻Elem
ents
20
901108
8234
332
1029
2119890minus12
1119890minus12
1119890minus12
7320094
5918
Delaunay
◻Elem
ents
1062
1112
31186
224
3898
2119890minus08
7119890minus09
9119890minus09
208
59589
6018
Delaunay
◻Elem
ents
05
58
1113
123122
224
15390
9119890minus09
5119890minus09
1119890minus08
807
2366
7061
18Delq
uad
998779Vo
lumes
40
minus210
1168
3224
332
403
1119890minus11
5119890minus12
2119890minus12
43117
8462
18Delq
uad
998779Vo
lumes
30
369
1150
6000
332
750
1119890minus11
6119890minus12
3119890minus12
5013502
6318
Delq
uad
998779Vo
lumes
20
312
1104
12316
332
1539
2119890minus11
8119890minus12
5119890minus12
6016395
6418
Delq
uad
998779Vo
lumes
1044
01094
48714
332
6089
9119890minus12
4119890minus12
4119890minus12
1183444
165
18Delq
uad
998779Vo
lumes
05
534
1081
193980
332
24247
5119890minus12
2119890minus12
4119890minus12
423
126768
6618
Delq
uad
998779Elem
ents
40
240
1090
1750
332
218
9119890minus12
4119890minus12
2119890minus12
43117
8667
18Delqu
ad998779
Elem
ents
30
150
1100
3184
332
398
1119890minus11
5119890minus12
2119890minus12
5013509
6818
Delq
uad
998779Elem
ents
20
89
1106
6426
332
803
5119890minus12
3119890minus12
2119890minus12
5916047
6918
Delq
uad
998779Elem
ents
1064
1111
24886
332
3110
2119890minus12
1119890minus12
1119890minus12
113
32788
7018
Delq
uad
998779Elem
ents
05
59
1113
98052
224
12256
1119890minus08
5119890minus09
8119890minus09
398
118153
7118
Delq
uad
◻Vo
lumes
40
minus398
1172
1650
332
206
1119890minus11
5119890minus12
2119890minus12
369907
7218
Delq
uad
◻Vo
lumes
30
minus249
1146
3058
332
382
4119890minus12
2119890minus12
1119890minus12
349349
7318
Delq
uad
◻Vo
lumes
20
minus131
1136
6228
224
778
3119890minus08
1119890minus08
5119890minus09
4010930
7418
Delq
uad
◻Vo
lumes
10minus51
1123
24536
224
3067
3119890minus08
2119890minus08
1119890minus08
7822716
7618
Delq
uad
◻Elem
ents
40
237
1090
1752
332
219
6119890minus12
3119890minus12
2119890minus12
4111213
7718
Delq
uad
◻Elem
ents
30
145
1101
3184
332
398
6119890minus12
3119890minus12
2119890minus12
43118
1178
18Delq
uad
◻Elem
ents
20
941107
6426
332
803
3119890minus12
1119890minus12
1119890minus12
6016220
7918
Delqu
ad◻
Elem
ents
1065
1111
24874
332
3109
1119890minus12
6119890minus13
8119890minus13
156
4400
080
18Delq
uad
◻Elem
ents
05
59
1113
9804
22
2412255
8119890minus09
4119890minus09
8119890minus09
567
162168
Science and Technology of Nuclear Installations 23
Tim
e con
sum
ed to
bui
ld th
e mat
rices
(sec
onds
)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay trianglesvolumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
10
30
100
300
1
3
Figure 23 Time needed to build the matrices versus number of unknowns
was considerably larger than the solution time itself Theeigenvalue problem was solved using a multilayer iterativeKrylov-Schur method [26] with a tolerance relative to thematrices norm
1003817100381710038171003817119877 sdot 120601 minus (1119896eff) sdot 119865 sdot 1206011003817100381710038171003817
119877 + (1119896eff) sdot 119865lt 10minus8 (11)
The reported residual norm and relative error are10038171003817100381710038171003817100381710038171003817119877 sdot 120601 minus
1
119896effsdot 119865 sdot 120601
10038171003817100381710038171003817100381710038171003817
1003817100381710038171003817119877 sdot 120601 minus (1119896eff) sdot 119865 sdot 1206011003817100381710038171003817
1003817100381710038171003817(1119896eff) sdot 1206011003817100381710038171003817
(12)
respectively The fields marked as outer linear and inneriterations refer to the number of steps needed to attainthe requested tolerance in each layer of the Krylov-Schuralgorithm The computer used to solve the problem has anIntel i7 920 267GHz processor with 4Gb of RAM runningDebian GNULinux Wheezy
When using a finite volumes-based scheme over anunstructured mesh the solver has to gather information
about which cells are neighbors and which are not Currentlygmsh does not write this kind of lists in its output filesso milonga has to explicitly solve the neighbors problemPerforming a linear search is an 119874(119873
2) task which is
unacceptable for problem sizes 119873 of interest The code usesa search based on a 119896-dimensional tree [27] which can inprinciple be performed in119874(119873) steps Still for large values of119873 the time needed to read and parse the mesh (number twopreviously mentioned) is the bottleneck of the solution Thisstep is not needed in finite elements although the construc-tion of the elementary matrices involves the computationof the multidimensional Jacobians and integrals which thenhave to be assembled Again for large 119873 this step (numberthree) takes up most of the time needed to solve the problem
The Delaunay algorithm is a standard method for gen-erating two-dimensional grids [16] by tessellating a planewith triangles If the mesh needs to be based on quadranglesinstead a recombination algorithm can be used to transformtwo adjacent triangles in one quadrangle whenever is possi-ble However for geometries which are based on rectangularshapes there exist other algorithms [25] both for meshingand for recombining the triangles that give rise to elementaryentities with right angles almost everywhere which may be a
24 Science and Technology of Nuclear Installations
Tim
e con
sum
ed to
solv
e the
eige
nval
ue p
robl
em (s
econ
ds)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
10
30
100
300
1
3
Figure 24 Time needed to solve the eigenvalue problem versus number of unknowns
desired property of the resulting grid For finite elements thesteps needed to build the matrices depend on the selectionof triangles or quadrangles as the basic elementary geometrybecause the shape functions change However the number ofunknowns is the same as the number of nodes that does notchange after a recombination procedure On the other handthe number of unknowns in the finite volumes schemes withtriangles is roughly twice as the number of unknowns withquadrangles for the same grid
Figure 19 shows how the computed static reactivity (ie1 minus 1119896eff) depends on the number of unknowns for eachof the sixteen combinations of geometry algorithm shapescheme The four accepted results published in the originalreference [10] are included for reference although it shouldbe taken into account that said reactivities were computedalmost forty years ago Figure 20 shows the total wall timeneeded to solve the problem as a function of 119873119866 whileFigures 21 22 23 and 24 show the times needed for eachof the first four steps involved in the solution maintainingthe same logarithmic scale for both the abscissas and theordinates Green data represent finite volumes whilst bluebullets correspond to finite elements Solid lines indicatequarter-core and dashed lines eighth-symmetry Fillet bullets
are results obtained by theDelaunay triangulation and emptybullets were computed with the delquad algorithm Finallytriangle-shaped data correspond to triangles and squares anddiamonds denote quadrangles as the basic geometry of thegrid
It can be seen that finite elements produce amuch smallerdispersion of eigenvalues 119896eff than finite volumes with therefinement of the mesh This can be explained because finitevolumes methods rely on a geometric condition of the meshwhich may change abruptly if the meshing algorithm decidesto allocate the cells in a rather different form for small changesin ℓ119888 Finite elements methods are less influenced by thesediscontinuous lay-out changes of the elements As expectedeighth-core symmetries give almost the same results as thequarter-core geometries with roughly half the unknownsFor small problems finite volumes run faster that finiteelements because the time needed to solve the neighborproblem is negligible When the problem size grows thistime increases and exceeds the overhead implied in the con-struction and assembly of the finite elements matrices Alsoat least for this configuration it is seen that the eigenvalueproblem is solved faster for finite volumes than for finiteelements
Science and Technology of Nuclear Installations 25
4 Conclusions
Unstructured grids provide the cognizant engineer witha wide variety of possibilities to deal with the design oranalysis of nuclear reactor cores These kinds of grids cansuccessfully reproduce continuous geometries commonlyfound in reactor cores such as cylinders and thereforenot only can the diffusion equation be better approximatedinside the domain of definition but also the fulfillment ofboundary conditions is improved A free computer codewas written from scratch that is able to completely solvethe 2D IAEA PWR Benchmark using unstructured gridsfor sixteen combinations of geometry meshing algorithmbasic shape and discretization scheme plus any value ofthe gridrsquos characteristic length by using a single input fileThe complete set of input files and codemdashexecutable andsourcemdashis available either online or upon request withcomments experiences suggestions and corrections beingmore than welcome Further development should includetackling full three-dimensional geometries with completethermal hydraulic feedback in order to analyze how thesolutions of the coupled neutronic-thermal problem dependon the spatial discretization scheme of the neutron leakageterm Parallelization of the computation and assembly of thematrices and of the solution of the eigenvalue problem andits implementation using GPUs are also desired features toimplement A problemwith direct applications that the futureversions of milonga ought to solve is the analysis of how thegeometry of the absorbing materials should be taken intoaccount in structured coarse grids in order to mitigate effectssuch as the rod-cusp problem
Suitable schemes for approximating the continuous dif-ferential operators by discrete matrix expressions includefinite volumes and finite elements families Finite volumesmethods compute cell mean values whilst finite elementsgive functional values at the gridrsquos nodes In general finiteelements are less sensitive to changes in the mesh so theresults they provide do not change significantly for differentmeshing algorithms or elementary shapes Small problemsare best solved by finite volumes as the neighbor-findingproblem is faster than the process of building and assemblingthe eigenvalue-problem matrices For a large number ofunknowns the process of finding which cell is neighbor ofwhichmdasheven based on a 119896-dimensional treemdashoverwhelmsthe computation and assembly of elementary matrices andfinite-element methods perform better
References
[1] G Glasstone and S Bell Nuclear Reactor Theory KriegerPublishing Company 1970
[2] J J Duderstadt and L J Hamilton Nuclear Reactor AnalysisJohn Wiley amp Sons New York NY USA 1976
[3] C Gho Reactor Physics Undergraduate Course Lecture NotesInstituto Balseiro 2006
[4] G G Theler and F J Bonetto ldquoOn the stability of the pointreactor kinetics equationsrdquoNuclear Engineering andDesign vol240 no 6 pp 1443ndash1449 2010
[5] E E Lewis andW F Miller Computational Methods of NeutronTransport John Wiley amp Sons 1984
[6] E Masiello ldquoAnalytical stability analysis of Coarse-Mesh finitedifference methodrdquo in Proceedings of the International Confer-ence on the Physics of Reactors (PHYSOR rsquo08) Nuclear Power ASustainable Resource pp 456ndash464 September 2008
[7] M L Zerkle Development of a polynomial nodal methodwith flux and current discontinuity factors [PhD thesis] Mas-sachusets Institute of Technology 1992
[8] J I Yoon andHG Joo ldquoTwo-level coarsemesh finite differenceformulation with multigroup source expansion nodal kernelsrdquoJournal of Nuclear Science andTechnology vol 45 no 7 pp 668ndash682 2008
[9] O Mazzantini M Schivo J Di Cesare R Garbero M Riveroand G Theler ldquoA coupled calculation suite for Atucha II oper-ational transients analysisrdquo Science and Technology of NuclearInstallations vol 2011 Article ID 785304 12 pages 2011
[10] Computational Benchmark Problem Comitee for the Mathe-matics and Computation Division of the American NuclearSociety ldquoArgonne Code Center Benchmark problem bookrdquoTech Rep ANL-7416 Supplement 2 Argonne National Labo-ratory 1977
[11] A F Henry Nuclear Reactor Analysis MIT Cambridge UK1975
[12] O C Zienkiewicz R L Taylor and J Z ZhuTheFinite ElementMethod Its Basis and Fundamentals vol 1 Elsevier 6th edition2005
[13] G Theler Difusion de neutrones en mallas no estructuradascomparacion entre volumenes y elementos finitos AcademicMonograph Universidad de Buenos Aires 2013
[14] G Theler ldquoMilonga a free nuclear reactor core analysis coderdquoAvailable online 2011
[15] G Theler F J Bonetto and A Clausse ldquoOptimizacion deparametros en reactores de potencia base de diseno delcodigo neutronico milongardquo in Reunion Anual de la AsociacionArgentina de Tecnologia Nuclear XXXVII 2010
[16] C Geuzaine and J-F Remacle ldquoGmsh a 3-D finite elementmesh generator with built-in pre- and post-processing facili-tiesrdquo International Journal for Numerical Methods in Engineer-ing vol 79 no 11 pp 1309ndash1331 2009
[17] S Balay J Brown K Buschelman et al ldquoPETSc users manualrdquoTech Rep ANL-9511-Revision 34 Argonne National Labora-tory 2013
[18] S Balay W D Gropp L Curfman McInnes and B F SmithldquoEfficient management of parallelism in object oriented numer-ical software librariesrdquo in Modern Software Tools in ScientificComputing E Arge A M Bruaset and H P Langtangen Edspp 163ndash202 Birkhauser 1997
[19] V Hernandez J E Roman and V Vidal ldquoSLEPC a scalable andflexible toolkit for the solution of eigenvalue problemsrdquo ACMTransactions on Mathematical Software vol 31 no 3 pp 351ndash362 2005
[20] M Galassi J Davies J Theiler et al GNU Scientific LibraryReference Manual 3rd edition 2009
[21] O C Zienkiewicz and R L Taylor The Finite Element MethodFor Solid and Structural Mechanics vol 3 Elsevier 6th edition2005
[22] A Imelda and K Doddy Evaluation of the IAEA 3-D PWRbenchmark problem using NESTLE code 2004
[23] R Mosteller ldquoStatic benchmarking of the NESTLE advancednodal coderdquo in Proceedings of the Joint International Conference
26 Science and Technology of Nuclear Installations
on Mathematical Methods and Supercomputing for NuclearApplications vol 2 pp 1596ndash1605 1997
[24] GTheler F J Bonetto andAClausse ldquoSolution of the 2D IAEAPWR Benchmark with the neutronic code milongardquo in Actasde la Reunion Anual de la Asociacion Argentina de TecnologıaNuclear XXXVIII 2011
[25] J F Remacle F Henrotte T Carrier-Baudouin et al ldquoAfrontal delaunay quad mesh generator using the 119871
infin normrdquoInternational Journal For Numerical Methods in Engineeringvol 94 no 5 pp 494ndash512 2013
[26] G W Stewart ldquoA Krylov-Schur algorithm for large eigenprob-lemsrdquo SIAM Journal on Matrix Analysis and Applications vol23 no 3 pp 601ndash614 2002
[27] J L Bentley ldquoMultidimensional binary search trees used forassociative searchingrdquo Communications of the ACM vol 18 no9 pp 509ndash517 1975
TribologyAdvances in
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Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Renewable Energy
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
EnergyJournal of
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Journal ofEngineeringVolume 2014
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Nuclear InstallationsScience and Technology of
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Nuclear EnergyInternational Journal of
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High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Science and Technology of Nuclear Installations 7
20 40 600995
09975
1
Analytical solutionFinite volumesFinite elements
keff
a985747c
Figure 7 The effective multiplication factor of a two-group barecircular reactor of radius 119886 as a function of the quotient 119886ℓ
119888between
the radius and the characteristic length of the cellelement
20 40 60
Finite volumesFinite elements
Erro
r ink
eff
10minus3
10minus4
10minus5
a985747c
Figure 8 Absolute value of the error committed in the computationof 119896eff versus 119886ℓ119888
Methods of the finite volumes family involve the inte-gration of the differential equation over each elementaryentity applying the divergence theorem to transform volumeintegrals into surface integrals and providing a mean to esti-mate the fluxes through the entityrsquos surface using informationcontained in its neighbors In this context each elementaryentity is called a cell and finite volumes methods give themean value of each of the group fluxes 120601119892(119894) at the 119894th cellwhich may be associated to the value of 120601119892(x119894) where x119894 isthe location of the cell barycenter Being an integral-basedmethod spatial-discontinuous parameters are handled moreefficiently than in finite differences and boundary conditionscan be easily incorporated as forced cell fluxes Nonethelesseven though these methods may be applied to unstructuredgrids the estimation of the fluxes on the cellsrsquo surfaces isusually performed by using some geometric approximationsthat may lose validity as the quality of the grid is worsened
10
70
90
130
150
170
10 30 50 70 90 110 130 150 170
4
1 38
36 37353
3 32 33 34312
26 27 28 29 3025
19 20 21 22 23 2418
10 11 12 13 14 15 16 17
2 331 4 5 6 7 8 9
x (cm)
y (cm
)
Figure 9 The 2D IAEA PWR benchmark geometry
Moreover the simple integral approach cannot take intoaccount discontinuous diffusion coefficients so when twoneighbors pertain to different materials the flux has to becomputed in a certain particular way to conserve the neutroncurrent according to (2)
Finally finite elements methods rely on a weak formula-tion of the differential problem similar to (3) that maintainsall the mathematical characteristics of the original strongformulation plus its boundary conditions The method isbased on shape functions that are local to each elementaryentitymdashnow called element defined by nodes as cornersmdashand on finding a set of nodal values such that a certaincondition is met which is usually that the residual of thesolution has to be orthogonal to each of the shape functionsThis condition is known as the Galerkin method and impliesthat the error committed in the approximate solution ofthe continuous problem is confined into a small subset ofthe original vector space of the continuous functions Thesemathematical properties make finite elements schemes veryattractive However these features depend on a large numberof integrations that ought to be performed numerically soa computational effortdesired accuracy tradeoff has to beconsidered Not only does the finite elements method givethe values of the flux 120601119892(x119894) at the 119894th node but also theshape functions provide explicit expressions to interpolateand to evaluate the unknown functions at any location x ofthe domain119880 Boundary conditions are divided into essentialand naturalThefirst group comprises theDirichlet boundaryconditions which are satisfied exactlymdashwithin the precisionof the eigenvalue problem solvermdashby the obtained solutionThe latter include the Neumann and the Robin conditionsthat are satisfied only approximately by the derivatives of theinterpolated solution through the shape functions with finermeshes giving better agreement with the prescribed values
The same unstructured grid may be used either for thefinite volumes or for the finite elements method In the first
8 Science and Technology of Nuclear Installations
Largest eigenvalue 1029690 (288336 pcm)1135 (3066 3079)1306 (5171 13076)
Number of unknowns 15740Outer iterations 3Linear iterations 32Inner iterations 1967Residual normRelative errorError estimateMemory used 49824 kBSoft page faults 13401Hard page faults 0Total CPU time 074 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
2483 times 10minus8
1223 times 10minus8
403 times 10minus9
keff
quarter-symmetry core meshed using Delaunay (triangles 985747c = 3) solved with finite volumesMilongarsquos 2D LWR IAEA benchmark problem case no 002
(a)
123456789
1011121314151617181920212223242526272829303132333435363738
k Pk 1206011k 1206012k075 3282 555133 4242 984147 4645 1090123 3921 911061 2677 453094 2995 694093 2927 689074 2011 549mdash 322 756146 4597 1078150 4730 1110133 4199 985107 3405 789104 3271 767095 2971 700072 1956 534mdash 306 733149 4695 1102136 4289 1007118 3733 876107 3367 792097 2892 718066 1628 491mdash 234 549120 3797 891097 3096 718090 2842 669084 2221 619mdash 568 1215mdash 067 257047 2042 348068 2055 504058 1414 430mdash 222 562057 1374 425mdash 388 820mdash 053 203mdash 060 229
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 10 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
case the unknowns are themean value of the fluxes over eachcell whilst in the latter the unknowns are the fluxes evaluatedat each node Therefore the number of unknowns 119873119866 isdifferent for each method even when using the same gridFigure 4 shows this situation and in Section 31 we further
illustrate these differences A fully detailed mathematicaldescription of the actual algorithms for both volumes andelements-based proposed discretizations can be found in anacademic monograph written by the author of this paper[13]
Science and Technology of Nuclear Installations 9
Largest eigenvalue 1029927 (290571 pcm)1131 (3175 2999)1220 (13097 5098)
Number of unknowns 15576Outer iterations 3Linear iterations 32Inner iterations 1947Residual normRelative errorError estimateMemory used 56256 kBSoft page faults 15645Hard page faults 0Total CPU time 09201 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
1394 times 10minus8
6865 times 10minus9
3425 times 10minus9
keff
quarter-symmetry core meshed using Delaunay (quads 985747c = 2) solved with finite volumesMilongarsquos 2D LWR IAEA benchmark problem case no 013
(a)
123456789
1011121314151617181920212223242526272829303132333435363738
k Pk 1206011k 1206012k074 3244 549133 4230 982147 4651 1092124 3946 917060 2645 443094 2997 696093 2938 691074 2005 545mdash 301 746145 4588 1076150 4720 1108133 4198 984109 3465 805104 3280 769095 2986 704071 1962 528mdash 293 715148 4680 1098136 4288 1007119 3755 881107 3385 796097 2905 721067 1653 493mdash 220 556121 3808 893098 3115 724092 2889 680083 2247 615mdash 543 1241mdash 067 260045 1977 330069 2076 509058 1460 428mdash 226 574056 1364 413mdash 371 821mdash 052 200mdash 060 238
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 11 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
3 Results
We now proceed to show two illustrative results that are tobe taken as an overview of the possibilities that unstructuredgrids can provide in order to tackle the multigroup neutron
diffusion problem The examples are two-dimensional prob-lems as they contain some of the complexities a real three-dimensional reactor posse yet the reported results are notso complicated as to be easily understood and analyzed Inparticular we state some basic differences between the finite
10 Science and Technology of Nuclear Installations
Largest eigenvalue 1029695(288388 pcm)1112 (3076 3032)838 (5000 13000)
Number of unknowns 15922Outer iterations 3Linear iterations 32Inner iterations 1990Residual normRelative errorError estimateMemory used 109964 kBSoft page faults 30591Hard page faults 0Total CPU time 158 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
1056 times 10minus8
5202 times 10minus9
5122 times 10minus9
keff
quarter-symmetry core meshed using Delaunay (quads 985747c = 2) solved with finite volumesMilongarsquos 2D LWR IAEA benchmark problem case no 018
(a)
123456789
1011121314151617181920212223242526272829303132333435363738
k Pk 1206011k 1206012k074 3220 549130 4152 961144 4651 1068120 3843 890061 2648 452094 2996 693094 2959 696072 2061 569mdash 358 862142 4495 1054147 4634 1088131 4127 968107 3409 789104 3275 768096 3003 708071 2014 554mdash 342 820146 4606 1081134 4222 991118 3711 871107 3375 794098 2929 727062 1692 522mdash 258 632119 3752 880096 3084 715091 2858 673080 2280 633mdash 611 1277mdash 080 318047 2043 350069 2088 510054 1463 450mdash 255 648051 1418 439mdash 410 854mdash 064 252mdash 071 285
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 12 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
volumes and the finite elements methods by solving a two-group homogeneous bare reactor with the Robin boundaryconditions over the very same grid although a rather coarseone so the differences can be observed directly into theresulting figuresWe then solve the classical two-dimensionalLWR problem also known as the 2D IAEA PWR benchmark
Not only do we show again the differences between the finitevolumes and elements formulation but also we solve theproblemusing different combinations ofmeshing algorithmsbasic shapes and characteristic lengths of the mesh
To solve the two examples shown below we employed themilonga code which was written from scratch by the author
Science and Technology of Nuclear Installations 11
Largest eigenvalue 1029159 (283326 pcm)1174 (2851 3224)1560 (5200 13200)
Number of unknowns 3132Outer iterations 3Linear iterations 32Inner iterations 391Residual normRelative errorError estimateMemory used 26888 kBSoft page faults 7322Hard page faults 0Total CPU time 0308 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
5408 times 10minus8
2665 times 10minus8
8424 times 10minus9
keff
Milongarsquos 2D LWR IAEA benchmark problem case no 031quarter-symmetry core meshed using Delquad (quads 985747c = 4) solved with finite volumes
(a)
123456789
1011121314151617181920212223242526272829303132333435363738
k Pk 1206011k 1206012k077 3422 570140 4468 1040153 4836 1135128 4070 948060 2704 448095 3017 702091 2861 673069 1918 508mdash 264 689151 4785 1122155 4897 1149137 4334 1017111 3535 823104 3272 768092 2908 685067 1876 496mdash 254 658154 4854 1139139 4392 1031120 3792 890107 3360 790095 2831 702061 1574 451mdash 197 515123 3868 907099 3132 730090 2818 663078 2168 579mdash 481 1191mdash 055 217044 1984 328067 2027 498052 1360 388mdash 190 520052 1329 383mdash 321 792mdash 043 168mdash 048 195
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 13 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
of this paper and is currently still under development withinhis ongoing PhD thesis There exists a first public release[14] under the terms of the GNU General Public Licensemdashthat is it is a free softwaremdashthat can only handle structuredgrids There is a second release being prepared whose
main relevance is that it can work with nonstructured gridsas well which is the main feature of the code It uses ageneral mathematical frameworkmdashcoded from scratch aswellmdashwhich provides input file parsing algebraic expres-sions evaluation one- and multidimensional interpolation of
12 Science and Technology of Nuclear Installations
Largest eigenvalue 1029788 (289260 pcm)1120 (3149 3011)1091 (13038 5114)
Number of unknowns 15776Outer iterations 3Linear iterations 32Inner iterations 1972Residual normRelative errorError estimateMemory used 56996 kBSoft page faults 15436Hard page faults 0Total CPU time 09121 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
2296 times 10minus12
1131 times 10minus12
6877 times 10minus13
keff
Milongarsquos 2D LWR IAEA benchmark problem case no 043eighth-symmetry core meshed using Delaunaya (triangs985747c = 2) solved with finite volumes
(a)
1 074 3241 5482 131 4188 9713 145 4582 10764 122 3877 9005 060 2644 4456 093 2986 6927 093 2941 6928 075 2030 5529 mdash 322 779
10 143 4523 106111 148 4667 109512 131 4145 97213 107 3423 79414 103 3269 76715 095 2993 70616 073 1988 53917 mdash 312 75118 147 4637 108819 134 4243 99620 118 3721 87321 107 3370 79322 098 2914 72323 067 1651 49924 mdash 235 57125 119 3763 88226 096 3077 71527 091 2847 67128 083 2255 61729 mdash 576 122530 mdash 068 26631 046 2016 34132 068 2058 50533 059 1443 43534 mdash 233 58935 057 1381 42036 mdash 381 82337 mdash 054 20838 mdash 062 244
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 14 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
scattered data shared-memory access numerical integrationfacilities and so forth The milonga code was designed withfour design basis vectors in mindmdashwhich are thoroughlydiscussed in the documentation [15]mdashwhich includes thetype of problems it can handle the code scalability which
features are expected and what to do with the obtainedresults
It works by first reading an input file that using plain-text English keywords and arguments defines the number119898 of spatial dimensions the number 119866 of group energies
Science and Technology of Nuclear Installations 13
Largest eigenvalue 1029726 (288675 pcm)1104 (3069 3069)864 (13000 5000)
Number of unknowns 4040Outer iterations 2Linear iterations 24Inner iterations 505Residual normRelative errorError estimateMemory used 42940 kBSoft page faults 11437Hard page faults 0Total CPU time 1064 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
2019 times 10minus8
9946 times 10minus9
7138 times 10minus9
keff
Milongarsquos 2D LWR IAEA benchmark problem case no 047eighth-symmetry core meshed using Delaunay (triangles 985747c = 3) solved with finite volumes
(a)
1 074 3194 5452 129 4117 9533 143 4515 10604 119 3814 8835 061 2634 4506 093 2984 6907 094 2954 6958 071 2064 5719 mdash 362 865
10 141 4458 104511 146 4599 107912 130 4097 96113 106 3388 78414 103 3263 76515 095 2998 70716 068 2015 55717 mdash 346 82318 145 4572 107319 133 4194 98420 117 3691 86621 107 3363 79222 098 2925 72623 059 1696 52824 mdash 263 63525 118 3728 87426 096 3068 71127 091 2850 67228 079 2280 63529 mdash 619 127630 mdash 081 32131 047 2036 34932 068 2085 50933 051 1465 45434 mdash 258 65035 049 1419 44336 mdash 416 85437 mdash 065 25438 mdash 071 288
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 15 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
and optionally a mesh file Currently only grids generatedwith the code gmsh [16] are only supported mainly becauseit is also a free software and it suits perfectly well milongarsquosdesign basis in the sense that the continuous geometry can bedefined as a function of a number of parameters Afterwardthe physical entities defined in the grid are mapped into
materials with macroscopic cross sections which maydepend on the spatial coordinates x bymeans of intermediatefunctions such as burn-up or temperatures distributionswhich in turn may be defined by algebraic expressions byinterpolating data located in files by reading shared-memoryobjects or by a combination of them In the same sense
14 Science and Technology of Nuclear Installations
Largest eigenvalue 1029695 (288387 pcm)1108 (3045 3045)833 (13000 5000)
Number of unknowns 8234Outer iterations 3Linear iterations 32Inner iterations 1029Residual normRelative error
Max 1206012(x y) coreMax 1206012(x y) reflector
2028 times 10minus12
9993 times 10minus13
1012 times 10minus12
keff
Relative errorError estimateMemory used 75468 kBSoft page faults 20094Hard page faults 0Total CPU time 1256 seconds
9993 times 10
1012 times 10minus12
Milongarsquos 2D LWR IAEA benchmark problem case no 058eighth-symmetry core meshed using Delaunay (quads 985747c = 2) solved with finite volumes
(a)
1 074 3207 5472 129 4136 9573 144 4534 10644 120 3828 8875 061 2638 4506 093 2985 6907 094 2948 6948 072 2053 5669 mdash 357 859
10 142 4478 105011 146 4617 108412 130 4111 96413 106 3396 78614 103 3263 76515 095 2991 70616 071 2005 55217 mdash 340 81618 145 4588 107719 133 4206 98720 117 3698 86821 107 3362 79122 098 2918 72523 062 1686 52024 mdash 258 62925 118 3737 87626 096 3073 71227 091 2848 67128 080 2272 63129 mdash 609 127230 mdash 080 31731 047 2035 34832 069 2081 50933 054 1458 44834 mdash 255 64635 052 1413 43736 mdash 409 85137 mdash 064 25138 mdash 071 284
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 16 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
boundary conditions are applied to appropriate physicalentities of dimension119898minus1The problemmatrices119877 and119865 arethen built and stored in an appropriate sparse format usingthe free PETSc [17 18] library and the eigenvalue problemis solved using the free SLEPc [19] library The results arestored into milongarsquos variables and functions which may be
evaluated integratedmdashusually using the free GNU ScientificLibrary [20] routinesmdashand of course written into appropriateoutputs Milonga can also solve problems parametrically andbe used to solve optimization problems As stated above thecode is a free software so corrections and contributions aremore than welcome
Science and Technology of Nuclear Installations 15
Largest eigenvalue 1029462 (286185 pcm)1136 (3093 2953)1233 (13100 5100)
Number of unknowns 6228Outer iterations 2Linear iterations 24Inner iterations 778Residual normR l ti
Max 1206012(x y) coreMax 1206012(x y) reflector
26 times 10minus8
1281 times 10minus8
4996 times 10minus9
keff
Relative errorError estimateMemory used 41460 kBSoft page faults 10930Hard page faults 0Total CPU time 0604 seconds
1281 times 10 8
4996 times 10minus9
Milongarsquos 2D LWR IAEA benchmark problem case no 073eighth-symmetry core meshed using Delaunay (quads 985747c = 2) solved with finite volumes
(a)
1 075 3297 5552 134 4268 9923 148 4660 10944 123 3926 9145 059 2645 4406 094 2997 6977 093 2914 6858 072 2002 5349 mdash 301 773
10 146 4606 108011 150 4735 111112 133 4212 98813 108 3448 80214 104 3271 76715 094 2954 69616 070 1954 52117 mdash 288 73518 149 4691 110119 136 4289 100720 119 3748 88021 107 3359 79022 096 2880 71423 064 1636 47524 mdash 217 56325 121 3810 89426 098 3110 72427 090 2829 66628 081 2227 59929 mdash 536 125730 mdash 062 25031 045 2002 33632 068 2040 50033 055 1414 40934 mdash 215 57935 055 1359 40836 mdash 359 83437 mdash 050 19938 mdash 061 236
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 17 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
31 A Coarse Bare Homogeneous Circle Figures 5 and 6 showthe results of solving a bare two-dimensional circular homo-geneous reactor of radius 119886 over an unstructured grid whichis deliberately coarse using the finite volumes method andthe finite elements method respectively Two group energies
were used and a null-flux boundary conditionwas fixed at theexternal surface Figures 5(a) and 6(a) compare the obtainedfast flux distributions In the first case the numerical solutionprovidesmean values for each neutron flux group in each cellwhile in the latter the solution is computed at the nodes and
16 Science and Technology of Nuclear Installations
Largest eigenvalue 1029851 (289858 pcm)1090 (3182 3182)851 (13000 5000)
Number of unknowns 1752Outer iterations 3Linear iterations 32Inner iterations 219Residual norm
Max 1206012(x y) coreMax 1206012(x y) reflector
566 times 10minus12
2789 times 10minus12
1595 times 10minus12
keff
Residual normRelative errorError estimateMemory used 42564 kBSoft page faults 11213Hard page faults 0Total CPU time 09921 seconds
566 times 10
2789 times 10minus12
1595 times 10minus12
Milongarsquos 2D LWR IAEA benchmark problem case no 076eighth-symmetry core meshed using Delaunay (quads 985747c = 4) solved with finite volumes
(a)
1 073 3151 5402 127 4059 9393 141 4457 10464 118 3768 8715 061 2615 4506 093 2979 6887 094 2971 6998 067 2092 5829 mdash 375 885
10 139 4397 103111 144 4541 106612 128 4052 95013 105 3365 77814 103 3259 76415 096 3016 71216 066 2045 56817 mdash 359 84118 143 4520 106119 132 4155 97520 116 3672 86221 107 3366 79222 098 2951 73323 054 1728 54324 mdash 273 65125 117 3700 86826 095 3057 70827 091 2860 67428 074 2311 64829 mdash 642 130030 mdash 084 33231 048 2041 35332 069 2103 51333 046 1493 46734 mdash 269 66735 046 1444 45536 mdash 432 87137 mdash 067 26338 mdash 074 297
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 18 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
continuous functions are evaluated by means of the shapefunctions used in the formulation Figures 5(b) and 6(b)illustrate the fast fluxes unknowns and its relative position inspace It can be noted that the mesh coarseness gives resultsthat differ substantially in both cases Finally the structure ofthe sparse eigenvalue problemmatrices is shown for each case
with blue red and cyan representing positive negative andexplicitly inserted zero values In the finite volume case thereare 184 unknowns (92 cells times 2 groups) while in the secondcase there are 218 unknowns (109 nodes times 2 groups) Thevolumesrsquo fissionmatrix is almost diagonal it has a bandwidthequal to the number of energy groups which in this case is
Science and Technology of Nuclear Installations 17
500 1000 2000 5000 10000 20000 50000Number of unknowns
2840
2860
2880
2900
2920
ORNLRisoslashKWUOntarioQuarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumes
Quarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elementsEighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumes
Eighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
120588(p
cm)
105 2 times 105 5 times 105
Figure 19 Static reactivity versus number of unknowns The four original solutions as published in 1977 [10] are included as reference
twoThe off-diagonal values appear right next to the diagonalelements because of the chosen ordering of the unknownsin the flux vector 120601 isin R184 Other orderings may be usedbut the rate of convergence of the eigenvalue problem can bedeteriorated On the other hand the elementsrsquo fission matrixhas a nontrivial structure because the grid is unstructuredand the net fission rate at each element depends on the fluxesat the nodes whose location inside thematrix depends in turnon how the gridwas generatedThis effect is similar to the onethat appears in structural analysis where mass matrices needto be lumped in order to simplify transient computations [21]The diagonal block of elementrsquos 119877 matrix and the null blockin 119865 correspond to the discrete equations that set the null-flux boundary conditions on the nodes located at the externalsurface Other types of boundary conditions lead to differentkinds of structures within the problem matrices
In case a part of the domain contained a nonmultiplicativematerial such as a reflector then there would appear sectionsof the fission matrix with null values in both methodsrendering 119865 singular in both methods Care should be takenwhen dealing with the numerical schemes for the eigenvalueproblem solution It can be seen in the elementsrsquo matricesa particular structure that implements the boundary condi-tions at the external surfaces This structure does not appearin the volumesrsquo matrices because the boundary conditions
appear as flux terms which are summed up over all thesurfaces of each cell so they aremasked inside the volumetricdiscretization of the divergence and gradient operators
As the two-group neutron diffusion equation with uni-form cross sections over a circle subject to null-flux bound-ary conditions has an analytical solution it is adequate tocompare how the two proposed numerical schemes relate toit In the studied problem we ignored upscattering and fastfissions Then the analytical effective multiplication factor is
119896eff =]Σ1198912 sdot Σ1199041rarr2
[Σ1198861 + Σ1199041rarr2 + 1198631(]0119886)2] [Σ1198862 + 1198632(]0119886)
2]
(7)
being ]0 = 24048 the smallest root of Besselrsquos first-kindfunction of order zero 1198690(119903)
Figure 7 shows how the two numerical effective multi-plication factors compare to the analytical solution givenby (7) as a function of the mesh refinement indicated bythe quotient 119886ℓ119888 between the radius of the circle and thecharacteristic length of the cellelements We can see thatthe 119896eff computed by the finite volumes (elements) methodis always greater (less) than the analytical solution Indeedit can be proven that for a bare one-dimensional slab thisis always the case [13] However this result does not hold
18 Science and Technology of Nuclear Installations
2000 3000 6000 10000 20000 30000 60000
01
003
001
03
1
3
10
30
100
300To
tal w
all t
ime (
seco
nds)
Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
Figure 20 Total wall time versus number of unknowns
for problems with nonuniform cross sections even in simpleone-dimensional reflected reactors
We may draw two other conclusions from Figure 7 Firstthat the finite element method seems to provide a bettersolution than the volumes-based scheme and second thateven though the error committed tends to decrease with finergrids its behavior is not monotonic for finite volumes Infact Figure 8 shows the difference between the numericaland analytical solutions using a logarithmic vertical scalewhere both conclusions are even more evident We deferthe discussion of such differences between the discretizationscheme until the next section It is worth to note howeverthat the fact that a finite-element-based scheme throws betterresults for the particular bare homogeneous circular reactorunder study than the finite volumes does not imply thatfinite volumes ought to be discarded as a valid tool forsolving the neutron diffusion equation in general cases Thecombination of lattice and core-level computations is usuallyperformedusing cell-based resultswhichwhen fed into node-based methods to solve the few-group neutron diffusionequation may introduce errors which potentially can leadto unacceptable solutions Nevertheless this analysis is farbeyond the scope of this paper which focuses on solving amathematical equation over unstructured grids
32 The 2D IAEA PWR Benchmark Problem This is aclassical two-group neutron diffusion problem first designedin the early 1970s and taken as a reference benchmark forcomputational codes A number of codes were used to solveeither this problem or its three-dimensional formulation[22 23] including milonga using a structured grid [24] Theoriginal formulation can be found in the reference [10] andthere is a reproduction that may be easily found online inreference [24] The geometry consists of a one-quarter ofa PWR core depicted in Figure 9 and the homogeneousmacroscopic cross sections are listed in Table 1 An axialbuckling term 119861
2
119911119892= 08times10
minus4 should be taken into accountThe external surface should be subject to a zero incomingcurrent condition which may be written as
120597120601119892
120597119899= minus
04692
119863119892
sdot 120601119892 (8)
The expected results are as follows
(1) Maximum eigenvalue(2) Fundamental flux distributions
(a) Radial flux traverses 120601119892(119909 0) and 120601119892(119909 119909)
Science and Technology of Nuclear Installations 19
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
1
3
10
30
100
300Ti
me c
onsu
med
to m
esh
the g
eom
etry
(sec
onds
)
Figure 21 Time needed to mesh the geometry versus number of unknowns
Table 1 Macroscopic cross-sections (units are not stated in the original reference but they are assumed to be in cmminus1 or cm as appropriate)
1198631
1198632
Σ1rarr2
Σ1198861
Σ1198862
]Σ1198912
Material1 15 04 002 001 0080 0135 Fuel 12 15 04 002 001 0085 0135 Fuel 23 15 04 002 001 0130 0135 Fuel 2 + rod4 20 03 004 0 0010 0 Reflector
Note the fluxes shall be normalized such that
1
119881coreint119881core
sum
119892
]Σ119891119892 sdot 120601119892119889119881 = 1 (9)
(b) Value and location of maximum power densityThis corresponds to maximum of 1206012 in thecore It is recommended that the maximumvalues of 1206012 both in the inner core and at thecorereflector interface be given
(3) Average subassembly powers 119875119896
119875119896 =1
119881119896
int119881119896
sum
119892
]Σ119891119892 sdot 120601119892119889119881 (10)
where 119881119896 is the volume of the 119896th subassembly and119896 designates the fuel subassemblies as shown inFigure 9
(4) Number of unknowns in the problem number ofiterations and total and outer
(5) Total computing time iteration time IO-time andcomputer used
(6) Type and numerical values of convergence criteria(7) Table of average group fluxes for a square mesh grid
of 20 times 20 cm(8) Dependence of results on mesh spacing
Even though the original problem is based on a quarter-core situation the problem has an eighth-core symmetry
20 Science and Technology of Nuclear Installations
Quarter Delaunay triangles elements
Tim
e con
sum
ed to
read
the m
esh
(sec
onds
)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumes
Quarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
1
3
10
30
100
300
Figure 22 Time needed to read the mesh versus number of unknowns
which cannot be taken into account by structured gridswhich are the main target of the benchmark Howevernonstructured grids can take into consideration any kind ofsymmetry almost without loss of accuracy and at the sametime reducing roughly the number of unknowns by a halfand the associated computational effort needed to solve theproblem by a factor of four Answers to items (1)ndash(7) canbe given completely by milonga using a single input file(see Supplementary data in SupplementaryMaterial availableonline at httpdxdoiorg1011552013641863) As asked initem (8) how results depend not only on the mesh spacingbut also on the meshing algorithm on the gridrsquos elementarygeometric shape and on the discretization scheme may shedlights on the subject whichmay be evenmore interesting thanthe numerical results themselves
Taking advantage of milongarsquos capability of reading andparsing command-line arguments the selection of the coregeometry (quarter or eighth) the meshing algorithm (delau-nay [16] or delquad [25]) the shape of the elementaryentities (triangles or quadrangles) the discretization scheme(volumes or elements) and the characteristic length ℓ119888 ofthe mesh can be provided at run time Fixing five values forℓ119888 = 4 3 2 1 05 gives 2 times 2 times 2 times 2 times 5 = 80 possiblecombinations which we solve with successive invocations to
milonga with the same input file but with different argumentsfrom a simple script Figures 10 11 12 13 14 15 16 17 and18 show the results corresponding to items (1)ndash(7) for someillustrative cases The complete set of figures and the codeused to generate themmay be provided upon request Table 2compiles the answers to the problem for every case studied
As the milonga code is still under development itsnumerical routines are not yet fully optimized nor designedfor parallel computation Therefore the reported times areonly rough estimates and should be taken with care Thesolution comprises five steps
(1) generate the grid with the requested geometry mesh-ing algorithm basic shape and characteristic lengthby calling to gmsh
(2) read the generated mesh(3) build the matrices(4) solve the eigenvalue problem(5) compute the requested results
The CPU time reported in Figures 10ndash18 is thus thesum of all of these five steps but not the time needed togenerate the figuresmdashwhich in some caseswith coarsemeshes
Science and Technology of Nuclear Installations 21
Table2
Resultsofthe2
DIAEA
PWRBe
nchm
arkp
roblem
obtained
with
them
ilong
acod
efor
thee
ightyp
ropo
sedcombinatio
nsofsymmetrym
eshing
algorithm
basicshapediscretization
schemeandcharacteris
ticelem
entc
elllengthTh
ereference
solutio
nis120588=28749times10minus5w
hich
correspo
ndsto119896eff=10296
Case
no
Symm-
etry
Mesh
algorithm
Basic
shape
Solutio
nmetho
dℓ119888
(cm)
Δ120588
(pcm
)max1206012
(mdash)
Total
unkn
owns
Outer
iter
Linear
iter
Inner
iter
Resid
ual
norm
Relativ
eerror
Error
estim
ate
Mem
ory
(Mb)
Page
faults
114
Delaunay
998779Vo
lumes
40
minus120
1145
8264
332
1033
6119890minus09
3119890minus09
1119890minus09
349470
214
Delaunay
998779Vo
lumes
30
85
1135
15740
332
1967
2119890minus08
1119890minus08
4119890minus09
4813401
314
Delaunay
998779Vo
lumes
20
197
1124
31188
332
3898
1119890minus08
6119890minus09
3119890minus09
7621932
414
Delaunay
998779Vo
lumes
10153
1123
127476
332
15934
7119890minus09
3119890minus09
3119890minus09
273
81886
514
Delaunay
998779Vo
lumes
05
187
1119
510676
332
63834
2119890minus09
9119890minus10
1119890minus09
1148
352261
614
Delaunay
998779Elem
ents
40
173
1100
4308
332
538
2119890minus08
8119890minus09
4119890minus09
318794
714
Delaunay
998779Elem
ents
30
117
1107
8108
332
1013
5119890minus09
2119890minus09
2119890minus09
4712946
814
Delaunay
998779Elem
ents
20
84
1112
15936
332
1992
6119890minus09
3119890minus09
2119890minus09
7321347
914
Delaunay
998779Elem
ents
1062
1116
64478
332
8059
6119890minus09
3119890minus09
4119890minus09
262
77105
1014
Delaunay
998779Elem
ents
05
58
1117
256700
332
32087
1119890minus09
5119890minus10
1119890minus09
1091
331969
1114
Delaunay
◻Vo
lumes
40
53
1149
5042
332
630
2119890minus08
1119890minus08
5119890minus09
288007
1214
Delaunay
◻Vo
lumes
30
346
1133
8560
332
1070
7119890minus09
3119890minus09
2119890minus09
3910895
1314
Delaunay
◻Vo
lumes
20
308
1131
15576
332
1947
1119890minus08
7119890minus09
3119890minus09
541564
514
14Delaunay
◻Vo
lumes
10177
1122
60774
332
7596
8119890minus09
4119890minus09
3119890minus09
167
50115
1514
Delaunay
◻Vo
lumes
05
199
1120
244936
332
30617
4119890minus09
2119890minus09
2119890minus09
698
215678
1614
Delaunay
◻Elem
ents
40
196
1100
5222
332
652
3119890minus08
1119890minus08
8119890minus09
4713098
1714
Delaunay
◻Elem
ents
30
123
1107
8810
332
1101
2119890minus08
9119890minus09
7119890minus09
6918598
1814
Delaunay
◻Elem
ents
20
901112
15922
332
1990
1119890minus08
5119890minus09
5119890minus09
107
30591
1914
Delaunay
◻Elem
ents
1063
1116
61456
332
7682
5119890minus09
2119890minus09
4119890minus09
389
113237
2014
Delaunay
◻Elem
ents
05
58
1117
246332
332
30791
8119890minus10
4119890minus10
1119890minus09
1676
498192
2114
Delq
uad
998779Vo
lumes
40
minus45
1144
6228
332
778
4119890minus08
2119890minus08
7119890minus09
308495
2214
Delq
uad
998779Vo
lumes
30
570
1153
11820
332
1477
3119890minus08
1119890minus08
4119890minus09
4010879
2314
Delq
uad
998779Vo
lumes
20
459
1103
24100
332
3012
2119890minus08
1119890minus08
5119890minus09
6217715
2414
Delq
uad
998779Vo
lumes
10512
1104
9640
03
3212050
2119890minus08
1119890minus08
9119890minus09
207
61714
2514
Delq
uad
998779Vo
lumes
05
528
1106
385600
332
48200
2119890minus09
1119890minus09
2119890minus09
838
255735
2614
Delqu
ad998779
Elem
ents
40
220
1093
3288
332
411
3119890minus08
2119890minus08
6119890minus09
308495
2714
Delq
uad
998779Elem
ents
30
140
1104
6148
332
768
4119890minus08
2119890minus08
8119890minus09
39110
0028
14Delq
uad
998779Elem
ents
20
82
1110
12392
332
1549
2119890minus08
9119890minus09
6119890minus09
6117067
2914
Delqu
ad998779
Elem
ents
1063
1115
48882
332
6110
2119890minus09
1119890minus09
1119890minus09
197
5804
630
14Delq
uad
998779Elem
ents
05
58
1116
194162
332
24270
2119890minus09
9119890minus10
2119890minus09
790
238769
3114
Delq
uad
◻Vo
lumes
40
minus416
1174
3132
332
391
5119890minus08
3119890minus08
8119890minus09
267322
3214
Delq
uad
◻Vo
lumes
30
minus248
1151
5922
332
740
9119890minus09
4119890minus09
2119890minus09
318779
3314
Delq
uad
◻Vo
lumes
20
minus132
1139
12050
332
1506
1119890minus08
7119890minus09
4119890minus09
4412239
3414
Delq
uad
◻Vo
lumes
10minus49
1126
48200
332
6025
6119890minus09
3119890minus09
2119890minus09
122
36094
3514
Delq
uad
◻Vo
lumes
05
minus23
1123
192800
332
24100
5119890minus09
2119890minus09
3119890minus09
464
140228
3614
Delq
uad
◻Elem
ents
40
224
1093
3294
332
411
3119890minus08
1119890minus08
7119890minus09
359852
3714
Delq
uad
◻Elem
ents
30
141
1104
6144
332
768
3119890minus08
2119890minus08
9119890minus09
5114018
3814
Delq
uad
◻Elem
ents
20
921111
12392
332
1549
1119890minus08
6119890minus09
5119890minus09
8122526
3914
Delq
uad
◻Elem
ents
1065
1115
48882
332
6110
5119890minus09
3119890minus09
4119890minus09
276
78431
22 Science and Technology of Nuclear Installations
Table2Con
tinued
Case
no
Symm-
etry
Mesh
algorithm
Basic
shape
Solutio
nmetho
dℓ119888
(cm)
Δ120588
(pcm
)max1206012
(mdash)
Total
unkn
owns
Outer
iter
Linear
iter
Inner
iter
Resid
ual
norm
Relativ
eerror
Error
estim
ate
Mem
ory
(Mb)
Page
faults
4014
Delq
uad
◻Elem
ents
05
59
1117
194162
332
24270
1119890minus09
5119890minus10
1119890minus09
1104
318967
4118
Delaunay
998779Vo
lumes
40
minus99
1139
4228
332
528
4119890minus12
2119890minus12
1119890minus12
359630
4218
Delaunay
998779Vo
lumes
30
50
1132
7712
332
964
1119890minus11
6119890minus12
2119890minus12
4211578
4318
Delaunay
998779Vo
lumes
20
177
1120
15776
332
1972
2119890minus12
1119890minus12
7119890minus13
5515436
4418
Delaunay
998779Vo
lumes
10142
1118
63902
332
7987
3119890minus12
1119890minus12
1119890minus12
160
46691
4518
Delaunay
998779Vo
lumes
05
176
1115
254612
332
31826
2119890minus12
8119890minus13
1119890minus12
555
168526
4618
Delaunay
998779Elem
ents
40
183
1096
2252
332
281
6119890minus12
3119890minus12
2119890minus12
359589
4718
Delaunay
998779Elem
ents
30
119
1104
4040
224
505
2119890minus08
1119890minus08
7119890minus09
41114
3748
18Delaunay
998779Elem
ents
20
85
1108
8156
332
1019
2119890minus12
8119890minus13
6119890minus13
5515053
4918
Delaunay
998779Elem
ents
1063
1112
32480
332
4060
1119890minus12
6119890minus13
8119890minus13
151
43671
5018
Delaunay
998779Elem
ents
05
58
1113
128358
332
1604
47119890minus13
3119890minus13
7119890minus13
527
157581
5118
Delaunay
◻Vo
lumes
40
37
1144
2380
224
297
5119890minus08
2119890minus08
1119890minus08
5314173
5218
Delaunay
◻Vo
lumes
30
218
1120
4194
332
524
7119890minus12
4119890minus12
2119890minus12
3610034
5318
Delaunay
◻Vo
lumes
20
345
1110
7960
332
995
5119890minus12
3119890minus12
1119890minus12
4712879
5418
Delaunay
◻Vo
lumes
10165
1117
30652
332
3831
2119890minus12
1119890minus12
7119890minus13
100
28851
5518
Delaunay
◻Vo
lumes
05
191
1117
122066
224
15258
2119890minus08
8119890minus09
9119890minus09
343
103825
5618
Delaunay
◻Elem
ents
40
175
1097
2526
332
315
4119890minus12
2119890minus12
1119890minus12
6016159
5718
Delaunay
◻Elem
ents
30
114
1104
4386
332
548
4119890minus12
2119890minus12
2119890minus12
5013768
5818
Delaunay
◻Elem
ents
20
901108
8234
332
1029
2119890minus12
1119890minus12
1119890minus12
7320094
5918
Delaunay
◻Elem
ents
1062
1112
31186
224
3898
2119890minus08
7119890minus09
9119890minus09
208
59589
6018
Delaunay
◻Elem
ents
05
58
1113
123122
224
15390
9119890minus09
5119890minus09
1119890minus08
807
2366
7061
18Delq
uad
998779Vo
lumes
40
minus210
1168
3224
332
403
1119890minus11
5119890minus12
2119890minus12
43117
8462
18Delq
uad
998779Vo
lumes
30
369
1150
6000
332
750
1119890minus11
6119890minus12
3119890minus12
5013502
6318
Delq
uad
998779Vo
lumes
20
312
1104
12316
332
1539
2119890minus11
8119890minus12
5119890minus12
6016395
6418
Delq
uad
998779Vo
lumes
1044
01094
48714
332
6089
9119890minus12
4119890minus12
4119890minus12
1183444
165
18Delq
uad
998779Vo
lumes
05
534
1081
193980
332
24247
5119890minus12
2119890minus12
4119890minus12
423
126768
6618
Delq
uad
998779Elem
ents
40
240
1090
1750
332
218
9119890minus12
4119890minus12
2119890minus12
43117
8667
18Delqu
ad998779
Elem
ents
30
150
1100
3184
332
398
1119890minus11
5119890minus12
2119890minus12
5013509
6818
Delq
uad
998779Elem
ents
20
89
1106
6426
332
803
5119890minus12
3119890minus12
2119890minus12
5916047
6918
Delq
uad
998779Elem
ents
1064
1111
24886
332
3110
2119890minus12
1119890minus12
1119890minus12
113
32788
7018
Delq
uad
998779Elem
ents
05
59
1113
98052
224
12256
1119890minus08
5119890minus09
8119890minus09
398
118153
7118
Delq
uad
◻Vo
lumes
40
minus398
1172
1650
332
206
1119890minus11
5119890minus12
2119890minus12
369907
7218
Delq
uad
◻Vo
lumes
30
minus249
1146
3058
332
382
4119890minus12
2119890minus12
1119890minus12
349349
7318
Delq
uad
◻Vo
lumes
20
minus131
1136
6228
224
778
3119890minus08
1119890minus08
5119890minus09
4010930
7418
Delq
uad
◻Vo
lumes
10minus51
1123
24536
224
3067
3119890minus08
2119890minus08
1119890minus08
7822716
7618
Delq
uad
◻Elem
ents
40
237
1090
1752
332
219
6119890minus12
3119890minus12
2119890minus12
4111213
7718
Delq
uad
◻Elem
ents
30
145
1101
3184
332
398
6119890minus12
3119890minus12
2119890minus12
43118
1178
18Delq
uad
◻Elem
ents
20
941107
6426
332
803
3119890minus12
1119890minus12
1119890minus12
6016220
7918
Delqu
ad◻
Elem
ents
1065
1111
24874
332
3109
1119890minus12
6119890minus13
8119890minus13
156
4400
080
18Delq
uad
◻Elem
ents
05
59
1113
9804
22
2412255
8119890minus09
4119890minus09
8119890minus09
567
162168
Science and Technology of Nuclear Installations 23
Tim
e con
sum
ed to
bui
ld th
e mat
rices
(sec
onds
)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay trianglesvolumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
10
30
100
300
1
3
Figure 23 Time needed to build the matrices versus number of unknowns
was considerably larger than the solution time itself Theeigenvalue problem was solved using a multilayer iterativeKrylov-Schur method [26] with a tolerance relative to thematrices norm
1003817100381710038171003817119877 sdot 120601 minus (1119896eff) sdot 119865 sdot 1206011003817100381710038171003817
119877 + (1119896eff) sdot 119865lt 10minus8 (11)
The reported residual norm and relative error are10038171003817100381710038171003817100381710038171003817119877 sdot 120601 minus
1
119896effsdot 119865 sdot 120601
10038171003817100381710038171003817100381710038171003817
1003817100381710038171003817119877 sdot 120601 minus (1119896eff) sdot 119865 sdot 1206011003817100381710038171003817
1003817100381710038171003817(1119896eff) sdot 1206011003817100381710038171003817
(12)
respectively The fields marked as outer linear and inneriterations refer to the number of steps needed to attainthe requested tolerance in each layer of the Krylov-Schuralgorithm The computer used to solve the problem has anIntel i7 920 267GHz processor with 4Gb of RAM runningDebian GNULinux Wheezy
When using a finite volumes-based scheme over anunstructured mesh the solver has to gather information
about which cells are neighbors and which are not Currentlygmsh does not write this kind of lists in its output filesso milonga has to explicitly solve the neighbors problemPerforming a linear search is an 119874(119873
2) task which is
unacceptable for problem sizes 119873 of interest The code usesa search based on a 119896-dimensional tree [27] which can inprinciple be performed in119874(119873) steps Still for large values of119873 the time needed to read and parse the mesh (number twopreviously mentioned) is the bottleneck of the solution Thisstep is not needed in finite elements although the construc-tion of the elementary matrices involves the computationof the multidimensional Jacobians and integrals which thenhave to be assembled Again for large 119873 this step (numberthree) takes up most of the time needed to solve the problem
The Delaunay algorithm is a standard method for gen-erating two-dimensional grids [16] by tessellating a planewith triangles If the mesh needs to be based on quadranglesinstead a recombination algorithm can be used to transformtwo adjacent triangles in one quadrangle whenever is possi-ble However for geometries which are based on rectangularshapes there exist other algorithms [25] both for meshingand for recombining the triangles that give rise to elementaryentities with right angles almost everywhere which may be a
24 Science and Technology of Nuclear Installations
Tim
e con
sum
ed to
solv
e the
eige
nval
ue p
robl
em (s
econ
ds)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
10
30
100
300
1
3
Figure 24 Time needed to solve the eigenvalue problem versus number of unknowns
desired property of the resulting grid For finite elements thesteps needed to build the matrices depend on the selectionof triangles or quadrangles as the basic elementary geometrybecause the shape functions change However the number ofunknowns is the same as the number of nodes that does notchange after a recombination procedure On the other handthe number of unknowns in the finite volumes schemes withtriangles is roughly twice as the number of unknowns withquadrangles for the same grid
Figure 19 shows how the computed static reactivity (ie1 minus 1119896eff) depends on the number of unknowns for eachof the sixteen combinations of geometry algorithm shapescheme The four accepted results published in the originalreference [10] are included for reference although it shouldbe taken into account that said reactivities were computedalmost forty years ago Figure 20 shows the total wall timeneeded to solve the problem as a function of 119873119866 whileFigures 21 22 23 and 24 show the times needed for eachof the first four steps involved in the solution maintainingthe same logarithmic scale for both the abscissas and theordinates Green data represent finite volumes whilst bluebullets correspond to finite elements Solid lines indicatequarter-core and dashed lines eighth-symmetry Fillet bullets
are results obtained by theDelaunay triangulation and emptybullets were computed with the delquad algorithm Finallytriangle-shaped data correspond to triangles and squares anddiamonds denote quadrangles as the basic geometry of thegrid
It can be seen that finite elements produce amuch smallerdispersion of eigenvalues 119896eff than finite volumes with therefinement of the mesh This can be explained because finitevolumes methods rely on a geometric condition of the meshwhich may change abruptly if the meshing algorithm decidesto allocate the cells in a rather different form for small changesin ℓ119888 Finite elements methods are less influenced by thesediscontinuous lay-out changes of the elements As expectedeighth-core symmetries give almost the same results as thequarter-core geometries with roughly half the unknownsFor small problems finite volumes run faster that finiteelements because the time needed to solve the neighborproblem is negligible When the problem size grows thistime increases and exceeds the overhead implied in the con-struction and assembly of the finite elements matrices Alsoat least for this configuration it is seen that the eigenvalueproblem is solved faster for finite volumes than for finiteelements
Science and Technology of Nuclear Installations 25
4 Conclusions
Unstructured grids provide the cognizant engineer witha wide variety of possibilities to deal with the design oranalysis of nuclear reactor cores These kinds of grids cansuccessfully reproduce continuous geometries commonlyfound in reactor cores such as cylinders and thereforenot only can the diffusion equation be better approximatedinside the domain of definition but also the fulfillment ofboundary conditions is improved A free computer codewas written from scratch that is able to completely solvethe 2D IAEA PWR Benchmark using unstructured gridsfor sixteen combinations of geometry meshing algorithmbasic shape and discretization scheme plus any value ofthe gridrsquos characteristic length by using a single input fileThe complete set of input files and codemdashexecutable andsourcemdashis available either online or upon request withcomments experiences suggestions and corrections beingmore than welcome Further development should includetackling full three-dimensional geometries with completethermal hydraulic feedback in order to analyze how thesolutions of the coupled neutronic-thermal problem dependon the spatial discretization scheme of the neutron leakageterm Parallelization of the computation and assembly of thematrices and of the solution of the eigenvalue problem andits implementation using GPUs are also desired features toimplement A problemwith direct applications that the futureversions of milonga ought to solve is the analysis of how thegeometry of the absorbing materials should be taken intoaccount in structured coarse grids in order to mitigate effectssuch as the rod-cusp problem
Suitable schemes for approximating the continuous dif-ferential operators by discrete matrix expressions includefinite volumes and finite elements families Finite volumesmethods compute cell mean values whilst finite elementsgive functional values at the gridrsquos nodes In general finiteelements are less sensitive to changes in the mesh so theresults they provide do not change significantly for differentmeshing algorithms or elementary shapes Small problemsare best solved by finite volumes as the neighbor-findingproblem is faster than the process of building and assemblingthe eigenvalue-problem matrices For a large number ofunknowns the process of finding which cell is neighbor ofwhichmdasheven based on a 119896-dimensional treemdashoverwhelmsthe computation and assembly of elementary matrices andfinite-element methods perform better
References
[1] G Glasstone and S Bell Nuclear Reactor Theory KriegerPublishing Company 1970
[2] J J Duderstadt and L J Hamilton Nuclear Reactor AnalysisJohn Wiley amp Sons New York NY USA 1976
[3] C Gho Reactor Physics Undergraduate Course Lecture NotesInstituto Balseiro 2006
[4] G G Theler and F J Bonetto ldquoOn the stability of the pointreactor kinetics equationsrdquoNuclear Engineering andDesign vol240 no 6 pp 1443ndash1449 2010
[5] E E Lewis andW F Miller Computational Methods of NeutronTransport John Wiley amp Sons 1984
[6] E Masiello ldquoAnalytical stability analysis of Coarse-Mesh finitedifference methodrdquo in Proceedings of the International Confer-ence on the Physics of Reactors (PHYSOR rsquo08) Nuclear Power ASustainable Resource pp 456ndash464 September 2008
[7] M L Zerkle Development of a polynomial nodal methodwith flux and current discontinuity factors [PhD thesis] Mas-sachusets Institute of Technology 1992
[8] J I Yoon andHG Joo ldquoTwo-level coarsemesh finite differenceformulation with multigroup source expansion nodal kernelsrdquoJournal of Nuclear Science andTechnology vol 45 no 7 pp 668ndash682 2008
[9] O Mazzantini M Schivo J Di Cesare R Garbero M Riveroand G Theler ldquoA coupled calculation suite for Atucha II oper-ational transients analysisrdquo Science and Technology of NuclearInstallations vol 2011 Article ID 785304 12 pages 2011
[10] Computational Benchmark Problem Comitee for the Mathe-matics and Computation Division of the American NuclearSociety ldquoArgonne Code Center Benchmark problem bookrdquoTech Rep ANL-7416 Supplement 2 Argonne National Labo-ratory 1977
[11] A F Henry Nuclear Reactor Analysis MIT Cambridge UK1975
[12] O C Zienkiewicz R L Taylor and J Z ZhuTheFinite ElementMethod Its Basis and Fundamentals vol 1 Elsevier 6th edition2005
[13] G Theler Difusion de neutrones en mallas no estructuradascomparacion entre volumenes y elementos finitos AcademicMonograph Universidad de Buenos Aires 2013
[14] G Theler ldquoMilonga a free nuclear reactor core analysis coderdquoAvailable online 2011
[15] G Theler F J Bonetto and A Clausse ldquoOptimizacion deparametros en reactores de potencia base de diseno delcodigo neutronico milongardquo in Reunion Anual de la AsociacionArgentina de Tecnologia Nuclear XXXVII 2010
[16] C Geuzaine and J-F Remacle ldquoGmsh a 3-D finite elementmesh generator with built-in pre- and post-processing facili-tiesrdquo International Journal for Numerical Methods in Engineer-ing vol 79 no 11 pp 1309ndash1331 2009
[17] S Balay J Brown K Buschelman et al ldquoPETSc users manualrdquoTech Rep ANL-9511-Revision 34 Argonne National Labora-tory 2013
[18] S Balay W D Gropp L Curfman McInnes and B F SmithldquoEfficient management of parallelism in object oriented numer-ical software librariesrdquo in Modern Software Tools in ScientificComputing E Arge A M Bruaset and H P Langtangen Edspp 163ndash202 Birkhauser 1997
[19] V Hernandez J E Roman and V Vidal ldquoSLEPC a scalable andflexible toolkit for the solution of eigenvalue problemsrdquo ACMTransactions on Mathematical Software vol 31 no 3 pp 351ndash362 2005
[20] M Galassi J Davies J Theiler et al GNU Scientific LibraryReference Manual 3rd edition 2009
[21] O C Zienkiewicz and R L Taylor The Finite Element MethodFor Solid and Structural Mechanics vol 3 Elsevier 6th edition2005
[22] A Imelda and K Doddy Evaluation of the IAEA 3-D PWRbenchmark problem using NESTLE code 2004
[23] R Mosteller ldquoStatic benchmarking of the NESTLE advancednodal coderdquo in Proceedings of the Joint International Conference
26 Science and Technology of Nuclear Installations
on Mathematical Methods and Supercomputing for NuclearApplications vol 2 pp 1596ndash1605 1997
[24] GTheler F J Bonetto andAClausse ldquoSolution of the 2D IAEAPWR Benchmark with the neutronic code milongardquo in Actasde la Reunion Anual de la Asociacion Argentina de TecnologıaNuclear XXXVIII 2011
[25] J F Remacle F Henrotte T Carrier-Baudouin et al ldquoAfrontal delaunay quad mesh generator using the 119871
infin normrdquoInternational Journal For Numerical Methods in Engineeringvol 94 no 5 pp 494ndash512 2013
[26] G W Stewart ldquoA Krylov-Schur algorithm for large eigenprob-lemsrdquo SIAM Journal on Matrix Analysis and Applications vol23 no 3 pp 601ndash614 2002
[27] J L Bentley ldquoMultidimensional binary search trees used forassociative searchingrdquo Communications of the ACM vol 18 no9 pp 509ndash517 1975
TribologyAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FuelsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofPetroleum Engineering
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
CombustionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Renewable Energy
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
StructuresJournal of
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear InstallationsScience and Technology of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solar EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Wind EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear EnergyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
8 Science and Technology of Nuclear Installations
Largest eigenvalue 1029690 (288336 pcm)1135 (3066 3079)1306 (5171 13076)
Number of unknowns 15740Outer iterations 3Linear iterations 32Inner iterations 1967Residual normRelative errorError estimateMemory used 49824 kBSoft page faults 13401Hard page faults 0Total CPU time 074 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
2483 times 10minus8
1223 times 10minus8
403 times 10minus9
keff
quarter-symmetry core meshed using Delaunay (triangles 985747c = 3) solved with finite volumesMilongarsquos 2D LWR IAEA benchmark problem case no 002
(a)
123456789
1011121314151617181920212223242526272829303132333435363738
k Pk 1206011k 1206012k075 3282 555133 4242 984147 4645 1090123 3921 911061 2677 453094 2995 694093 2927 689074 2011 549mdash 322 756146 4597 1078150 4730 1110133 4199 985107 3405 789104 3271 767095 2971 700072 1956 534mdash 306 733149 4695 1102136 4289 1007118 3733 876107 3367 792097 2892 718066 1628 491mdash 234 549120 3797 891097 3096 718090 2842 669084 2221 619mdash 568 1215mdash 067 257047 2042 348068 2055 504058 1414 430mdash 222 562057 1374 425mdash 388 820mdash 053 203mdash 060 229
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 10 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
case the unknowns are themean value of the fluxes over eachcell whilst in the latter the unknowns are the fluxes evaluatedat each node Therefore the number of unknowns 119873119866 isdifferent for each method even when using the same gridFigure 4 shows this situation and in Section 31 we further
illustrate these differences A fully detailed mathematicaldescription of the actual algorithms for both volumes andelements-based proposed discretizations can be found in anacademic monograph written by the author of this paper[13]
Science and Technology of Nuclear Installations 9
Largest eigenvalue 1029927 (290571 pcm)1131 (3175 2999)1220 (13097 5098)
Number of unknowns 15576Outer iterations 3Linear iterations 32Inner iterations 1947Residual normRelative errorError estimateMemory used 56256 kBSoft page faults 15645Hard page faults 0Total CPU time 09201 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
1394 times 10minus8
6865 times 10minus9
3425 times 10minus9
keff
quarter-symmetry core meshed using Delaunay (quads 985747c = 2) solved with finite volumesMilongarsquos 2D LWR IAEA benchmark problem case no 013
(a)
123456789
1011121314151617181920212223242526272829303132333435363738
k Pk 1206011k 1206012k074 3244 549133 4230 982147 4651 1092124 3946 917060 2645 443094 2997 696093 2938 691074 2005 545mdash 301 746145 4588 1076150 4720 1108133 4198 984109 3465 805104 3280 769095 2986 704071 1962 528mdash 293 715148 4680 1098136 4288 1007119 3755 881107 3385 796097 2905 721067 1653 493mdash 220 556121 3808 893098 3115 724092 2889 680083 2247 615mdash 543 1241mdash 067 260045 1977 330069 2076 509058 1460 428mdash 226 574056 1364 413mdash 371 821mdash 052 200mdash 060 238
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 11 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
3 Results
We now proceed to show two illustrative results that are tobe taken as an overview of the possibilities that unstructuredgrids can provide in order to tackle the multigroup neutron
diffusion problem The examples are two-dimensional prob-lems as they contain some of the complexities a real three-dimensional reactor posse yet the reported results are notso complicated as to be easily understood and analyzed Inparticular we state some basic differences between the finite
10 Science and Technology of Nuclear Installations
Largest eigenvalue 1029695(288388 pcm)1112 (3076 3032)838 (5000 13000)
Number of unknowns 15922Outer iterations 3Linear iterations 32Inner iterations 1990Residual normRelative errorError estimateMemory used 109964 kBSoft page faults 30591Hard page faults 0Total CPU time 158 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
1056 times 10minus8
5202 times 10minus9
5122 times 10minus9
keff
quarter-symmetry core meshed using Delaunay (quads 985747c = 2) solved with finite volumesMilongarsquos 2D LWR IAEA benchmark problem case no 018
(a)
123456789
1011121314151617181920212223242526272829303132333435363738
k Pk 1206011k 1206012k074 3220 549130 4152 961144 4651 1068120 3843 890061 2648 452094 2996 693094 2959 696072 2061 569mdash 358 862142 4495 1054147 4634 1088131 4127 968107 3409 789104 3275 768096 3003 708071 2014 554mdash 342 820146 4606 1081134 4222 991118 3711 871107 3375 794098 2929 727062 1692 522mdash 258 632119 3752 880096 3084 715091 2858 673080 2280 633mdash 611 1277mdash 080 318047 2043 350069 2088 510054 1463 450mdash 255 648051 1418 439mdash 410 854mdash 064 252mdash 071 285
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 12 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
volumes and the finite elements methods by solving a two-group homogeneous bare reactor with the Robin boundaryconditions over the very same grid although a rather coarseone so the differences can be observed directly into theresulting figuresWe then solve the classical two-dimensionalLWR problem also known as the 2D IAEA PWR benchmark
Not only do we show again the differences between the finitevolumes and elements formulation but also we solve theproblemusing different combinations ofmeshing algorithmsbasic shapes and characteristic lengths of the mesh
To solve the two examples shown below we employed themilonga code which was written from scratch by the author
Science and Technology of Nuclear Installations 11
Largest eigenvalue 1029159 (283326 pcm)1174 (2851 3224)1560 (5200 13200)
Number of unknowns 3132Outer iterations 3Linear iterations 32Inner iterations 391Residual normRelative errorError estimateMemory used 26888 kBSoft page faults 7322Hard page faults 0Total CPU time 0308 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
5408 times 10minus8
2665 times 10minus8
8424 times 10minus9
keff
Milongarsquos 2D LWR IAEA benchmark problem case no 031quarter-symmetry core meshed using Delquad (quads 985747c = 4) solved with finite volumes
(a)
123456789
1011121314151617181920212223242526272829303132333435363738
k Pk 1206011k 1206012k077 3422 570140 4468 1040153 4836 1135128 4070 948060 2704 448095 3017 702091 2861 673069 1918 508mdash 264 689151 4785 1122155 4897 1149137 4334 1017111 3535 823104 3272 768092 2908 685067 1876 496mdash 254 658154 4854 1139139 4392 1031120 3792 890107 3360 790095 2831 702061 1574 451mdash 197 515123 3868 907099 3132 730090 2818 663078 2168 579mdash 481 1191mdash 055 217044 1984 328067 2027 498052 1360 388mdash 190 520052 1329 383mdash 321 792mdash 043 168mdash 048 195
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 13 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
of this paper and is currently still under development withinhis ongoing PhD thesis There exists a first public release[14] under the terms of the GNU General Public Licensemdashthat is it is a free softwaremdashthat can only handle structuredgrids There is a second release being prepared whose
main relevance is that it can work with nonstructured gridsas well which is the main feature of the code It uses ageneral mathematical frameworkmdashcoded from scratch aswellmdashwhich provides input file parsing algebraic expres-sions evaluation one- and multidimensional interpolation of
12 Science and Technology of Nuclear Installations
Largest eigenvalue 1029788 (289260 pcm)1120 (3149 3011)1091 (13038 5114)
Number of unknowns 15776Outer iterations 3Linear iterations 32Inner iterations 1972Residual normRelative errorError estimateMemory used 56996 kBSoft page faults 15436Hard page faults 0Total CPU time 09121 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
2296 times 10minus12
1131 times 10minus12
6877 times 10minus13
keff
Milongarsquos 2D LWR IAEA benchmark problem case no 043eighth-symmetry core meshed using Delaunaya (triangs985747c = 2) solved with finite volumes
(a)
1 074 3241 5482 131 4188 9713 145 4582 10764 122 3877 9005 060 2644 4456 093 2986 6927 093 2941 6928 075 2030 5529 mdash 322 779
10 143 4523 106111 148 4667 109512 131 4145 97213 107 3423 79414 103 3269 76715 095 2993 70616 073 1988 53917 mdash 312 75118 147 4637 108819 134 4243 99620 118 3721 87321 107 3370 79322 098 2914 72323 067 1651 49924 mdash 235 57125 119 3763 88226 096 3077 71527 091 2847 67128 083 2255 61729 mdash 576 122530 mdash 068 26631 046 2016 34132 068 2058 50533 059 1443 43534 mdash 233 58935 057 1381 42036 mdash 381 82337 mdash 054 20838 mdash 062 244
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 14 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
scattered data shared-memory access numerical integrationfacilities and so forth The milonga code was designed withfour design basis vectors in mindmdashwhich are thoroughlydiscussed in the documentation [15]mdashwhich includes thetype of problems it can handle the code scalability which
features are expected and what to do with the obtainedresults
It works by first reading an input file that using plain-text English keywords and arguments defines the number119898 of spatial dimensions the number 119866 of group energies
Science and Technology of Nuclear Installations 13
Largest eigenvalue 1029726 (288675 pcm)1104 (3069 3069)864 (13000 5000)
Number of unknowns 4040Outer iterations 2Linear iterations 24Inner iterations 505Residual normRelative errorError estimateMemory used 42940 kBSoft page faults 11437Hard page faults 0Total CPU time 1064 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
2019 times 10minus8
9946 times 10minus9
7138 times 10minus9
keff
Milongarsquos 2D LWR IAEA benchmark problem case no 047eighth-symmetry core meshed using Delaunay (triangles 985747c = 3) solved with finite volumes
(a)
1 074 3194 5452 129 4117 9533 143 4515 10604 119 3814 8835 061 2634 4506 093 2984 6907 094 2954 6958 071 2064 5719 mdash 362 865
10 141 4458 104511 146 4599 107912 130 4097 96113 106 3388 78414 103 3263 76515 095 2998 70716 068 2015 55717 mdash 346 82318 145 4572 107319 133 4194 98420 117 3691 86621 107 3363 79222 098 2925 72623 059 1696 52824 mdash 263 63525 118 3728 87426 096 3068 71127 091 2850 67228 079 2280 63529 mdash 619 127630 mdash 081 32131 047 2036 34932 068 2085 50933 051 1465 45434 mdash 258 65035 049 1419 44336 mdash 416 85437 mdash 065 25438 mdash 071 288
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 15 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
and optionally a mesh file Currently only grids generatedwith the code gmsh [16] are only supported mainly becauseit is also a free software and it suits perfectly well milongarsquosdesign basis in the sense that the continuous geometry can bedefined as a function of a number of parameters Afterwardthe physical entities defined in the grid are mapped into
materials with macroscopic cross sections which maydepend on the spatial coordinates x bymeans of intermediatefunctions such as burn-up or temperatures distributionswhich in turn may be defined by algebraic expressions byinterpolating data located in files by reading shared-memoryobjects or by a combination of them In the same sense
14 Science and Technology of Nuclear Installations
Largest eigenvalue 1029695 (288387 pcm)1108 (3045 3045)833 (13000 5000)
Number of unknowns 8234Outer iterations 3Linear iterations 32Inner iterations 1029Residual normRelative error
Max 1206012(x y) coreMax 1206012(x y) reflector
2028 times 10minus12
9993 times 10minus13
1012 times 10minus12
keff
Relative errorError estimateMemory used 75468 kBSoft page faults 20094Hard page faults 0Total CPU time 1256 seconds
9993 times 10
1012 times 10minus12
Milongarsquos 2D LWR IAEA benchmark problem case no 058eighth-symmetry core meshed using Delaunay (quads 985747c = 2) solved with finite volumes
(a)
1 074 3207 5472 129 4136 9573 144 4534 10644 120 3828 8875 061 2638 4506 093 2985 6907 094 2948 6948 072 2053 5669 mdash 357 859
10 142 4478 105011 146 4617 108412 130 4111 96413 106 3396 78614 103 3263 76515 095 2991 70616 071 2005 55217 mdash 340 81618 145 4588 107719 133 4206 98720 117 3698 86821 107 3362 79122 098 2918 72523 062 1686 52024 mdash 258 62925 118 3737 87626 096 3073 71227 091 2848 67128 080 2272 63129 mdash 609 127230 mdash 080 31731 047 2035 34832 069 2081 50933 054 1458 44834 mdash 255 64635 052 1413 43736 mdash 409 85137 mdash 064 25138 mdash 071 284
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 16 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
boundary conditions are applied to appropriate physicalentities of dimension119898minus1The problemmatrices119877 and119865 arethen built and stored in an appropriate sparse format usingthe free PETSc [17 18] library and the eigenvalue problemis solved using the free SLEPc [19] library The results arestored into milongarsquos variables and functions which may be
evaluated integratedmdashusually using the free GNU ScientificLibrary [20] routinesmdashand of course written into appropriateoutputs Milonga can also solve problems parametrically andbe used to solve optimization problems As stated above thecode is a free software so corrections and contributions aremore than welcome
Science and Technology of Nuclear Installations 15
Largest eigenvalue 1029462 (286185 pcm)1136 (3093 2953)1233 (13100 5100)
Number of unknowns 6228Outer iterations 2Linear iterations 24Inner iterations 778Residual normR l ti
Max 1206012(x y) coreMax 1206012(x y) reflector
26 times 10minus8
1281 times 10minus8
4996 times 10minus9
keff
Relative errorError estimateMemory used 41460 kBSoft page faults 10930Hard page faults 0Total CPU time 0604 seconds
1281 times 10 8
4996 times 10minus9
Milongarsquos 2D LWR IAEA benchmark problem case no 073eighth-symmetry core meshed using Delaunay (quads 985747c = 2) solved with finite volumes
(a)
1 075 3297 5552 134 4268 9923 148 4660 10944 123 3926 9145 059 2645 4406 094 2997 6977 093 2914 6858 072 2002 5349 mdash 301 773
10 146 4606 108011 150 4735 111112 133 4212 98813 108 3448 80214 104 3271 76715 094 2954 69616 070 1954 52117 mdash 288 73518 149 4691 110119 136 4289 100720 119 3748 88021 107 3359 79022 096 2880 71423 064 1636 47524 mdash 217 56325 121 3810 89426 098 3110 72427 090 2829 66628 081 2227 59929 mdash 536 125730 mdash 062 25031 045 2002 33632 068 2040 50033 055 1414 40934 mdash 215 57935 055 1359 40836 mdash 359 83437 mdash 050 19938 mdash 061 236
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 17 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
31 A Coarse Bare Homogeneous Circle Figures 5 and 6 showthe results of solving a bare two-dimensional circular homo-geneous reactor of radius 119886 over an unstructured grid whichis deliberately coarse using the finite volumes method andthe finite elements method respectively Two group energies
were used and a null-flux boundary conditionwas fixed at theexternal surface Figures 5(a) and 6(a) compare the obtainedfast flux distributions In the first case the numerical solutionprovidesmean values for each neutron flux group in each cellwhile in the latter the solution is computed at the nodes and
16 Science and Technology of Nuclear Installations
Largest eigenvalue 1029851 (289858 pcm)1090 (3182 3182)851 (13000 5000)
Number of unknowns 1752Outer iterations 3Linear iterations 32Inner iterations 219Residual norm
Max 1206012(x y) coreMax 1206012(x y) reflector
566 times 10minus12
2789 times 10minus12
1595 times 10minus12
keff
Residual normRelative errorError estimateMemory used 42564 kBSoft page faults 11213Hard page faults 0Total CPU time 09921 seconds
566 times 10
2789 times 10minus12
1595 times 10minus12
Milongarsquos 2D LWR IAEA benchmark problem case no 076eighth-symmetry core meshed using Delaunay (quads 985747c = 4) solved with finite volumes
(a)
1 073 3151 5402 127 4059 9393 141 4457 10464 118 3768 8715 061 2615 4506 093 2979 6887 094 2971 6998 067 2092 5829 mdash 375 885
10 139 4397 103111 144 4541 106612 128 4052 95013 105 3365 77814 103 3259 76415 096 3016 71216 066 2045 56817 mdash 359 84118 143 4520 106119 132 4155 97520 116 3672 86221 107 3366 79222 098 2951 73323 054 1728 54324 mdash 273 65125 117 3700 86826 095 3057 70827 091 2860 67428 074 2311 64829 mdash 642 130030 mdash 084 33231 048 2041 35332 069 2103 51333 046 1493 46734 mdash 269 66735 046 1444 45536 mdash 432 87137 mdash 067 26338 mdash 074 297
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 18 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
continuous functions are evaluated by means of the shapefunctions used in the formulation Figures 5(b) and 6(b)illustrate the fast fluxes unknowns and its relative position inspace It can be noted that the mesh coarseness gives resultsthat differ substantially in both cases Finally the structure ofthe sparse eigenvalue problemmatrices is shown for each case
with blue red and cyan representing positive negative andexplicitly inserted zero values In the finite volume case thereare 184 unknowns (92 cells times 2 groups) while in the secondcase there are 218 unknowns (109 nodes times 2 groups) Thevolumesrsquo fissionmatrix is almost diagonal it has a bandwidthequal to the number of energy groups which in this case is
Science and Technology of Nuclear Installations 17
500 1000 2000 5000 10000 20000 50000Number of unknowns
2840
2860
2880
2900
2920
ORNLRisoslashKWUOntarioQuarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumes
Quarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elementsEighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumes
Eighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
120588(p
cm)
105 2 times 105 5 times 105
Figure 19 Static reactivity versus number of unknowns The four original solutions as published in 1977 [10] are included as reference
twoThe off-diagonal values appear right next to the diagonalelements because of the chosen ordering of the unknownsin the flux vector 120601 isin R184 Other orderings may be usedbut the rate of convergence of the eigenvalue problem can bedeteriorated On the other hand the elementsrsquo fission matrixhas a nontrivial structure because the grid is unstructuredand the net fission rate at each element depends on the fluxesat the nodes whose location inside thematrix depends in turnon how the gridwas generatedThis effect is similar to the onethat appears in structural analysis where mass matrices needto be lumped in order to simplify transient computations [21]The diagonal block of elementrsquos 119877 matrix and the null blockin 119865 correspond to the discrete equations that set the null-flux boundary conditions on the nodes located at the externalsurface Other types of boundary conditions lead to differentkinds of structures within the problem matrices
In case a part of the domain contained a nonmultiplicativematerial such as a reflector then there would appear sectionsof the fission matrix with null values in both methodsrendering 119865 singular in both methods Care should be takenwhen dealing with the numerical schemes for the eigenvalueproblem solution It can be seen in the elementsrsquo matricesa particular structure that implements the boundary condi-tions at the external surfaces This structure does not appearin the volumesrsquo matrices because the boundary conditions
appear as flux terms which are summed up over all thesurfaces of each cell so they aremasked inside the volumetricdiscretization of the divergence and gradient operators
As the two-group neutron diffusion equation with uni-form cross sections over a circle subject to null-flux bound-ary conditions has an analytical solution it is adequate tocompare how the two proposed numerical schemes relate toit In the studied problem we ignored upscattering and fastfissions Then the analytical effective multiplication factor is
119896eff =]Σ1198912 sdot Σ1199041rarr2
[Σ1198861 + Σ1199041rarr2 + 1198631(]0119886)2] [Σ1198862 + 1198632(]0119886)
2]
(7)
being ]0 = 24048 the smallest root of Besselrsquos first-kindfunction of order zero 1198690(119903)
Figure 7 shows how the two numerical effective multi-plication factors compare to the analytical solution givenby (7) as a function of the mesh refinement indicated bythe quotient 119886ℓ119888 between the radius of the circle and thecharacteristic length of the cellelements We can see thatthe 119896eff computed by the finite volumes (elements) methodis always greater (less) than the analytical solution Indeedit can be proven that for a bare one-dimensional slab thisis always the case [13] However this result does not hold
18 Science and Technology of Nuclear Installations
2000 3000 6000 10000 20000 30000 60000
01
003
001
03
1
3
10
30
100
300To
tal w
all t
ime (
seco
nds)
Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
Figure 20 Total wall time versus number of unknowns
for problems with nonuniform cross sections even in simpleone-dimensional reflected reactors
We may draw two other conclusions from Figure 7 Firstthat the finite element method seems to provide a bettersolution than the volumes-based scheme and second thateven though the error committed tends to decrease with finergrids its behavior is not monotonic for finite volumes Infact Figure 8 shows the difference between the numericaland analytical solutions using a logarithmic vertical scalewhere both conclusions are even more evident We deferthe discussion of such differences between the discretizationscheme until the next section It is worth to note howeverthat the fact that a finite-element-based scheme throws betterresults for the particular bare homogeneous circular reactorunder study than the finite volumes does not imply thatfinite volumes ought to be discarded as a valid tool forsolving the neutron diffusion equation in general cases Thecombination of lattice and core-level computations is usuallyperformedusing cell-based resultswhichwhen fed into node-based methods to solve the few-group neutron diffusionequation may introduce errors which potentially can leadto unacceptable solutions Nevertheless this analysis is farbeyond the scope of this paper which focuses on solving amathematical equation over unstructured grids
32 The 2D IAEA PWR Benchmark Problem This is aclassical two-group neutron diffusion problem first designedin the early 1970s and taken as a reference benchmark forcomputational codes A number of codes were used to solveeither this problem or its three-dimensional formulation[22 23] including milonga using a structured grid [24] Theoriginal formulation can be found in the reference [10] andthere is a reproduction that may be easily found online inreference [24] The geometry consists of a one-quarter ofa PWR core depicted in Figure 9 and the homogeneousmacroscopic cross sections are listed in Table 1 An axialbuckling term 119861
2
119911119892= 08times10
minus4 should be taken into accountThe external surface should be subject to a zero incomingcurrent condition which may be written as
120597120601119892
120597119899= minus
04692
119863119892
sdot 120601119892 (8)
The expected results are as follows
(1) Maximum eigenvalue(2) Fundamental flux distributions
(a) Radial flux traverses 120601119892(119909 0) and 120601119892(119909 119909)
Science and Technology of Nuclear Installations 19
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
1
3
10
30
100
300Ti
me c
onsu
med
to m
esh
the g
eom
etry
(sec
onds
)
Figure 21 Time needed to mesh the geometry versus number of unknowns
Table 1 Macroscopic cross-sections (units are not stated in the original reference but they are assumed to be in cmminus1 or cm as appropriate)
1198631
1198632
Σ1rarr2
Σ1198861
Σ1198862
]Σ1198912
Material1 15 04 002 001 0080 0135 Fuel 12 15 04 002 001 0085 0135 Fuel 23 15 04 002 001 0130 0135 Fuel 2 + rod4 20 03 004 0 0010 0 Reflector
Note the fluxes shall be normalized such that
1
119881coreint119881core
sum
119892
]Σ119891119892 sdot 120601119892119889119881 = 1 (9)
(b) Value and location of maximum power densityThis corresponds to maximum of 1206012 in thecore It is recommended that the maximumvalues of 1206012 both in the inner core and at thecorereflector interface be given
(3) Average subassembly powers 119875119896
119875119896 =1
119881119896
int119881119896
sum
119892
]Σ119891119892 sdot 120601119892119889119881 (10)
where 119881119896 is the volume of the 119896th subassembly and119896 designates the fuel subassemblies as shown inFigure 9
(4) Number of unknowns in the problem number ofiterations and total and outer
(5) Total computing time iteration time IO-time andcomputer used
(6) Type and numerical values of convergence criteria(7) Table of average group fluxes for a square mesh grid
of 20 times 20 cm(8) Dependence of results on mesh spacing
Even though the original problem is based on a quarter-core situation the problem has an eighth-core symmetry
20 Science and Technology of Nuclear Installations
Quarter Delaunay triangles elements
Tim
e con
sum
ed to
read
the m
esh
(sec
onds
)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumes
Quarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
1
3
10
30
100
300
Figure 22 Time needed to read the mesh versus number of unknowns
which cannot be taken into account by structured gridswhich are the main target of the benchmark Howevernonstructured grids can take into consideration any kind ofsymmetry almost without loss of accuracy and at the sametime reducing roughly the number of unknowns by a halfand the associated computational effort needed to solve theproblem by a factor of four Answers to items (1)ndash(7) canbe given completely by milonga using a single input file(see Supplementary data in SupplementaryMaterial availableonline at httpdxdoiorg1011552013641863) As asked initem (8) how results depend not only on the mesh spacingbut also on the meshing algorithm on the gridrsquos elementarygeometric shape and on the discretization scheme may shedlights on the subject whichmay be evenmore interesting thanthe numerical results themselves
Taking advantage of milongarsquos capability of reading andparsing command-line arguments the selection of the coregeometry (quarter or eighth) the meshing algorithm (delau-nay [16] or delquad [25]) the shape of the elementaryentities (triangles or quadrangles) the discretization scheme(volumes or elements) and the characteristic length ℓ119888 ofthe mesh can be provided at run time Fixing five values forℓ119888 = 4 3 2 1 05 gives 2 times 2 times 2 times 2 times 5 = 80 possiblecombinations which we solve with successive invocations to
milonga with the same input file but with different argumentsfrom a simple script Figures 10 11 12 13 14 15 16 17 and18 show the results corresponding to items (1)ndash(7) for someillustrative cases The complete set of figures and the codeused to generate themmay be provided upon request Table 2compiles the answers to the problem for every case studied
As the milonga code is still under development itsnumerical routines are not yet fully optimized nor designedfor parallel computation Therefore the reported times areonly rough estimates and should be taken with care Thesolution comprises five steps
(1) generate the grid with the requested geometry mesh-ing algorithm basic shape and characteristic lengthby calling to gmsh
(2) read the generated mesh(3) build the matrices(4) solve the eigenvalue problem(5) compute the requested results
The CPU time reported in Figures 10ndash18 is thus thesum of all of these five steps but not the time needed togenerate the figuresmdashwhich in some caseswith coarsemeshes
Science and Technology of Nuclear Installations 21
Table2
Resultsofthe2
DIAEA
PWRBe
nchm
arkp
roblem
obtained
with
them
ilong
acod
efor
thee
ightyp
ropo
sedcombinatio
nsofsymmetrym
eshing
algorithm
basicshapediscretization
schemeandcharacteris
ticelem
entc
elllengthTh
ereference
solutio
nis120588=28749times10minus5w
hich
correspo
ndsto119896eff=10296
Case
no
Symm-
etry
Mesh
algorithm
Basic
shape
Solutio
nmetho
dℓ119888
(cm)
Δ120588
(pcm
)max1206012
(mdash)
Total
unkn
owns
Outer
iter
Linear
iter
Inner
iter
Resid
ual
norm
Relativ
eerror
Error
estim
ate
Mem
ory
(Mb)
Page
faults
114
Delaunay
998779Vo
lumes
40
minus120
1145
8264
332
1033
6119890minus09
3119890minus09
1119890minus09
349470
214
Delaunay
998779Vo
lumes
30
85
1135
15740
332
1967
2119890minus08
1119890minus08
4119890minus09
4813401
314
Delaunay
998779Vo
lumes
20
197
1124
31188
332
3898
1119890minus08
6119890minus09
3119890minus09
7621932
414
Delaunay
998779Vo
lumes
10153
1123
127476
332
15934
7119890minus09
3119890minus09
3119890minus09
273
81886
514
Delaunay
998779Vo
lumes
05
187
1119
510676
332
63834
2119890minus09
9119890minus10
1119890minus09
1148
352261
614
Delaunay
998779Elem
ents
40
173
1100
4308
332
538
2119890minus08
8119890minus09
4119890minus09
318794
714
Delaunay
998779Elem
ents
30
117
1107
8108
332
1013
5119890minus09
2119890minus09
2119890minus09
4712946
814
Delaunay
998779Elem
ents
20
84
1112
15936
332
1992
6119890minus09
3119890minus09
2119890minus09
7321347
914
Delaunay
998779Elem
ents
1062
1116
64478
332
8059
6119890minus09
3119890minus09
4119890minus09
262
77105
1014
Delaunay
998779Elem
ents
05
58
1117
256700
332
32087
1119890minus09
5119890minus10
1119890minus09
1091
331969
1114
Delaunay
◻Vo
lumes
40
53
1149
5042
332
630
2119890minus08
1119890minus08
5119890minus09
288007
1214
Delaunay
◻Vo
lumes
30
346
1133
8560
332
1070
7119890minus09
3119890minus09
2119890minus09
3910895
1314
Delaunay
◻Vo
lumes
20
308
1131
15576
332
1947
1119890minus08
7119890minus09
3119890minus09
541564
514
14Delaunay
◻Vo
lumes
10177
1122
60774
332
7596
8119890minus09
4119890minus09
3119890minus09
167
50115
1514
Delaunay
◻Vo
lumes
05
199
1120
244936
332
30617
4119890minus09
2119890minus09
2119890minus09
698
215678
1614
Delaunay
◻Elem
ents
40
196
1100
5222
332
652
3119890minus08
1119890minus08
8119890minus09
4713098
1714
Delaunay
◻Elem
ents
30
123
1107
8810
332
1101
2119890minus08
9119890minus09
7119890minus09
6918598
1814
Delaunay
◻Elem
ents
20
901112
15922
332
1990
1119890minus08
5119890minus09
5119890minus09
107
30591
1914
Delaunay
◻Elem
ents
1063
1116
61456
332
7682
5119890minus09
2119890minus09
4119890minus09
389
113237
2014
Delaunay
◻Elem
ents
05
58
1117
246332
332
30791
8119890minus10
4119890minus10
1119890minus09
1676
498192
2114
Delq
uad
998779Vo
lumes
40
minus45
1144
6228
332
778
4119890minus08
2119890minus08
7119890minus09
308495
2214
Delq
uad
998779Vo
lumes
30
570
1153
11820
332
1477
3119890minus08
1119890minus08
4119890minus09
4010879
2314
Delq
uad
998779Vo
lumes
20
459
1103
24100
332
3012
2119890minus08
1119890minus08
5119890minus09
6217715
2414
Delq
uad
998779Vo
lumes
10512
1104
9640
03
3212050
2119890minus08
1119890minus08
9119890minus09
207
61714
2514
Delq
uad
998779Vo
lumes
05
528
1106
385600
332
48200
2119890minus09
1119890minus09
2119890minus09
838
255735
2614
Delqu
ad998779
Elem
ents
40
220
1093
3288
332
411
3119890minus08
2119890minus08
6119890minus09
308495
2714
Delq
uad
998779Elem
ents
30
140
1104
6148
332
768
4119890minus08
2119890minus08
8119890minus09
39110
0028
14Delq
uad
998779Elem
ents
20
82
1110
12392
332
1549
2119890minus08
9119890minus09
6119890minus09
6117067
2914
Delqu
ad998779
Elem
ents
1063
1115
48882
332
6110
2119890minus09
1119890minus09
1119890minus09
197
5804
630
14Delq
uad
998779Elem
ents
05
58
1116
194162
332
24270
2119890minus09
9119890minus10
2119890minus09
790
238769
3114
Delq
uad
◻Vo
lumes
40
minus416
1174
3132
332
391
5119890minus08
3119890minus08
8119890minus09
267322
3214
Delq
uad
◻Vo
lumes
30
minus248
1151
5922
332
740
9119890minus09
4119890minus09
2119890minus09
318779
3314
Delq
uad
◻Vo
lumes
20
minus132
1139
12050
332
1506
1119890minus08
7119890minus09
4119890minus09
4412239
3414
Delq
uad
◻Vo
lumes
10minus49
1126
48200
332
6025
6119890minus09
3119890minus09
2119890minus09
122
36094
3514
Delq
uad
◻Vo
lumes
05
minus23
1123
192800
332
24100
5119890minus09
2119890minus09
3119890minus09
464
140228
3614
Delq
uad
◻Elem
ents
40
224
1093
3294
332
411
3119890minus08
1119890minus08
7119890minus09
359852
3714
Delq
uad
◻Elem
ents
30
141
1104
6144
332
768
3119890minus08
2119890minus08
9119890minus09
5114018
3814
Delq
uad
◻Elem
ents
20
921111
12392
332
1549
1119890minus08
6119890minus09
5119890minus09
8122526
3914
Delq
uad
◻Elem
ents
1065
1115
48882
332
6110
5119890minus09
3119890minus09
4119890minus09
276
78431
22 Science and Technology of Nuclear Installations
Table2Con
tinued
Case
no
Symm-
etry
Mesh
algorithm
Basic
shape
Solutio
nmetho
dℓ119888
(cm)
Δ120588
(pcm
)max1206012
(mdash)
Total
unkn
owns
Outer
iter
Linear
iter
Inner
iter
Resid
ual
norm
Relativ
eerror
Error
estim
ate
Mem
ory
(Mb)
Page
faults
4014
Delq
uad
◻Elem
ents
05
59
1117
194162
332
24270
1119890minus09
5119890minus10
1119890minus09
1104
318967
4118
Delaunay
998779Vo
lumes
40
minus99
1139
4228
332
528
4119890minus12
2119890minus12
1119890minus12
359630
4218
Delaunay
998779Vo
lumes
30
50
1132
7712
332
964
1119890minus11
6119890minus12
2119890minus12
4211578
4318
Delaunay
998779Vo
lumes
20
177
1120
15776
332
1972
2119890minus12
1119890minus12
7119890minus13
5515436
4418
Delaunay
998779Vo
lumes
10142
1118
63902
332
7987
3119890minus12
1119890minus12
1119890minus12
160
46691
4518
Delaunay
998779Vo
lumes
05
176
1115
254612
332
31826
2119890minus12
8119890minus13
1119890minus12
555
168526
4618
Delaunay
998779Elem
ents
40
183
1096
2252
332
281
6119890minus12
3119890minus12
2119890minus12
359589
4718
Delaunay
998779Elem
ents
30
119
1104
4040
224
505
2119890minus08
1119890minus08
7119890minus09
41114
3748
18Delaunay
998779Elem
ents
20
85
1108
8156
332
1019
2119890minus12
8119890minus13
6119890minus13
5515053
4918
Delaunay
998779Elem
ents
1063
1112
32480
332
4060
1119890minus12
6119890minus13
8119890minus13
151
43671
5018
Delaunay
998779Elem
ents
05
58
1113
128358
332
1604
47119890minus13
3119890minus13
7119890minus13
527
157581
5118
Delaunay
◻Vo
lumes
40
37
1144
2380
224
297
5119890minus08
2119890minus08
1119890minus08
5314173
5218
Delaunay
◻Vo
lumes
30
218
1120
4194
332
524
7119890minus12
4119890minus12
2119890minus12
3610034
5318
Delaunay
◻Vo
lumes
20
345
1110
7960
332
995
5119890minus12
3119890minus12
1119890minus12
4712879
5418
Delaunay
◻Vo
lumes
10165
1117
30652
332
3831
2119890minus12
1119890minus12
7119890minus13
100
28851
5518
Delaunay
◻Vo
lumes
05
191
1117
122066
224
15258
2119890minus08
8119890minus09
9119890minus09
343
103825
5618
Delaunay
◻Elem
ents
40
175
1097
2526
332
315
4119890minus12
2119890minus12
1119890minus12
6016159
5718
Delaunay
◻Elem
ents
30
114
1104
4386
332
548
4119890minus12
2119890minus12
2119890minus12
5013768
5818
Delaunay
◻Elem
ents
20
901108
8234
332
1029
2119890minus12
1119890minus12
1119890minus12
7320094
5918
Delaunay
◻Elem
ents
1062
1112
31186
224
3898
2119890minus08
7119890minus09
9119890minus09
208
59589
6018
Delaunay
◻Elem
ents
05
58
1113
123122
224
15390
9119890minus09
5119890minus09
1119890minus08
807
2366
7061
18Delq
uad
998779Vo
lumes
40
minus210
1168
3224
332
403
1119890minus11
5119890minus12
2119890minus12
43117
8462
18Delq
uad
998779Vo
lumes
30
369
1150
6000
332
750
1119890minus11
6119890minus12
3119890minus12
5013502
6318
Delq
uad
998779Vo
lumes
20
312
1104
12316
332
1539
2119890minus11
8119890minus12
5119890minus12
6016395
6418
Delq
uad
998779Vo
lumes
1044
01094
48714
332
6089
9119890minus12
4119890minus12
4119890minus12
1183444
165
18Delq
uad
998779Vo
lumes
05
534
1081
193980
332
24247
5119890minus12
2119890minus12
4119890minus12
423
126768
6618
Delq
uad
998779Elem
ents
40
240
1090
1750
332
218
9119890minus12
4119890minus12
2119890minus12
43117
8667
18Delqu
ad998779
Elem
ents
30
150
1100
3184
332
398
1119890minus11
5119890minus12
2119890minus12
5013509
6818
Delq
uad
998779Elem
ents
20
89
1106
6426
332
803
5119890minus12
3119890minus12
2119890minus12
5916047
6918
Delq
uad
998779Elem
ents
1064
1111
24886
332
3110
2119890minus12
1119890minus12
1119890minus12
113
32788
7018
Delq
uad
998779Elem
ents
05
59
1113
98052
224
12256
1119890minus08
5119890minus09
8119890minus09
398
118153
7118
Delq
uad
◻Vo
lumes
40
minus398
1172
1650
332
206
1119890minus11
5119890minus12
2119890minus12
369907
7218
Delq
uad
◻Vo
lumes
30
minus249
1146
3058
332
382
4119890minus12
2119890minus12
1119890minus12
349349
7318
Delq
uad
◻Vo
lumes
20
minus131
1136
6228
224
778
3119890minus08
1119890minus08
5119890minus09
4010930
7418
Delq
uad
◻Vo
lumes
10minus51
1123
24536
224
3067
3119890minus08
2119890minus08
1119890minus08
7822716
7618
Delq
uad
◻Elem
ents
40
237
1090
1752
332
219
6119890minus12
3119890minus12
2119890minus12
4111213
7718
Delq
uad
◻Elem
ents
30
145
1101
3184
332
398
6119890minus12
3119890minus12
2119890minus12
43118
1178
18Delq
uad
◻Elem
ents
20
941107
6426
332
803
3119890minus12
1119890minus12
1119890minus12
6016220
7918
Delqu
ad◻
Elem
ents
1065
1111
24874
332
3109
1119890minus12
6119890minus13
8119890minus13
156
4400
080
18Delq
uad
◻Elem
ents
05
59
1113
9804
22
2412255
8119890minus09
4119890minus09
8119890minus09
567
162168
Science and Technology of Nuclear Installations 23
Tim
e con
sum
ed to
bui
ld th
e mat
rices
(sec
onds
)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay trianglesvolumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
10
30
100
300
1
3
Figure 23 Time needed to build the matrices versus number of unknowns
was considerably larger than the solution time itself Theeigenvalue problem was solved using a multilayer iterativeKrylov-Schur method [26] with a tolerance relative to thematrices norm
1003817100381710038171003817119877 sdot 120601 minus (1119896eff) sdot 119865 sdot 1206011003817100381710038171003817
119877 + (1119896eff) sdot 119865lt 10minus8 (11)
The reported residual norm and relative error are10038171003817100381710038171003817100381710038171003817119877 sdot 120601 minus
1
119896effsdot 119865 sdot 120601
10038171003817100381710038171003817100381710038171003817
1003817100381710038171003817119877 sdot 120601 minus (1119896eff) sdot 119865 sdot 1206011003817100381710038171003817
1003817100381710038171003817(1119896eff) sdot 1206011003817100381710038171003817
(12)
respectively The fields marked as outer linear and inneriterations refer to the number of steps needed to attainthe requested tolerance in each layer of the Krylov-Schuralgorithm The computer used to solve the problem has anIntel i7 920 267GHz processor with 4Gb of RAM runningDebian GNULinux Wheezy
When using a finite volumes-based scheme over anunstructured mesh the solver has to gather information
about which cells are neighbors and which are not Currentlygmsh does not write this kind of lists in its output filesso milonga has to explicitly solve the neighbors problemPerforming a linear search is an 119874(119873
2) task which is
unacceptable for problem sizes 119873 of interest The code usesa search based on a 119896-dimensional tree [27] which can inprinciple be performed in119874(119873) steps Still for large values of119873 the time needed to read and parse the mesh (number twopreviously mentioned) is the bottleneck of the solution Thisstep is not needed in finite elements although the construc-tion of the elementary matrices involves the computationof the multidimensional Jacobians and integrals which thenhave to be assembled Again for large 119873 this step (numberthree) takes up most of the time needed to solve the problem
The Delaunay algorithm is a standard method for gen-erating two-dimensional grids [16] by tessellating a planewith triangles If the mesh needs to be based on quadranglesinstead a recombination algorithm can be used to transformtwo adjacent triangles in one quadrangle whenever is possi-ble However for geometries which are based on rectangularshapes there exist other algorithms [25] both for meshingand for recombining the triangles that give rise to elementaryentities with right angles almost everywhere which may be a
24 Science and Technology of Nuclear Installations
Tim
e con
sum
ed to
solv
e the
eige
nval
ue p
robl
em (s
econ
ds)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
10
30
100
300
1
3
Figure 24 Time needed to solve the eigenvalue problem versus number of unknowns
desired property of the resulting grid For finite elements thesteps needed to build the matrices depend on the selectionof triangles or quadrangles as the basic elementary geometrybecause the shape functions change However the number ofunknowns is the same as the number of nodes that does notchange after a recombination procedure On the other handthe number of unknowns in the finite volumes schemes withtriangles is roughly twice as the number of unknowns withquadrangles for the same grid
Figure 19 shows how the computed static reactivity (ie1 minus 1119896eff) depends on the number of unknowns for eachof the sixteen combinations of geometry algorithm shapescheme The four accepted results published in the originalreference [10] are included for reference although it shouldbe taken into account that said reactivities were computedalmost forty years ago Figure 20 shows the total wall timeneeded to solve the problem as a function of 119873119866 whileFigures 21 22 23 and 24 show the times needed for eachof the first four steps involved in the solution maintainingthe same logarithmic scale for both the abscissas and theordinates Green data represent finite volumes whilst bluebullets correspond to finite elements Solid lines indicatequarter-core and dashed lines eighth-symmetry Fillet bullets
are results obtained by theDelaunay triangulation and emptybullets were computed with the delquad algorithm Finallytriangle-shaped data correspond to triangles and squares anddiamonds denote quadrangles as the basic geometry of thegrid
It can be seen that finite elements produce amuch smallerdispersion of eigenvalues 119896eff than finite volumes with therefinement of the mesh This can be explained because finitevolumes methods rely on a geometric condition of the meshwhich may change abruptly if the meshing algorithm decidesto allocate the cells in a rather different form for small changesin ℓ119888 Finite elements methods are less influenced by thesediscontinuous lay-out changes of the elements As expectedeighth-core symmetries give almost the same results as thequarter-core geometries with roughly half the unknownsFor small problems finite volumes run faster that finiteelements because the time needed to solve the neighborproblem is negligible When the problem size grows thistime increases and exceeds the overhead implied in the con-struction and assembly of the finite elements matrices Alsoat least for this configuration it is seen that the eigenvalueproblem is solved faster for finite volumes than for finiteelements
Science and Technology of Nuclear Installations 25
4 Conclusions
Unstructured grids provide the cognizant engineer witha wide variety of possibilities to deal with the design oranalysis of nuclear reactor cores These kinds of grids cansuccessfully reproduce continuous geometries commonlyfound in reactor cores such as cylinders and thereforenot only can the diffusion equation be better approximatedinside the domain of definition but also the fulfillment ofboundary conditions is improved A free computer codewas written from scratch that is able to completely solvethe 2D IAEA PWR Benchmark using unstructured gridsfor sixteen combinations of geometry meshing algorithmbasic shape and discretization scheme plus any value ofthe gridrsquos characteristic length by using a single input fileThe complete set of input files and codemdashexecutable andsourcemdashis available either online or upon request withcomments experiences suggestions and corrections beingmore than welcome Further development should includetackling full three-dimensional geometries with completethermal hydraulic feedback in order to analyze how thesolutions of the coupled neutronic-thermal problem dependon the spatial discretization scheme of the neutron leakageterm Parallelization of the computation and assembly of thematrices and of the solution of the eigenvalue problem andits implementation using GPUs are also desired features toimplement A problemwith direct applications that the futureversions of milonga ought to solve is the analysis of how thegeometry of the absorbing materials should be taken intoaccount in structured coarse grids in order to mitigate effectssuch as the rod-cusp problem
Suitable schemes for approximating the continuous dif-ferential operators by discrete matrix expressions includefinite volumes and finite elements families Finite volumesmethods compute cell mean values whilst finite elementsgive functional values at the gridrsquos nodes In general finiteelements are less sensitive to changes in the mesh so theresults they provide do not change significantly for differentmeshing algorithms or elementary shapes Small problemsare best solved by finite volumes as the neighbor-findingproblem is faster than the process of building and assemblingthe eigenvalue-problem matrices For a large number ofunknowns the process of finding which cell is neighbor ofwhichmdasheven based on a 119896-dimensional treemdashoverwhelmsthe computation and assembly of elementary matrices andfinite-element methods perform better
References
[1] G Glasstone and S Bell Nuclear Reactor Theory KriegerPublishing Company 1970
[2] J J Duderstadt and L J Hamilton Nuclear Reactor AnalysisJohn Wiley amp Sons New York NY USA 1976
[3] C Gho Reactor Physics Undergraduate Course Lecture NotesInstituto Balseiro 2006
[4] G G Theler and F J Bonetto ldquoOn the stability of the pointreactor kinetics equationsrdquoNuclear Engineering andDesign vol240 no 6 pp 1443ndash1449 2010
[5] E E Lewis andW F Miller Computational Methods of NeutronTransport John Wiley amp Sons 1984
[6] E Masiello ldquoAnalytical stability analysis of Coarse-Mesh finitedifference methodrdquo in Proceedings of the International Confer-ence on the Physics of Reactors (PHYSOR rsquo08) Nuclear Power ASustainable Resource pp 456ndash464 September 2008
[7] M L Zerkle Development of a polynomial nodal methodwith flux and current discontinuity factors [PhD thesis] Mas-sachusets Institute of Technology 1992
[8] J I Yoon andHG Joo ldquoTwo-level coarsemesh finite differenceformulation with multigroup source expansion nodal kernelsrdquoJournal of Nuclear Science andTechnology vol 45 no 7 pp 668ndash682 2008
[9] O Mazzantini M Schivo J Di Cesare R Garbero M Riveroand G Theler ldquoA coupled calculation suite for Atucha II oper-ational transients analysisrdquo Science and Technology of NuclearInstallations vol 2011 Article ID 785304 12 pages 2011
[10] Computational Benchmark Problem Comitee for the Mathe-matics and Computation Division of the American NuclearSociety ldquoArgonne Code Center Benchmark problem bookrdquoTech Rep ANL-7416 Supplement 2 Argonne National Labo-ratory 1977
[11] A F Henry Nuclear Reactor Analysis MIT Cambridge UK1975
[12] O C Zienkiewicz R L Taylor and J Z ZhuTheFinite ElementMethod Its Basis and Fundamentals vol 1 Elsevier 6th edition2005
[13] G Theler Difusion de neutrones en mallas no estructuradascomparacion entre volumenes y elementos finitos AcademicMonograph Universidad de Buenos Aires 2013
[14] G Theler ldquoMilonga a free nuclear reactor core analysis coderdquoAvailable online 2011
[15] G Theler F J Bonetto and A Clausse ldquoOptimizacion deparametros en reactores de potencia base de diseno delcodigo neutronico milongardquo in Reunion Anual de la AsociacionArgentina de Tecnologia Nuclear XXXVII 2010
[16] C Geuzaine and J-F Remacle ldquoGmsh a 3-D finite elementmesh generator with built-in pre- and post-processing facili-tiesrdquo International Journal for Numerical Methods in Engineer-ing vol 79 no 11 pp 1309ndash1331 2009
[17] S Balay J Brown K Buschelman et al ldquoPETSc users manualrdquoTech Rep ANL-9511-Revision 34 Argonne National Labora-tory 2013
[18] S Balay W D Gropp L Curfman McInnes and B F SmithldquoEfficient management of parallelism in object oriented numer-ical software librariesrdquo in Modern Software Tools in ScientificComputing E Arge A M Bruaset and H P Langtangen Edspp 163ndash202 Birkhauser 1997
[19] V Hernandez J E Roman and V Vidal ldquoSLEPC a scalable andflexible toolkit for the solution of eigenvalue problemsrdquo ACMTransactions on Mathematical Software vol 31 no 3 pp 351ndash362 2005
[20] M Galassi J Davies J Theiler et al GNU Scientific LibraryReference Manual 3rd edition 2009
[21] O C Zienkiewicz and R L Taylor The Finite Element MethodFor Solid and Structural Mechanics vol 3 Elsevier 6th edition2005
[22] A Imelda and K Doddy Evaluation of the IAEA 3-D PWRbenchmark problem using NESTLE code 2004
[23] R Mosteller ldquoStatic benchmarking of the NESTLE advancednodal coderdquo in Proceedings of the Joint International Conference
26 Science and Technology of Nuclear Installations
on Mathematical Methods and Supercomputing for NuclearApplications vol 2 pp 1596ndash1605 1997
[24] GTheler F J Bonetto andAClausse ldquoSolution of the 2D IAEAPWR Benchmark with the neutronic code milongardquo in Actasde la Reunion Anual de la Asociacion Argentina de TecnologıaNuclear XXXVIII 2011
[25] J F Remacle F Henrotte T Carrier-Baudouin et al ldquoAfrontal delaunay quad mesh generator using the 119871
infin normrdquoInternational Journal For Numerical Methods in Engineeringvol 94 no 5 pp 494ndash512 2013
[26] G W Stewart ldquoA Krylov-Schur algorithm for large eigenprob-lemsrdquo SIAM Journal on Matrix Analysis and Applications vol23 no 3 pp 601ndash614 2002
[27] J L Bentley ldquoMultidimensional binary search trees used forassociative searchingrdquo Communications of the ACM vol 18 no9 pp 509ndash517 1975
TribologyAdvances in
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International Journal of
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Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Renewable Energy
Submit your manuscripts athttpwwwhindawicom
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StructuresJournal of
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
EnergyJournal of
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Journal ofEngineeringVolume 2014
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Nuclear InstallationsScience and Technology of
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Wind EnergyJournal of
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Nuclear EnergyInternational Journal of
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High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Science and Technology of Nuclear Installations 9
Largest eigenvalue 1029927 (290571 pcm)1131 (3175 2999)1220 (13097 5098)
Number of unknowns 15576Outer iterations 3Linear iterations 32Inner iterations 1947Residual normRelative errorError estimateMemory used 56256 kBSoft page faults 15645Hard page faults 0Total CPU time 09201 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
1394 times 10minus8
6865 times 10minus9
3425 times 10minus9
keff
quarter-symmetry core meshed using Delaunay (quads 985747c = 2) solved with finite volumesMilongarsquos 2D LWR IAEA benchmark problem case no 013
(a)
123456789
1011121314151617181920212223242526272829303132333435363738
k Pk 1206011k 1206012k074 3244 549133 4230 982147 4651 1092124 3946 917060 2645 443094 2997 696093 2938 691074 2005 545mdash 301 746145 4588 1076150 4720 1108133 4198 984109 3465 805104 3280 769095 2986 704071 1962 528mdash 293 715148 4680 1098136 4288 1007119 3755 881107 3385 796097 2905 721067 1653 493mdash 220 556121 3808 893098 3115 724092 2889 680083 2247 615mdash 543 1241mdash 067 260045 1977 330069 2076 509058 1460 428mdash 226 574056 1364 413mdash 371 821mdash 052 200mdash 060 238
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 11 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
3 Results
We now proceed to show two illustrative results that are tobe taken as an overview of the possibilities that unstructuredgrids can provide in order to tackle the multigroup neutron
diffusion problem The examples are two-dimensional prob-lems as they contain some of the complexities a real three-dimensional reactor posse yet the reported results are notso complicated as to be easily understood and analyzed Inparticular we state some basic differences between the finite
10 Science and Technology of Nuclear Installations
Largest eigenvalue 1029695(288388 pcm)1112 (3076 3032)838 (5000 13000)
Number of unknowns 15922Outer iterations 3Linear iterations 32Inner iterations 1990Residual normRelative errorError estimateMemory used 109964 kBSoft page faults 30591Hard page faults 0Total CPU time 158 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
1056 times 10minus8
5202 times 10minus9
5122 times 10minus9
keff
quarter-symmetry core meshed using Delaunay (quads 985747c = 2) solved with finite volumesMilongarsquos 2D LWR IAEA benchmark problem case no 018
(a)
123456789
1011121314151617181920212223242526272829303132333435363738
k Pk 1206011k 1206012k074 3220 549130 4152 961144 4651 1068120 3843 890061 2648 452094 2996 693094 2959 696072 2061 569mdash 358 862142 4495 1054147 4634 1088131 4127 968107 3409 789104 3275 768096 3003 708071 2014 554mdash 342 820146 4606 1081134 4222 991118 3711 871107 3375 794098 2929 727062 1692 522mdash 258 632119 3752 880096 3084 715091 2858 673080 2280 633mdash 611 1277mdash 080 318047 2043 350069 2088 510054 1463 450mdash 255 648051 1418 439mdash 410 854mdash 064 252mdash 071 285
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 12 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
volumes and the finite elements methods by solving a two-group homogeneous bare reactor with the Robin boundaryconditions over the very same grid although a rather coarseone so the differences can be observed directly into theresulting figuresWe then solve the classical two-dimensionalLWR problem also known as the 2D IAEA PWR benchmark
Not only do we show again the differences between the finitevolumes and elements formulation but also we solve theproblemusing different combinations ofmeshing algorithmsbasic shapes and characteristic lengths of the mesh
To solve the two examples shown below we employed themilonga code which was written from scratch by the author
Science and Technology of Nuclear Installations 11
Largest eigenvalue 1029159 (283326 pcm)1174 (2851 3224)1560 (5200 13200)
Number of unknowns 3132Outer iterations 3Linear iterations 32Inner iterations 391Residual normRelative errorError estimateMemory used 26888 kBSoft page faults 7322Hard page faults 0Total CPU time 0308 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
5408 times 10minus8
2665 times 10minus8
8424 times 10minus9
keff
Milongarsquos 2D LWR IAEA benchmark problem case no 031quarter-symmetry core meshed using Delquad (quads 985747c = 4) solved with finite volumes
(a)
123456789
1011121314151617181920212223242526272829303132333435363738
k Pk 1206011k 1206012k077 3422 570140 4468 1040153 4836 1135128 4070 948060 2704 448095 3017 702091 2861 673069 1918 508mdash 264 689151 4785 1122155 4897 1149137 4334 1017111 3535 823104 3272 768092 2908 685067 1876 496mdash 254 658154 4854 1139139 4392 1031120 3792 890107 3360 790095 2831 702061 1574 451mdash 197 515123 3868 907099 3132 730090 2818 663078 2168 579mdash 481 1191mdash 055 217044 1984 328067 2027 498052 1360 388mdash 190 520052 1329 383mdash 321 792mdash 043 168mdash 048 195
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 13 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
of this paper and is currently still under development withinhis ongoing PhD thesis There exists a first public release[14] under the terms of the GNU General Public Licensemdashthat is it is a free softwaremdashthat can only handle structuredgrids There is a second release being prepared whose
main relevance is that it can work with nonstructured gridsas well which is the main feature of the code It uses ageneral mathematical frameworkmdashcoded from scratch aswellmdashwhich provides input file parsing algebraic expres-sions evaluation one- and multidimensional interpolation of
12 Science and Technology of Nuclear Installations
Largest eigenvalue 1029788 (289260 pcm)1120 (3149 3011)1091 (13038 5114)
Number of unknowns 15776Outer iterations 3Linear iterations 32Inner iterations 1972Residual normRelative errorError estimateMemory used 56996 kBSoft page faults 15436Hard page faults 0Total CPU time 09121 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
2296 times 10minus12
1131 times 10minus12
6877 times 10minus13
keff
Milongarsquos 2D LWR IAEA benchmark problem case no 043eighth-symmetry core meshed using Delaunaya (triangs985747c = 2) solved with finite volumes
(a)
1 074 3241 5482 131 4188 9713 145 4582 10764 122 3877 9005 060 2644 4456 093 2986 6927 093 2941 6928 075 2030 5529 mdash 322 779
10 143 4523 106111 148 4667 109512 131 4145 97213 107 3423 79414 103 3269 76715 095 2993 70616 073 1988 53917 mdash 312 75118 147 4637 108819 134 4243 99620 118 3721 87321 107 3370 79322 098 2914 72323 067 1651 49924 mdash 235 57125 119 3763 88226 096 3077 71527 091 2847 67128 083 2255 61729 mdash 576 122530 mdash 068 26631 046 2016 34132 068 2058 50533 059 1443 43534 mdash 233 58935 057 1381 42036 mdash 381 82337 mdash 054 20838 mdash 062 244
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 14 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
scattered data shared-memory access numerical integrationfacilities and so forth The milonga code was designed withfour design basis vectors in mindmdashwhich are thoroughlydiscussed in the documentation [15]mdashwhich includes thetype of problems it can handle the code scalability which
features are expected and what to do with the obtainedresults
It works by first reading an input file that using plain-text English keywords and arguments defines the number119898 of spatial dimensions the number 119866 of group energies
Science and Technology of Nuclear Installations 13
Largest eigenvalue 1029726 (288675 pcm)1104 (3069 3069)864 (13000 5000)
Number of unknowns 4040Outer iterations 2Linear iterations 24Inner iterations 505Residual normRelative errorError estimateMemory used 42940 kBSoft page faults 11437Hard page faults 0Total CPU time 1064 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
2019 times 10minus8
9946 times 10minus9
7138 times 10minus9
keff
Milongarsquos 2D LWR IAEA benchmark problem case no 047eighth-symmetry core meshed using Delaunay (triangles 985747c = 3) solved with finite volumes
(a)
1 074 3194 5452 129 4117 9533 143 4515 10604 119 3814 8835 061 2634 4506 093 2984 6907 094 2954 6958 071 2064 5719 mdash 362 865
10 141 4458 104511 146 4599 107912 130 4097 96113 106 3388 78414 103 3263 76515 095 2998 70716 068 2015 55717 mdash 346 82318 145 4572 107319 133 4194 98420 117 3691 86621 107 3363 79222 098 2925 72623 059 1696 52824 mdash 263 63525 118 3728 87426 096 3068 71127 091 2850 67228 079 2280 63529 mdash 619 127630 mdash 081 32131 047 2036 34932 068 2085 50933 051 1465 45434 mdash 258 65035 049 1419 44336 mdash 416 85437 mdash 065 25438 mdash 071 288
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 15 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
and optionally a mesh file Currently only grids generatedwith the code gmsh [16] are only supported mainly becauseit is also a free software and it suits perfectly well milongarsquosdesign basis in the sense that the continuous geometry can bedefined as a function of a number of parameters Afterwardthe physical entities defined in the grid are mapped into
materials with macroscopic cross sections which maydepend on the spatial coordinates x bymeans of intermediatefunctions such as burn-up or temperatures distributionswhich in turn may be defined by algebraic expressions byinterpolating data located in files by reading shared-memoryobjects or by a combination of them In the same sense
14 Science and Technology of Nuclear Installations
Largest eigenvalue 1029695 (288387 pcm)1108 (3045 3045)833 (13000 5000)
Number of unknowns 8234Outer iterations 3Linear iterations 32Inner iterations 1029Residual normRelative error
Max 1206012(x y) coreMax 1206012(x y) reflector
2028 times 10minus12
9993 times 10minus13
1012 times 10minus12
keff
Relative errorError estimateMemory used 75468 kBSoft page faults 20094Hard page faults 0Total CPU time 1256 seconds
9993 times 10
1012 times 10minus12
Milongarsquos 2D LWR IAEA benchmark problem case no 058eighth-symmetry core meshed using Delaunay (quads 985747c = 2) solved with finite volumes
(a)
1 074 3207 5472 129 4136 9573 144 4534 10644 120 3828 8875 061 2638 4506 093 2985 6907 094 2948 6948 072 2053 5669 mdash 357 859
10 142 4478 105011 146 4617 108412 130 4111 96413 106 3396 78614 103 3263 76515 095 2991 70616 071 2005 55217 mdash 340 81618 145 4588 107719 133 4206 98720 117 3698 86821 107 3362 79122 098 2918 72523 062 1686 52024 mdash 258 62925 118 3737 87626 096 3073 71227 091 2848 67128 080 2272 63129 mdash 609 127230 mdash 080 31731 047 2035 34832 069 2081 50933 054 1458 44834 mdash 255 64635 052 1413 43736 mdash 409 85137 mdash 064 25138 mdash 071 284
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 16 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
boundary conditions are applied to appropriate physicalentities of dimension119898minus1The problemmatrices119877 and119865 arethen built and stored in an appropriate sparse format usingthe free PETSc [17 18] library and the eigenvalue problemis solved using the free SLEPc [19] library The results arestored into milongarsquos variables and functions which may be
evaluated integratedmdashusually using the free GNU ScientificLibrary [20] routinesmdashand of course written into appropriateoutputs Milonga can also solve problems parametrically andbe used to solve optimization problems As stated above thecode is a free software so corrections and contributions aremore than welcome
Science and Technology of Nuclear Installations 15
Largest eigenvalue 1029462 (286185 pcm)1136 (3093 2953)1233 (13100 5100)
Number of unknowns 6228Outer iterations 2Linear iterations 24Inner iterations 778Residual normR l ti
Max 1206012(x y) coreMax 1206012(x y) reflector
26 times 10minus8
1281 times 10minus8
4996 times 10minus9
keff
Relative errorError estimateMemory used 41460 kBSoft page faults 10930Hard page faults 0Total CPU time 0604 seconds
1281 times 10 8
4996 times 10minus9
Milongarsquos 2D LWR IAEA benchmark problem case no 073eighth-symmetry core meshed using Delaunay (quads 985747c = 2) solved with finite volumes
(a)
1 075 3297 5552 134 4268 9923 148 4660 10944 123 3926 9145 059 2645 4406 094 2997 6977 093 2914 6858 072 2002 5349 mdash 301 773
10 146 4606 108011 150 4735 111112 133 4212 98813 108 3448 80214 104 3271 76715 094 2954 69616 070 1954 52117 mdash 288 73518 149 4691 110119 136 4289 100720 119 3748 88021 107 3359 79022 096 2880 71423 064 1636 47524 mdash 217 56325 121 3810 89426 098 3110 72427 090 2829 66628 081 2227 59929 mdash 536 125730 mdash 062 25031 045 2002 33632 068 2040 50033 055 1414 40934 mdash 215 57935 055 1359 40836 mdash 359 83437 mdash 050 19938 mdash 061 236
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 17 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
31 A Coarse Bare Homogeneous Circle Figures 5 and 6 showthe results of solving a bare two-dimensional circular homo-geneous reactor of radius 119886 over an unstructured grid whichis deliberately coarse using the finite volumes method andthe finite elements method respectively Two group energies
were used and a null-flux boundary conditionwas fixed at theexternal surface Figures 5(a) and 6(a) compare the obtainedfast flux distributions In the first case the numerical solutionprovidesmean values for each neutron flux group in each cellwhile in the latter the solution is computed at the nodes and
16 Science and Technology of Nuclear Installations
Largest eigenvalue 1029851 (289858 pcm)1090 (3182 3182)851 (13000 5000)
Number of unknowns 1752Outer iterations 3Linear iterations 32Inner iterations 219Residual norm
Max 1206012(x y) coreMax 1206012(x y) reflector
566 times 10minus12
2789 times 10minus12
1595 times 10minus12
keff
Residual normRelative errorError estimateMemory used 42564 kBSoft page faults 11213Hard page faults 0Total CPU time 09921 seconds
566 times 10
2789 times 10minus12
1595 times 10minus12
Milongarsquos 2D LWR IAEA benchmark problem case no 076eighth-symmetry core meshed using Delaunay (quads 985747c = 4) solved with finite volumes
(a)
1 073 3151 5402 127 4059 9393 141 4457 10464 118 3768 8715 061 2615 4506 093 2979 6887 094 2971 6998 067 2092 5829 mdash 375 885
10 139 4397 103111 144 4541 106612 128 4052 95013 105 3365 77814 103 3259 76415 096 3016 71216 066 2045 56817 mdash 359 84118 143 4520 106119 132 4155 97520 116 3672 86221 107 3366 79222 098 2951 73323 054 1728 54324 mdash 273 65125 117 3700 86826 095 3057 70827 091 2860 67428 074 2311 64829 mdash 642 130030 mdash 084 33231 048 2041 35332 069 2103 51333 046 1493 46734 mdash 269 66735 046 1444 45536 mdash 432 87137 mdash 067 26338 mdash 074 297
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 18 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
continuous functions are evaluated by means of the shapefunctions used in the formulation Figures 5(b) and 6(b)illustrate the fast fluxes unknowns and its relative position inspace It can be noted that the mesh coarseness gives resultsthat differ substantially in both cases Finally the structure ofthe sparse eigenvalue problemmatrices is shown for each case
with blue red and cyan representing positive negative andexplicitly inserted zero values In the finite volume case thereare 184 unknowns (92 cells times 2 groups) while in the secondcase there are 218 unknowns (109 nodes times 2 groups) Thevolumesrsquo fissionmatrix is almost diagonal it has a bandwidthequal to the number of energy groups which in this case is
Science and Technology of Nuclear Installations 17
500 1000 2000 5000 10000 20000 50000Number of unknowns
2840
2860
2880
2900
2920
ORNLRisoslashKWUOntarioQuarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumes
Quarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elementsEighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumes
Eighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
120588(p
cm)
105 2 times 105 5 times 105
Figure 19 Static reactivity versus number of unknowns The four original solutions as published in 1977 [10] are included as reference
twoThe off-diagonal values appear right next to the diagonalelements because of the chosen ordering of the unknownsin the flux vector 120601 isin R184 Other orderings may be usedbut the rate of convergence of the eigenvalue problem can bedeteriorated On the other hand the elementsrsquo fission matrixhas a nontrivial structure because the grid is unstructuredand the net fission rate at each element depends on the fluxesat the nodes whose location inside thematrix depends in turnon how the gridwas generatedThis effect is similar to the onethat appears in structural analysis where mass matrices needto be lumped in order to simplify transient computations [21]The diagonal block of elementrsquos 119877 matrix and the null blockin 119865 correspond to the discrete equations that set the null-flux boundary conditions on the nodes located at the externalsurface Other types of boundary conditions lead to differentkinds of structures within the problem matrices
In case a part of the domain contained a nonmultiplicativematerial such as a reflector then there would appear sectionsof the fission matrix with null values in both methodsrendering 119865 singular in both methods Care should be takenwhen dealing with the numerical schemes for the eigenvalueproblem solution It can be seen in the elementsrsquo matricesa particular structure that implements the boundary condi-tions at the external surfaces This structure does not appearin the volumesrsquo matrices because the boundary conditions
appear as flux terms which are summed up over all thesurfaces of each cell so they aremasked inside the volumetricdiscretization of the divergence and gradient operators
As the two-group neutron diffusion equation with uni-form cross sections over a circle subject to null-flux bound-ary conditions has an analytical solution it is adequate tocompare how the two proposed numerical schemes relate toit In the studied problem we ignored upscattering and fastfissions Then the analytical effective multiplication factor is
119896eff =]Σ1198912 sdot Σ1199041rarr2
[Σ1198861 + Σ1199041rarr2 + 1198631(]0119886)2] [Σ1198862 + 1198632(]0119886)
2]
(7)
being ]0 = 24048 the smallest root of Besselrsquos first-kindfunction of order zero 1198690(119903)
Figure 7 shows how the two numerical effective multi-plication factors compare to the analytical solution givenby (7) as a function of the mesh refinement indicated bythe quotient 119886ℓ119888 between the radius of the circle and thecharacteristic length of the cellelements We can see thatthe 119896eff computed by the finite volumes (elements) methodis always greater (less) than the analytical solution Indeedit can be proven that for a bare one-dimensional slab thisis always the case [13] However this result does not hold
18 Science and Technology of Nuclear Installations
2000 3000 6000 10000 20000 30000 60000
01
003
001
03
1
3
10
30
100
300To
tal w
all t
ime (
seco
nds)
Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
Figure 20 Total wall time versus number of unknowns
for problems with nonuniform cross sections even in simpleone-dimensional reflected reactors
We may draw two other conclusions from Figure 7 Firstthat the finite element method seems to provide a bettersolution than the volumes-based scheme and second thateven though the error committed tends to decrease with finergrids its behavior is not monotonic for finite volumes Infact Figure 8 shows the difference between the numericaland analytical solutions using a logarithmic vertical scalewhere both conclusions are even more evident We deferthe discussion of such differences between the discretizationscheme until the next section It is worth to note howeverthat the fact that a finite-element-based scheme throws betterresults for the particular bare homogeneous circular reactorunder study than the finite volumes does not imply thatfinite volumes ought to be discarded as a valid tool forsolving the neutron diffusion equation in general cases Thecombination of lattice and core-level computations is usuallyperformedusing cell-based resultswhichwhen fed into node-based methods to solve the few-group neutron diffusionequation may introduce errors which potentially can leadto unacceptable solutions Nevertheless this analysis is farbeyond the scope of this paper which focuses on solving amathematical equation over unstructured grids
32 The 2D IAEA PWR Benchmark Problem This is aclassical two-group neutron diffusion problem first designedin the early 1970s and taken as a reference benchmark forcomputational codes A number of codes were used to solveeither this problem or its three-dimensional formulation[22 23] including milonga using a structured grid [24] Theoriginal formulation can be found in the reference [10] andthere is a reproduction that may be easily found online inreference [24] The geometry consists of a one-quarter ofa PWR core depicted in Figure 9 and the homogeneousmacroscopic cross sections are listed in Table 1 An axialbuckling term 119861
2
119911119892= 08times10
minus4 should be taken into accountThe external surface should be subject to a zero incomingcurrent condition which may be written as
120597120601119892
120597119899= minus
04692
119863119892
sdot 120601119892 (8)
The expected results are as follows
(1) Maximum eigenvalue(2) Fundamental flux distributions
(a) Radial flux traverses 120601119892(119909 0) and 120601119892(119909 119909)
Science and Technology of Nuclear Installations 19
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
1
3
10
30
100
300Ti
me c
onsu
med
to m
esh
the g
eom
etry
(sec
onds
)
Figure 21 Time needed to mesh the geometry versus number of unknowns
Table 1 Macroscopic cross-sections (units are not stated in the original reference but they are assumed to be in cmminus1 or cm as appropriate)
1198631
1198632
Σ1rarr2
Σ1198861
Σ1198862
]Σ1198912
Material1 15 04 002 001 0080 0135 Fuel 12 15 04 002 001 0085 0135 Fuel 23 15 04 002 001 0130 0135 Fuel 2 + rod4 20 03 004 0 0010 0 Reflector
Note the fluxes shall be normalized such that
1
119881coreint119881core
sum
119892
]Σ119891119892 sdot 120601119892119889119881 = 1 (9)
(b) Value and location of maximum power densityThis corresponds to maximum of 1206012 in thecore It is recommended that the maximumvalues of 1206012 both in the inner core and at thecorereflector interface be given
(3) Average subassembly powers 119875119896
119875119896 =1
119881119896
int119881119896
sum
119892
]Σ119891119892 sdot 120601119892119889119881 (10)
where 119881119896 is the volume of the 119896th subassembly and119896 designates the fuel subassemblies as shown inFigure 9
(4) Number of unknowns in the problem number ofiterations and total and outer
(5) Total computing time iteration time IO-time andcomputer used
(6) Type and numerical values of convergence criteria(7) Table of average group fluxes for a square mesh grid
of 20 times 20 cm(8) Dependence of results on mesh spacing
Even though the original problem is based on a quarter-core situation the problem has an eighth-core symmetry
20 Science and Technology of Nuclear Installations
Quarter Delaunay triangles elements
Tim
e con
sum
ed to
read
the m
esh
(sec
onds
)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumes
Quarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
1
3
10
30
100
300
Figure 22 Time needed to read the mesh versus number of unknowns
which cannot be taken into account by structured gridswhich are the main target of the benchmark Howevernonstructured grids can take into consideration any kind ofsymmetry almost without loss of accuracy and at the sametime reducing roughly the number of unknowns by a halfand the associated computational effort needed to solve theproblem by a factor of four Answers to items (1)ndash(7) canbe given completely by milonga using a single input file(see Supplementary data in SupplementaryMaterial availableonline at httpdxdoiorg1011552013641863) As asked initem (8) how results depend not only on the mesh spacingbut also on the meshing algorithm on the gridrsquos elementarygeometric shape and on the discretization scheme may shedlights on the subject whichmay be evenmore interesting thanthe numerical results themselves
Taking advantage of milongarsquos capability of reading andparsing command-line arguments the selection of the coregeometry (quarter or eighth) the meshing algorithm (delau-nay [16] or delquad [25]) the shape of the elementaryentities (triangles or quadrangles) the discretization scheme(volumes or elements) and the characteristic length ℓ119888 ofthe mesh can be provided at run time Fixing five values forℓ119888 = 4 3 2 1 05 gives 2 times 2 times 2 times 2 times 5 = 80 possiblecombinations which we solve with successive invocations to
milonga with the same input file but with different argumentsfrom a simple script Figures 10 11 12 13 14 15 16 17 and18 show the results corresponding to items (1)ndash(7) for someillustrative cases The complete set of figures and the codeused to generate themmay be provided upon request Table 2compiles the answers to the problem for every case studied
As the milonga code is still under development itsnumerical routines are not yet fully optimized nor designedfor parallel computation Therefore the reported times areonly rough estimates and should be taken with care Thesolution comprises five steps
(1) generate the grid with the requested geometry mesh-ing algorithm basic shape and characteristic lengthby calling to gmsh
(2) read the generated mesh(3) build the matrices(4) solve the eigenvalue problem(5) compute the requested results
The CPU time reported in Figures 10ndash18 is thus thesum of all of these five steps but not the time needed togenerate the figuresmdashwhich in some caseswith coarsemeshes
Science and Technology of Nuclear Installations 21
Table2
Resultsofthe2
DIAEA
PWRBe
nchm
arkp
roblem
obtained
with
them
ilong
acod
efor
thee
ightyp
ropo
sedcombinatio
nsofsymmetrym
eshing
algorithm
basicshapediscretization
schemeandcharacteris
ticelem
entc
elllengthTh
ereference
solutio
nis120588=28749times10minus5w
hich
correspo
ndsto119896eff=10296
Case
no
Symm-
etry
Mesh
algorithm
Basic
shape
Solutio
nmetho
dℓ119888
(cm)
Δ120588
(pcm
)max1206012
(mdash)
Total
unkn
owns
Outer
iter
Linear
iter
Inner
iter
Resid
ual
norm
Relativ
eerror
Error
estim
ate
Mem
ory
(Mb)
Page
faults
114
Delaunay
998779Vo
lumes
40
minus120
1145
8264
332
1033
6119890minus09
3119890minus09
1119890minus09
349470
214
Delaunay
998779Vo
lumes
30
85
1135
15740
332
1967
2119890minus08
1119890minus08
4119890minus09
4813401
314
Delaunay
998779Vo
lumes
20
197
1124
31188
332
3898
1119890minus08
6119890minus09
3119890minus09
7621932
414
Delaunay
998779Vo
lumes
10153
1123
127476
332
15934
7119890minus09
3119890minus09
3119890minus09
273
81886
514
Delaunay
998779Vo
lumes
05
187
1119
510676
332
63834
2119890minus09
9119890minus10
1119890minus09
1148
352261
614
Delaunay
998779Elem
ents
40
173
1100
4308
332
538
2119890minus08
8119890minus09
4119890minus09
318794
714
Delaunay
998779Elem
ents
30
117
1107
8108
332
1013
5119890minus09
2119890minus09
2119890minus09
4712946
814
Delaunay
998779Elem
ents
20
84
1112
15936
332
1992
6119890minus09
3119890minus09
2119890minus09
7321347
914
Delaunay
998779Elem
ents
1062
1116
64478
332
8059
6119890minus09
3119890minus09
4119890minus09
262
77105
1014
Delaunay
998779Elem
ents
05
58
1117
256700
332
32087
1119890minus09
5119890minus10
1119890minus09
1091
331969
1114
Delaunay
◻Vo
lumes
40
53
1149
5042
332
630
2119890minus08
1119890minus08
5119890minus09
288007
1214
Delaunay
◻Vo
lumes
30
346
1133
8560
332
1070
7119890minus09
3119890minus09
2119890minus09
3910895
1314
Delaunay
◻Vo
lumes
20
308
1131
15576
332
1947
1119890minus08
7119890minus09
3119890minus09
541564
514
14Delaunay
◻Vo
lumes
10177
1122
60774
332
7596
8119890minus09
4119890minus09
3119890minus09
167
50115
1514
Delaunay
◻Vo
lumes
05
199
1120
244936
332
30617
4119890minus09
2119890minus09
2119890minus09
698
215678
1614
Delaunay
◻Elem
ents
40
196
1100
5222
332
652
3119890minus08
1119890minus08
8119890minus09
4713098
1714
Delaunay
◻Elem
ents
30
123
1107
8810
332
1101
2119890minus08
9119890minus09
7119890minus09
6918598
1814
Delaunay
◻Elem
ents
20
901112
15922
332
1990
1119890minus08
5119890minus09
5119890minus09
107
30591
1914
Delaunay
◻Elem
ents
1063
1116
61456
332
7682
5119890minus09
2119890minus09
4119890minus09
389
113237
2014
Delaunay
◻Elem
ents
05
58
1117
246332
332
30791
8119890minus10
4119890minus10
1119890minus09
1676
498192
2114
Delq
uad
998779Vo
lumes
40
minus45
1144
6228
332
778
4119890minus08
2119890minus08
7119890minus09
308495
2214
Delq
uad
998779Vo
lumes
30
570
1153
11820
332
1477
3119890minus08
1119890minus08
4119890minus09
4010879
2314
Delq
uad
998779Vo
lumes
20
459
1103
24100
332
3012
2119890minus08
1119890minus08
5119890minus09
6217715
2414
Delq
uad
998779Vo
lumes
10512
1104
9640
03
3212050
2119890minus08
1119890minus08
9119890minus09
207
61714
2514
Delq
uad
998779Vo
lumes
05
528
1106
385600
332
48200
2119890minus09
1119890minus09
2119890minus09
838
255735
2614
Delqu
ad998779
Elem
ents
40
220
1093
3288
332
411
3119890minus08
2119890minus08
6119890minus09
308495
2714
Delq
uad
998779Elem
ents
30
140
1104
6148
332
768
4119890minus08
2119890minus08
8119890minus09
39110
0028
14Delq
uad
998779Elem
ents
20
82
1110
12392
332
1549
2119890minus08
9119890minus09
6119890minus09
6117067
2914
Delqu
ad998779
Elem
ents
1063
1115
48882
332
6110
2119890minus09
1119890minus09
1119890minus09
197
5804
630
14Delq
uad
998779Elem
ents
05
58
1116
194162
332
24270
2119890minus09
9119890minus10
2119890minus09
790
238769
3114
Delq
uad
◻Vo
lumes
40
minus416
1174
3132
332
391
5119890minus08
3119890minus08
8119890minus09
267322
3214
Delq
uad
◻Vo
lumes
30
minus248
1151
5922
332
740
9119890minus09
4119890minus09
2119890minus09
318779
3314
Delq
uad
◻Vo
lumes
20
minus132
1139
12050
332
1506
1119890minus08
7119890minus09
4119890minus09
4412239
3414
Delq
uad
◻Vo
lumes
10minus49
1126
48200
332
6025
6119890minus09
3119890minus09
2119890minus09
122
36094
3514
Delq
uad
◻Vo
lumes
05
minus23
1123
192800
332
24100
5119890minus09
2119890minus09
3119890minus09
464
140228
3614
Delq
uad
◻Elem
ents
40
224
1093
3294
332
411
3119890minus08
1119890minus08
7119890minus09
359852
3714
Delq
uad
◻Elem
ents
30
141
1104
6144
332
768
3119890minus08
2119890minus08
9119890minus09
5114018
3814
Delq
uad
◻Elem
ents
20
921111
12392
332
1549
1119890minus08
6119890minus09
5119890minus09
8122526
3914
Delq
uad
◻Elem
ents
1065
1115
48882
332
6110
5119890minus09
3119890minus09
4119890minus09
276
78431
22 Science and Technology of Nuclear Installations
Table2Con
tinued
Case
no
Symm-
etry
Mesh
algorithm
Basic
shape
Solutio
nmetho
dℓ119888
(cm)
Δ120588
(pcm
)max1206012
(mdash)
Total
unkn
owns
Outer
iter
Linear
iter
Inner
iter
Resid
ual
norm
Relativ
eerror
Error
estim
ate
Mem
ory
(Mb)
Page
faults
4014
Delq
uad
◻Elem
ents
05
59
1117
194162
332
24270
1119890minus09
5119890minus10
1119890minus09
1104
318967
4118
Delaunay
998779Vo
lumes
40
minus99
1139
4228
332
528
4119890minus12
2119890minus12
1119890minus12
359630
4218
Delaunay
998779Vo
lumes
30
50
1132
7712
332
964
1119890minus11
6119890minus12
2119890minus12
4211578
4318
Delaunay
998779Vo
lumes
20
177
1120
15776
332
1972
2119890minus12
1119890minus12
7119890minus13
5515436
4418
Delaunay
998779Vo
lumes
10142
1118
63902
332
7987
3119890minus12
1119890minus12
1119890minus12
160
46691
4518
Delaunay
998779Vo
lumes
05
176
1115
254612
332
31826
2119890minus12
8119890minus13
1119890minus12
555
168526
4618
Delaunay
998779Elem
ents
40
183
1096
2252
332
281
6119890minus12
3119890minus12
2119890minus12
359589
4718
Delaunay
998779Elem
ents
30
119
1104
4040
224
505
2119890minus08
1119890minus08
7119890minus09
41114
3748
18Delaunay
998779Elem
ents
20
85
1108
8156
332
1019
2119890minus12
8119890minus13
6119890minus13
5515053
4918
Delaunay
998779Elem
ents
1063
1112
32480
332
4060
1119890minus12
6119890minus13
8119890minus13
151
43671
5018
Delaunay
998779Elem
ents
05
58
1113
128358
332
1604
47119890minus13
3119890minus13
7119890minus13
527
157581
5118
Delaunay
◻Vo
lumes
40
37
1144
2380
224
297
5119890minus08
2119890minus08
1119890minus08
5314173
5218
Delaunay
◻Vo
lumes
30
218
1120
4194
332
524
7119890minus12
4119890minus12
2119890minus12
3610034
5318
Delaunay
◻Vo
lumes
20
345
1110
7960
332
995
5119890minus12
3119890minus12
1119890minus12
4712879
5418
Delaunay
◻Vo
lumes
10165
1117
30652
332
3831
2119890minus12
1119890minus12
7119890minus13
100
28851
5518
Delaunay
◻Vo
lumes
05
191
1117
122066
224
15258
2119890minus08
8119890minus09
9119890minus09
343
103825
5618
Delaunay
◻Elem
ents
40
175
1097
2526
332
315
4119890minus12
2119890minus12
1119890minus12
6016159
5718
Delaunay
◻Elem
ents
30
114
1104
4386
332
548
4119890minus12
2119890minus12
2119890minus12
5013768
5818
Delaunay
◻Elem
ents
20
901108
8234
332
1029
2119890minus12
1119890minus12
1119890minus12
7320094
5918
Delaunay
◻Elem
ents
1062
1112
31186
224
3898
2119890minus08
7119890minus09
9119890minus09
208
59589
6018
Delaunay
◻Elem
ents
05
58
1113
123122
224
15390
9119890minus09
5119890minus09
1119890minus08
807
2366
7061
18Delq
uad
998779Vo
lumes
40
minus210
1168
3224
332
403
1119890minus11
5119890minus12
2119890minus12
43117
8462
18Delq
uad
998779Vo
lumes
30
369
1150
6000
332
750
1119890minus11
6119890minus12
3119890minus12
5013502
6318
Delq
uad
998779Vo
lumes
20
312
1104
12316
332
1539
2119890minus11
8119890minus12
5119890minus12
6016395
6418
Delq
uad
998779Vo
lumes
1044
01094
48714
332
6089
9119890minus12
4119890minus12
4119890minus12
1183444
165
18Delq
uad
998779Vo
lumes
05
534
1081
193980
332
24247
5119890minus12
2119890minus12
4119890minus12
423
126768
6618
Delq
uad
998779Elem
ents
40
240
1090
1750
332
218
9119890minus12
4119890minus12
2119890minus12
43117
8667
18Delqu
ad998779
Elem
ents
30
150
1100
3184
332
398
1119890minus11
5119890minus12
2119890minus12
5013509
6818
Delq
uad
998779Elem
ents
20
89
1106
6426
332
803
5119890minus12
3119890minus12
2119890minus12
5916047
6918
Delq
uad
998779Elem
ents
1064
1111
24886
332
3110
2119890minus12
1119890minus12
1119890minus12
113
32788
7018
Delq
uad
998779Elem
ents
05
59
1113
98052
224
12256
1119890minus08
5119890minus09
8119890minus09
398
118153
7118
Delq
uad
◻Vo
lumes
40
minus398
1172
1650
332
206
1119890minus11
5119890minus12
2119890minus12
369907
7218
Delq
uad
◻Vo
lumes
30
minus249
1146
3058
332
382
4119890minus12
2119890minus12
1119890minus12
349349
7318
Delq
uad
◻Vo
lumes
20
minus131
1136
6228
224
778
3119890minus08
1119890minus08
5119890minus09
4010930
7418
Delq
uad
◻Vo
lumes
10minus51
1123
24536
224
3067
3119890minus08
2119890minus08
1119890minus08
7822716
7618
Delq
uad
◻Elem
ents
40
237
1090
1752
332
219
6119890minus12
3119890minus12
2119890minus12
4111213
7718
Delq
uad
◻Elem
ents
30
145
1101
3184
332
398
6119890minus12
3119890minus12
2119890minus12
43118
1178
18Delq
uad
◻Elem
ents
20
941107
6426
332
803
3119890minus12
1119890minus12
1119890minus12
6016220
7918
Delqu
ad◻
Elem
ents
1065
1111
24874
332
3109
1119890minus12
6119890minus13
8119890minus13
156
4400
080
18Delq
uad
◻Elem
ents
05
59
1113
9804
22
2412255
8119890minus09
4119890minus09
8119890minus09
567
162168
Science and Technology of Nuclear Installations 23
Tim
e con
sum
ed to
bui
ld th
e mat
rices
(sec
onds
)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay trianglesvolumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
10
30
100
300
1
3
Figure 23 Time needed to build the matrices versus number of unknowns
was considerably larger than the solution time itself Theeigenvalue problem was solved using a multilayer iterativeKrylov-Schur method [26] with a tolerance relative to thematrices norm
1003817100381710038171003817119877 sdot 120601 minus (1119896eff) sdot 119865 sdot 1206011003817100381710038171003817
119877 + (1119896eff) sdot 119865lt 10minus8 (11)
The reported residual norm and relative error are10038171003817100381710038171003817100381710038171003817119877 sdot 120601 minus
1
119896effsdot 119865 sdot 120601
10038171003817100381710038171003817100381710038171003817
1003817100381710038171003817119877 sdot 120601 minus (1119896eff) sdot 119865 sdot 1206011003817100381710038171003817
1003817100381710038171003817(1119896eff) sdot 1206011003817100381710038171003817
(12)
respectively The fields marked as outer linear and inneriterations refer to the number of steps needed to attainthe requested tolerance in each layer of the Krylov-Schuralgorithm The computer used to solve the problem has anIntel i7 920 267GHz processor with 4Gb of RAM runningDebian GNULinux Wheezy
When using a finite volumes-based scheme over anunstructured mesh the solver has to gather information
about which cells are neighbors and which are not Currentlygmsh does not write this kind of lists in its output filesso milonga has to explicitly solve the neighbors problemPerforming a linear search is an 119874(119873
2) task which is
unacceptable for problem sizes 119873 of interest The code usesa search based on a 119896-dimensional tree [27] which can inprinciple be performed in119874(119873) steps Still for large values of119873 the time needed to read and parse the mesh (number twopreviously mentioned) is the bottleneck of the solution Thisstep is not needed in finite elements although the construc-tion of the elementary matrices involves the computationof the multidimensional Jacobians and integrals which thenhave to be assembled Again for large 119873 this step (numberthree) takes up most of the time needed to solve the problem
The Delaunay algorithm is a standard method for gen-erating two-dimensional grids [16] by tessellating a planewith triangles If the mesh needs to be based on quadranglesinstead a recombination algorithm can be used to transformtwo adjacent triangles in one quadrangle whenever is possi-ble However for geometries which are based on rectangularshapes there exist other algorithms [25] both for meshingand for recombining the triangles that give rise to elementaryentities with right angles almost everywhere which may be a
24 Science and Technology of Nuclear Installations
Tim
e con
sum
ed to
solv
e the
eige
nval
ue p
robl
em (s
econ
ds)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
10
30
100
300
1
3
Figure 24 Time needed to solve the eigenvalue problem versus number of unknowns
desired property of the resulting grid For finite elements thesteps needed to build the matrices depend on the selectionof triangles or quadrangles as the basic elementary geometrybecause the shape functions change However the number ofunknowns is the same as the number of nodes that does notchange after a recombination procedure On the other handthe number of unknowns in the finite volumes schemes withtriangles is roughly twice as the number of unknowns withquadrangles for the same grid
Figure 19 shows how the computed static reactivity (ie1 minus 1119896eff) depends on the number of unknowns for eachof the sixteen combinations of geometry algorithm shapescheme The four accepted results published in the originalreference [10] are included for reference although it shouldbe taken into account that said reactivities were computedalmost forty years ago Figure 20 shows the total wall timeneeded to solve the problem as a function of 119873119866 whileFigures 21 22 23 and 24 show the times needed for eachof the first four steps involved in the solution maintainingthe same logarithmic scale for both the abscissas and theordinates Green data represent finite volumes whilst bluebullets correspond to finite elements Solid lines indicatequarter-core and dashed lines eighth-symmetry Fillet bullets
are results obtained by theDelaunay triangulation and emptybullets were computed with the delquad algorithm Finallytriangle-shaped data correspond to triangles and squares anddiamonds denote quadrangles as the basic geometry of thegrid
It can be seen that finite elements produce amuch smallerdispersion of eigenvalues 119896eff than finite volumes with therefinement of the mesh This can be explained because finitevolumes methods rely on a geometric condition of the meshwhich may change abruptly if the meshing algorithm decidesto allocate the cells in a rather different form for small changesin ℓ119888 Finite elements methods are less influenced by thesediscontinuous lay-out changes of the elements As expectedeighth-core symmetries give almost the same results as thequarter-core geometries with roughly half the unknownsFor small problems finite volumes run faster that finiteelements because the time needed to solve the neighborproblem is negligible When the problem size grows thistime increases and exceeds the overhead implied in the con-struction and assembly of the finite elements matrices Alsoat least for this configuration it is seen that the eigenvalueproblem is solved faster for finite volumes than for finiteelements
Science and Technology of Nuclear Installations 25
4 Conclusions
Unstructured grids provide the cognizant engineer witha wide variety of possibilities to deal with the design oranalysis of nuclear reactor cores These kinds of grids cansuccessfully reproduce continuous geometries commonlyfound in reactor cores such as cylinders and thereforenot only can the diffusion equation be better approximatedinside the domain of definition but also the fulfillment ofboundary conditions is improved A free computer codewas written from scratch that is able to completely solvethe 2D IAEA PWR Benchmark using unstructured gridsfor sixteen combinations of geometry meshing algorithmbasic shape and discretization scheme plus any value ofthe gridrsquos characteristic length by using a single input fileThe complete set of input files and codemdashexecutable andsourcemdashis available either online or upon request withcomments experiences suggestions and corrections beingmore than welcome Further development should includetackling full three-dimensional geometries with completethermal hydraulic feedback in order to analyze how thesolutions of the coupled neutronic-thermal problem dependon the spatial discretization scheme of the neutron leakageterm Parallelization of the computation and assembly of thematrices and of the solution of the eigenvalue problem andits implementation using GPUs are also desired features toimplement A problemwith direct applications that the futureversions of milonga ought to solve is the analysis of how thegeometry of the absorbing materials should be taken intoaccount in structured coarse grids in order to mitigate effectssuch as the rod-cusp problem
Suitable schemes for approximating the continuous dif-ferential operators by discrete matrix expressions includefinite volumes and finite elements families Finite volumesmethods compute cell mean values whilst finite elementsgive functional values at the gridrsquos nodes In general finiteelements are less sensitive to changes in the mesh so theresults they provide do not change significantly for differentmeshing algorithms or elementary shapes Small problemsare best solved by finite volumes as the neighbor-findingproblem is faster than the process of building and assemblingthe eigenvalue-problem matrices For a large number ofunknowns the process of finding which cell is neighbor ofwhichmdasheven based on a 119896-dimensional treemdashoverwhelmsthe computation and assembly of elementary matrices andfinite-element methods perform better
References
[1] G Glasstone and S Bell Nuclear Reactor Theory KriegerPublishing Company 1970
[2] J J Duderstadt and L J Hamilton Nuclear Reactor AnalysisJohn Wiley amp Sons New York NY USA 1976
[3] C Gho Reactor Physics Undergraduate Course Lecture NotesInstituto Balseiro 2006
[4] G G Theler and F J Bonetto ldquoOn the stability of the pointreactor kinetics equationsrdquoNuclear Engineering andDesign vol240 no 6 pp 1443ndash1449 2010
[5] E E Lewis andW F Miller Computational Methods of NeutronTransport John Wiley amp Sons 1984
[6] E Masiello ldquoAnalytical stability analysis of Coarse-Mesh finitedifference methodrdquo in Proceedings of the International Confer-ence on the Physics of Reactors (PHYSOR rsquo08) Nuclear Power ASustainable Resource pp 456ndash464 September 2008
[7] M L Zerkle Development of a polynomial nodal methodwith flux and current discontinuity factors [PhD thesis] Mas-sachusets Institute of Technology 1992
[8] J I Yoon andHG Joo ldquoTwo-level coarsemesh finite differenceformulation with multigroup source expansion nodal kernelsrdquoJournal of Nuclear Science andTechnology vol 45 no 7 pp 668ndash682 2008
[9] O Mazzantini M Schivo J Di Cesare R Garbero M Riveroand G Theler ldquoA coupled calculation suite for Atucha II oper-ational transients analysisrdquo Science and Technology of NuclearInstallations vol 2011 Article ID 785304 12 pages 2011
[10] Computational Benchmark Problem Comitee for the Mathe-matics and Computation Division of the American NuclearSociety ldquoArgonne Code Center Benchmark problem bookrdquoTech Rep ANL-7416 Supplement 2 Argonne National Labo-ratory 1977
[11] A F Henry Nuclear Reactor Analysis MIT Cambridge UK1975
[12] O C Zienkiewicz R L Taylor and J Z ZhuTheFinite ElementMethod Its Basis and Fundamentals vol 1 Elsevier 6th edition2005
[13] G Theler Difusion de neutrones en mallas no estructuradascomparacion entre volumenes y elementos finitos AcademicMonograph Universidad de Buenos Aires 2013
[14] G Theler ldquoMilonga a free nuclear reactor core analysis coderdquoAvailable online 2011
[15] G Theler F J Bonetto and A Clausse ldquoOptimizacion deparametros en reactores de potencia base de diseno delcodigo neutronico milongardquo in Reunion Anual de la AsociacionArgentina de Tecnologia Nuclear XXXVII 2010
[16] C Geuzaine and J-F Remacle ldquoGmsh a 3-D finite elementmesh generator with built-in pre- and post-processing facili-tiesrdquo International Journal for Numerical Methods in Engineer-ing vol 79 no 11 pp 1309ndash1331 2009
[17] S Balay J Brown K Buschelman et al ldquoPETSc users manualrdquoTech Rep ANL-9511-Revision 34 Argonne National Labora-tory 2013
[18] S Balay W D Gropp L Curfman McInnes and B F SmithldquoEfficient management of parallelism in object oriented numer-ical software librariesrdquo in Modern Software Tools in ScientificComputing E Arge A M Bruaset and H P Langtangen Edspp 163ndash202 Birkhauser 1997
[19] V Hernandez J E Roman and V Vidal ldquoSLEPC a scalable andflexible toolkit for the solution of eigenvalue problemsrdquo ACMTransactions on Mathematical Software vol 31 no 3 pp 351ndash362 2005
[20] M Galassi J Davies J Theiler et al GNU Scientific LibraryReference Manual 3rd edition 2009
[21] O C Zienkiewicz and R L Taylor The Finite Element MethodFor Solid and Structural Mechanics vol 3 Elsevier 6th edition2005
[22] A Imelda and K Doddy Evaluation of the IAEA 3-D PWRbenchmark problem using NESTLE code 2004
[23] R Mosteller ldquoStatic benchmarking of the NESTLE advancednodal coderdquo in Proceedings of the Joint International Conference
26 Science and Technology of Nuclear Installations
on Mathematical Methods and Supercomputing for NuclearApplications vol 2 pp 1596ndash1605 1997
[24] GTheler F J Bonetto andAClausse ldquoSolution of the 2D IAEAPWR Benchmark with the neutronic code milongardquo in Actasde la Reunion Anual de la Asociacion Argentina de TecnologıaNuclear XXXVIII 2011
[25] J F Remacle F Henrotte T Carrier-Baudouin et al ldquoAfrontal delaunay quad mesh generator using the 119871
infin normrdquoInternational Journal For Numerical Methods in Engineeringvol 94 no 5 pp 494ndash512 2013
[26] G W Stewart ldquoA Krylov-Schur algorithm for large eigenprob-lemsrdquo SIAM Journal on Matrix Analysis and Applications vol23 no 3 pp 601ndash614 2002
[27] J L Bentley ldquoMultidimensional binary search trees used forassociative searchingrdquo Communications of the ACM vol 18 no9 pp 509ndash517 1975
TribologyAdvances in
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Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Renewable Energy
Submit your manuscripts athttpwwwhindawicom
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StructuresJournal of
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RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
EnergyJournal of
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Journal ofEngineeringVolume 2014
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Nuclear EnergyInternational Journal of
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High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
10 Science and Technology of Nuclear Installations
Largest eigenvalue 1029695(288388 pcm)1112 (3076 3032)838 (5000 13000)
Number of unknowns 15922Outer iterations 3Linear iterations 32Inner iterations 1990Residual normRelative errorError estimateMemory used 109964 kBSoft page faults 30591Hard page faults 0Total CPU time 158 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
1056 times 10minus8
5202 times 10minus9
5122 times 10minus9
keff
quarter-symmetry core meshed using Delaunay (quads 985747c = 2) solved with finite volumesMilongarsquos 2D LWR IAEA benchmark problem case no 018
(a)
123456789
1011121314151617181920212223242526272829303132333435363738
k Pk 1206011k 1206012k074 3220 549130 4152 961144 4651 1068120 3843 890061 2648 452094 2996 693094 2959 696072 2061 569mdash 358 862142 4495 1054147 4634 1088131 4127 968107 3409 789104 3275 768096 3003 708071 2014 554mdash 342 820146 4606 1081134 4222 991118 3711 871107 3375 794098 2929 727062 1692 522mdash 258 632119 3752 880096 3084 715091 2858 673080 2280 633mdash 611 1277mdash 080 318047 2043 350069 2088 510054 1463 450mdash 255 648051 1418 439mdash 410 854mdash 064 252mdash 071 285
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 12 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
volumes and the finite elements methods by solving a two-group homogeneous bare reactor with the Robin boundaryconditions over the very same grid although a rather coarseone so the differences can be observed directly into theresulting figuresWe then solve the classical two-dimensionalLWR problem also known as the 2D IAEA PWR benchmark
Not only do we show again the differences between the finitevolumes and elements formulation but also we solve theproblemusing different combinations ofmeshing algorithmsbasic shapes and characteristic lengths of the mesh
To solve the two examples shown below we employed themilonga code which was written from scratch by the author
Science and Technology of Nuclear Installations 11
Largest eigenvalue 1029159 (283326 pcm)1174 (2851 3224)1560 (5200 13200)
Number of unknowns 3132Outer iterations 3Linear iterations 32Inner iterations 391Residual normRelative errorError estimateMemory used 26888 kBSoft page faults 7322Hard page faults 0Total CPU time 0308 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
5408 times 10minus8
2665 times 10minus8
8424 times 10minus9
keff
Milongarsquos 2D LWR IAEA benchmark problem case no 031quarter-symmetry core meshed using Delquad (quads 985747c = 4) solved with finite volumes
(a)
123456789
1011121314151617181920212223242526272829303132333435363738
k Pk 1206011k 1206012k077 3422 570140 4468 1040153 4836 1135128 4070 948060 2704 448095 3017 702091 2861 673069 1918 508mdash 264 689151 4785 1122155 4897 1149137 4334 1017111 3535 823104 3272 768092 2908 685067 1876 496mdash 254 658154 4854 1139139 4392 1031120 3792 890107 3360 790095 2831 702061 1574 451mdash 197 515123 3868 907099 3132 730090 2818 663078 2168 579mdash 481 1191mdash 055 217044 1984 328067 2027 498052 1360 388mdash 190 520052 1329 383mdash 321 792mdash 043 168mdash 048 195
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 13 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
of this paper and is currently still under development withinhis ongoing PhD thesis There exists a first public release[14] under the terms of the GNU General Public Licensemdashthat is it is a free softwaremdashthat can only handle structuredgrids There is a second release being prepared whose
main relevance is that it can work with nonstructured gridsas well which is the main feature of the code It uses ageneral mathematical frameworkmdashcoded from scratch aswellmdashwhich provides input file parsing algebraic expres-sions evaluation one- and multidimensional interpolation of
12 Science and Technology of Nuclear Installations
Largest eigenvalue 1029788 (289260 pcm)1120 (3149 3011)1091 (13038 5114)
Number of unknowns 15776Outer iterations 3Linear iterations 32Inner iterations 1972Residual normRelative errorError estimateMemory used 56996 kBSoft page faults 15436Hard page faults 0Total CPU time 09121 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
2296 times 10minus12
1131 times 10minus12
6877 times 10minus13
keff
Milongarsquos 2D LWR IAEA benchmark problem case no 043eighth-symmetry core meshed using Delaunaya (triangs985747c = 2) solved with finite volumes
(a)
1 074 3241 5482 131 4188 9713 145 4582 10764 122 3877 9005 060 2644 4456 093 2986 6927 093 2941 6928 075 2030 5529 mdash 322 779
10 143 4523 106111 148 4667 109512 131 4145 97213 107 3423 79414 103 3269 76715 095 2993 70616 073 1988 53917 mdash 312 75118 147 4637 108819 134 4243 99620 118 3721 87321 107 3370 79322 098 2914 72323 067 1651 49924 mdash 235 57125 119 3763 88226 096 3077 71527 091 2847 67128 083 2255 61729 mdash 576 122530 mdash 068 26631 046 2016 34132 068 2058 50533 059 1443 43534 mdash 233 58935 057 1381 42036 mdash 381 82337 mdash 054 20838 mdash 062 244
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 14 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
scattered data shared-memory access numerical integrationfacilities and so forth The milonga code was designed withfour design basis vectors in mindmdashwhich are thoroughlydiscussed in the documentation [15]mdashwhich includes thetype of problems it can handle the code scalability which
features are expected and what to do with the obtainedresults
It works by first reading an input file that using plain-text English keywords and arguments defines the number119898 of spatial dimensions the number 119866 of group energies
Science and Technology of Nuclear Installations 13
Largest eigenvalue 1029726 (288675 pcm)1104 (3069 3069)864 (13000 5000)
Number of unknowns 4040Outer iterations 2Linear iterations 24Inner iterations 505Residual normRelative errorError estimateMemory used 42940 kBSoft page faults 11437Hard page faults 0Total CPU time 1064 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
2019 times 10minus8
9946 times 10minus9
7138 times 10minus9
keff
Milongarsquos 2D LWR IAEA benchmark problem case no 047eighth-symmetry core meshed using Delaunay (triangles 985747c = 3) solved with finite volumes
(a)
1 074 3194 5452 129 4117 9533 143 4515 10604 119 3814 8835 061 2634 4506 093 2984 6907 094 2954 6958 071 2064 5719 mdash 362 865
10 141 4458 104511 146 4599 107912 130 4097 96113 106 3388 78414 103 3263 76515 095 2998 70716 068 2015 55717 mdash 346 82318 145 4572 107319 133 4194 98420 117 3691 86621 107 3363 79222 098 2925 72623 059 1696 52824 mdash 263 63525 118 3728 87426 096 3068 71127 091 2850 67228 079 2280 63529 mdash 619 127630 mdash 081 32131 047 2036 34932 068 2085 50933 051 1465 45434 mdash 258 65035 049 1419 44336 mdash 416 85437 mdash 065 25438 mdash 071 288
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 15 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
and optionally a mesh file Currently only grids generatedwith the code gmsh [16] are only supported mainly becauseit is also a free software and it suits perfectly well milongarsquosdesign basis in the sense that the continuous geometry can bedefined as a function of a number of parameters Afterwardthe physical entities defined in the grid are mapped into
materials with macroscopic cross sections which maydepend on the spatial coordinates x bymeans of intermediatefunctions such as burn-up or temperatures distributionswhich in turn may be defined by algebraic expressions byinterpolating data located in files by reading shared-memoryobjects or by a combination of them In the same sense
14 Science and Technology of Nuclear Installations
Largest eigenvalue 1029695 (288387 pcm)1108 (3045 3045)833 (13000 5000)
Number of unknowns 8234Outer iterations 3Linear iterations 32Inner iterations 1029Residual normRelative error
Max 1206012(x y) coreMax 1206012(x y) reflector
2028 times 10minus12
9993 times 10minus13
1012 times 10minus12
keff
Relative errorError estimateMemory used 75468 kBSoft page faults 20094Hard page faults 0Total CPU time 1256 seconds
9993 times 10
1012 times 10minus12
Milongarsquos 2D LWR IAEA benchmark problem case no 058eighth-symmetry core meshed using Delaunay (quads 985747c = 2) solved with finite volumes
(a)
1 074 3207 5472 129 4136 9573 144 4534 10644 120 3828 8875 061 2638 4506 093 2985 6907 094 2948 6948 072 2053 5669 mdash 357 859
10 142 4478 105011 146 4617 108412 130 4111 96413 106 3396 78614 103 3263 76515 095 2991 70616 071 2005 55217 mdash 340 81618 145 4588 107719 133 4206 98720 117 3698 86821 107 3362 79122 098 2918 72523 062 1686 52024 mdash 258 62925 118 3737 87626 096 3073 71227 091 2848 67128 080 2272 63129 mdash 609 127230 mdash 080 31731 047 2035 34832 069 2081 50933 054 1458 44834 mdash 255 64635 052 1413 43736 mdash 409 85137 mdash 064 25138 mdash 071 284
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 16 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
boundary conditions are applied to appropriate physicalentities of dimension119898minus1The problemmatrices119877 and119865 arethen built and stored in an appropriate sparse format usingthe free PETSc [17 18] library and the eigenvalue problemis solved using the free SLEPc [19] library The results arestored into milongarsquos variables and functions which may be
evaluated integratedmdashusually using the free GNU ScientificLibrary [20] routinesmdashand of course written into appropriateoutputs Milonga can also solve problems parametrically andbe used to solve optimization problems As stated above thecode is a free software so corrections and contributions aremore than welcome
Science and Technology of Nuclear Installations 15
Largest eigenvalue 1029462 (286185 pcm)1136 (3093 2953)1233 (13100 5100)
Number of unknowns 6228Outer iterations 2Linear iterations 24Inner iterations 778Residual normR l ti
Max 1206012(x y) coreMax 1206012(x y) reflector
26 times 10minus8
1281 times 10minus8
4996 times 10minus9
keff
Relative errorError estimateMemory used 41460 kBSoft page faults 10930Hard page faults 0Total CPU time 0604 seconds
1281 times 10 8
4996 times 10minus9
Milongarsquos 2D LWR IAEA benchmark problem case no 073eighth-symmetry core meshed using Delaunay (quads 985747c = 2) solved with finite volumes
(a)
1 075 3297 5552 134 4268 9923 148 4660 10944 123 3926 9145 059 2645 4406 094 2997 6977 093 2914 6858 072 2002 5349 mdash 301 773
10 146 4606 108011 150 4735 111112 133 4212 98813 108 3448 80214 104 3271 76715 094 2954 69616 070 1954 52117 mdash 288 73518 149 4691 110119 136 4289 100720 119 3748 88021 107 3359 79022 096 2880 71423 064 1636 47524 mdash 217 56325 121 3810 89426 098 3110 72427 090 2829 66628 081 2227 59929 mdash 536 125730 mdash 062 25031 045 2002 33632 068 2040 50033 055 1414 40934 mdash 215 57935 055 1359 40836 mdash 359 83437 mdash 050 19938 mdash 061 236
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 17 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
31 A Coarse Bare Homogeneous Circle Figures 5 and 6 showthe results of solving a bare two-dimensional circular homo-geneous reactor of radius 119886 over an unstructured grid whichis deliberately coarse using the finite volumes method andthe finite elements method respectively Two group energies
were used and a null-flux boundary conditionwas fixed at theexternal surface Figures 5(a) and 6(a) compare the obtainedfast flux distributions In the first case the numerical solutionprovidesmean values for each neutron flux group in each cellwhile in the latter the solution is computed at the nodes and
16 Science and Technology of Nuclear Installations
Largest eigenvalue 1029851 (289858 pcm)1090 (3182 3182)851 (13000 5000)
Number of unknowns 1752Outer iterations 3Linear iterations 32Inner iterations 219Residual norm
Max 1206012(x y) coreMax 1206012(x y) reflector
566 times 10minus12
2789 times 10minus12
1595 times 10minus12
keff
Residual normRelative errorError estimateMemory used 42564 kBSoft page faults 11213Hard page faults 0Total CPU time 09921 seconds
566 times 10
2789 times 10minus12
1595 times 10minus12
Milongarsquos 2D LWR IAEA benchmark problem case no 076eighth-symmetry core meshed using Delaunay (quads 985747c = 4) solved with finite volumes
(a)
1 073 3151 5402 127 4059 9393 141 4457 10464 118 3768 8715 061 2615 4506 093 2979 6887 094 2971 6998 067 2092 5829 mdash 375 885
10 139 4397 103111 144 4541 106612 128 4052 95013 105 3365 77814 103 3259 76415 096 3016 71216 066 2045 56817 mdash 359 84118 143 4520 106119 132 4155 97520 116 3672 86221 107 3366 79222 098 2951 73323 054 1728 54324 mdash 273 65125 117 3700 86826 095 3057 70827 091 2860 67428 074 2311 64829 mdash 642 130030 mdash 084 33231 048 2041 35332 069 2103 51333 046 1493 46734 mdash 269 66735 046 1444 45536 mdash 432 87137 mdash 067 26338 mdash 074 297
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 18 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
continuous functions are evaluated by means of the shapefunctions used in the formulation Figures 5(b) and 6(b)illustrate the fast fluxes unknowns and its relative position inspace It can be noted that the mesh coarseness gives resultsthat differ substantially in both cases Finally the structure ofthe sparse eigenvalue problemmatrices is shown for each case
with blue red and cyan representing positive negative andexplicitly inserted zero values In the finite volume case thereare 184 unknowns (92 cells times 2 groups) while in the secondcase there are 218 unknowns (109 nodes times 2 groups) Thevolumesrsquo fissionmatrix is almost diagonal it has a bandwidthequal to the number of energy groups which in this case is
Science and Technology of Nuclear Installations 17
500 1000 2000 5000 10000 20000 50000Number of unknowns
2840
2860
2880
2900
2920
ORNLRisoslashKWUOntarioQuarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumes
Quarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elementsEighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumes
Eighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
120588(p
cm)
105 2 times 105 5 times 105
Figure 19 Static reactivity versus number of unknowns The four original solutions as published in 1977 [10] are included as reference
twoThe off-diagonal values appear right next to the diagonalelements because of the chosen ordering of the unknownsin the flux vector 120601 isin R184 Other orderings may be usedbut the rate of convergence of the eigenvalue problem can bedeteriorated On the other hand the elementsrsquo fission matrixhas a nontrivial structure because the grid is unstructuredand the net fission rate at each element depends on the fluxesat the nodes whose location inside thematrix depends in turnon how the gridwas generatedThis effect is similar to the onethat appears in structural analysis where mass matrices needto be lumped in order to simplify transient computations [21]The diagonal block of elementrsquos 119877 matrix and the null blockin 119865 correspond to the discrete equations that set the null-flux boundary conditions on the nodes located at the externalsurface Other types of boundary conditions lead to differentkinds of structures within the problem matrices
In case a part of the domain contained a nonmultiplicativematerial such as a reflector then there would appear sectionsof the fission matrix with null values in both methodsrendering 119865 singular in both methods Care should be takenwhen dealing with the numerical schemes for the eigenvalueproblem solution It can be seen in the elementsrsquo matricesa particular structure that implements the boundary condi-tions at the external surfaces This structure does not appearin the volumesrsquo matrices because the boundary conditions
appear as flux terms which are summed up over all thesurfaces of each cell so they aremasked inside the volumetricdiscretization of the divergence and gradient operators
As the two-group neutron diffusion equation with uni-form cross sections over a circle subject to null-flux bound-ary conditions has an analytical solution it is adequate tocompare how the two proposed numerical schemes relate toit In the studied problem we ignored upscattering and fastfissions Then the analytical effective multiplication factor is
119896eff =]Σ1198912 sdot Σ1199041rarr2
[Σ1198861 + Σ1199041rarr2 + 1198631(]0119886)2] [Σ1198862 + 1198632(]0119886)
2]
(7)
being ]0 = 24048 the smallest root of Besselrsquos first-kindfunction of order zero 1198690(119903)
Figure 7 shows how the two numerical effective multi-plication factors compare to the analytical solution givenby (7) as a function of the mesh refinement indicated bythe quotient 119886ℓ119888 between the radius of the circle and thecharacteristic length of the cellelements We can see thatthe 119896eff computed by the finite volumes (elements) methodis always greater (less) than the analytical solution Indeedit can be proven that for a bare one-dimensional slab thisis always the case [13] However this result does not hold
18 Science and Technology of Nuclear Installations
2000 3000 6000 10000 20000 30000 60000
01
003
001
03
1
3
10
30
100
300To
tal w
all t
ime (
seco
nds)
Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
Figure 20 Total wall time versus number of unknowns
for problems with nonuniform cross sections even in simpleone-dimensional reflected reactors
We may draw two other conclusions from Figure 7 Firstthat the finite element method seems to provide a bettersolution than the volumes-based scheme and second thateven though the error committed tends to decrease with finergrids its behavior is not monotonic for finite volumes Infact Figure 8 shows the difference between the numericaland analytical solutions using a logarithmic vertical scalewhere both conclusions are even more evident We deferthe discussion of such differences between the discretizationscheme until the next section It is worth to note howeverthat the fact that a finite-element-based scheme throws betterresults for the particular bare homogeneous circular reactorunder study than the finite volumes does not imply thatfinite volumes ought to be discarded as a valid tool forsolving the neutron diffusion equation in general cases Thecombination of lattice and core-level computations is usuallyperformedusing cell-based resultswhichwhen fed into node-based methods to solve the few-group neutron diffusionequation may introduce errors which potentially can leadto unacceptable solutions Nevertheless this analysis is farbeyond the scope of this paper which focuses on solving amathematical equation over unstructured grids
32 The 2D IAEA PWR Benchmark Problem This is aclassical two-group neutron diffusion problem first designedin the early 1970s and taken as a reference benchmark forcomputational codes A number of codes were used to solveeither this problem or its three-dimensional formulation[22 23] including milonga using a structured grid [24] Theoriginal formulation can be found in the reference [10] andthere is a reproduction that may be easily found online inreference [24] The geometry consists of a one-quarter ofa PWR core depicted in Figure 9 and the homogeneousmacroscopic cross sections are listed in Table 1 An axialbuckling term 119861
2
119911119892= 08times10
minus4 should be taken into accountThe external surface should be subject to a zero incomingcurrent condition which may be written as
120597120601119892
120597119899= minus
04692
119863119892
sdot 120601119892 (8)
The expected results are as follows
(1) Maximum eigenvalue(2) Fundamental flux distributions
(a) Radial flux traverses 120601119892(119909 0) and 120601119892(119909 119909)
Science and Technology of Nuclear Installations 19
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
1
3
10
30
100
300Ti
me c
onsu
med
to m
esh
the g
eom
etry
(sec
onds
)
Figure 21 Time needed to mesh the geometry versus number of unknowns
Table 1 Macroscopic cross-sections (units are not stated in the original reference but they are assumed to be in cmminus1 or cm as appropriate)
1198631
1198632
Σ1rarr2
Σ1198861
Σ1198862
]Σ1198912
Material1 15 04 002 001 0080 0135 Fuel 12 15 04 002 001 0085 0135 Fuel 23 15 04 002 001 0130 0135 Fuel 2 + rod4 20 03 004 0 0010 0 Reflector
Note the fluxes shall be normalized such that
1
119881coreint119881core
sum
119892
]Σ119891119892 sdot 120601119892119889119881 = 1 (9)
(b) Value and location of maximum power densityThis corresponds to maximum of 1206012 in thecore It is recommended that the maximumvalues of 1206012 both in the inner core and at thecorereflector interface be given
(3) Average subassembly powers 119875119896
119875119896 =1
119881119896
int119881119896
sum
119892
]Σ119891119892 sdot 120601119892119889119881 (10)
where 119881119896 is the volume of the 119896th subassembly and119896 designates the fuel subassemblies as shown inFigure 9
(4) Number of unknowns in the problem number ofiterations and total and outer
(5) Total computing time iteration time IO-time andcomputer used
(6) Type and numerical values of convergence criteria(7) Table of average group fluxes for a square mesh grid
of 20 times 20 cm(8) Dependence of results on mesh spacing
Even though the original problem is based on a quarter-core situation the problem has an eighth-core symmetry
20 Science and Technology of Nuclear Installations
Quarter Delaunay triangles elements
Tim
e con
sum
ed to
read
the m
esh
(sec
onds
)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumes
Quarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
1
3
10
30
100
300
Figure 22 Time needed to read the mesh versus number of unknowns
which cannot be taken into account by structured gridswhich are the main target of the benchmark Howevernonstructured grids can take into consideration any kind ofsymmetry almost without loss of accuracy and at the sametime reducing roughly the number of unknowns by a halfand the associated computational effort needed to solve theproblem by a factor of four Answers to items (1)ndash(7) canbe given completely by milonga using a single input file(see Supplementary data in SupplementaryMaterial availableonline at httpdxdoiorg1011552013641863) As asked initem (8) how results depend not only on the mesh spacingbut also on the meshing algorithm on the gridrsquos elementarygeometric shape and on the discretization scheme may shedlights on the subject whichmay be evenmore interesting thanthe numerical results themselves
Taking advantage of milongarsquos capability of reading andparsing command-line arguments the selection of the coregeometry (quarter or eighth) the meshing algorithm (delau-nay [16] or delquad [25]) the shape of the elementaryentities (triangles or quadrangles) the discretization scheme(volumes or elements) and the characteristic length ℓ119888 ofthe mesh can be provided at run time Fixing five values forℓ119888 = 4 3 2 1 05 gives 2 times 2 times 2 times 2 times 5 = 80 possiblecombinations which we solve with successive invocations to
milonga with the same input file but with different argumentsfrom a simple script Figures 10 11 12 13 14 15 16 17 and18 show the results corresponding to items (1)ndash(7) for someillustrative cases The complete set of figures and the codeused to generate themmay be provided upon request Table 2compiles the answers to the problem for every case studied
As the milonga code is still under development itsnumerical routines are not yet fully optimized nor designedfor parallel computation Therefore the reported times areonly rough estimates and should be taken with care Thesolution comprises five steps
(1) generate the grid with the requested geometry mesh-ing algorithm basic shape and characteristic lengthby calling to gmsh
(2) read the generated mesh(3) build the matrices(4) solve the eigenvalue problem(5) compute the requested results
The CPU time reported in Figures 10ndash18 is thus thesum of all of these five steps but not the time needed togenerate the figuresmdashwhich in some caseswith coarsemeshes
Science and Technology of Nuclear Installations 21
Table2
Resultsofthe2
DIAEA
PWRBe
nchm
arkp
roblem
obtained
with
them
ilong
acod
efor
thee
ightyp
ropo
sedcombinatio
nsofsymmetrym
eshing
algorithm
basicshapediscretization
schemeandcharacteris
ticelem
entc
elllengthTh
ereference
solutio
nis120588=28749times10minus5w
hich
correspo
ndsto119896eff=10296
Case
no
Symm-
etry
Mesh
algorithm
Basic
shape
Solutio
nmetho
dℓ119888
(cm)
Δ120588
(pcm
)max1206012
(mdash)
Total
unkn
owns
Outer
iter
Linear
iter
Inner
iter
Resid
ual
norm
Relativ
eerror
Error
estim
ate
Mem
ory
(Mb)
Page
faults
114
Delaunay
998779Vo
lumes
40
minus120
1145
8264
332
1033
6119890minus09
3119890minus09
1119890minus09
349470
214
Delaunay
998779Vo
lumes
30
85
1135
15740
332
1967
2119890minus08
1119890minus08
4119890minus09
4813401
314
Delaunay
998779Vo
lumes
20
197
1124
31188
332
3898
1119890minus08
6119890minus09
3119890minus09
7621932
414
Delaunay
998779Vo
lumes
10153
1123
127476
332
15934
7119890minus09
3119890minus09
3119890minus09
273
81886
514
Delaunay
998779Vo
lumes
05
187
1119
510676
332
63834
2119890minus09
9119890minus10
1119890minus09
1148
352261
614
Delaunay
998779Elem
ents
40
173
1100
4308
332
538
2119890minus08
8119890minus09
4119890minus09
318794
714
Delaunay
998779Elem
ents
30
117
1107
8108
332
1013
5119890minus09
2119890minus09
2119890minus09
4712946
814
Delaunay
998779Elem
ents
20
84
1112
15936
332
1992
6119890minus09
3119890minus09
2119890minus09
7321347
914
Delaunay
998779Elem
ents
1062
1116
64478
332
8059
6119890minus09
3119890minus09
4119890minus09
262
77105
1014
Delaunay
998779Elem
ents
05
58
1117
256700
332
32087
1119890minus09
5119890minus10
1119890minus09
1091
331969
1114
Delaunay
◻Vo
lumes
40
53
1149
5042
332
630
2119890minus08
1119890minus08
5119890minus09
288007
1214
Delaunay
◻Vo
lumes
30
346
1133
8560
332
1070
7119890minus09
3119890minus09
2119890minus09
3910895
1314
Delaunay
◻Vo
lumes
20
308
1131
15576
332
1947
1119890minus08
7119890minus09
3119890minus09
541564
514
14Delaunay
◻Vo
lumes
10177
1122
60774
332
7596
8119890minus09
4119890minus09
3119890minus09
167
50115
1514
Delaunay
◻Vo
lumes
05
199
1120
244936
332
30617
4119890minus09
2119890minus09
2119890minus09
698
215678
1614
Delaunay
◻Elem
ents
40
196
1100
5222
332
652
3119890minus08
1119890minus08
8119890minus09
4713098
1714
Delaunay
◻Elem
ents
30
123
1107
8810
332
1101
2119890minus08
9119890minus09
7119890minus09
6918598
1814
Delaunay
◻Elem
ents
20
901112
15922
332
1990
1119890minus08
5119890minus09
5119890minus09
107
30591
1914
Delaunay
◻Elem
ents
1063
1116
61456
332
7682
5119890minus09
2119890minus09
4119890minus09
389
113237
2014
Delaunay
◻Elem
ents
05
58
1117
246332
332
30791
8119890minus10
4119890minus10
1119890minus09
1676
498192
2114
Delq
uad
998779Vo
lumes
40
minus45
1144
6228
332
778
4119890minus08
2119890minus08
7119890minus09
308495
2214
Delq
uad
998779Vo
lumes
30
570
1153
11820
332
1477
3119890minus08
1119890minus08
4119890minus09
4010879
2314
Delq
uad
998779Vo
lumes
20
459
1103
24100
332
3012
2119890minus08
1119890minus08
5119890minus09
6217715
2414
Delq
uad
998779Vo
lumes
10512
1104
9640
03
3212050
2119890minus08
1119890minus08
9119890minus09
207
61714
2514
Delq
uad
998779Vo
lumes
05
528
1106
385600
332
48200
2119890minus09
1119890minus09
2119890minus09
838
255735
2614
Delqu
ad998779
Elem
ents
40
220
1093
3288
332
411
3119890minus08
2119890minus08
6119890minus09
308495
2714
Delq
uad
998779Elem
ents
30
140
1104
6148
332
768
4119890minus08
2119890minus08
8119890minus09
39110
0028
14Delq
uad
998779Elem
ents
20
82
1110
12392
332
1549
2119890minus08
9119890minus09
6119890minus09
6117067
2914
Delqu
ad998779
Elem
ents
1063
1115
48882
332
6110
2119890minus09
1119890minus09
1119890minus09
197
5804
630
14Delq
uad
998779Elem
ents
05
58
1116
194162
332
24270
2119890minus09
9119890minus10
2119890minus09
790
238769
3114
Delq
uad
◻Vo
lumes
40
minus416
1174
3132
332
391
5119890minus08
3119890minus08
8119890minus09
267322
3214
Delq
uad
◻Vo
lumes
30
minus248
1151
5922
332
740
9119890minus09
4119890minus09
2119890minus09
318779
3314
Delq
uad
◻Vo
lumes
20
minus132
1139
12050
332
1506
1119890minus08
7119890minus09
4119890minus09
4412239
3414
Delq
uad
◻Vo
lumes
10minus49
1126
48200
332
6025
6119890minus09
3119890minus09
2119890minus09
122
36094
3514
Delq
uad
◻Vo
lumes
05
minus23
1123
192800
332
24100
5119890minus09
2119890minus09
3119890minus09
464
140228
3614
Delq
uad
◻Elem
ents
40
224
1093
3294
332
411
3119890minus08
1119890minus08
7119890minus09
359852
3714
Delq
uad
◻Elem
ents
30
141
1104
6144
332
768
3119890minus08
2119890minus08
9119890minus09
5114018
3814
Delq
uad
◻Elem
ents
20
921111
12392
332
1549
1119890minus08
6119890minus09
5119890minus09
8122526
3914
Delq
uad
◻Elem
ents
1065
1115
48882
332
6110
5119890minus09
3119890minus09
4119890minus09
276
78431
22 Science and Technology of Nuclear Installations
Table2Con
tinued
Case
no
Symm-
etry
Mesh
algorithm
Basic
shape
Solutio
nmetho
dℓ119888
(cm)
Δ120588
(pcm
)max1206012
(mdash)
Total
unkn
owns
Outer
iter
Linear
iter
Inner
iter
Resid
ual
norm
Relativ
eerror
Error
estim
ate
Mem
ory
(Mb)
Page
faults
4014
Delq
uad
◻Elem
ents
05
59
1117
194162
332
24270
1119890minus09
5119890minus10
1119890minus09
1104
318967
4118
Delaunay
998779Vo
lumes
40
minus99
1139
4228
332
528
4119890minus12
2119890minus12
1119890minus12
359630
4218
Delaunay
998779Vo
lumes
30
50
1132
7712
332
964
1119890minus11
6119890minus12
2119890minus12
4211578
4318
Delaunay
998779Vo
lumes
20
177
1120
15776
332
1972
2119890minus12
1119890minus12
7119890minus13
5515436
4418
Delaunay
998779Vo
lumes
10142
1118
63902
332
7987
3119890minus12
1119890minus12
1119890minus12
160
46691
4518
Delaunay
998779Vo
lumes
05
176
1115
254612
332
31826
2119890minus12
8119890minus13
1119890minus12
555
168526
4618
Delaunay
998779Elem
ents
40
183
1096
2252
332
281
6119890minus12
3119890minus12
2119890minus12
359589
4718
Delaunay
998779Elem
ents
30
119
1104
4040
224
505
2119890minus08
1119890minus08
7119890minus09
41114
3748
18Delaunay
998779Elem
ents
20
85
1108
8156
332
1019
2119890minus12
8119890minus13
6119890minus13
5515053
4918
Delaunay
998779Elem
ents
1063
1112
32480
332
4060
1119890minus12
6119890minus13
8119890minus13
151
43671
5018
Delaunay
998779Elem
ents
05
58
1113
128358
332
1604
47119890minus13
3119890minus13
7119890minus13
527
157581
5118
Delaunay
◻Vo
lumes
40
37
1144
2380
224
297
5119890minus08
2119890minus08
1119890minus08
5314173
5218
Delaunay
◻Vo
lumes
30
218
1120
4194
332
524
7119890minus12
4119890minus12
2119890minus12
3610034
5318
Delaunay
◻Vo
lumes
20
345
1110
7960
332
995
5119890minus12
3119890minus12
1119890minus12
4712879
5418
Delaunay
◻Vo
lumes
10165
1117
30652
332
3831
2119890minus12
1119890minus12
7119890minus13
100
28851
5518
Delaunay
◻Vo
lumes
05
191
1117
122066
224
15258
2119890minus08
8119890minus09
9119890minus09
343
103825
5618
Delaunay
◻Elem
ents
40
175
1097
2526
332
315
4119890minus12
2119890minus12
1119890minus12
6016159
5718
Delaunay
◻Elem
ents
30
114
1104
4386
332
548
4119890minus12
2119890minus12
2119890minus12
5013768
5818
Delaunay
◻Elem
ents
20
901108
8234
332
1029
2119890minus12
1119890minus12
1119890minus12
7320094
5918
Delaunay
◻Elem
ents
1062
1112
31186
224
3898
2119890minus08
7119890minus09
9119890minus09
208
59589
6018
Delaunay
◻Elem
ents
05
58
1113
123122
224
15390
9119890minus09
5119890minus09
1119890minus08
807
2366
7061
18Delq
uad
998779Vo
lumes
40
minus210
1168
3224
332
403
1119890minus11
5119890minus12
2119890minus12
43117
8462
18Delq
uad
998779Vo
lumes
30
369
1150
6000
332
750
1119890minus11
6119890minus12
3119890minus12
5013502
6318
Delq
uad
998779Vo
lumes
20
312
1104
12316
332
1539
2119890minus11
8119890minus12
5119890minus12
6016395
6418
Delq
uad
998779Vo
lumes
1044
01094
48714
332
6089
9119890minus12
4119890minus12
4119890minus12
1183444
165
18Delq
uad
998779Vo
lumes
05
534
1081
193980
332
24247
5119890minus12
2119890minus12
4119890minus12
423
126768
6618
Delq
uad
998779Elem
ents
40
240
1090
1750
332
218
9119890minus12
4119890minus12
2119890minus12
43117
8667
18Delqu
ad998779
Elem
ents
30
150
1100
3184
332
398
1119890minus11
5119890minus12
2119890minus12
5013509
6818
Delq
uad
998779Elem
ents
20
89
1106
6426
332
803
5119890minus12
3119890minus12
2119890minus12
5916047
6918
Delq
uad
998779Elem
ents
1064
1111
24886
332
3110
2119890minus12
1119890minus12
1119890minus12
113
32788
7018
Delq
uad
998779Elem
ents
05
59
1113
98052
224
12256
1119890minus08
5119890minus09
8119890minus09
398
118153
7118
Delq
uad
◻Vo
lumes
40
minus398
1172
1650
332
206
1119890minus11
5119890minus12
2119890minus12
369907
7218
Delq
uad
◻Vo
lumes
30
minus249
1146
3058
332
382
4119890minus12
2119890minus12
1119890minus12
349349
7318
Delq
uad
◻Vo
lumes
20
minus131
1136
6228
224
778
3119890minus08
1119890minus08
5119890minus09
4010930
7418
Delq
uad
◻Vo
lumes
10minus51
1123
24536
224
3067
3119890minus08
2119890minus08
1119890minus08
7822716
7618
Delq
uad
◻Elem
ents
40
237
1090
1752
332
219
6119890minus12
3119890minus12
2119890minus12
4111213
7718
Delq
uad
◻Elem
ents
30
145
1101
3184
332
398
6119890minus12
3119890minus12
2119890minus12
43118
1178
18Delq
uad
◻Elem
ents
20
941107
6426
332
803
3119890minus12
1119890minus12
1119890minus12
6016220
7918
Delqu
ad◻
Elem
ents
1065
1111
24874
332
3109
1119890minus12
6119890minus13
8119890minus13
156
4400
080
18Delq
uad
◻Elem
ents
05
59
1113
9804
22
2412255
8119890minus09
4119890minus09
8119890minus09
567
162168
Science and Technology of Nuclear Installations 23
Tim
e con
sum
ed to
bui
ld th
e mat
rices
(sec
onds
)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay trianglesvolumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
10
30
100
300
1
3
Figure 23 Time needed to build the matrices versus number of unknowns
was considerably larger than the solution time itself Theeigenvalue problem was solved using a multilayer iterativeKrylov-Schur method [26] with a tolerance relative to thematrices norm
1003817100381710038171003817119877 sdot 120601 minus (1119896eff) sdot 119865 sdot 1206011003817100381710038171003817
119877 + (1119896eff) sdot 119865lt 10minus8 (11)
The reported residual norm and relative error are10038171003817100381710038171003817100381710038171003817119877 sdot 120601 minus
1
119896effsdot 119865 sdot 120601
10038171003817100381710038171003817100381710038171003817
1003817100381710038171003817119877 sdot 120601 minus (1119896eff) sdot 119865 sdot 1206011003817100381710038171003817
1003817100381710038171003817(1119896eff) sdot 1206011003817100381710038171003817
(12)
respectively The fields marked as outer linear and inneriterations refer to the number of steps needed to attainthe requested tolerance in each layer of the Krylov-Schuralgorithm The computer used to solve the problem has anIntel i7 920 267GHz processor with 4Gb of RAM runningDebian GNULinux Wheezy
When using a finite volumes-based scheme over anunstructured mesh the solver has to gather information
about which cells are neighbors and which are not Currentlygmsh does not write this kind of lists in its output filesso milonga has to explicitly solve the neighbors problemPerforming a linear search is an 119874(119873
2) task which is
unacceptable for problem sizes 119873 of interest The code usesa search based on a 119896-dimensional tree [27] which can inprinciple be performed in119874(119873) steps Still for large values of119873 the time needed to read and parse the mesh (number twopreviously mentioned) is the bottleneck of the solution Thisstep is not needed in finite elements although the construc-tion of the elementary matrices involves the computationof the multidimensional Jacobians and integrals which thenhave to be assembled Again for large 119873 this step (numberthree) takes up most of the time needed to solve the problem
The Delaunay algorithm is a standard method for gen-erating two-dimensional grids [16] by tessellating a planewith triangles If the mesh needs to be based on quadranglesinstead a recombination algorithm can be used to transformtwo adjacent triangles in one quadrangle whenever is possi-ble However for geometries which are based on rectangularshapes there exist other algorithms [25] both for meshingand for recombining the triangles that give rise to elementaryentities with right angles almost everywhere which may be a
24 Science and Technology of Nuclear Installations
Tim
e con
sum
ed to
solv
e the
eige
nval
ue p
robl
em (s
econ
ds)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
10
30
100
300
1
3
Figure 24 Time needed to solve the eigenvalue problem versus number of unknowns
desired property of the resulting grid For finite elements thesteps needed to build the matrices depend on the selectionof triangles or quadrangles as the basic elementary geometrybecause the shape functions change However the number ofunknowns is the same as the number of nodes that does notchange after a recombination procedure On the other handthe number of unknowns in the finite volumes schemes withtriangles is roughly twice as the number of unknowns withquadrangles for the same grid
Figure 19 shows how the computed static reactivity (ie1 minus 1119896eff) depends on the number of unknowns for eachof the sixteen combinations of geometry algorithm shapescheme The four accepted results published in the originalreference [10] are included for reference although it shouldbe taken into account that said reactivities were computedalmost forty years ago Figure 20 shows the total wall timeneeded to solve the problem as a function of 119873119866 whileFigures 21 22 23 and 24 show the times needed for eachof the first four steps involved in the solution maintainingthe same logarithmic scale for both the abscissas and theordinates Green data represent finite volumes whilst bluebullets correspond to finite elements Solid lines indicatequarter-core and dashed lines eighth-symmetry Fillet bullets
are results obtained by theDelaunay triangulation and emptybullets were computed with the delquad algorithm Finallytriangle-shaped data correspond to triangles and squares anddiamonds denote quadrangles as the basic geometry of thegrid
It can be seen that finite elements produce amuch smallerdispersion of eigenvalues 119896eff than finite volumes with therefinement of the mesh This can be explained because finitevolumes methods rely on a geometric condition of the meshwhich may change abruptly if the meshing algorithm decidesto allocate the cells in a rather different form for small changesin ℓ119888 Finite elements methods are less influenced by thesediscontinuous lay-out changes of the elements As expectedeighth-core symmetries give almost the same results as thequarter-core geometries with roughly half the unknownsFor small problems finite volumes run faster that finiteelements because the time needed to solve the neighborproblem is negligible When the problem size grows thistime increases and exceeds the overhead implied in the con-struction and assembly of the finite elements matrices Alsoat least for this configuration it is seen that the eigenvalueproblem is solved faster for finite volumes than for finiteelements
Science and Technology of Nuclear Installations 25
4 Conclusions
Unstructured grids provide the cognizant engineer witha wide variety of possibilities to deal with the design oranalysis of nuclear reactor cores These kinds of grids cansuccessfully reproduce continuous geometries commonlyfound in reactor cores such as cylinders and thereforenot only can the diffusion equation be better approximatedinside the domain of definition but also the fulfillment ofboundary conditions is improved A free computer codewas written from scratch that is able to completely solvethe 2D IAEA PWR Benchmark using unstructured gridsfor sixteen combinations of geometry meshing algorithmbasic shape and discretization scheme plus any value ofthe gridrsquos characteristic length by using a single input fileThe complete set of input files and codemdashexecutable andsourcemdashis available either online or upon request withcomments experiences suggestions and corrections beingmore than welcome Further development should includetackling full three-dimensional geometries with completethermal hydraulic feedback in order to analyze how thesolutions of the coupled neutronic-thermal problem dependon the spatial discretization scheme of the neutron leakageterm Parallelization of the computation and assembly of thematrices and of the solution of the eigenvalue problem andits implementation using GPUs are also desired features toimplement A problemwith direct applications that the futureversions of milonga ought to solve is the analysis of how thegeometry of the absorbing materials should be taken intoaccount in structured coarse grids in order to mitigate effectssuch as the rod-cusp problem
Suitable schemes for approximating the continuous dif-ferential operators by discrete matrix expressions includefinite volumes and finite elements families Finite volumesmethods compute cell mean values whilst finite elementsgive functional values at the gridrsquos nodes In general finiteelements are less sensitive to changes in the mesh so theresults they provide do not change significantly for differentmeshing algorithms or elementary shapes Small problemsare best solved by finite volumes as the neighbor-findingproblem is faster than the process of building and assemblingthe eigenvalue-problem matrices For a large number ofunknowns the process of finding which cell is neighbor ofwhichmdasheven based on a 119896-dimensional treemdashoverwhelmsthe computation and assembly of elementary matrices andfinite-element methods perform better
References
[1] G Glasstone and S Bell Nuclear Reactor Theory KriegerPublishing Company 1970
[2] J J Duderstadt and L J Hamilton Nuclear Reactor AnalysisJohn Wiley amp Sons New York NY USA 1976
[3] C Gho Reactor Physics Undergraduate Course Lecture NotesInstituto Balseiro 2006
[4] G G Theler and F J Bonetto ldquoOn the stability of the pointreactor kinetics equationsrdquoNuclear Engineering andDesign vol240 no 6 pp 1443ndash1449 2010
[5] E E Lewis andW F Miller Computational Methods of NeutronTransport John Wiley amp Sons 1984
[6] E Masiello ldquoAnalytical stability analysis of Coarse-Mesh finitedifference methodrdquo in Proceedings of the International Confer-ence on the Physics of Reactors (PHYSOR rsquo08) Nuclear Power ASustainable Resource pp 456ndash464 September 2008
[7] M L Zerkle Development of a polynomial nodal methodwith flux and current discontinuity factors [PhD thesis] Mas-sachusets Institute of Technology 1992
[8] J I Yoon andHG Joo ldquoTwo-level coarsemesh finite differenceformulation with multigroup source expansion nodal kernelsrdquoJournal of Nuclear Science andTechnology vol 45 no 7 pp 668ndash682 2008
[9] O Mazzantini M Schivo J Di Cesare R Garbero M Riveroand G Theler ldquoA coupled calculation suite for Atucha II oper-ational transients analysisrdquo Science and Technology of NuclearInstallations vol 2011 Article ID 785304 12 pages 2011
[10] Computational Benchmark Problem Comitee for the Mathe-matics and Computation Division of the American NuclearSociety ldquoArgonne Code Center Benchmark problem bookrdquoTech Rep ANL-7416 Supplement 2 Argonne National Labo-ratory 1977
[11] A F Henry Nuclear Reactor Analysis MIT Cambridge UK1975
[12] O C Zienkiewicz R L Taylor and J Z ZhuTheFinite ElementMethod Its Basis and Fundamentals vol 1 Elsevier 6th edition2005
[13] G Theler Difusion de neutrones en mallas no estructuradascomparacion entre volumenes y elementos finitos AcademicMonograph Universidad de Buenos Aires 2013
[14] G Theler ldquoMilonga a free nuclear reactor core analysis coderdquoAvailable online 2011
[15] G Theler F J Bonetto and A Clausse ldquoOptimizacion deparametros en reactores de potencia base de diseno delcodigo neutronico milongardquo in Reunion Anual de la AsociacionArgentina de Tecnologia Nuclear XXXVII 2010
[16] C Geuzaine and J-F Remacle ldquoGmsh a 3-D finite elementmesh generator with built-in pre- and post-processing facili-tiesrdquo International Journal for Numerical Methods in Engineer-ing vol 79 no 11 pp 1309ndash1331 2009
[17] S Balay J Brown K Buschelman et al ldquoPETSc users manualrdquoTech Rep ANL-9511-Revision 34 Argonne National Labora-tory 2013
[18] S Balay W D Gropp L Curfman McInnes and B F SmithldquoEfficient management of parallelism in object oriented numer-ical software librariesrdquo in Modern Software Tools in ScientificComputing E Arge A M Bruaset and H P Langtangen Edspp 163ndash202 Birkhauser 1997
[19] V Hernandez J E Roman and V Vidal ldquoSLEPC a scalable andflexible toolkit for the solution of eigenvalue problemsrdquo ACMTransactions on Mathematical Software vol 31 no 3 pp 351ndash362 2005
[20] M Galassi J Davies J Theiler et al GNU Scientific LibraryReference Manual 3rd edition 2009
[21] O C Zienkiewicz and R L Taylor The Finite Element MethodFor Solid and Structural Mechanics vol 3 Elsevier 6th edition2005
[22] A Imelda and K Doddy Evaluation of the IAEA 3-D PWRbenchmark problem using NESTLE code 2004
[23] R Mosteller ldquoStatic benchmarking of the NESTLE advancednodal coderdquo in Proceedings of the Joint International Conference
26 Science and Technology of Nuclear Installations
on Mathematical Methods and Supercomputing for NuclearApplications vol 2 pp 1596ndash1605 1997
[24] GTheler F J Bonetto andAClausse ldquoSolution of the 2D IAEAPWR Benchmark with the neutronic code milongardquo in Actasde la Reunion Anual de la Asociacion Argentina de TecnologıaNuclear XXXVIII 2011
[25] J F Remacle F Henrotte T Carrier-Baudouin et al ldquoAfrontal delaunay quad mesh generator using the 119871
infin normrdquoInternational Journal For Numerical Methods in Engineeringvol 94 no 5 pp 494ndash512 2013
[26] G W Stewart ldquoA Krylov-Schur algorithm for large eigenprob-lemsrdquo SIAM Journal on Matrix Analysis and Applications vol23 no 3 pp 601ndash614 2002
[27] J L Bentley ldquoMultidimensional binary search trees used forassociative searchingrdquo Communications of the ACM vol 18 no9 pp 509ndash517 1975
TribologyAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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FuelsJournal of
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Renewable Energy
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
StructuresJournal of
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear InstallationsScience and Technology of
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Solar EnergyJournal of
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Wind EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear EnergyInternational Journal of
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High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Science and Technology of Nuclear Installations 11
Largest eigenvalue 1029159 (283326 pcm)1174 (2851 3224)1560 (5200 13200)
Number of unknowns 3132Outer iterations 3Linear iterations 32Inner iterations 391Residual normRelative errorError estimateMemory used 26888 kBSoft page faults 7322Hard page faults 0Total CPU time 0308 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
5408 times 10minus8
2665 times 10minus8
8424 times 10minus9
keff
Milongarsquos 2D LWR IAEA benchmark problem case no 031quarter-symmetry core meshed using Delquad (quads 985747c = 4) solved with finite volumes
(a)
123456789
1011121314151617181920212223242526272829303132333435363738
k Pk 1206011k 1206012k077 3422 570140 4468 1040153 4836 1135128 4070 948060 2704 448095 3017 702091 2861 673069 1918 508mdash 264 689151 4785 1122155 4897 1149137 4334 1017111 3535 823104 3272 768092 2908 685067 1876 496mdash 254 658154 4854 1139139 4392 1031120 3792 890107 3360 790095 2831 702061 1574 451mdash 197 515123 3868 907099 3132 730090 2818 663078 2168 579mdash 481 1191mdash 055 217044 1984 328067 2027 498052 1360 388mdash 190 520052 1329 383mdash 321 792mdash 043 168mdash 048 195
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 13 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
of this paper and is currently still under development withinhis ongoing PhD thesis There exists a first public release[14] under the terms of the GNU General Public Licensemdashthat is it is a free softwaremdashthat can only handle structuredgrids There is a second release being prepared whose
main relevance is that it can work with nonstructured gridsas well which is the main feature of the code It uses ageneral mathematical frameworkmdashcoded from scratch aswellmdashwhich provides input file parsing algebraic expres-sions evaluation one- and multidimensional interpolation of
12 Science and Technology of Nuclear Installations
Largest eigenvalue 1029788 (289260 pcm)1120 (3149 3011)1091 (13038 5114)
Number of unknowns 15776Outer iterations 3Linear iterations 32Inner iterations 1972Residual normRelative errorError estimateMemory used 56996 kBSoft page faults 15436Hard page faults 0Total CPU time 09121 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
2296 times 10minus12
1131 times 10minus12
6877 times 10minus13
keff
Milongarsquos 2D LWR IAEA benchmark problem case no 043eighth-symmetry core meshed using Delaunaya (triangs985747c = 2) solved with finite volumes
(a)
1 074 3241 5482 131 4188 9713 145 4582 10764 122 3877 9005 060 2644 4456 093 2986 6927 093 2941 6928 075 2030 5529 mdash 322 779
10 143 4523 106111 148 4667 109512 131 4145 97213 107 3423 79414 103 3269 76715 095 2993 70616 073 1988 53917 mdash 312 75118 147 4637 108819 134 4243 99620 118 3721 87321 107 3370 79322 098 2914 72323 067 1651 49924 mdash 235 57125 119 3763 88226 096 3077 71527 091 2847 67128 083 2255 61729 mdash 576 122530 mdash 068 26631 046 2016 34132 068 2058 50533 059 1443 43534 mdash 233 58935 057 1381 42036 mdash 381 82337 mdash 054 20838 mdash 062 244
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 14 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
scattered data shared-memory access numerical integrationfacilities and so forth The milonga code was designed withfour design basis vectors in mindmdashwhich are thoroughlydiscussed in the documentation [15]mdashwhich includes thetype of problems it can handle the code scalability which
features are expected and what to do with the obtainedresults
It works by first reading an input file that using plain-text English keywords and arguments defines the number119898 of spatial dimensions the number 119866 of group energies
Science and Technology of Nuclear Installations 13
Largest eigenvalue 1029726 (288675 pcm)1104 (3069 3069)864 (13000 5000)
Number of unknowns 4040Outer iterations 2Linear iterations 24Inner iterations 505Residual normRelative errorError estimateMemory used 42940 kBSoft page faults 11437Hard page faults 0Total CPU time 1064 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
2019 times 10minus8
9946 times 10minus9
7138 times 10minus9
keff
Milongarsquos 2D LWR IAEA benchmark problem case no 047eighth-symmetry core meshed using Delaunay (triangles 985747c = 3) solved with finite volumes
(a)
1 074 3194 5452 129 4117 9533 143 4515 10604 119 3814 8835 061 2634 4506 093 2984 6907 094 2954 6958 071 2064 5719 mdash 362 865
10 141 4458 104511 146 4599 107912 130 4097 96113 106 3388 78414 103 3263 76515 095 2998 70716 068 2015 55717 mdash 346 82318 145 4572 107319 133 4194 98420 117 3691 86621 107 3363 79222 098 2925 72623 059 1696 52824 mdash 263 63525 118 3728 87426 096 3068 71127 091 2850 67228 079 2280 63529 mdash 619 127630 mdash 081 32131 047 2036 34932 068 2085 50933 051 1465 45434 mdash 258 65035 049 1419 44336 mdash 416 85437 mdash 065 25438 mdash 071 288
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 15 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
and optionally a mesh file Currently only grids generatedwith the code gmsh [16] are only supported mainly becauseit is also a free software and it suits perfectly well milongarsquosdesign basis in the sense that the continuous geometry can bedefined as a function of a number of parameters Afterwardthe physical entities defined in the grid are mapped into
materials with macroscopic cross sections which maydepend on the spatial coordinates x bymeans of intermediatefunctions such as burn-up or temperatures distributionswhich in turn may be defined by algebraic expressions byinterpolating data located in files by reading shared-memoryobjects or by a combination of them In the same sense
14 Science and Technology of Nuclear Installations
Largest eigenvalue 1029695 (288387 pcm)1108 (3045 3045)833 (13000 5000)
Number of unknowns 8234Outer iterations 3Linear iterations 32Inner iterations 1029Residual normRelative error
Max 1206012(x y) coreMax 1206012(x y) reflector
2028 times 10minus12
9993 times 10minus13
1012 times 10minus12
keff
Relative errorError estimateMemory used 75468 kBSoft page faults 20094Hard page faults 0Total CPU time 1256 seconds
9993 times 10
1012 times 10minus12
Milongarsquos 2D LWR IAEA benchmark problem case no 058eighth-symmetry core meshed using Delaunay (quads 985747c = 2) solved with finite volumes
(a)
1 074 3207 5472 129 4136 9573 144 4534 10644 120 3828 8875 061 2638 4506 093 2985 6907 094 2948 6948 072 2053 5669 mdash 357 859
10 142 4478 105011 146 4617 108412 130 4111 96413 106 3396 78614 103 3263 76515 095 2991 70616 071 2005 55217 mdash 340 81618 145 4588 107719 133 4206 98720 117 3698 86821 107 3362 79122 098 2918 72523 062 1686 52024 mdash 258 62925 118 3737 87626 096 3073 71227 091 2848 67128 080 2272 63129 mdash 609 127230 mdash 080 31731 047 2035 34832 069 2081 50933 054 1458 44834 mdash 255 64635 052 1413 43736 mdash 409 85137 mdash 064 25138 mdash 071 284
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 16 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
boundary conditions are applied to appropriate physicalentities of dimension119898minus1The problemmatrices119877 and119865 arethen built and stored in an appropriate sparse format usingthe free PETSc [17 18] library and the eigenvalue problemis solved using the free SLEPc [19] library The results arestored into milongarsquos variables and functions which may be
evaluated integratedmdashusually using the free GNU ScientificLibrary [20] routinesmdashand of course written into appropriateoutputs Milonga can also solve problems parametrically andbe used to solve optimization problems As stated above thecode is a free software so corrections and contributions aremore than welcome
Science and Technology of Nuclear Installations 15
Largest eigenvalue 1029462 (286185 pcm)1136 (3093 2953)1233 (13100 5100)
Number of unknowns 6228Outer iterations 2Linear iterations 24Inner iterations 778Residual normR l ti
Max 1206012(x y) coreMax 1206012(x y) reflector
26 times 10minus8
1281 times 10minus8
4996 times 10minus9
keff
Relative errorError estimateMemory used 41460 kBSoft page faults 10930Hard page faults 0Total CPU time 0604 seconds
1281 times 10 8
4996 times 10minus9
Milongarsquos 2D LWR IAEA benchmark problem case no 073eighth-symmetry core meshed using Delaunay (quads 985747c = 2) solved with finite volumes
(a)
1 075 3297 5552 134 4268 9923 148 4660 10944 123 3926 9145 059 2645 4406 094 2997 6977 093 2914 6858 072 2002 5349 mdash 301 773
10 146 4606 108011 150 4735 111112 133 4212 98813 108 3448 80214 104 3271 76715 094 2954 69616 070 1954 52117 mdash 288 73518 149 4691 110119 136 4289 100720 119 3748 88021 107 3359 79022 096 2880 71423 064 1636 47524 mdash 217 56325 121 3810 89426 098 3110 72427 090 2829 66628 081 2227 59929 mdash 536 125730 mdash 062 25031 045 2002 33632 068 2040 50033 055 1414 40934 mdash 215 57935 055 1359 40836 mdash 359 83437 mdash 050 19938 mdash 061 236
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 17 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
31 A Coarse Bare Homogeneous Circle Figures 5 and 6 showthe results of solving a bare two-dimensional circular homo-geneous reactor of radius 119886 over an unstructured grid whichis deliberately coarse using the finite volumes method andthe finite elements method respectively Two group energies
were used and a null-flux boundary conditionwas fixed at theexternal surface Figures 5(a) and 6(a) compare the obtainedfast flux distributions In the first case the numerical solutionprovidesmean values for each neutron flux group in each cellwhile in the latter the solution is computed at the nodes and
16 Science and Technology of Nuclear Installations
Largest eigenvalue 1029851 (289858 pcm)1090 (3182 3182)851 (13000 5000)
Number of unknowns 1752Outer iterations 3Linear iterations 32Inner iterations 219Residual norm
Max 1206012(x y) coreMax 1206012(x y) reflector
566 times 10minus12
2789 times 10minus12
1595 times 10minus12
keff
Residual normRelative errorError estimateMemory used 42564 kBSoft page faults 11213Hard page faults 0Total CPU time 09921 seconds
566 times 10
2789 times 10minus12
1595 times 10minus12
Milongarsquos 2D LWR IAEA benchmark problem case no 076eighth-symmetry core meshed using Delaunay (quads 985747c = 4) solved with finite volumes
(a)
1 073 3151 5402 127 4059 9393 141 4457 10464 118 3768 8715 061 2615 4506 093 2979 6887 094 2971 6998 067 2092 5829 mdash 375 885
10 139 4397 103111 144 4541 106612 128 4052 95013 105 3365 77814 103 3259 76415 096 3016 71216 066 2045 56817 mdash 359 84118 143 4520 106119 132 4155 97520 116 3672 86221 107 3366 79222 098 2951 73323 054 1728 54324 mdash 273 65125 117 3700 86826 095 3057 70827 091 2860 67428 074 2311 64829 mdash 642 130030 mdash 084 33231 048 2041 35332 069 2103 51333 046 1493 46734 mdash 269 66735 046 1444 45536 mdash 432 87137 mdash 067 26338 mdash 074 297
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 18 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
continuous functions are evaluated by means of the shapefunctions used in the formulation Figures 5(b) and 6(b)illustrate the fast fluxes unknowns and its relative position inspace It can be noted that the mesh coarseness gives resultsthat differ substantially in both cases Finally the structure ofthe sparse eigenvalue problemmatrices is shown for each case
with blue red and cyan representing positive negative andexplicitly inserted zero values In the finite volume case thereare 184 unknowns (92 cells times 2 groups) while in the secondcase there are 218 unknowns (109 nodes times 2 groups) Thevolumesrsquo fissionmatrix is almost diagonal it has a bandwidthequal to the number of energy groups which in this case is
Science and Technology of Nuclear Installations 17
500 1000 2000 5000 10000 20000 50000Number of unknowns
2840
2860
2880
2900
2920
ORNLRisoslashKWUOntarioQuarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumes
Quarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elementsEighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumes
Eighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
120588(p
cm)
105 2 times 105 5 times 105
Figure 19 Static reactivity versus number of unknowns The four original solutions as published in 1977 [10] are included as reference
twoThe off-diagonal values appear right next to the diagonalelements because of the chosen ordering of the unknownsin the flux vector 120601 isin R184 Other orderings may be usedbut the rate of convergence of the eigenvalue problem can bedeteriorated On the other hand the elementsrsquo fission matrixhas a nontrivial structure because the grid is unstructuredand the net fission rate at each element depends on the fluxesat the nodes whose location inside thematrix depends in turnon how the gridwas generatedThis effect is similar to the onethat appears in structural analysis where mass matrices needto be lumped in order to simplify transient computations [21]The diagonal block of elementrsquos 119877 matrix and the null blockin 119865 correspond to the discrete equations that set the null-flux boundary conditions on the nodes located at the externalsurface Other types of boundary conditions lead to differentkinds of structures within the problem matrices
In case a part of the domain contained a nonmultiplicativematerial such as a reflector then there would appear sectionsof the fission matrix with null values in both methodsrendering 119865 singular in both methods Care should be takenwhen dealing with the numerical schemes for the eigenvalueproblem solution It can be seen in the elementsrsquo matricesa particular structure that implements the boundary condi-tions at the external surfaces This structure does not appearin the volumesrsquo matrices because the boundary conditions
appear as flux terms which are summed up over all thesurfaces of each cell so they aremasked inside the volumetricdiscretization of the divergence and gradient operators
As the two-group neutron diffusion equation with uni-form cross sections over a circle subject to null-flux bound-ary conditions has an analytical solution it is adequate tocompare how the two proposed numerical schemes relate toit In the studied problem we ignored upscattering and fastfissions Then the analytical effective multiplication factor is
119896eff =]Σ1198912 sdot Σ1199041rarr2
[Σ1198861 + Σ1199041rarr2 + 1198631(]0119886)2] [Σ1198862 + 1198632(]0119886)
2]
(7)
being ]0 = 24048 the smallest root of Besselrsquos first-kindfunction of order zero 1198690(119903)
Figure 7 shows how the two numerical effective multi-plication factors compare to the analytical solution givenby (7) as a function of the mesh refinement indicated bythe quotient 119886ℓ119888 between the radius of the circle and thecharacteristic length of the cellelements We can see thatthe 119896eff computed by the finite volumes (elements) methodis always greater (less) than the analytical solution Indeedit can be proven that for a bare one-dimensional slab thisis always the case [13] However this result does not hold
18 Science and Technology of Nuclear Installations
2000 3000 6000 10000 20000 30000 60000
01
003
001
03
1
3
10
30
100
300To
tal w
all t
ime (
seco
nds)
Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
Figure 20 Total wall time versus number of unknowns
for problems with nonuniform cross sections even in simpleone-dimensional reflected reactors
We may draw two other conclusions from Figure 7 Firstthat the finite element method seems to provide a bettersolution than the volumes-based scheme and second thateven though the error committed tends to decrease with finergrids its behavior is not monotonic for finite volumes Infact Figure 8 shows the difference between the numericaland analytical solutions using a logarithmic vertical scalewhere both conclusions are even more evident We deferthe discussion of such differences between the discretizationscheme until the next section It is worth to note howeverthat the fact that a finite-element-based scheme throws betterresults for the particular bare homogeneous circular reactorunder study than the finite volumes does not imply thatfinite volumes ought to be discarded as a valid tool forsolving the neutron diffusion equation in general cases Thecombination of lattice and core-level computations is usuallyperformedusing cell-based resultswhichwhen fed into node-based methods to solve the few-group neutron diffusionequation may introduce errors which potentially can leadto unacceptable solutions Nevertheless this analysis is farbeyond the scope of this paper which focuses on solving amathematical equation over unstructured grids
32 The 2D IAEA PWR Benchmark Problem This is aclassical two-group neutron diffusion problem first designedin the early 1970s and taken as a reference benchmark forcomputational codes A number of codes were used to solveeither this problem or its three-dimensional formulation[22 23] including milonga using a structured grid [24] Theoriginal formulation can be found in the reference [10] andthere is a reproduction that may be easily found online inreference [24] The geometry consists of a one-quarter ofa PWR core depicted in Figure 9 and the homogeneousmacroscopic cross sections are listed in Table 1 An axialbuckling term 119861
2
119911119892= 08times10
minus4 should be taken into accountThe external surface should be subject to a zero incomingcurrent condition which may be written as
120597120601119892
120597119899= minus
04692
119863119892
sdot 120601119892 (8)
The expected results are as follows
(1) Maximum eigenvalue(2) Fundamental flux distributions
(a) Radial flux traverses 120601119892(119909 0) and 120601119892(119909 119909)
Science and Technology of Nuclear Installations 19
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
1
3
10
30
100
300Ti
me c
onsu
med
to m
esh
the g
eom
etry
(sec
onds
)
Figure 21 Time needed to mesh the geometry versus number of unknowns
Table 1 Macroscopic cross-sections (units are not stated in the original reference but they are assumed to be in cmminus1 or cm as appropriate)
1198631
1198632
Σ1rarr2
Σ1198861
Σ1198862
]Σ1198912
Material1 15 04 002 001 0080 0135 Fuel 12 15 04 002 001 0085 0135 Fuel 23 15 04 002 001 0130 0135 Fuel 2 + rod4 20 03 004 0 0010 0 Reflector
Note the fluxes shall be normalized such that
1
119881coreint119881core
sum
119892
]Σ119891119892 sdot 120601119892119889119881 = 1 (9)
(b) Value and location of maximum power densityThis corresponds to maximum of 1206012 in thecore It is recommended that the maximumvalues of 1206012 both in the inner core and at thecorereflector interface be given
(3) Average subassembly powers 119875119896
119875119896 =1
119881119896
int119881119896
sum
119892
]Σ119891119892 sdot 120601119892119889119881 (10)
where 119881119896 is the volume of the 119896th subassembly and119896 designates the fuel subassemblies as shown inFigure 9
(4) Number of unknowns in the problem number ofiterations and total and outer
(5) Total computing time iteration time IO-time andcomputer used
(6) Type and numerical values of convergence criteria(7) Table of average group fluxes for a square mesh grid
of 20 times 20 cm(8) Dependence of results on mesh spacing
Even though the original problem is based on a quarter-core situation the problem has an eighth-core symmetry
20 Science and Technology of Nuclear Installations
Quarter Delaunay triangles elements
Tim
e con
sum
ed to
read
the m
esh
(sec
onds
)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumes
Quarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
1
3
10
30
100
300
Figure 22 Time needed to read the mesh versus number of unknowns
which cannot be taken into account by structured gridswhich are the main target of the benchmark Howevernonstructured grids can take into consideration any kind ofsymmetry almost without loss of accuracy and at the sametime reducing roughly the number of unknowns by a halfand the associated computational effort needed to solve theproblem by a factor of four Answers to items (1)ndash(7) canbe given completely by milonga using a single input file(see Supplementary data in SupplementaryMaterial availableonline at httpdxdoiorg1011552013641863) As asked initem (8) how results depend not only on the mesh spacingbut also on the meshing algorithm on the gridrsquos elementarygeometric shape and on the discretization scheme may shedlights on the subject whichmay be evenmore interesting thanthe numerical results themselves
Taking advantage of milongarsquos capability of reading andparsing command-line arguments the selection of the coregeometry (quarter or eighth) the meshing algorithm (delau-nay [16] or delquad [25]) the shape of the elementaryentities (triangles or quadrangles) the discretization scheme(volumes or elements) and the characteristic length ℓ119888 ofthe mesh can be provided at run time Fixing five values forℓ119888 = 4 3 2 1 05 gives 2 times 2 times 2 times 2 times 5 = 80 possiblecombinations which we solve with successive invocations to
milonga with the same input file but with different argumentsfrom a simple script Figures 10 11 12 13 14 15 16 17 and18 show the results corresponding to items (1)ndash(7) for someillustrative cases The complete set of figures and the codeused to generate themmay be provided upon request Table 2compiles the answers to the problem for every case studied
As the milonga code is still under development itsnumerical routines are not yet fully optimized nor designedfor parallel computation Therefore the reported times areonly rough estimates and should be taken with care Thesolution comprises five steps
(1) generate the grid with the requested geometry mesh-ing algorithm basic shape and characteristic lengthby calling to gmsh
(2) read the generated mesh(3) build the matrices(4) solve the eigenvalue problem(5) compute the requested results
The CPU time reported in Figures 10ndash18 is thus thesum of all of these five steps but not the time needed togenerate the figuresmdashwhich in some caseswith coarsemeshes
Science and Technology of Nuclear Installations 21
Table2
Resultsofthe2
DIAEA
PWRBe
nchm
arkp
roblem
obtained
with
them
ilong
acod
efor
thee
ightyp
ropo
sedcombinatio
nsofsymmetrym
eshing
algorithm
basicshapediscretization
schemeandcharacteris
ticelem
entc
elllengthTh
ereference
solutio
nis120588=28749times10minus5w
hich
correspo
ndsto119896eff=10296
Case
no
Symm-
etry
Mesh
algorithm
Basic
shape
Solutio
nmetho
dℓ119888
(cm)
Δ120588
(pcm
)max1206012
(mdash)
Total
unkn
owns
Outer
iter
Linear
iter
Inner
iter
Resid
ual
norm
Relativ
eerror
Error
estim
ate
Mem
ory
(Mb)
Page
faults
114
Delaunay
998779Vo
lumes
40
minus120
1145
8264
332
1033
6119890minus09
3119890minus09
1119890minus09
349470
214
Delaunay
998779Vo
lumes
30
85
1135
15740
332
1967
2119890minus08
1119890minus08
4119890minus09
4813401
314
Delaunay
998779Vo
lumes
20
197
1124
31188
332
3898
1119890minus08
6119890minus09
3119890minus09
7621932
414
Delaunay
998779Vo
lumes
10153
1123
127476
332
15934
7119890minus09
3119890minus09
3119890minus09
273
81886
514
Delaunay
998779Vo
lumes
05
187
1119
510676
332
63834
2119890minus09
9119890minus10
1119890minus09
1148
352261
614
Delaunay
998779Elem
ents
40
173
1100
4308
332
538
2119890minus08
8119890minus09
4119890minus09
318794
714
Delaunay
998779Elem
ents
30
117
1107
8108
332
1013
5119890minus09
2119890minus09
2119890minus09
4712946
814
Delaunay
998779Elem
ents
20
84
1112
15936
332
1992
6119890minus09
3119890minus09
2119890minus09
7321347
914
Delaunay
998779Elem
ents
1062
1116
64478
332
8059
6119890minus09
3119890minus09
4119890minus09
262
77105
1014
Delaunay
998779Elem
ents
05
58
1117
256700
332
32087
1119890minus09
5119890minus10
1119890minus09
1091
331969
1114
Delaunay
◻Vo
lumes
40
53
1149
5042
332
630
2119890minus08
1119890minus08
5119890minus09
288007
1214
Delaunay
◻Vo
lumes
30
346
1133
8560
332
1070
7119890minus09
3119890minus09
2119890minus09
3910895
1314
Delaunay
◻Vo
lumes
20
308
1131
15576
332
1947
1119890minus08
7119890minus09
3119890minus09
541564
514
14Delaunay
◻Vo
lumes
10177
1122
60774
332
7596
8119890minus09
4119890minus09
3119890minus09
167
50115
1514
Delaunay
◻Vo
lumes
05
199
1120
244936
332
30617
4119890minus09
2119890minus09
2119890minus09
698
215678
1614
Delaunay
◻Elem
ents
40
196
1100
5222
332
652
3119890minus08
1119890minus08
8119890minus09
4713098
1714
Delaunay
◻Elem
ents
30
123
1107
8810
332
1101
2119890minus08
9119890minus09
7119890minus09
6918598
1814
Delaunay
◻Elem
ents
20
901112
15922
332
1990
1119890minus08
5119890minus09
5119890minus09
107
30591
1914
Delaunay
◻Elem
ents
1063
1116
61456
332
7682
5119890minus09
2119890minus09
4119890minus09
389
113237
2014
Delaunay
◻Elem
ents
05
58
1117
246332
332
30791
8119890minus10
4119890minus10
1119890minus09
1676
498192
2114
Delq
uad
998779Vo
lumes
40
minus45
1144
6228
332
778
4119890minus08
2119890minus08
7119890minus09
308495
2214
Delq
uad
998779Vo
lumes
30
570
1153
11820
332
1477
3119890minus08
1119890minus08
4119890minus09
4010879
2314
Delq
uad
998779Vo
lumes
20
459
1103
24100
332
3012
2119890minus08
1119890minus08
5119890minus09
6217715
2414
Delq
uad
998779Vo
lumes
10512
1104
9640
03
3212050
2119890minus08
1119890minus08
9119890minus09
207
61714
2514
Delq
uad
998779Vo
lumes
05
528
1106
385600
332
48200
2119890minus09
1119890minus09
2119890minus09
838
255735
2614
Delqu
ad998779
Elem
ents
40
220
1093
3288
332
411
3119890minus08
2119890minus08
6119890minus09
308495
2714
Delq
uad
998779Elem
ents
30
140
1104
6148
332
768
4119890minus08
2119890minus08
8119890minus09
39110
0028
14Delq
uad
998779Elem
ents
20
82
1110
12392
332
1549
2119890minus08
9119890minus09
6119890minus09
6117067
2914
Delqu
ad998779
Elem
ents
1063
1115
48882
332
6110
2119890minus09
1119890minus09
1119890minus09
197
5804
630
14Delq
uad
998779Elem
ents
05
58
1116
194162
332
24270
2119890minus09
9119890minus10
2119890minus09
790
238769
3114
Delq
uad
◻Vo
lumes
40
minus416
1174
3132
332
391
5119890minus08
3119890minus08
8119890minus09
267322
3214
Delq
uad
◻Vo
lumes
30
minus248
1151
5922
332
740
9119890minus09
4119890minus09
2119890minus09
318779
3314
Delq
uad
◻Vo
lumes
20
minus132
1139
12050
332
1506
1119890minus08
7119890minus09
4119890minus09
4412239
3414
Delq
uad
◻Vo
lumes
10minus49
1126
48200
332
6025
6119890minus09
3119890minus09
2119890minus09
122
36094
3514
Delq
uad
◻Vo
lumes
05
minus23
1123
192800
332
24100
5119890minus09
2119890minus09
3119890minus09
464
140228
3614
Delq
uad
◻Elem
ents
40
224
1093
3294
332
411
3119890minus08
1119890minus08
7119890minus09
359852
3714
Delq
uad
◻Elem
ents
30
141
1104
6144
332
768
3119890minus08
2119890minus08
9119890minus09
5114018
3814
Delq
uad
◻Elem
ents
20
921111
12392
332
1549
1119890minus08
6119890minus09
5119890minus09
8122526
3914
Delq
uad
◻Elem
ents
1065
1115
48882
332
6110
5119890minus09
3119890minus09
4119890minus09
276
78431
22 Science and Technology of Nuclear Installations
Table2Con
tinued
Case
no
Symm-
etry
Mesh
algorithm
Basic
shape
Solutio
nmetho
dℓ119888
(cm)
Δ120588
(pcm
)max1206012
(mdash)
Total
unkn
owns
Outer
iter
Linear
iter
Inner
iter
Resid
ual
norm
Relativ
eerror
Error
estim
ate
Mem
ory
(Mb)
Page
faults
4014
Delq
uad
◻Elem
ents
05
59
1117
194162
332
24270
1119890minus09
5119890minus10
1119890minus09
1104
318967
4118
Delaunay
998779Vo
lumes
40
minus99
1139
4228
332
528
4119890minus12
2119890minus12
1119890minus12
359630
4218
Delaunay
998779Vo
lumes
30
50
1132
7712
332
964
1119890minus11
6119890minus12
2119890minus12
4211578
4318
Delaunay
998779Vo
lumes
20
177
1120
15776
332
1972
2119890minus12
1119890minus12
7119890minus13
5515436
4418
Delaunay
998779Vo
lumes
10142
1118
63902
332
7987
3119890minus12
1119890minus12
1119890minus12
160
46691
4518
Delaunay
998779Vo
lumes
05
176
1115
254612
332
31826
2119890minus12
8119890minus13
1119890minus12
555
168526
4618
Delaunay
998779Elem
ents
40
183
1096
2252
332
281
6119890minus12
3119890minus12
2119890minus12
359589
4718
Delaunay
998779Elem
ents
30
119
1104
4040
224
505
2119890minus08
1119890minus08
7119890minus09
41114
3748
18Delaunay
998779Elem
ents
20
85
1108
8156
332
1019
2119890minus12
8119890minus13
6119890minus13
5515053
4918
Delaunay
998779Elem
ents
1063
1112
32480
332
4060
1119890minus12
6119890minus13
8119890minus13
151
43671
5018
Delaunay
998779Elem
ents
05
58
1113
128358
332
1604
47119890minus13
3119890minus13
7119890minus13
527
157581
5118
Delaunay
◻Vo
lumes
40
37
1144
2380
224
297
5119890minus08
2119890minus08
1119890minus08
5314173
5218
Delaunay
◻Vo
lumes
30
218
1120
4194
332
524
7119890minus12
4119890minus12
2119890minus12
3610034
5318
Delaunay
◻Vo
lumes
20
345
1110
7960
332
995
5119890minus12
3119890minus12
1119890minus12
4712879
5418
Delaunay
◻Vo
lumes
10165
1117
30652
332
3831
2119890minus12
1119890minus12
7119890minus13
100
28851
5518
Delaunay
◻Vo
lumes
05
191
1117
122066
224
15258
2119890minus08
8119890minus09
9119890minus09
343
103825
5618
Delaunay
◻Elem
ents
40
175
1097
2526
332
315
4119890minus12
2119890minus12
1119890minus12
6016159
5718
Delaunay
◻Elem
ents
30
114
1104
4386
332
548
4119890minus12
2119890minus12
2119890minus12
5013768
5818
Delaunay
◻Elem
ents
20
901108
8234
332
1029
2119890minus12
1119890minus12
1119890minus12
7320094
5918
Delaunay
◻Elem
ents
1062
1112
31186
224
3898
2119890minus08
7119890minus09
9119890minus09
208
59589
6018
Delaunay
◻Elem
ents
05
58
1113
123122
224
15390
9119890minus09
5119890minus09
1119890minus08
807
2366
7061
18Delq
uad
998779Vo
lumes
40
minus210
1168
3224
332
403
1119890minus11
5119890minus12
2119890minus12
43117
8462
18Delq
uad
998779Vo
lumes
30
369
1150
6000
332
750
1119890minus11
6119890minus12
3119890minus12
5013502
6318
Delq
uad
998779Vo
lumes
20
312
1104
12316
332
1539
2119890minus11
8119890minus12
5119890minus12
6016395
6418
Delq
uad
998779Vo
lumes
1044
01094
48714
332
6089
9119890minus12
4119890minus12
4119890minus12
1183444
165
18Delq
uad
998779Vo
lumes
05
534
1081
193980
332
24247
5119890minus12
2119890minus12
4119890minus12
423
126768
6618
Delq
uad
998779Elem
ents
40
240
1090
1750
332
218
9119890minus12
4119890minus12
2119890minus12
43117
8667
18Delqu
ad998779
Elem
ents
30
150
1100
3184
332
398
1119890minus11
5119890minus12
2119890minus12
5013509
6818
Delq
uad
998779Elem
ents
20
89
1106
6426
332
803
5119890minus12
3119890minus12
2119890minus12
5916047
6918
Delq
uad
998779Elem
ents
1064
1111
24886
332
3110
2119890minus12
1119890minus12
1119890minus12
113
32788
7018
Delq
uad
998779Elem
ents
05
59
1113
98052
224
12256
1119890minus08
5119890minus09
8119890minus09
398
118153
7118
Delq
uad
◻Vo
lumes
40
minus398
1172
1650
332
206
1119890minus11
5119890minus12
2119890minus12
369907
7218
Delq
uad
◻Vo
lumes
30
minus249
1146
3058
332
382
4119890minus12
2119890minus12
1119890minus12
349349
7318
Delq
uad
◻Vo
lumes
20
minus131
1136
6228
224
778
3119890minus08
1119890minus08
5119890minus09
4010930
7418
Delq
uad
◻Vo
lumes
10minus51
1123
24536
224
3067
3119890minus08
2119890minus08
1119890minus08
7822716
7618
Delq
uad
◻Elem
ents
40
237
1090
1752
332
219
6119890minus12
3119890minus12
2119890minus12
4111213
7718
Delq
uad
◻Elem
ents
30
145
1101
3184
332
398
6119890minus12
3119890minus12
2119890minus12
43118
1178
18Delq
uad
◻Elem
ents
20
941107
6426
332
803
3119890minus12
1119890minus12
1119890minus12
6016220
7918
Delqu
ad◻
Elem
ents
1065
1111
24874
332
3109
1119890minus12
6119890minus13
8119890minus13
156
4400
080
18Delq
uad
◻Elem
ents
05
59
1113
9804
22
2412255
8119890minus09
4119890minus09
8119890minus09
567
162168
Science and Technology of Nuclear Installations 23
Tim
e con
sum
ed to
bui
ld th
e mat
rices
(sec
onds
)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay trianglesvolumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
10
30
100
300
1
3
Figure 23 Time needed to build the matrices versus number of unknowns
was considerably larger than the solution time itself Theeigenvalue problem was solved using a multilayer iterativeKrylov-Schur method [26] with a tolerance relative to thematrices norm
1003817100381710038171003817119877 sdot 120601 minus (1119896eff) sdot 119865 sdot 1206011003817100381710038171003817
119877 + (1119896eff) sdot 119865lt 10minus8 (11)
The reported residual norm and relative error are10038171003817100381710038171003817100381710038171003817119877 sdot 120601 minus
1
119896effsdot 119865 sdot 120601
10038171003817100381710038171003817100381710038171003817
1003817100381710038171003817119877 sdot 120601 minus (1119896eff) sdot 119865 sdot 1206011003817100381710038171003817
1003817100381710038171003817(1119896eff) sdot 1206011003817100381710038171003817
(12)
respectively The fields marked as outer linear and inneriterations refer to the number of steps needed to attainthe requested tolerance in each layer of the Krylov-Schuralgorithm The computer used to solve the problem has anIntel i7 920 267GHz processor with 4Gb of RAM runningDebian GNULinux Wheezy
When using a finite volumes-based scheme over anunstructured mesh the solver has to gather information
about which cells are neighbors and which are not Currentlygmsh does not write this kind of lists in its output filesso milonga has to explicitly solve the neighbors problemPerforming a linear search is an 119874(119873
2) task which is
unacceptable for problem sizes 119873 of interest The code usesa search based on a 119896-dimensional tree [27] which can inprinciple be performed in119874(119873) steps Still for large values of119873 the time needed to read and parse the mesh (number twopreviously mentioned) is the bottleneck of the solution Thisstep is not needed in finite elements although the construc-tion of the elementary matrices involves the computationof the multidimensional Jacobians and integrals which thenhave to be assembled Again for large 119873 this step (numberthree) takes up most of the time needed to solve the problem
The Delaunay algorithm is a standard method for gen-erating two-dimensional grids [16] by tessellating a planewith triangles If the mesh needs to be based on quadranglesinstead a recombination algorithm can be used to transformtwo adjacent triangles in one quadrangle whenever is possi-ble However for geometries which are based on rectangularshapes there exist other algorithms [25] both for meshingand for recombining the triangles that give rise to elementaryentities with right angles almost everywhere which may be a
24 Science and Technology of Nuclear Installations
Tim
e con
sum
ed to
solv
e the
eige
nval
ue p
robl
em (s
econ
ds)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
10
30
100
300
1
3
Figure 24 Time needed to solve the eigenvalue problem versus number of unknowns
desired property of the resulting grid For finite elements thesteps needed to build the matrices depend on the selectionof triangles or quadrangles as the basic elementary geometrybecause the shape functions change However the number ofunknowns is the same as the number of nodes that does notchange after a recombination procedure On the other handthe number of unknowns in the finite volumes schemes withtriangles is roughly twice as the number of unknowns withquadrangles for the same grid
Figure 19 shows how the computed static reactivity (ie1 minus 1119896eff) depends on the number of unknowns for eachof the sixteen combinations of geometry algorithm shapescheme The four accepted results published in the originalreference [10] are included for reference although it shouldbe taken into account that said reactivities were computedalmost forty years ago Figure 20 shows the total wall timeneeded to solve the problem as a function of 119873119866 whileFigures 21 22 23 and 24 show the times needed for eachof the first four steps involved in the solution maintainingthe same logarithmic scale for both the abscissas and theordinates Green data represent finite volumes whilst bluebullets correspond to finite elements Solid lines indicatequarter-core and dashed lines eighth-symmetry Fillet bullets
are results obtained by theDelaunay triangulation and emptybullets were computed with the delquad algorithm Finallytriangle-shaped data correspond to triangles and squares anddiamonds denote quadrangles as the basic geometry of thegrid
It can be seen that finite elements produce amuch smallerdispersion of eigenvalues 119896eff than finite volumes with therefinement of the mesh This can be explained because finitevolumes methods rely on a geometric condition of the meshwhich may change abruptly if the meshing algorithm decidesto allocate the cells in a rather different form for small changesin ℓ119888 Finite elements methods are less influenced by thesediscontinuous lay-out changes of the elements As expectedeighth-core symmetries give almost the same results as thequarter-core geometries with roughly half the unknownsFor small problems finite volumes run faster that finiteelements because the time needed to solve the neighborproblem is negligible When the problem size grows thistime increases and exceeds the overhead implied in the con-struction and assembly of the finite elements matrices Alsoat least for this configuration it is seen that the eigenvalueproblem is solved faster for finite volumes than for finiteelements
Science and Technology of Nuclear Installations 25
4 Conclusions
Unstructured grids provide the cognizant engineer witha wide variety of possibilities to deal with the design oranalysis of nuclear reactor cores These kinds of grids cansuccessfully reproduce continuous geometries commonlyfound in reactor cores such as cylinders and thereforenot only can the diffusion equation be better approximatedinside the domain of definition but also the fulfillment ofboundary conditions is improved A free computer codewas written from scratch that is able to completely solvethe 2D IAEA PWR Benchmark using unstructured gridsfor sixteen combinations of geometry meshing algorithmbasic shape and discretization scheme plus any value ofthe gridrsquos characteristic length by using a single input fileThe complete set of input files and codemdashexecutable andsourcemdashis available either online or upon request withcomments experiences suggestions and corrections beingmore than welcome Further development should includetackling full three-dimensional geometries with completethermal hydraulic feedback in order to analyze how thesolutions of the coupled neutronic-thermal problem dependon the spatial discretization scheme of the neutron leakageterm Parallelization of the computation and assembly of thematrices and of the solution of the eigenvalue problem andits implementation using GPUs are also desired features toimplement A problemwith direct applications that the futureversions of milonga ought to solve is the analysis of how thegeometry of the absorbing materials should be taken intoaccount in structured coarse grids in order to mitigate effectssuch as the rod-cusp problem
Suitable schemes for approximating the continuous dif-ferential operators by discrete matrix expressions includefinite volumes and finite elements families Finite volumesmethods compute cell mean values whilst finite elementsgive functional values at the gridrsquos nodes In general finiteelements are less sensitive to changes in the mesh so theresults they provide do not change significantly for differentmeshing algorithms or elementary shapes Small problemsare best solved by finite volumes as the neighbor-findingproblem is faster than the process of building and assemblingthe eigenvalue-problem matrices For a large number ofunknowns the process of finding which cell is neighbor ofwhichmdasheven based on a 119896-dimensional treemdashoverwhelmsthe computation and assembly of elementary matrices andfinite-element methods perform better
References
[1] G Glasstone and S Bell Nuclear Reactor Theory KriegerPublishing Company 1970
[2] J J Duderstadt and L J Hamilton Nuclear Reactor AnalysisJohn Wiley amp Sons New York NY USA 1976
[3] C Gho Reactor Physics Undergraduate Course Lecture NotesInstituto Balseiro 2006
[4] G G Theler and F J Bonetto ldquoOn the stability of the pointreactor kinetics equationsrdquoNuclear Engineering andDesign vol240 no 6 pp 1443ndash1449 2010
[5] E E Lewis andW F Miller Computational Methods of NeutronTransport John Wiley amp Sons 1984
[6] E Masiello ldquoAnalytical stability analysis of Coarse-Mesh finitedifference methodrdquo in Proceedings of the International Confer-ence on the Physics of Reactors (PHYSOR rsquo08) Nuclear Power ASustainable Resource pp 456ndash464 September 2008
[7] M L Zerkle Development of a polynomial nodal methodwith flux and current discontinuity factors [PhD thesis] Mas-sachusets Institute of Technology 1992
[8] J I Yoon andHG Joo ldquoTwo-level coarsemesh finite differenceformulation with multigroup source expansion nodal kernelsrdquoJournal of Nuclear Science andTechnology vol 45 no 7 pp 668ndash682 2008
[9] O Mazzantini M Schivo J Di Cesare R Garbero M Riveroand G Theler ldquoA coupled calculation suite for Atucha II oper-ational transients analysisrdquo Science and Technology of NuclearInstallations vol 2011 Article ID 785304 12 pages 2011
[10] Computational Benchmark Problem Comitee for the Mathe-matics and Computation Division of the American NuclearSociety ldquoArgonne Code Center Benchmark problem bookrdquoTech Rep ANL-7416 Supplement 2 Argonne National Labo-ratory 1977
[11] A F Henry Nuclear Reactor Analysis MIT Cambridge UK1975
[12] O C Zienkiewicz R L Taylor and J Z ZhuTheFinite ElementMethod Its Basis and Fundamentals vol 1 Elsevier 6th edition2005
[13] G Theler Difusion de neutrones en mallas no estructuradascomparacion entre volumenes y elementos finitos AcademicMonograph Universidad de Buenos Aires 2013
[14] G Theler ldquoMilonga a free nuclear reactor core analysis coderdquoAvailable online 2011
[15] G Theler F J Bonetto and A Clausse ldquoOptimizacion deparametros en reactores de potencia base de diseno delcodigo neutronico milongardquo in Reunion Anual de la AsociacionArgentina de Tecnologia Nuclear XXXVII 2010
[16] C Geuzaine and J-F Remacle ldquoGmsh a 3-D finite elementmesh generator with built-in pre- and post-processing facili-tiesrdquo International Journal for Numerical Methods in Engineer-ing vol 79 no 11 pp 1309ndash1331 2009
[17] S Balay J Brown K Buschelman et al ldquoPETSc users manualrdquoTech Rep ANL-9511-Revision 34 Argonne National Labora-tory 2013
[18] S Balay W D Gropp L Curfman McInnes and B F SmithldquoEfficient management of parallelism in object oriented numer-ical software librariesrdquo in Modern Software Tools in ScientificComputing E Arge A M Bruaset and H P Langtangen Edspp 163ndash202 Birkhauser 1997
[19] V Hernandez J E Roman and V Vidal ldquoSLEPC a scalable andflexible toolkit for the solution of eigenvalue problemsrdquo ACMTransactions on Mathematical Software vol 31 no 3 pp 351ndash362 2005
[20] M Galassi J Davies J Theiler et al GNU Scientific LibraryReference Manual 3rd edition 2009
[21] O C Zienkiewicz and R L Taylor The Finite Element MethodFor Solid and Structural Mechanics vol 3 Elsevier 6th edition2005
[22] A Imelda and K Doddy Evaluation of the IAEA 3-D PWRbenchmark problem using NESTLE code 2004
[23] R Mosteller ldquoStatic benchmarking of the NESTLE advancednodal coderdquo in Proceedings of the Joint International Conference
26 Science and Technology of Nuclear Installations
on Mathematical Methods and Supercomputing for NuclearApplications vol 2 pp 1596ndash1605 1997
[24] GTheler F J Bonetto andAClausse ldquoSolution of the 2D IAEAPWR Benchmark with the neutronic code milongardquo in Actasde la Reunion Anual de la Asociacion Argentina de TecnologıaNuclear XXXVIII 2011
[25] J F Remacle F Henrotte T Carrier-Baudouin et al ldquoAfrontal delaunay quad mesh generator using the 119871
infin normrdquoInternational Journal For Numerical Methods in Engineeringvol 94 no 5 pp 494ndash512 2013
[26] G W Stewart ldquoA Krylov-Schur algorithm for large eigenprob-lemsrdquo SIAM Journal on Matrix Analysis and Applications vol23 no 3 pp 601ndash614 2002
[27] J L Bentley ldquoMultidimensional binary search trees used forassociative searchingrdquo Communications of the ACM vol 18 no9 pp 509ndash517 1975
TribologyAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FuelsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofPetroleum Engineering
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
CombustionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Renewable Energy
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
StructuresJournal of
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear InstallationsScience and Technology of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solar EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Wind EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear EnergyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
12 Science and Technology of Nuclear Installations
Largest eigenvalue 1029788 (289260 pcm)1120 (3149 3011)1091 (13038 5114)
Number of unknowns 15776Outer iterations 3Linear iterations 32Inner iterations 1972Residual normRelative errorError estimateMemory used 56996 kBSoft page faults 15436Hard page faults 0Total CPU time 09121 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
2296 times 10minus12
1131 times 10minus12
6877 times 10minus13
keff
Milongarsquos 2D LWR IAEA benchmark problem case no 043eighth-symmetry core meshed using Delaunaya (triangs985747c = 2) solved with finite volumes
(a)
1 074 3241 5482 131 4188 9713 145 4582 10764 122 3877 9005 060 2644 4456 093 2986 6927 093 2941 6928 075 2030 5529 mdash 322 779
10 143 4523 106111 148 4667 109512 131 4145 97213 107 3423 79414 103 3269 76715 095 2993 70616 073 1988 53917 mdash 312 75118 147 4637 108819 134 4243 99620 118 3721 87321 107 3370 79322 098 2914 72323 067 1651 49924 mdash 235 57125 119 3763 88226 096 3077 71527 091 2847 67128 083 2255 61729 mdash 576 122530 mdash 068 26631 046 2016 34132 068 2058 50533 059 1443 43534 mdash 233 58935 057 1381 42036 mdash 381 82337 mdash 054 20838 mdash 062 244
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 14 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
scattered data shared-memory access numerical integrationfacilities and so forth The milonga code was designed withfour design basis vectors in mindmdashwhich are thoroughlydiscussed in the documentation [15]mdashwhich includes thetype of problems it can handle the code scalability which
features are expected and what to do with the obtainedresults
It works by first reading an input file that using plain-text English keywords and arguments defines the number119898 of spatial dimensions the number 119866 of group energies
Science and Technology of Nuclear Installations 13
Largest eigenvalue 1029726 (288675 pcm)1104 (3069 3069)864 (13000 5000)
Number of unknowns 4040Outer iterations 2Linear iterations 24Inner iterations 505Residual normRelative errorError estimateMemory used 42940 kBSoft page faults 11437Hard page faults 0Total CPU time 1064 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
2019 times 10minus8
9946 times 10minus9
7138 times 10minus9
keff
Milongarsquos 2D LWR IAEA benchmark problem case no 047eighth-symmetry core meshed using Delaunay (triangles 985747c = 3) solved with finite volumes
(a)
1 074 3194 5452 129 4117 9533 143 4515 10604 119 3814 8835 061 2634 4506 093 2984 6907 094 2954 6958 071 2064 5719 mdash 362 865
10 141 4458 104511 146 4599 107912 130 4097 96113 106 3388 78414 103 3263 76515 095 2998 70716 068 2015 55717 mdash 346 82318 145 4572 107319 133 4194 98420 117 3691 86621 107 3363 79222 098 2925 72623 059 1696 52824 mdash 263 63525 118 3728 87426 096 3068 71127 091 2850 67228 079 2280 63529 mdash 619 127630 mdash 081 32131 047 2036 34932 068 2085 50933 051 1465 45434 mdash 258 65035 049 1419 44336 mdash 416 85437 mdash 065 25438 mdash 071 288
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 15 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
and optionally a mesh file Currently only grids generatedwith the code gmsh [16] are only supported mainly becauseit is also a free software and it suits perfectly well milongarsquosdesign basis in the sense that the continuous geometry can bedefined as a function of a number of parameters Afterwardthe physical entities defined in the grid are mapped into
materials with macroscopic cross sections which maydepend on the spatial coordinates x bymeans of intermediatefunctions such as burn-up or temperatures distributionswhich in turn may be defined by algebraic expressions byinterpolating data located in files by reading shared-memoryobjects or by a combination of them In the same sense
14 Science and Technology of Nuclear Installations
Largest eigenvalue 1029695 (288387 pcm)1108 (3045 3045)833 (13000 5000)
Number of unknowns 8234Outer iterations 3Linear iterations 32Inner iterations 1029Residual normRelative error
Max 1206012(x y) coreMax 1206012(x y) reflector
2028 times 10minus12
9993 times 10minus13
1012 times 10minus12
keff
Relative errorError estimateMemory used 75468 kBSoft page faults 20094Hard page faults 0Total CPU time 1256 seconds
9993 times 10
1012 times 10minus12
Milongarsquos 2D LWR IAEA benchmark problem case no 058eighth-symmetry core meshed using Delaunay (quads 985747c = 2) solved with finite volumes
(a)
1 074 3207 5472 129 4136 9573 144 4534 10644 120 3828 8875 061 2638 4506 093 2985 6907 094 2948 6948 072 2053 5669 mdash 357 859
10 142 4478 105011 146 4617 108412 130 4111 96413 106 3396 78614 103 3263 76515 095 2991 70616 071 2005 55217 mdash 340 81618 145 4588 107719 133 4206 98720 117 3698 86821 107 3362 79122 098 2918 72523 062 1686 52024 mdash 258 62925 118 3737 87626 096 3073 71227 091 2848 67128 080 2272 63129 mdash 609 127230 mdash 080 31731 047 2035 34832 069 2081 50933 054 1458 44834 mdash 255 64635 052 1413 43736 mdash 409 85137 mdash 064 25138 mdash 071 284
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 16 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
boundary conditions are applied to appropriate physicalentities of dimension119898minus1The problemmatrices119877 and119865 arethen built and stored in an appropriate sparse format usingthe free PETSc [17 18] library and the eigenvalue problemis solved using the free SLEPc [19] library The results arestored into milongarsquos variables and functions which may be
evaluated integratedmdashusually using the free GNU ScientificLibrary [20] routinesmdashand of course written into appropriateoutputs Milonga can also solve problems parametrically andbe used to solve optimization problems As stated above thecode is a free software so corrections and contributions aremore than welcome
Science and Technology of Nuclear Installations 15
Largest eigenvalue 1029462 (286185 pcm)1136 (3093 2953)1233 (13100 5100)
Number of unknowns 6228Outer iterations 2Linear iterations 24Inner iterations 778Residual normR l ti
Max 1206012(x y) coreMax 1206012(x y) reflector
26 times 10minus8
1281 times 10minus8
4996 times 10minus9
keff
Relative errorError estimateMemory used 41460 kBSoft page faults 10930Hard page faults 0Total CPU time 0604 seconds
1281 times 10 8
4996 times 10minus9
Milongarsquos 2D LWR IAEA benchmark problem case no 073eighth-symmetry core meshed using Delaunay (quads 985747c = 2) solved with finite volumes
(a)
1 075 3297 5552 134 4268 9923 148 4660 10944 123 3926 9145 059 2645 4406 094 2997 6977 093 2914 6858 072 2002 5349 mdash 301 773
10 146 4606 108011 150 4735 111112 133 4212 98813 108 3448 80214 104 3271 76715 094 2954 69616 070 1954 52117 mdash 288 73518 149 4691 110119 136 4289 100720 119 3748 88021 107 3359 79022 096 2880 71423 064 1636 47524 mdash 217 56325 121 3810 89426 098 3110 72427 090 2829 66628 081 2227 59929 mdash 536 125730 mdash 062 25031 045 2002 33632 068 2040 50033 055 1414 40934 mdash 215 57935 055 1359 40836 mdash 359 83437 mdash 050 19938 mdash 061 236
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 17 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
31 A Coarse Bare Homogeneous Circle Figures 5 and 6 showthe results of solving a bare two-dimensional circular homo-geneous reactor of radius 119886 over an unstructured grid whichis deliberately coarse using the finite volumes method andthe finite elements method respectively Two group energies
were used and a null-flux boundary conditionwas fixed at theexternal surface Figures 5(a) and 6(a) compare the obtainedfast flux distributions In the first case the numerical solutionprovidesmean values for each neutron flux group in each cellwhile in the latter the solution is computed at the nodes and
16 Science and Technology of Nuclear Installations
Largest eigenvalue 1029851 (289858 pcm)1090 (3182 3182)851 (13000 5000)
Number of unknowns 1752Outer iterations 3Linear iterations 32Inner iterations 219Residual norm
Max 1206012(x y) coreMax 1206012(x y) reflector
566 times 10minus12
2789 times 10minus12
1595 times 10minus12
keff
Residual normRelative errorError estimateMemory used 42564 kBSoft page faults 11213Hard page faults 0Total CPU time 09921 seconds
566 times 10
2789 times 10minus12
1595 times 10minus12
Milongarsquos 2D LWR IAEA benchmark problem case no 076eighth-symmetry core meshed using Delaunay (quads 985747c = 4) solved with finite volumes
(a)
1 073 3151 5402 127 4059 9393 141 4457 10464 118 3768 8715 061 2615 4506 093 2979 6887 094 2971 6998 067 2092 5829 mdash 375 885
10 139 4397 103111 144 4541 106612 128 4052 95013 105 3365 77814 103 3259 76415 096 3016 71216 066 2045 56817 mdash 359 84118 143 4520 106119 132 4155 97520 116 3672 86221 107 3366 79222 098 2951 73323 054 1728 54324 mdash 273 65125 117 3700 86826 095 3057 70827 091 2860 67428 074 2311 64829 mdash 642 130030 mdash 084 33231 048 2041 35332 069 2103 51333 046 1493 46734 mdash 269 66735 046 1444 45536 mdash 432 87137 mdash 067 26338 mdash 074 297
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 18 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
continuous functions are evaluated by means of the shapefunctions used in the formulation Figures 5(b) and 6(b)illustrate the fast fluxes unknowns and its relative position inspace It can be noted that the mesh coarseness gives resultsthat differ substantially in both cases Finally the structure ofthe sparse eigenvalue problemmatrices is shown for each case
with blue red and cyan representing positive negative andexplicitly inserted zero values In the finite volume case thereare 184 unknowns (92 cells times 2 groups) while in the secondcase there are 218 unknowns (109 nodes times 2 groups) Thevolumesrsquo fissionmatrix is almost diagonal it has a bandwidthequal to the number of energy groups which in this case is
Science and Technology of Nuclear Installations 17
500 1000 2000 5000 10000 20000 50000Number of unknowns
2840
2860
2880
2900
2920
ORNLRisoslashKWUOntarioQuarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumes
Quarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elementsEighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumes
Eighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
120588(p
cm)
105 2 times 105 5 times 105
Figure 19 Static reactivity versus number of unknowns The four original solutions as published in 1977 [10] are included as reference
twoThe off-diagonal values appear right next to the diagonalelements because of the chosen ordering of the unknownsin the flux vector 120601 isin R184 Other orderings may be usedbut the rate of convergence of the eigenvalue problem can bedeteriorated On the other hand the elementsrsquo fission matrixhas a nontrivial structure because the grid is unstructuredand the net fission rate at each element depends on the fluxesat the nodes whose location inside thematrix depends in turnon how the gridwas generatedThis effect is similar to the onethat appears in structural analysis where mass matrices needto be lumped in order to simplify transient computations [21]The diagonal block of elementrsquos 119877 matrix and the null blockin 119865 correspond to the discrete equations that set the null-flux boundary conditions on the nodes located at the externalsurface Other types of boundary conditions lead to differentkinds of structures within the problem matrices
In case a part of the domain contained a nonmultiplicativematerial such as a reflector then there would appear sectionsof the fission matrix with null values in both methodsrendering 119865 singular in both methods Care should be takenwhen dealing with the numerical schemes for the eigenvalueproblem solution It can be seen in the elementsrsquo matricesa particular structure that implements the boundary condi-tions at the external surfaces This structure does not appearin the volumesrsquo matrices because the boundary conditions
appear as flux terms which are summed up over all thesurfaces of each cell so they aremasked inside the volumetricdiscretization of the divergence and gradient operators
As the two-group neutron diffusion equation with uni-form cross sections over a circle subject to null-flux bound-ary conditions has an analytical solution it is adequate tocompare how the two proposed numerical schemes relate toit In the studied problem we ignored upscattering and fastfissions Then the analytical effective multiplication factor is
119896eff =]Σ1198912 sdot Σ1199041rarr2
[Σ1198861 + Σ1199041rarr2 + 1198631(]0119886)2] [Σ1198862 + 1198632(]0119886)
2]
(7)
being ]0 = 24048 the smallest root of Besselrsquos first-kindfunction of order zero 1198690(119903)
Figure 7 shows how the two numerical effective multi-plication factors compare to the analytical solution givenby (7) as a function of the mesh refinement indicated bythe quotient 119886ℓ119888 between the radius of the circle and thecharacteristic length of the cellelements We can see thatthe 119896eff computed by the finite volumes (elements) methodis always greater (less) than the analytical solution Indeedit can be proven that for a bare one-dimensional slab thisis always the case [13] However this result does not hold
18 Science and Technology of Nuclear Installations
2000 3000 6000 10000 20000 30000 60000
01
003
001
03
1
3
10
30
100
300To
tal w
all t
ime (
seco
nds)
Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
Figure 20 Total wall time versus number of unknowns
for problems with nonuniform cross sections even in simpleone-dimensional reflected reactors
We may draw two other conclusions from Figure 7 Firstthat the finite element method seems to provide a bettersolution than the volumes-based scheme and second thateven though the error committed tends to decrease with finergrids its behavior is not monotonic for finite volumes Infact Figure 8 shows the difference between the numericaland analytical solutions using a logarithmic vertical scalewhere both conclusions are even more evident We deferthe discussion of such differences between the discretizationscheme until the next section It is worth to note howeverthat the fact that a finite-element-based scheme throws betterresults for the particular bare homogeneous circular reactorunder study than the finite volumes does not imply thatfinite volumes ought to be discarded as a valid tool forsolving the neutron diffusion equation in general cases Thecombination of lattice and core-level computations is usuallyperformedusing cell-based resultswhichwhen fed into node-based methods to solve the few-group neutron diffusionequation may introduce errors which potentially can leadto unacceptable solutions Nevertheless this analysis is farbeyond the scope of this paper which focuses on solving amathematical equation over unstructured grids
32 The 2D IAEA PWR Benchmark Problem This is aclassical two-group neutron diffusion problem first designedin the early 1970s and taken as a reference benchmark forcomputational codes A number of codes were used to solveeither this problem or its three-dimensional formulation[22 23] including milonga using a structured grid [24] Theoriginal formulation can be found in the reference [10] andthere is a reproduction that may be easily found online inreference [24] The geometry consists of a one-quarter ofa PWR core depicted in Figure 9 and the homogeneousmacroscopic cross sections are listed in Table 1 An axialbuckling term 119861
2
119911119892= 08times10
minus4 should be taken into accountThe external surface should be subject to a zero incomingcurrent condition which may be written as
120597120601119892
120597119899= minus
04692
119863119892
sdot 120601119892 (8)
The expected results are as follows
(1) Maximum eigenvalue(2) Fundamental flux distributions
(a) Radial flux traverses 120601119892(119909 0) and 120601119892(119909 119909)
Science and Technology of Nuclear Installations 19
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
1
3
10
30
100
300Ti
me c
onsu
med
to m
esh
the g
eom
etry
(sec
onds
)
Figure 21 Time needed to mesh the geometry versus number of unknowns
Table 1 Macroscopic cross-sections (units are not stated in the original reference but they are assumed to be in cmminus1 or cm as appropriate)
1198631
1198632
Σ1rarr2
Σ1198861
Σ1198862
]Σ1198912
Material1 15 04 002 001 0080 0135 Fuel 12 15 04 002 001 0085 0135 Fuel 23 15 04 002 001 0130 0135 Fuel 2 + rod4 20 03 004 0 0010 0 Reflector
Note the fluxes shall be normalized such that
1
119881coreint119881core
sum
119892
]Σ119891119892 sdot 120601119892119889119881 = 1 (9)
(b) Value and location of maximum power densityThis corresponds to maximum of 1206012 in thecore It is recommended that the maximumvalues of 1206012 both in the inner core and at thecorereflector interface be given
(3) Average subassembly powers 119875119896
119875119896 =1
119881119896
int119881119896
sum
119892
]Σ119891119892 sdot 120601119892119889119881 (10)
where 119881119896 is the volume of the 119896th subassembly and119896 designates the fuel subassemblies as shown inFigure 9
(4) Number of unknowns in the problem number ofiterations and total and outer
(5) Total computing time iteration time IO-time andcomputer used
(6) Type and numerical values of convergence criteria(7) Table of average group fluxes for a square mesh grid
of 20 times 20 cm(8) Dependence of results on mesh spacing
Even though the original problem is based on a quarter-core situation the problem has an eighth-core symmetry
20 Science and Technology of Nuclear Installations
Quarter Delaunay triangles elements
Tim
e con
sum
ed to
read
the m
esh
(sec
onds
)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumes
Quarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
1
3
10
30
100
300
Figure 22 Time needed to read the mesh versus number of unknowns
which cannot be taken into account by structured gridswhich are the main target of the benchmark Howevernonstructured grids can take into consideration any kind ofsymmetry almost without loss of accuracy and at the sametime reducing roughly the number of unknowns by a halfand the associated computational effort needed to solve theproblem by a factor of four Answers to items (1)ndash(7) canbe given completely by milonga using a single input file(see Supplementary data in SupplementaryMaterial availableonline at httpdxdoiorg1011552013641863) As asked initem (8) how results depend not only on the mesh spacingbut also on the meshing algorithm on the gridrsquos elementarygeometric shape and on the discretization scheme may shedlights on the subject whichmay be evenmore interesting thanthe numerical results themselves
Taking advantage of milongarsquos capability of reading andparsing command-line arguments the selection of the coregeometry (quarter or eighth) the meshing algorithm (delau-nay [16] or delquad [25]) the shape of the elementaryentities (triangles or quadrangles) the discretization scheme(volumes or elements) and the characteristic length ℓ119888 ofthe mesh can be provided at run time Fixing five values forℓ119888 = 4 3 2 1 05 gives 2 times 2 times 2 times 2 times 5 = 80 possiblecombinations which we solve with successive invocations to
milonga with the same input file but with different argumentsfrom a simple script Figures 10 11 12 13 14 15 16 17 and18 show the results corresponding to items (1)ndash(7) for someillustrative cases The complete set of figures and the codeused to generate themmay be provided upon request Table 2compiles the answers to the problem for every case studied
As the milonga code is still under development itsnumerical routines are not yet fully optimized nor designedfor parallel computation Therefore the reported times areonly rough estimates and should be taken with care Thesolution comprises five steps
(1) generate the grid with the requested geometry mesh-ing algorithm basic shape and characteristic lengthby calling to gmsh
(2) read the generated mesh(3) build the matrices(4) solve the eigenvalue problem(5) compute the requested results
The CPU time reported in Figures 10ndash18 is thus thesum of all of these five steps but not the time needed togenerate the figuresmdashwhich in some caseswith coarsemeshes
Science and Technology of Nuclear Installations 21
Table2
Resultsofthe2
DIAEA
PWRBe
nchm
arkp
roblem
obtained
with
them
ilong
acod
efor
thee
ightyp
ropo
sedcombinatio
nsofsymmetrym
eshing
algorithm
basicshapediscretization
schemeandcharacteris
ticelem
entc
elllengthTh
ereference
solutio
nis120588=28749times10minus5w
hich
correspo
ndsto119896eff=10296
Case
no
Symm-
etry
Mesh
algorithm
Basic
shape
Solutio
nmetho
dℓ119888
(cm)
Δ120588
(pcm
)max1206012
(mdash)
Total
unkn
owns
Outer
iter
Linear
iter
Inner
iter
Resid
ual
norm
Relativ
eerror
Error
estim
ate
Mem
ory
(Mb)
Page
faults
114
Delaunay
998779Vo
lumes
40
minus120
1145
8264
332
1033
6119890minus09
3119890minus09
1119890minus09
349470
214
Delaunay
998779Vo
lumes
30
85
1135
15740
332
1967
2119890minus08
1119890minus08
4119890minus09
4813401
314
Delaunay
998779Vo
lumes
20
197
1124
31188
332
3898
1119890minus08
6119890minus09
3119890minus09
7621932
414
Delaunay
998779Vo
lumes
10153
1123
127476
332
15934
7119890minus09
3119890minus09
3119890minus09
273
81886
514
Delaunay
998779Vo
lumes
05
187
1119
510676
332
63834
2119890minus09
9119890minus10
1119890minus09
1148
352261
614
Delaunay
998779Elem
ents
40
173
1100
4308
332
538
2119890minus08
8119890minus09
4119890minus09
318794
714
Delaunay
998779Elem
ents
30
117
1107
8108
332
1013
5119890minus09
2119890minus09
2119890minus09
4712946
814
Delaunay
998779Elem
ents
20
84
1112
15936
332
1992
6119890minus09
3119890minus09
2119890minus09
7321347
914
Delaunay
998779Elem
ents
1062
1116
64478
332
8059
6119890minus09
3119890minus09
4119890minus09
262
77105
1014
Delaunay
998779Elem
ents
05
58
1117
256700
332
32087
1119890minus09
5119890minus10
1119890minus09
1091
331969
1114
Delaunay
◻Vo
lumes
40
53
1149
5042
332
630
2119890minus08
1119890minus08
5119890minus09
288007
1214
Delaunay
◻Vo
lumes
30
346
1133
8560
332
1070
7119890minus09
3119890minus09
2119890minus09
3910895
1314
Delaunay
◻Vo
lumes
20
308
1131
15576
332
1947
1119890minus08
7119890minus09
3119890minus09
541564
514
14Delaunay
◻Vo
lumes
10177
1122
60774
332
7596
8119890minus09
4119890minus09
3119890minus09
167
50115
1514
Delaunay
◻Vo
lumes
05
199
1120
244936
332
30617
4119890minus09
2119890minus09
2119890minus09
698
215678
1614
Delaunay
◻Elem
ents
40
196
1100
5222
332
652
3119890minus08
1119890minus08
8119890minus09
4713098
1714
Delaunay
◻Elem
ents
30
123
1107
8810
332
1101
2119890minus08
9119890minus09
7119890minus09
6918598
1814
Delaunay
◻Elem
ents
20
901112
15922
332
1990
1119890minus08
5119890minus09
5119890minus09
107
30591
1914
Delaunay
◻Elem
ents
1063
1116
61456
332
7682
5119890minus09
2119890minus09
4119890minus09
389
113237
2014
Delaunay
◻Elem
ents
05
58
1117
246332
332
30791
8119890minus10
4119890minus10
1119890minus09
1676
498192
2114
Delq
uad
998779Vo
lumes
40
minus45
1144
6228
332
778
4119890minus08
2119890minus08
7119890minus09
308495
2214
Delq
uad
998779Vo
lumes
30
570
1153
11820
332
1477
3119890minus08
1119890minus08
4119890minus09
4010879
2314
Delq
uad
998779Vo
lumes
20
459
1103
24100
332
3012
2119890minus08
1119890minus08
5119890minus09
6217715
2414
Delq
uad
998779Vo
lumes
10512
1104
9640
03
3212050
2119890minus08
1119890minus08
9119890minus09
207
61714
2514
Delq
uad
998779Vo
lumes
05
528
1106
385600
332
48200
2119890minus09
1119890minus09
2119890minus09
838
255735
2614
Delqu
ad998779
Elem
ents
40
220
1093
3288
332
411
3119890minus08
2119890minus08
6119890minus09
308495
2714
Delq
uad
998779Elem
ents
30
140
1104
6148
332
768
4119890minus08
2119890minus08
8119890minus09
39110
0028
14Delq
uad
998779Elem
ents
20
82
1110
12392
332
1549
2119890minus08
9119890minus09
6119890minus09
6117067
2914
Delqu
ad998779
Elem
ents
1063
1115
48882
332
6110
2119890minus09
1119890minus09
1119890minus09
197
5804
630
14Delq
uad
998779Elem
ents
05
58
1116
194162
332
24270
2119890minus09
9119890minus10
2119890minus09
790
238769
3114
Delq
uad
◻Vo
lumes
40
minus416
1174
3132
332
391
5119890minus08
3119890minus08
8119890minus09
267322
3214
Delq
uad
◻Vo
lumes
30
minus248
1151
5922
332
740
9119890minus09
4119890minus09
2119890minus09
318779
3314
Delq
uad
◻Vo
lumes
20
minus132
1139
12050
332
1506
1119890minus08
7119890minus09
4119890minus09
4412239
3414
Delq
uad
◻Vo
lumes
10minus49
1126
48200
332
6025
6119890minus09
3119890minus09
2119890minus09
122
36094
3514
Delq
uad
◻Vo
lumes
05
minus23
1123
192800
332
24100
5119890minus09
2119890minus09
3119890minus09
464
140228
3614
Delq
uad
◻Elem
ents
40
224
1093
3294
332
411
3119890minus08
1119890minus08
7119890minus09
359852
3714
Delq
uad
◻Elem
ents
30
141
1104
6144
332
768
3119890minus08
2119890minus08
9119890minus09
5114018
3814
Delq
uad
◻Elem
ents
20
921111
12392
332
1549
1119890minus08
6119890minus09
5119890minus09
8122526
3914
Delq
uad
◻Elem
ents
1065
1115
48882
332
6110
5119890minus09
3119890minus09
4119890minus09
276
78431
22 Science and Technology of Nuclear Installations
Table2Con
tinued
Case
no
Symm-
etry
Mesh
algorithm
Basic
shape
Solutio
nmetho
dℓ119888
(cm)
Δ120588
(pcm
)max1206012
(mdash)
Total
unkn
owns
Outer
iter
Linear
iter
Inner
iter
Resid
ual
norm
Relativ
eerror
Error
estim
ate
Mem
ory
(Mb)
Page
faults
4014
Delq
uad
◻Elem
ents
05
59
1117
194162
332
24270
1119890minus09
5119890minus10
1119890minus09
1104
318967
4118
Delaunay
998779Vo
lumes
40
minus99
1139
4228
332
528
4119890minus12
2119890minus12
1119890minus12
359630
4218
Delaunay
998779Vo
lumes
30
50
1132
7712
332
964
1119890minus11
6119890minus12
2119890minus12
4211578
4318
Delaunay
998779Vo
lumes
20
177
1120
15776
332
1972
2119890minus12
1119890minus12
7119890minus13
5515436
4418
Delaunay
998779Vo
lumes
10142
1118
63902
332
7987
3119890minus12
1119890minus12
1119890minus12
160
46691
4518
Delaunay
998779Vo
lumes
05
176
1115
254612
332
31826
2119890minus12
8119890minus13
1119890minus12
555
168526
4618
Delaunay
998779Elem
ents
40
183
1096
2252
332
281
6119890minus12
3119890minus12
2119890minus12
359589
4718
Delaunay
998779Elem
ents
30
119
1104
4040
224
505
2119890minus08
1119890minus08
7119890minus09
41114
3748
18Delaunay
998779Elem
ents
20
85
1108
8156
332
1019
2119890minus12
8119890minus13
6119890minus13
5515053
4918
Delaunay
998779Elem
ents
1063
1112
32480
332
4060
1119890minus12
6119890minus13
8119890minus13
151
43671
5018
Delaunay
998779Elem
ents
05
58
1113
128358
332
1604
47119890minus13
3119890minus13
7119890minus13
527
157581
5118
Delaunay
◻Vo
lumes
40
37
1144
2380
224
297
5119890minus08
2119890minus08
1119890minus08
5314173
5218
Delaunay
◻Vo
lumes
30
218
1120
4194
332
524
7119890minus12
4119890minus12
2119890minus12
3610034
5318
Delaunay
◻Vo
lumes
20
345
1110
7960
332
995
5119890minus12
3119890minus12
1119890minus12
4712879
5418
Delaunay
◻Vo
lumes
10165
1117
30652
332
3831
2119890minus12
1119890minus12
7119890minus13
100
28851
5518
Delaunay
◻Vo
lumes
05
191
1117
122066
224
15258
2119890minus08
8119890minus09
9119890minus09
343
103825
5618
Delaunay
◻Elem
ents
40
175
1097
2526
332
315
4119890minus12
2119890minus12
1119890minus12
6016159
5718
Delaunay
◻Elem
ents
30
114
1104
4386
332
548
4119890minus12
2119890minus12
2119890minus12
5013768
5818
Delaunay
◻Elem
ents
20
901108
8234
332
1029
2119890minus12
1119890minus12
1119890minus12
7320094
5918
Delaunay
◻Elem
ents
1062
1112
31186
224
3898
2119890minus08
7119890minus09
9119890minus09
208
59589
6018
Delaunay
◻Elem
ents
05
58
1113
123122
224
15390
9119890minus09
5119890minus09
1119890minus08
807
2366
7061
18Delq
uad
998779Vo
lumes
40
minus210
1168
3224
332
403
1119890minus11
5119890minus12
2119890minus12
43117
8462
18Delq
uad
998779Vo
lumes
30
369
1150
6000
332
750
1119890minus11
6119890minus12
3119890minus12
5013502
6318
Delq
uad
998779Vo
lumes
20
312
1104
12316
332
1539
2119890minus11
8119890minus12
5119890minus12
6016395
6418
Delq
uad
998779Vo
lumes
1044
01094
48714
332
6089
9119890minus12
4119890minus12
4119890minus12
1183444
165
18Delq
uad
998779Vo
lumes
05
534
1081
193980
332
24247
5119890minus12
2119890minus12
4119890minus12
423
126768
6618
Delq
uad
998779Elem
ents
40
240
1090
1750
332
218
9119890minus12
4119890minus12
2119890minus12
43117
8667
18Delqu
ad998779
Elem
ents
30
150
1100
3184
332
398
1119890minus11
5119890minus12
2119890minus12
5013509
6818
Delq
uad
998779Elem
ents
20
89
1106
6426
332
803
5119890minus12
3119890minus12
2119890minus12
5916047
6918
Delq
uad
998779Elem
ents
1064
1111
24886
332
3110
2119890minus12
1119890minus12
1119890minus12
113
32788
7018
Delq
uad
998779Elem
ents
05
59
1113
98052
224
12256
1119890minus08
5119890minus09
8119890minus09
398
118153
7118
Delq
uad
◻Vo
lumes
40
minus398
1172
1650
332
206
1119890minus11
5119890minus12
2119890minus12
369907
7218
Delq
uad
◻Vo
lumes
30
minus249
1146
3058
332
382
4119890minus12
2119890minus12
1119890minus12
349349
7318
Delq
uad
◻Vo
lumes
20
minus131
1136
6228
224
778
3119890minus08
1119890minus08
5119890minus09
4010930
7418
Delq
uad
◻Vo
lumes
10minus51
1123
24536
224
3067
3119890minus08
2119890minus08
1119890minus08
7822716
7618
Delq
uad
◻Elem
ents
40
237
1090
1752
332
219
6119890minus12
3119890minus12
2119890minus12
4111213
7718
Delq
uad
◻Elem
ents
30
145
1101
3184
332
398
6119890minus12
3119890minus12
2119890minus12
43118
1178
18Delq
uad
◻Elem
ents
20
941107
6426
332
803
3119890minus12
1119890minus12
1119890minus12
6016220
7918
Delqu
ad◻
Elem
ents
1065
1111
24874
332
3109
1119890minus12
6119890minus13
8119890minus13
156
4400
080
18Delq
uad
◻Elem
ents
05
59
1113
9804
22
2412255
8119890minus09
4119890minus09
8119890minus09
567
162168
Science and Technology of Nuclear Installations 23
Tim
e con
sum
ed to
bui
ld th
e mat
rices
(sec
onds
)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay trianglesvolumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
10
30
100
300
1
3
Figure 23 Time needed to build the matrices versus number of unknowns
was considerably larger than the solution time itself Theeigenvalue problem was solved using a multilayer iterativeKrylov-Schur method [26] with a tolerance relative to thematrices norm
1003817100381710038171003817119877 sdot 120601 minus (1119896eff) sdot 119865 sdot 1206011003817100381710038171003817
119877 + (1119896eff) sdot 119865lt 10minus8 (11)
The reported residual norm and relative error are10038171003817100381710038171003817100381710038171003817119877 sdot 120601 minus
1
119896effsdot 119865 sdot 120601
10038171003817100381710038171003817100381710038171003817
1003817100381710038171003817119877 sdot 120601 minus (1119896eff) sdot 119865 sdot 1206011003817100381710038171003817
1003817100381710038171003817(1119896eff) sdot 1206011003817100381710038171003817
(12)
respectively The fields marked as outer linear and inneriterations refer to the number of steps needed to attainthe requested tolerance in each layer of the Krylov-Schuralgorithm The computer used to solve the problem has anIntel i7 920 267GHz processor with 4Gb of RAM runningDebian GNULinux Wheezy
When using a finite volumes-based scheme over anunstructured mesh the solver has to gather information
about which cells are neighbors and which are not Currentlygmsh does not write this kind of lists in its output filesso milonga has to explicitly solve the neighbors problemPerforming a linear search is an 119874(119873
2) task which is
unacceptable for problem sizes 119873 of interest The code usesa search based on a 119896-dimensional tree [27] which can inprinciple be performed in119874(119873) steps Still for large values of119873 the time needed to read and parse the mesh (number twopreviously mentioned) is the bottleneck of the solution Thisstep is not needed in finite elements although the construc-tion of the elementary matrices involves the computationof the multidimensional Jacobians and integrals which thenhave to be assembled Again for large 119873 this step (numberthree) takes up most of the time needed to solve the problem
The Delaunay algorithm is a standard method for gen-erating two-dimensional grids [16] by tessellating a planewith triangles If the mesh needs to be based on quadranglesinstead a recombination algorithm can be used to transformtwo adjacent triangles in one quadrangle whenever is possi-ble However for geometries which are based on rectangularshapes there exist other algorithms [25] both for meshingand for recombining the triangles that give rise to elementaryentities with right angles almost everywhere which may be a
24 Science and Technology of Nuclear Installations
Tim
e con
sum
ed to
solv
e the
eige
nval
ue p
robl
em (s
econ
ds)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
10
30
100
300
1
3
Figure 24 Time needed to solve the eigenvalue problem versus number of unknowns
desired property of the resulting grid For finite elements thesteps needed to build the matrices depend on the selectionof triangles or quadrangles as the basic elementary geometrybecause the shape functions change However the number ofunknowns is the same as the number of nodes that does notchange after a recombination procedure On the other handthe number of unknowns in the finite volumes schemes withtriangles is roughly twice as the number of unknowns withquadrangles for the same grid
Figure 19 shows how the computed static reactivity (ie1 minus 1119896eff) depends on the number of unknowns for eachof the sixteen combinations of geometry algorithm shapescheme The four accepted results published in the originalreference [10] are included for reference although it shouldbe taken into account that said reactivities were computedalmost forty years ago Figure 20 shows the total wall timeneeded to solve the problem as a function of 119873119866 whileFigures 21 22 23 and 24 show the times needed for eachof the first four steps involved in the solution maintainingthe same logarithmic scale for both the abscissas and theordinates Green data represent finite volumes whilst bluebullets correspond to finite elements Solid lines indicatequarter-core and dashed lines eighth-symmetry Fillet bullets
are results obtained by theDelaunay triangulation and emptybullets were computed with the delquad algorithm Finallytriangle-shaped data correspond to triangles and squares anddiamonds denote quadrangles as the basic geometry of thegrid
It can be seen that finite elements produce amuch smallerdispersion of eigenvalues 119896eff than finite volumes with therefinement of the mesh This can be explained because finitevolumes methods rely on a geometric condition of the meshwhich may change abruptly if the meshing algorithm decidesto allocate the cells in a rather different form for small changesin ℓ119888 Finite elements methods are less influenced by thesediscontinuous lay-out changes of the elements As expectedeighth-core symmetries give almost the same results as thequarter-core geometries with roughly half the unknownsFor small problems finite volumes run faster that finiteelements because the time needed to solve the neighborproblem is negligible When the problem size grows thistime increases and exceeds the overhead implied in the con-struction and assembly of the finite elements matrices Alsoat least for this configuration it is seen that the eigenvalueproblem is solved faster for finite volumes than for finiteelements
Science and Technology of Nuclear Installations 25
4 Conclusions
Unstructured grids provide the cognizant engineer witha wide variety of possibilities to deal with the design oranalysis of nuclear reactor cores These kinds of grids cansuccessfully reproduce continuous geometries commonlyfound in reactor cores such as cylinders and thereforenot only can the diffusion equation be better approximatedinside the domain of definition but also the fulfillment ofboundary conditions is improved A free computer codewas written from scratch that is able to completely solvethe 2D IAEA PWR Benchmark using unstructured gridsfor sixteen combinations of geometry meshing algorithmbasic shape and discretization scheme plus any value ofthe gridrsquos characteristic length by using a single input fileThe complete set of input files and codemdashexecutable andsourcemdashis available either online or upon request withcomments experiences suggestions and corrections beingmore than welcome Further development should includetackling full three-dimensional geometries with completethermal hydraulic feedback in order to analyze how thesolutions of the coupled neutronic-thermal problem dependon the spatial discretization scheme of the neutron leakageterm Parallelization of the computation and assembly of thematrices and of the solution of the eigenvalue problem andits implementation using GPUs are also desired features toimplement A problemwith direct applications that the futureversions of milonga ought to solve is the analysis of how thegeometry of the absorbing materials should be taken intoaccount in structured coarse grids in order to mitigate effectssuch as the rod-cusp problem
Suitable schemes for approximating the continuous dif-ferential operators by discrete matrix expressions includefinite volumes and finite elements families Finite volumesmethods compute cell mean values whilst finite elementsgive functional values at the gridrsquos nodes In general finiteelements are less sensitive to changes in the mesh so theresults they provide do not change significantly for differentmeshing algorithms or elementary shapes Small problemsare best solved by finite volumes as the neighbor-findingproblem is faster than the process of building and assemblingthe eigenvalue-problem matrices For a large number ofunknowns the process of finding which cell is neighbor ofwhichmdasheven based on a 119896-dimensional treemdashoverwhelmsthe computation and assembly of elementary matrices andfinite-element methods perform better
References
[1] G Glasstone and S Bell Nuclear Reactor Theory KriegerPublishing Company 1970
[2] J J Duderstadt and L J Hamilton Nuclear Reactor AnalysisJohn Wiley amp Sons New York NY USA 1976
[3] C Gho Reactor Physics Undergraduate Course Lecture NotesInstituto Balseiro 2006
[4] G G Theler and F J Bonetto ldquoOn the stability of the pointreactor kinetics equationsrdquoNuclear Engineering andDesign vol240 no 6 pp 1443ndash1449 2010
[5] E E Lewis andW F Miller Computational Methods of NeutronTransport John Wiley amp Sons 1984
[6] E Masiello ldquoAnalytical stability analysis of Coarse-Mesh finitedifference methodrdquo in Proceedings of the International Confer-ence on the Physics of Reactors (PHYSOR rsquo08) Nuclear Power ASustainable Resource pp 456ndash464 September 2008
[7] M L Zerkle Development of a polynomial nodal methodwith flux and current discontinuity factors [PhD thesis] Mas-sachusets Institute of Technology 1992
[8] J I Yoon andHG Joo ldquoTwo-level coarsemesh finite differenceformulation with multigroup source expansion nodal kernelsrdquoJournal of Nuclear Science andTechnology vol 45 no 7 pp 668ndash682 2008
[9] O Mazzantini M Schivo J Di Cesare R Garbero M Riveroand G Theler ldquoA coupled calculation suite for Atucha II oper-ational transients analysisrdquo Science and Technology of NuclearInstallations vol 2011 Article ID 785304 12 pages 2011
[10] Computational Benchmark Problem Comitee for the Mathe-matics and Computation Division of the American NuclearSociety ldquoArgonne Code Center Benchmark problem bookrdquoTech Rep ANL-7416 Supplement 2 Argonne National Labo-ratory 1977
[11] A F Henry Nuclear Reactor Analysis MIT Cambridge UK1975
[12] O C Zienkiewicz R L Taylor and J Z ZhuTheFinite ElementMethod Its Basis and Fundamentals vol 1 Elsevier 6th edition2005
[13] G Theler Difusion de neutrones en mallas no estructuradascomparacion entre volumenes y elementos finitos AcademicMonograph Universidad de Buenos Aires 2013
[14] G Theler ldquoMilonga a free nuclear reactor core analysis coderdquoAvailable online 2011
[15] G Theler F J Bonetto and A Clausse ldquoOptimizacion deparametros en reactores de potencia base de diseno delcodigo neutronico milongardquo in Reunion Anual de la AsociacionArgentina de Tecnologia Nuclear XXXVII 2010
[16] C Geuzaine and J-F Remacle ldquoGmsh a 3-D finite elementmesh generator with built-in pre- and post-processing facili-tiesrdquo International Journal for Numerical Methods in Engineer-ing vol 79 no 11 pp 1309ndash1331 2009
[17] S Balay J Brown K Buschelman et al ldquoPETSc users manualrdquoTech Rep ANL-9511-Revision 34 Argonne National Labora-tory 2013
[18] S Balay W D Gropp L Curfman McInnes and B F SmithldquoEfficient management of parallelism in object oriented numer-ical software librariesrdquo in Modern Software Tools in ScientificComputing E Arge A M Bruaset and H P Langtangen Edspp 163ndash202 Birkhauser 1997
[19] V Hernandez J E Roman and V Vidal ldquoSLEPC a scalable andflexible toolkit for the solution of eigenvalue problemsrdquo ACMTransactions on Mathematical Software vol 31 no 3 pp 351ndash362 2005
[20] M Galassi J Davies J Theiler et al GNU Scientific LibraryReference Manual 3rd edition 2009
[21] O C Zienkiewicz and R L Taylor The Finite Element MethodFor Solid and Structural Mechanics vol 3 Elsevier 6th edition2005
[22] A Imelda and K Doddy Evaluation of the IAEA 3-D PWRbenchmark problem using NESTLE code 2004
[23] R Mosteller ldquoStatic benchmarking of the NESTLE advancednodal coderdquo in Proceedings of the Joint International Conference
26 Science and Technology of Nuclear Installations
on Mathematical Methods and Supercomputing for NuclearApplications vol 2 pp 1596ndash1605 1997
[24] GTheler F J Bonetto andAClausse ldquoSolution of the 2D IAEAPWR Benchmark with the neutronic code milongardquo in Actasde la Reunion Anual de la Asociacion Argentina de TecnologıaNuclear XXXVIII 2011
[25] J F Remacle F Henrotte T Carrier-Baudouin et al ldquoAfrontal delaunay quad mesh generator using the 119871
infin normrdquoInternational Journal For Numerical Methods in Engineeringvol 94 no 5 pp 494ndash512 2013
[26] G W Stewart ldquoA Krylov-Schur algorithm for large eigenprob-lemsrdquo SIAM Journal on Matrix Analysis and Applications vol23 no 3 pp 601ndash614 2002
[27] J L Bentley ldquoMultidimensional binary search trees used forassociative searchingrdquo Communications of the ACM vol 18 no9 pp 509ndash517 1975
TribologyAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FuelsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofPetroleum Engineering
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
CombustionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Renewable Energy
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
StructuresJournal of
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear InstallationsScience and Technology of
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Solar EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Wind EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear EnergyInternational Journal of
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High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Science and Technology of Nuclear Installations 13
Largest eigenvalue 1029726 (288675 pcm)1104 (3069 3069)864 (13000 5000)
Number of unknowns 4040Outer iterations 2Linear iterations 24Inner iterations 505Residual normRelative errorError estimateMemory used 42940 kBSoft page faults 11437Hard page faults 0Total CPU time 1064 seconds
Max 1206012(x y) coreMax 1206012(x y) reflector
2019 times 10minus8
9946 times 10minus9
7138 times 10minus9
keff
Milongarsquos 2D LWR IAEA benchmark problem case no 047eighth-symmetry core meshed using Delaunay (triangles 985747c = 3) solved with finite volumes
(a)
1 074 3194 5452 129 4117 9533 143 4515 10604 119 3814 8835 061 2634 4506 093 2984 6907 094 2954 6958 071 2064 5719 mdash 362 865
10 141 4458 104511 146 4599 107912 130 4097 96113 106 3388 78414 103 3263 76515 095 2998 70716 068 2015 55717 mdash 346 82318 145 4572 107319 133 4194 98420 117 3691 86621 107 3363 79222 098 2925 72623 059 1696 52824 mdash 263 63525 118 3728 87426 096 3068 71127 091 2850 67228 079 2280 63529 mdash 619 127630 mdash 081 32131 047 2036 34932 068 2085 50933 051 1465 45434 mdash 258 65035 049 1419 44336 mdash 416 85437 mdash 065 25438 mdash 071 288
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 15 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
and optionally a mesh file Currently only grids generatedwith the code gmsh [16] are only supported mainly becauseit is also a free software and it suits perfectly well milongarsquosdesign basis in the sense that the continuous geometry can bedefined as a function of a number of parameters Afterwardthe physical entities defined in the grid are mapped into
materials with macroscopic cross sections which maydepend on the spatial coordinates x bymeans of intermediatefunctions such as burn-up or temperatures distributionswhich in turn may be defined by algebraic expressions byinterpolating data located in files by reading shared-memoryobjects or by a combination of them In the same sense
14 Science and Technology of Nuclear Installations
Largest eigenvalue 1029695 (288387 pcm)1108 (3045 3045)833 (13000 5000)
Number of unknowns 8234Outer iterations 3Linear iterations 32Inner iterations 1029Residual normRelative error
Max 1206012(x y) coreMax 1206012(x y) reflector
2028 times 10minus12
9993 times 10minus13
1012 times 10minus12
keff
Relative errorError estimateMemory used 75468 kBSoft page faults 20094Hard page faults 0Total CPU time 1256 seconds
9993 times 10
1012 times 10minus12
Milongarsquos 2D LWR IAEA benchmark problem case no 058eighth-symmetry core meshed using Delaunay (quads 985747c = 2) solved with finite volumes
(a)
1 074 3207 5472 129 4136 9573 144 4534 10644 120 3828 8875 061 2638 4506 093 2985 6907 094 2948 6948 072 2053 5669 mdash 357 859
10 142 4478 105011 146 4617 108412 130 4111 96413 106 3396 78614 103 3263 76515 095 2991 70616 071 2005 55217 mdash 340 81618 145 4588 107719 133 4206 98720 117 3698 86821 107 3362 79122 098 2918 72523 062 1686 52024 mdash 258 62925 118 3737 87626 096 3073 71227 091 2848 67128 080 2272 63129 mdash 609 127230 mdash 080 31731 047 2035 34832 069 2081 50933 054 1458 44834 mdash 255 64635 052 1413 43736 mdash 409 85137 mdash 064 25138 mdash 071 284
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 16 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
boundary conditions are applied to appropriate physicalentities of dimension119898minus1The problemmatrices119877 and119865 arethen built and stored in an appropriate sparse format usingthe free PETSc [17 18] library and the eigenvalue problemis solved using the free SLEPc [19] library The results arestored into milongarsquos variables and functions which may be
evaluated integratedmdashusually using the free GNU ScientificLibrary [20] routinesmdashand of course written into appropriateoutputs Milonga can also solve problems parametrically andbe used to solve optimization problems As stated above thecode is a free software so corrections and contributions aremore than welcome
Science and Technology of Nuclear Installations 15
Largest eigenvalue 1029462 (286185 pcm)1136 (3093 2953)1233 (13100 5100)
Number of unknowns 6228Outer iterations 2Linear iterations 24Inner iterations 778Residual normR l ti
Max 1206012(x y) coreMax 1206012(x y) reflector
26 times 10minus8
1281 times 10minus8
4996 times 10minus9
keff
Relative errorError estimateMemory used 41460 kBSoft page faults 10930Hard page faults 0Total CPU time 0604 seconds
1281 times 10 8
4996 times 10minus9
Milongarsquos 2D LWR IAEA benchmark problem case no 073eighth-symmetry core meshed using Delaunay (quads 985747c = 2) solved with finite volumes
(a)
1 075 3297 5552 134 4268 9923 148 4660 10944 123 3926 9145 059 2645 4406 094 2997 6977 093 2914 6858 072 2002 5349 mdash 301 773
10 146 4606 108011 150 4735 111112 133 4212 98813 108 3448 80214 104 3271 76715 094 2954 69616 070 1954 52117 mdash 288 73518 149 4691 110119 136 4289 100720 119 3748 88021 107 3359 79022 096 2880 71423 064 1636 47524 mdash 217 56325 121 3810 89426 098 3110 72427 090 2829 66628 081 2227 59929 mdash 536 125730 mdash 062 25031 045 2002 33632 068 2040 50033 055 1414 40934 mdash 215 57935 055 1359 40836 mdash 359 83437 mdash 050 19938 mdash 061 236
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 17 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
31 A Coarse Bare Homogeneous Circle Figures 5 and 6 showthe results of solving a bare two-dimensional circular homo-geneous reactor of radius 119886 over an unstructured grid whichis deliberately coarse using the finite volumes method andthe finite elements method respectively Two group energies
were used and a null-flux boundary conditionwas fixed at theexternal surface Figures 5(a) and 6(a) compare the obtainedfast flux distributions In the first case the numerical solutionprovidesmean values for each neutron flux group in each cellwhile in the latter the solution is computed at the nodes and
16 Science and Technology of Nuclear Installations
Largest eigenvalue 1029851 (289858 pcm)1090 (3182 3182)851 (13000 5000)
Number of unknowns 1752Outer iterations 3Linear iterations 32Inner iterations 219Residual norm
Max 1206012(x y) coreMax 1206012(x y) reflector
566 times 10minus12
2789 times 10minus12
1595 times 10minus12
keff
Residual normRelative errorError estimateMemory used 42564 kBSoft page faults 11213Hard page faults 0Total CPU time 09921 seconds
566 times 10
2789 times 10minus12
1595 times 10minus12
Milongarsquos 2D LWR IAEA benchmark problem case no 076eighth-symmetry core meshed using Delaunay (quads 985747c = 4) solved with finite volumes
(a)
1 073 3151 5402 127 4059 9393 141 4457 10464 118 3768 8715 061 2615 4506 093 2979 6887 094 2971 6998 067 2092 5829 mdash 375 885
10 139 4397 103111 144 4541 106612 128 4052 95013 105 3365 77814 103 3259 76415 096 3016 71216 066 2045 56817 mdash 359 84118 143 4520 106119 132 4155 97520 116 3672 86221 107 3366 79222 098 2951 73323 054 1728 54324 mdash 273 65125 117 3700 86826 095 3057 70827 091 2860 67428 074 2311 64829 mdash 642 130030 mdash 084 33231 048 2041 35332 069 2103 51333 046 1493 46734 mdash 269 66735 046 1444 45536 mdash 432 87137 mdash 067 26338 mdash 074 297
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 18 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
continuous functions are evaluated by means of the shapefunctions used in the formulation Figures 5(b) and 6(b)illustrate the fast fluxes unknowns and its relative position inspace It can be noted that the mesh coarseness gives resultsthat differ substantially in both cases Finally the structure ofthe sparse eigenvalue problemmatrices is shown for each case
with blue red and cyan representing positive negative andexplicitly inserted zero values In the finite volume case thereare 184 unknowns (92 cells times 2 groups) while in the secondcase there are 218 unknowns (109 nodes times 2 groups) Thevolumesrsquo fissionmatrix is almost diagonal it has a bandwidthequal to the number of energy groups which in this case is
Science and Technology of Nuclear Installations 17
500 1000 2000 5000 10000 20000 50000Number of unknowns
2840
2860
2880
2900
2920
ORNLRisoslashKWUOntarioQuarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumes
Quarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elementsEighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumes
Eighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
120588(p
cm)
105 2 times 105 5 times 105
Figure 19 Static reactivity versus number of unknowns The four original solutions as published in 1977 [10] are included as reference
twoThe off-diagonal values appear right next to the diagonalelements because of the chosen ordering of the unknownsin the flux vector 120601 isin R184 Other orderings may be usedbut the rate of convergence of the eigenvalue problem can bedeteriorated On the other hand the elementsrsquo fission matrixhas a nontrivial structure because the grid is unstructuredand the net fission rate at each element depends on the fluxesat the nodes whose location inside thematrix depends in turnon how the gridwas generatedThis effect is similar to the onethat appears in structural analysis where mass matrices needto be lumped in order to simplify transient computations [21]The diagonal block of elementrsquos 119877 matrix and the null blockin 119865 correspond to the discrete equations that set the null-flux boundary conditions on the nodes located at the externalsurface Other types of boundary conditions lead to differentkinds of structures within the problem matrices
In case a part of the domain contained a nonmultiplicativematerial such as a reflector then there would appear sectionsof the fission matrix with null values in both methodsrendering 119865 singular in both methods Care should be takenwhen dealing with the numerical schemes for the eigenvalueproblem solution It can be seen in the elementsrsquo matricesa particular structure that implements the boundary condi-tions at the external surfaces This structure does not appearin the volumesrsquo matrices because the boundary conditions
appear as flux terms which are summed up over all thesurfaces of each cell so they aremasked inside the volumetricdiscretization of the divergence and gradient operators
As the two-group neutron diffusion equation with uni-form cross sections over a circle subject to null-flux bound-ary conditions has an analytical solution it is adequate tocompare how the two proposed numerical schemes relate toit In the studied problem we ignored upscattering and fastfissions Then the analytical effective multiplication factor is
119896eff =]Σ1198912 sdot Σ1199041rarr2
[Σ1198861 + Σ1199041rarr2 + 1198631(]0119886)2] [Σ1198862 + 1198632(]0119886)
2]
(7)
being ]0 = 24048 the smallest root of Besselrsquos first-kindfunction of order zero 1198690(119903)
Figure 7 shows how the two numerical effective multi-plication factors compare to the analytical solution givenby (7) as a function of the mesh refinement indicated bythe quotient 119886ℓ119888 between the radius of the circle and thecharacteristic length of the cellelements We can see thatthe 119896eff computed by the finite volumes (elements) methodis always greater (less) than the analytical solution Indeedit can be proven that for a bare one-dimensional slab thisis always the case [13] However this result does not hold
18 Science and Technology of Nuclear Installations
2000 3000 6000 10000 20000 30000 60000
01
003
001
03
1
3
10
30
100
300To
tal w
all t
ime (
seco
nds)
Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
Figure 20 Total wall time versus number of unknowns
for problems with nonuniform cross sections even in simpleone-dimensional reflected reactors
We may draw two other conclusions from Figure 7 Firstthat the finite element method seems to provide a bettersolution than the volumes-based scheme and second thateven though the error committed tends to decrease with finergrids its behavior is not monotonic for finite volumes Infact Figure 8 shows the difference between the numericaland analytical solutions using a logarithmic vertical scalewhere both conclusions are even more evident We deferthe discussion of such differences between the discretizationscheme until the next section It is worth to note howeverthat the fact that a finite-element-based scheme throws betterresults for the particular bare homogeneous circular reactorunder study than the finite volumes does not imply thatfinite volumes ought to be discarded as a valid tool forsolving the neutron diffusion equation in general cases Thecombination of lattice and core-level computations is usuallyperformedusing cell-based resultswhichwhen fed into node-based methods to solve the few-group neutron diffusionequation may introduce errors which potentially can leadto unacceptable solutions Nevertheless this analysis is farbeyond the scope of this paper which focuses on solving amathematical equation over unstructured grids
32 The 2D IAEA PWR Benchmark Problem This is aclassical two-group neutron diffusion problem first designedin the early 1970s and taken as a reference benchmark forcomputational codes A number of codes were used to solveeither this problem or its three-dimensional formulation[22 23] including milonga using a structured grid [24] Theoriginal formulation can be found in the reference [10] andthere is a reproduction that may be easily found online inreference [24] The geometry consists of a one-quarter ofa PWR core depicted in Figure 9 and the homogeneousmacroscopic cross sections are listed in Table 1 An axialbuckling term 119861
2
119911119892= 08times10
minus4 should be taken into accountThe external surface should be subject to a zero incomingcurrent condition which may be written as
120597120601119892
120597119899= minus
04692
119863119892
sdot 120601119892 (8)
The expected results are as follows
(1) Maximum eigenvalue(2) Fundamental flux distributions
(a) Radial flux traverses 120601119892(119909 0) and 120601119892(119909 119909)
Science and Technology of Nuclear Installations 19
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
1
3
10
30
100
300Ti
me c
onsu
med
to m
esh
the g
eom
etry
(sec
onds
)
Figure 21 Time needed to mesh the geometry versus number of unknowns
Table 1 Macroscopic cross-sections (units are not stated in the original reference but they are assumed to be in cmminus1 or cm as appropriate)
1198631
1198632
Σ1rarr2
Σ1198861
Σ1198862
]Σ1198912
Material1 15 04 002 001 0080 0135 Fuel 12 15 04 002 001 0085 0135 Fuel 23 15 04 002 001 0130 0135 Fuel 2 + rod4 20 03 004 0 0010 0 Reflector
Note the fluxes shall be normalized such that
1
119881coreint119881core
sum
119892
]Σ119891119892 sdot 120601119892119889119881 = 1 (9)
(b) Value and location of maximum power densityThis corresponds to maximum of 1206012 in thecore It is recommended that the maximumvalues of 1206012 both in the inner core and at thecorereflector interface be given
(3) Average subassembly powers 119875119896
119875119896 =1
119881119896
int119881119896
sum
119892
]Σ119891119892 sdot 120601119892119889119881 (10)
where 119881119896 is the volume of the 119896th subassembly and119896 designates the fuel subassemblies as shown inFigure 9
(4) Number of unknowns in the problem number ofiterations and total and outer
(5) Total computing time iteration time IO-time andcomputer used
(6) Type and numerical values of convergence criteria(7) Table of average group fluxes for a square mesh grid
of 20 times 20 cm(8) Dependence of results on mesh spacing
Even though the original problem is based on a quarter-core situation the problem has an eighth-core symmetry
20 Science and Technology of Nuclear Installations
Quarter Delaunay triangles elements
Tim
e con
sum
ed to
read
the m
esh
(sec
onds
)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumes
Quarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
1
3
10
30
100
300
Figure 22 Time needed to read the mesh versus number of unknowns
which cannot be taken into account by structured gridswhich are the main target of the benchmark Howevernonstructured grids can take into consideration any kind ofsymmetry almost without loss of accuracy and at the sametime reducing roughly the number of unknowns by a halfand the associated computational effort needed to solve theproblem by a factor of four Answers to items (1)ndash(7) canbe given completely by milonga using a single input file(see Supplementary data in SupplementaryMaterial availableonline at httpdxdoiorg1011552013641863) As asked initem (8) how results depend not only on the mesh spacingbut also on the meshing algorithm on the gridrsquos elementarygeometric shape and on the discretization scheme may shedlights on the subject whichmay be evenmore interesting thanthe numerical results themselves
Taking advantage of milongarsquos capability of reading andparsing command-line arguments the selection of the coregeometry (quarter or eighth) the meshing algorithm (delau-nay [16] or delquad [25]) the shape of the elementaryentities (triangles or quadrangles) the discretization scheme(volumes or elements) and the characteristic length ℓ119888 ofthe mesh can be provided at run time Fixing five values forℓ119888 = 4 3 2 1 05 gives 2 times 2 times 2 times 2 times 5 = 80 possiblecombinations which we solve with successive invocations to
milonga with the same input file but with different argumentsfrom a simple script Figures 10 11 12 13 14 15 16 17 and18 show the results corresponding to items (1)ndash(7) for someillustrative cases The complete set of figures and the codeused to generate themmay be provided upon request Table 2compiles the answers to the problem for every case studied
As the milonga code is still under development itsnumerical routines are not yet fully optimized nor designedfor parallel computation Therefore the reported times areonly rough estimates and should be taken with care Thesolution comprises five steps
(1) generate the grid with the requested geometry mesh-ing algorithm basic shape and characteristic lengthby calling to gmsh
(2) read the generated mesh(3) build the matrices(4) solve the eigenvalue problem(5) compute the requested results
The CPU time reported in Figures 10ndash18 is thus thesum of all of these five steps but not the time needed togenerate the figuresmdashwhich in some caseswith coarsemeshes
Science and Technology of Nuclear Installations 21
Table2
Resultsofthe2
DIAEA
PWRBe
nchm
arkp
roblem
obtained
with
them
ilong
acod
efor
thee
ightyp
ropo
sedcombinatio
nsofsymmetrym
eshing
algorithm
basicshapediscretization
schemeandcharacteris
ticelem
entc
elllengthTh
ereference
solutio
nis120588=28749times10minus5w
hich
correspo
ndsto119896eff=10296
Case
no
Symm-
etry
Mesh
algorithm
Basic
shape
Solutio
nmetho
dℓ119888
(cm)
Δ120588
(pcm
)max1206012
(mdash)
Total
unkn
owns
Outer
iter
Linear
iter
Inner
iter
Resid
ual
norm
Relativ
eerror
Error
estim
ate
Mem
ory
(Mb)
Page
faults
114
Delaunay
998779Vo
lumes
40
minus120
1145
8264
332
1033
6119890minus09
3119890minus09
1119890minus09
349470
214
Delaunay
998779Vo
lumes
30
85
1135
15740
332
1967
2119890minus08
1119890minus08
4119890minus09
4813401
314
Delaunay
998779Vo
lumes
20
197
1124
31188
332
3898
1119890minus08
6119890minus09
3119890minus09
7621932
414
Delaunay
998779Vo
lumes
10153
1123
127476
332
15934
7119890minus09
3119890minus09
3119890minus09
273
81886
514
Delaunay
998779Vo
lumes
05
187
1119
510676
332
63834
2119890minus09
9119890minus10
1119890minus09
1148
352261
614
Delaunay
998779Elem
ents
40
173
1100
4308
332
538
2119890minus08
8119890minus09
4119890minus09
318794
714
Delaunay
998779Elem
ents
30
117
1107
8108
332
1013
5119890minus09
2119890minus09
2119890minus09
4712946
814
Delaunay
998779Elem
ents
20
84
1112
15936
332
1992
6119890minus09
3119890minus09
2119890minus09
7321347
914
Delaunay
998779Elem
ents
1062
1116
64478
332
8059
6119890minus09
3119890minus09
4119890minus09
262
77105
1014
Delaunay
998779Elem
ents
05
58
1117
256700
332
32087
1119890minus09
5119890minus10
1119890minus09
1091
331969
1114
Delaunay
◻Vo
lumes
40
53
1149
5042
332
630
2119890minus08
1119890minus08
5119890minus09
288007
1214
Delaunay
◻Vo
lumes
30
346
1133
8560
332
1070
7119890minus09
3119890minus09
2119890minus09
3910895
1314
Delaunay
◻Vo
lumes
20
308
1131
15576
332
1947
1119890minus08
7119890minus09
3119890minus09
541564
514
14Delaunay
◻Vo
lumes
10177
1122
60774
332
7596
8119890minus09
4119890minus09
3119890minus09
167
50115
1514
Delaunay
◻Vo
lumes
05
199
1120
244936
332
30617
4119890minus09
2119890minus09
2119890minus09
698
215678
1614
Delaunay
◻Elem
ents
40
196
1100
5222
332
652
3119890minus08
1119890minus08
8119890minus09
4713098
1714
Delaunay
◻Elem
ents
30
123
1107
8810
332
1101
2119890minus08
9119890minus09
7119890minus09
6918598
1814
Delaunay
◻Elem
ents
20
901112
15922
332
1990
1119890minus08
5119890minus09
5119890minus09
107
30591
1914
Delaunay
◻Elem
ents
1063
1116
61456
332
7682
5119890minus09
2119890minus09
4119890minus09
389
113237
2014
Delaunay
◻Elem
ents
05
58
1117
246332
332
30791
8119890minus10
4119890minus10
1119890minus09
1676
498192
2114
Delq
uad
998779Vo
lumes
40
minus45
1144
6228
332
778
4119890minus08
2119890minus08
7119890minus09
308495
2214
Delq
uad
998779Vo
lumes
30
570
1153
11820
332
1477
3119890minus08
1119890minus08
4119890minus09
4010879
2314
Delq
uad
998779Vo
lumes
20
459
1103
24100
332
3012
2119890minus08
1119890minus08
5119890minus09
6217715
2414
Delq
uad
998779Vo
lumes
10512
1104
9640
03
3212050
2119890minus08
1119890minus08
9119890minus09
207
61714
2514
Delq
uad
998779Vo
lumes
05
528
1106
385600
332
48200
2119890minus09
1119890minus09
2119890minus09
838
255735
2614
Delqu
ad998779
Elem
ents
40
220
1093
3288
332
411
3119890minus08
2119890minus08
6119890minus09
308495
2714
Delq
uad
998779Elem
ents
30
140
1104
6148
332
768
4119890minus08
2119890minus08
8119890minus09
39110
0028
14Delq
uad
998779Elem
ents
20
82
1110
12392
332
1549
2119890minus08
9119890minus09
6119890minus09
6117067
2914
Delqu
ad998779
Elem
ents
1063
1115
48882
332
6110
2119890minus09
1119890minus09
1119890minus09
197
5804
630
14Delq
uad
998779Elem
ents
05
58
1116
194162
332
24270
2119890minus09
9119890minus10
2119890minus09
790
238769
3114
Delq
uad
◻Vo
lumes
40
minus416
1174
3132
332
391
5119890minus08
3119890minus08
8119890minus09
267322
3214
Delq
uad
◻Vo
lumes
30
minus248
1151
5922
332
740
9119890minus09
4119890minus09
2119890minus09
318779
3314
Delq
uad
◻Vo
lumes
20
minus132
1139
12050
332
1506
1119890minus08
7119890minus09
4119890minus09
4412239
3414
Delq
uad
◻Vo
lumes
10minus49
1126
48200
332
6025
6119890minus09
3119890minus09
2119890minus09
122
36094
3514
Delq
uad
◻Vo
lumes
05
minus23
1123
192800
332
24100
5119890minus09
2119890minus09
3119890minus09
464
140228
3614
Delq
uad
◻Elem
ents
40
224
1093
3294
332
411
3119890minus08
1119890minus08
7119890minus09
359852
3714
Delq
uad
◻Elem
ents
30
141
1104
6144
332
768
3119890minus08
2119890minus08
9119890minus09
5114018
3814
Delq
uad
◻Elem
ents
20
921111
12392
332
1549
1119890minus08
6119890minus09
5119890minus09
8122526
3914
Delq
uad
◻Elem
ents
1065
1115
48882
332
6110
5119890minus09
3119890minus09
4119890minus09
276
78431
22 Science and Technology of Nuclear Installations
Table2Con
tinued
Case
no
Symm-
etry
Mesh
algorithm
Basic
shape
Solutio
nmetho
dℓ119888
(cm)
Δ120588
(pcm
)max1206012
(mdash)
Total
unkn
owns
Outer
iter
Linear
iter
Inner
iter
Resid
ual
norm
Relativ
eerror
Error
estim
ate
Mem
ory
(Mb)
Page
faults
4014
Delq
uad
◻Elem
ents
05
59
1117
194162
332
24270
1119890minus09
5119890minus10
1119890minus09
1104
318967
4118
Delaunay
998779Vo
lumes
40
minus99
1139
4228
332
528
4119890minus12
2119890minus12
1119890minus12
359630
4218
Delaunay
998779Vo
lumes
30
50
1132
7712
332
964
1119890minus11
6119890minus12
2119890minus12
4211578
4318
Delaunay
998779Vo
lumes
20
177
1120
15776
332
1972
2119890minus12
1119890minus12
7119890minus13
5515436
4418
Delaunay
998779Vo
lumes
10142
1118
63902
332
7987
3119890minus12
1119890minus12
1119890minus12
160
46691
4518
Delaunay
998779Vo
lumes
05
176
1115
254612
332
31826
2119890minus12
8119890minus13
1119890minus12
555
168526
4618
Delaunay
998779Elem
ents
40
183
1096
2252
332
281
6119890minus12
3119890minus12
2119890minus12
359589
4718
Delaunay
998779Elem
ents
30
119
1104
4040
224
505
2119890minus08
1119890minus08
7119890minus09
41114
3748
18Delaunay
998779Elem
ents
20
85
1108
8156
332
1019
2119890minus12
8119890minus13
6119890minus13
5515053
4918
Delaunay
998779Elem
ents
1063
1112
32480
332
4060
1119890minus12
6119890minus13
8119890minus13
151
43671
5018
Delaunay
998779Elem
ents
05
58
1113
128358
332
1604
47119890minus13
3119890minus13
7119890minus13
527
157581
5118
Delaunay
◻Vo
lumes
40
37
1144
2380
224
297
5119890minus08
2119890minus08
1119890minus08
5314173
5218
Delaunay
◻Vo
lumes
30
218
1120
4194
332
524
7119890minus12
4119890minus12
2119890minus12
3610034
5318
Delaunay
◻Vo
lumes
20
345
1110
7960
332
995
5119890minus12
3119890minus12
1119890minus12
4712879
5418
Delaunay
◻Vo
lumes
10165
1117
30652
332
3831
2119890minus12
1119890minus12
7119890minus13
100
28851
5518
Delaunay
◻Vo
lumes
05
191
1117
122066
224
15258
2119890minus08
8119890minus09
9119890minus09
343
103825
5618
Delaunay
◻Elem
ents
40
175
1097
2526
332
315
4119890minus12
2119890minus12
1119890minus12
6016159
5718
Delaunay
◻Elem
ents
30
114
1104
4386
332
548
4119890minus12
2119890minus12
2119890minus12
5013768
5818
Delaunay
◻Elem
ents
20
901108
8234
332
1029
2119890minus12
1119890minus12
1119890minus12
7320094
5918
Delaunay
◻Elem
ents
1062
1112
31186
224
3898
2119890minus08
7119890minus09
9119890minus09
208
59589
6018
Delaunay
◻Elem
ents
05
58
1113
123122
224
15390
9119890minus09
5119890minus09
1119890minus08
807
2366
7061
18Delq
uad
998779Vo
lumes
40
minus210
1168
3224
332
403
1119890minus11
5119890minus12
2119890minus12
43117
8462
18Delq
uad
998779Vo
lumes
30
369
1150
6000
332
750
1119890minus11
6119890minus12
3119890minus12
5013502
6318
Delq
uad
998779Vo
lumes
20
312
1104
12316
332
1539
2119890minus11
8119890minus12
5119890minus12
6016395
6418
Delq
uad
998779Vo
lumes
1044
01094
48714
332
6089
9119890minus12
4119890minus12
4119890minus12
1183444
165
18Delq
uad
998779Vo
lumes
05
534
1081
193980
332
24247
5119890minus12
2119890minus12
4119890minus12
423
126768
6618
Delq
uad
998779Elem
ents
40
240
1090
1750
332
218
9119890minus12
4119890minus12
2119890minus12
43117
8667
18Delqu
ad998779
Elem
ents
30
150
1100
3184
332
398
1119890minus11
5119890minus12
2119890minus12
5013509
6818
Delq
uad
998779Elem
ents
20
89
1106
6426
332
803
5119890minus12
3119890minus12
2119890minus12
5916047
6918
Delq
uad
998779Elem
ents
1064
1111
24886
332
3110
2119890minus12
1119890minus12
1119890minus12
113
32788
7018
Delq
uad
998779Elem
ents
05
59
1113
98052
224
12256
1119890minus08
5119890minus09
8119890minus09
398
118153
7118
Delq
uad
◻Vo
lumes
40
minus398
1172
1650
332
206
1119890minus11
5119890minus12
2119890minus12
369907
7218
Delq
uad
◻Vo
lumes
30
minus249
1146
3058
332
382
4119890minus12
2119890minus12
1119890minus12
349349
7318
Delq
uad
◻Vo
lumes
20
minus131
1136
6228
224
778
3119890minus08
1119890minus08
5119890minus09
4010930
7418
Delq
uad
◻Vo
lumes
10minus51
1123
24536
224
3067
3119890minus08
2119890minus08
1119890minus08
7822716
7618
Delq
uad
◻Elem
ents
40
237
1090
1752
332
219
6119890minus12
3119890minus12
2119890minus12
4111213
7718
Delq
uad
◻Elem
ents
30
145
1101
3184
332
398
6119890minus12
3119890minus12
2119890minus12
43118
1178
18Delq
uad
◻Elem
ents
20
941107
6426
332
803
3119890minus12
1119890minus12
1119890minus12
6016220
7918
Delqu
ad◻
Elem
ents
1065
1111
24874
332
3109
1119890minus12
6119890minus13
8119890minus13
156
4400
080
18Delq
uad
◻Elem
ents
05
59
1113
9804
22
2412255
8119890minus09
4119890minus09
8119890minus09
567
162168
Science and Technology of Nuclear Installations 23
Tim
e con
sum
ed to
bui
ld th
e mat
rices
(sec
onds
)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay trianglesvolumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
10
30
100
300
1
3
Figure 23 Time needed to build the matrices versus number of unknowns
was considerably larger than the solution time itself Theeigenvalue problem was solved using a multilayer iterativeKrylov-Schur method [26] with a tolerance relative to thematrices norm
1003817100381710038171003817119877 sdot 120601 minus (1119896eff) sdot 119865 sdot 1206011003817100381710038171003817
119877 + (1119896eff) sdot 119865lt 10minus8 (11)
The reported residual norm and relative error are10038171003817100381710038171003817100381710038171003817119877 sdot 120601 minus
1
119896effsdot 119865 sdot 120601
10038171003817100381710038171003817100381710038171003817
1003817100381710038171003817119877 sdot 120601 minus (1119896eff) sdot 119865 sdot 1206011003817100381710038171003817
1003817100381710038171003817(1119896eff) sdot 1206011003817100381710038171003817
(12)
respectively The fields marked as outer linear and inneriterations refer to the number of steps needed to attainthe requested tolerance in each layer of the Krylov-Schuralgorithm The computer used to solve the problem has anIntel i7 920 267GHz processor with 4Gb of RAM runningDebian GNULinux Wheezy
When using a finite volumes-based scheme over anunstructured mesh the solver has to gather information
about which cells are neighbors and which are not Currentlygmsh does not write this kind of lists in its output filesso milonga has to explicitly solve the neighbors problemPerforming a linear search is an 119874(119873
2) task which is
unacceptable for problem sizes 119873 of interest The code usesa search based on a 119896-dimensional tree [27] which can inprinciple be performed in119874(119873) steps Still for large values of119873 the time needed to read and parse the mesh (number twopreviously mentioned) is the bottleneck of the solution Thisstep is not needed in finite elements although the construc-tion of the elementary matrices involves the computationof the multidimensional Jacobians and integrals which thenhave to be assembled Again for large 119873 this step (numberthree) takes up most of the time needed to solve the problem
The Delaunay algorithm is a standard method for gen-erating two-dimensional grids [16] by tessellating a planewith triangles If the mesh needs to be based on quadranglesinstead a recombination algorithm can be used to transformtwo adjacent triangles in one quadrangle whenever is possi-ble However for geometries which are based on rectangularshapes there exist other algorithms [25] both for meshingand for recombining the triangles that give rise to elementaryentities with right angles almost everywhere which may be a
24 Science and Technology of Nuclear Installations
Tim
e con
sum
ed to
solv
e the
eige
nval
ue p
robl
em (s
econ
ds)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
10
30
100
300
1
3
Figure 24 Time needed to solve the eigenvalue problem versus number of unknowns
desired property of the resulting grid For finite elements thesteps needed to build the matrices depend on the selectionof triangles or quadrangles as the basic elementary geometrybecause the shape functions change However the number ofunknowns is the same as the number of nodes that does notchange after a recombination procedure On the other handthe number of unknowns in the finite volumes schemes withtriangles is roughly twice as the number of unknowns withquadrangles for the same grid
Figure 19 shows how the computed static reactivity (ie1 minus 1119896eff) depends on the number of unknowns for eachof the sixteen combinations of geometry algorithm shapescheme The four accepted results published in the originalreference [10] are included for reference although it shouldbe taken into account that said reactivities were computedalmost forty years ago Figure 20 shows the total wall timeneeded to solve the problem as a function of 119873119866 whileFigures 21 22 23 and 24 show the times needed for eachof the first four steps involved in the solution maintainingthe same logarithmic scale for both the abscissas and theordinates Green data represent finite volumes whilst bluebullets correspond to finite elements Solid lines indicatequarter-core and dashed lines eighth-symmetry Fillet bullets
are results obtained by theDelaunay triangulation and emptybullets were computed with the delquad algorithm Finallytriangle-shaped data correspond to triangles and squares anddiamonds denote quadrangles as the basic geometry of thegrid
It can be seen that finite elements produce amuch smallerdispersion of eigenvalues 119896eff than finite volumes with therefinement of the mesh This can be explained because finitevolumes methods rely on a geometric condition of the meshwhich may change abruptly if the meshing algorithm decidesto allocate the cells in a rather different form for small changesin ℓ119888 Finite elements methods are less influenced by thesediscontinuous lay-out changes of the elements As expectedeighth-core symmetries give almost the same results as thequarter-core geometries with roughly half the unknownsFor small problems finite volumes run faster that finiteelements because the time needed to solve the neighborproblem is negligible When the problem size grows thistime increases and exceeds the overhead implied in the con-struction and assembly of the finite elements matrices Alsoat least for this configuration it is seen that the eigenvalueproblem is solved faster for finite volumes than for finiteelements
Science and Technology of Nuclear Installations 25
4 Conclusions
Unstructured grids provide the cognizant engineer witha wide variety of possibilities to deal with the design oranalysis of nuclear reactor cores These kinds of grids cansuccessfully reproduce continuous geometries commonlyfound in reactor cores such as cylinders and thereforenot only can the diffusion equation be better approximatedinside the domain of definition but also the fulfillment ofboundary conditions is improved A free computer codewas written from scratch that is able to completely solvethe 2D IAEA PWR Benchmark using unstructured gridsfor sixteen combinations of geometry meshing algorithmbasic shape and discretization scheme plus any value ofthe gridrsquos characteristic length by using a single input fileThe complete set of input files and codemdashexecutable andsourcemdashis available either online or upon request withcomments experiences suggestions and corrections beingmore than welcome Further development should includetackling full three-dimensional geometries with completethermal hydraulic feedback in order to analyze how thesolutions of the coupled neutronic-thermal problem dependon the spatial discretization scheme of the neutron leakageterm Parallelization of the computation and assembly of thematrices and of the solution of the eigenvalue problem andits implementation using GPUs are also desired features toimplement A problemwith direct applications that the futureversions of milonga ought to solve is the analysis of how thegeometry of the absorbing materials should be taken intoaccount in structured coarse grids in order to mitigate effectssuch as the rod-cusp problem
Suitable schemes for approximating the continuous dif-ferential operators by discrete matrix expressions includefinite volumes and finite elements families Finite volumesmethods compute cell mean values whilst finite elementsgive functional values at the gridrsquos nodes In general finiteelements are less sensitive to changes in the mesh so theresults they provide do not change significantly for differentmeshing algorithms or elementary shapes Small problemsare best solved by finite volumes as the neighbor-findingproblem is faster than the process of building and assemblingthe eigenvalue-problem matrices For a large number ofunknowns the process of finding which cell is neighbor ofwhichmdasheven based on a 119896-dimensional treemdashoverwhelmsthe computation and assembly of elementary matrices andfinite-element methods perform better
References
[1] G Glasstone and S Bell Nuclear Reactor Theory KriegerPublishing Company 1970
[2] J J Duderstadt and L J Hamilton Nuclear Reactor AnalysisJohn Wiley amp Sons New York NY USA 1976
[3] C Gho Reactor Physics Undergraduate Course Lecture NotesInstituto Balseiro 2006
[4] G G Theler and F J Bonetto ldquoOn the stability of the pointreactor kinetics equationsrdquoNuclear Engineering andDesign vol240 no 6 pp 1443ndash1449 2010
[5] E E Lewis andW F Miller Computational Methods of NeutronTransport John Wiley amp Sons 1984
[6] E Masiello ldquoAnalytical stability analysis of Coarse-Mesh finitedifference methodrdquo in Proceedings of the International Confer-ence on the Physics of Reactors (PHYSOR rsquo08) Nuclear Power ASustainable Resource pp 456ndash464 September 2008
[7] M L Zerkle Development of a polynomial nodal methodwith flux and current discontinuity factors [PhD thesis] Mas-sachusets Institute of Technology 1992
[8] J I Yoon andHG Joo ldquoTwo-level coarsemesh finite differenceformulation with multigroup source expansion nodal kernelsrdquoJournal of Nuclear Science andTechnology vol 45 no 7 pp 668ndash682 2008
[9] O Mazzantini M Schivo J Di Cesare R Garbero M Riveroand G Theler ldquoA coupled calculation suite for Atucha II oper-ational transients analysisrdquo Science and Technology of NuclearInstallations vol 2011 Article ID 785304 12 pages 2011
[10] Computational Benchmark Problem Comitee for the Mathe-matics and Computation Division of the American NuclearSociety ldquoArgonne Code Center Benchmark problem bookrdquoTech Rep ANL-7416 Supplement 2 Argonne National Labo-ratory 1977
[11] A F Henry Nuclear Reactor Analysis MIT Cambridge UK1975
[12] O C Zienkiewicz R L Taylor and J Z ZhuTheFinite ElementMethod Its Basis and Fundamentals vol 1 Elsevier 6th edition2005
[13] G Theler Difusion de neutrones en mallas no estructuradascomparacion entre volumenes y elementos finitos AcademicMonograph Universidad de Buenos Aires 2013
[14] G Theler ldquoMilonga a free nuclear reactor core analysis coderdquoAvailable online 2011
[15] G Theler F J Bonetto and A Clausse ldquoOptimizacion deparametros en reactores de potencia base de diseno delcodigo neutronico milongardquo in Reunion Anual de la AsociacionArgentina de Tecnologia Nuclear XXXVII 2010
[16] C Geuzaine and J-F Remacle ldquoGmsh a 3-D finite elementmesh generator with built-in pre- and post-processing facili-tiesrdquo International Journal for Numerical Methods in Engineer-ing vol 79 no 11 pp 1309ndash1331 2009
[17] S Balay J Brown K Buschelman et al ldquoPETSc users manualrdquoTech Rep ANL-9511-Revision 34 Argonne National Labora-tory 2013
[18] S Balay W D Gropp L Curfman McInnes and B F SmithldquoEfficient management of parallelism in object oriented numer-ical software librariesrdquo in Modern Software Tools in ScientificComputing E Arge A M Bruaset and H P Langtangen Edspp 163ndash202 Birkhauser 1997
[19] V Hernandez J E Roman and V Vidal ldquoSLEPC a scalable andflexible toolkit for the solution of eigenvalue problemsrdquo ACMTransactions on Mathematical Software vol 31 no 3 pp 351ndash362 2005
[20] M Galassi J Davies J Theiler et al GNU Scientific LibraryReference Manual 3rd edition 2009
[21] O C Zienkiewicz and R L Taylor The Finite Element MethodFor Solid and Structural Mechanics vol 3 Elsevier 6th edition2005
[22] A Imelda and K Doddy Evaluation of the IAEA 3-D PWRbenchmark problem using NESTLE code 2004
[23] R Mosteller ldquoStatic benchmarking of the NESTLE advancednodal coderdquo in Proceedings of the Joint International Conference
26 Science and Technology of Nuclear Installations
on Mathematical Methods and Supercomputing for NuclearApplications vol 2 pp 1596ndash1605 1997
[24] GTheler F J Bonetto andAClausse ldquoSolution of the 2D IAEAPWR Benchmark with the neutronic code milongardquo in Actasde la Reunion Anual de la Asociacion Argentina de TecnologıaNuclear XXXVIII 2011
[25] J F Remacle F Henrotte T Carrier-Baudouin et al ldquoAfrontal delaunay quad mesh generator using the 119871
infin normrdquoInternational Journal For Numerical Methods in Engineeringvol 94 no 5 pp 494ndash512 2013
[26] G W Stewart ldquoA Krylov-Schur algorithm for large eigenprob-lemsrdquo SIAM Journal on Matrix Analysis and Applications vol23 no 3 pp 601ndash614 2002
[27] J L Bentley ldquoMultidimensional binary search trees used forassociative searchingrdquo Communications of the ACM vol 18 no9 pp 509ndash517 1975
TribologyAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FuelsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofPetroleum Engineering
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
CombustionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Renewable Energy
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
StructuresJournal of
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear InstallationsScience and Technology of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solar EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Wind EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear EnergyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
14 Science and Technology of Nuclear Installations
Largest eigenvalue 1029695 (288387 pcm)1108 (3045 3045)833 (13000 5000)
Number of unknowns 8234Outer iterations 3Linear iterations 32Inner iterations 1029Residual normRelative error
Max 1206012(x y) coreMax 1206012(x y) reflector
2028 times 10minus12
9993 times 10minus13
1012 times 10minus12
keff
Relative errorError estimateMemory used 75468 kBSoft page faults 20094Hard page faults 0Total CPU time 1256 seconds
9993 times 10
1012 times 10minus12
Milongarsquos 2D LWR IAEA benchmark problem case no 058eighth-symmetry core meshed using Delaunay (quads 985747c = 2) solved with finite volumes
(a)
1 074 3207 5472 129 4136 9573 144 4534 10644 120 3828 8875 061 2638 4506 093 2985 6907 094 2948 6948 072 2053 5669 mdash 357 859
10 142 4478 105011 146 4617 108412 130 4111 96413 106 3396 78614 103 3263 76515 095 2991 70616 071 2005 55217 mdash 340 81618 145 4588 107719 133 4206 98720 117 3698 86821 107 3362 79122 098 2918 72523 062 1686 52024 mdash 258 62925 118 3737 87626 096 3073 71227 091 2848 67128 080 2272 63129 mdash 609 127230 mdash 080 31731 047 2035 34832 069 2081 50933 054 1458 44834 mdash 255 64635 052 1413 43736 mdash 409 85137 mdash 064 25138 mdash 071 284
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 16 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
boundary conditions are applied to appropriate physicalentities of dimension119898minus1The problemmatrices119877 and119865 arethen built and stored in an appropriate sparse format usingthe free PETSc [17 18] library and the eigenvalue problemis solved using the free SLEPc [19] library The results arestored into milongarsquos variables and functions which may be
evaluated integratedmdashusually using the free GNU ScientificLibrary [20] routinesmdashand of course written into appropriateoutputs Milonga can also solve problems parametrically andbe used to solve optimization problems As stated above thecode is a free software so corrections and contributions aremore than welcome
Science and Technology of Nuclear Installations 15
Largest eigenvalue 1029462 (286185 pcm)1136 (3093 2953)1233 (13100 5100)
Number of unknowns 6228Outer iterations 2Linear iterations 24Inner iterations 778Residual normR l ti
Max 1206012(x y) coreMax 1206012(x y) reflector
26 times 10minus8
1281 times 10minus8
4996 times 10minus9
keff
Relative errorError estimateMemory used 41460 kBSoft page faults 10930Hard page faults 0Total CPU time 0604 seconds
1281 times 10 8
4996 times 10minus9
Milongarsquos 2D LWR IAEA benchmark problem case no 073eighth-symmetry core meshed using Delaunay (quads 985747c = 2) solved with finite volumes
(a)
1 075 3297 5552 134 4268 9923 148 4660 10944 123 3926 9145 059 2645 4406 094 2997 6977 093 2914 6858 072 2002 5349 mdash 301 773
10 146 4606 108011 150 4735 111112 133 4212 98813 108 3448 80214 104 3271 76715 094 2954 69616 070 1954 52117 mdash 288 73518 149 4691 110119 136 4289 100720 119 3748 88021 107 3359 79022 096 2880 71423 064 1636 47524 mdash 217 56325 121 3810 89426 098 3110 72427 090 2829 66628 081 2227 59929 mdash 536 125730 mdash 062 25031 045 2002 33632 068 2040 50033 055 1414 40934 mdash 215 57935 055 1359 40836 mdash 359 83437 mdash 050 19938 mdash 061 236
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 17 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
31 A Coarse Bare Homogeneous Circle Figures 5 and 6 showthe results of solving a bare two-dimensional circular homo-geneous reactor of radius 119886 over an unstructured grid whichis deliberately coarse using the finite volumes method andthe finite elements method respectively Two group energies
were used and a null-flux boundary conditionwas fixed at theexternal surface Figures 5(a) and 6(a) compare the obtainedfast flux distributions In the first case the numerical solutionprovidesmean values for each neutron flux group in each cellwhile in the latter the solution is computed at the nodes and
16 Science and Technology of Nuclear Installations
Largest eigenvalue 1029851 (289858 pcm)1090 (3182 3182)851 (13000 5000)
Number of unknowns 1752Outer iterations 3Linear iterations 32Inner iterations 219Residual norm
Max 1206012(x y) coreMax 1206012(x y) reflector
566 times 10minus12
2789 times 10minus12
1595 times 10minus12
keff
Residual normRelative errorError estimateMemory used 42564 kBSoft page faults 11213Hard page faults 0Total CPU time 09921 seconds
566 times 10
2789 times 10minus12
1595 times 10minus12
Milongarsquos 2D LWR IAEA benchmark problem case no 076eighth-symmetry core meshed using Delaunay (quads 985747c = 4) solved with finite volumes
(a)
1 073 3151 5402 127 4059 9393 141 4457 10464 118 3768 8715 061 2615 4506 093 2979 6887 094 2971 6998 067 2092 5829 mdash 375 885
10 139 4397 103111 144 4541 106612 128 4052 95013 105 3365 77814 103 3259 76415 096 3016 71216 066 2045 56817 mdash 359 84118 143 4520 106119 132 4155 97520 116 3672 86221 107 3366 79222 098 2951 73323 054 1728 54324 mdash 273 65125 117 3700 86826 095 3057 70827 091 2860 67428 074 2311 64829 mdash 642 130030 mdash 084 33231 048 2041 35332 069 2103 51333 046 1493 46734 mdash 269 66735 046 1444 45536 mdash 432 87137 mdash 067 26338 mdash 074 297
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 18 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
continuous functions are evaluated by means of the shapefunctions used in the formulation Figures 5(b) and 6(b)illustrate the fast fluxes unknowns and its relative position inspace It can be noted that the mesh coarseness gives resultsthat differ substantially in both cases Finally the structure ofthe sparse eigenvalue problemmatrices is shown for each case
with blue red and cyan representing positive negative andexplicitly inserted zero values In the finite volume case thereare 184 unknowns (92 cells times 2 groups) while in the secondcase there are 218 unknowns (109 nodes times 2 groups) Thevolumesrsquo fissionmatrix is almost diagonal it has a bandwidthequal to the number of energy groups which in this case is
Science and Technology of Nuclear Installations 17
500 1000 2000 5000 10000 20000 50000Number of unknowns
2840
2860
2880
2900
2920
ORNLRisoslashKWUOntarioQuarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumes
Quarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elementsEighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumes
Eighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
120588(p
cm)
105 2 times 105 5 times 105
Figure 19 Static reactivity versus number of unknowns The four original solutions as published in 1977 [10] are included as reference
twoThe off-diagonal values appear right next to the diagonalelements because of the chosen ordering of the unknownsin the flux vector 120601 isin R184 Other orderings may be usedbut the rate of convergence of the eigenvalue problem can bedeteriorated On the other hand the elementsrsquo fission matrixhas a nontrivial structure because the grid is unstructuredand the net fission rate at each element depends on the fluxesat the nodes whose location inside thematrix depends in turnon how the gridwas generatedThis effect is similar to the onethat appears in structural analysis where mass matrices needto be lumped in order to simplify transient computations [21]The diagonal block of elementrsquos 119877 matrix and the null blockin 119865 correspond to the discrete equations that set the null-flux boundary conditions on the nodes located at the externalsurface Other types of boundary conditions lead to differentkinds of structures within the problem matrices
In case a part of the domain contained a nonmultiplicativematerial such as a reflector then there would appear sectionsof the fission matrix with null values in both methodsrendering 119865 singular in both methods Care should be takenwhen dealing with the numerical schemes for the eigenvalueproblem solution It can be seen in the elementsrsquo matricesa particular structure that implements the boundary condi-tions at the external surfaces This structure does not appearin the volumesrsquo matrices because the boundary conditions
appear as flux terms which are summed up over all thesurfaces of each cell so they aremasked inside the volumetricdiscretization of the divergence and gradient operators
As the two-group neutron diffusion equation with uni-form cross sections over a circle subject to null-flux bound-ary conditions has an analytical solution it is adequate tocompare how the two proposed numerical schemes relate toit In the studied problem we ignored upscattering and fastfissions Then the analytical effective multiplication factor is
119896eff =]Σ1198912 sdot Σ1199041rarr2
[Σ1198861 + Σ1199041rarr2 + 1198631(]0119886)2] [Σ1198862 + 1198632(]0119886)
2]
(7)
being ]0 = 24048 the smallest root of Besselrsquos first-kindfunction of order zero 1198690(119903)
Figure 7 shows how the two numerical effective multi-plication factors compare to the analytical solution givenby (7) as a function of the mesh refinement indicated bythe quotient 119886ℓ119888 between the radius of the circle and thecharacteristic length of the cellelements We can see thatthe 119896eff computed by the finite volumes (elements) methodis always greater (less) than the analytical solution Indeedit can be proven that for a bare one-dimensional slab thisis always the case [13] However this result does not hold
18 Science and Technology of Nuclear Installations
2000 3000 6000 10000 20000 30000 60000
01
003
001
03
1
3
10
30
100
300To
tal w
all t
ime (
seco
nds)
Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
Figure 20 Total wall time versus number of unknowns
for problems with nonuniform cross sections even in simpleone-dimensional reflected reactors
We may draw two other conclusions from Figure 7 Firstthat the finite element method seems to provide a bettersolution than the volumes-based scheme and second thateven though the error committed tends to decrease with finergrids its behavior is not monotonic for finite volumes Infact Figure 8 shows the difference between the numericaland analytical solutions using a logarithmic vertical scalewhere both conclusions are even more evident We deferthe discussion of such differences between the discretizationscheme until the next section It is worth to note howeverthat the fact that a finite-element-based scheme throws betterresults for the particular bare homogeneous circular reactorunder study than the finite volumes does not imply thatfinite volumes ought to be discarded as a valid tool forsolving the neutron diffusion equation in general cases Thecombination of lattice and core-level computations is usuallyperformedusing cell-based resultswhichwhen fed into node-based methods to solve the few-group neutron diffusionequation may introduce errors which potentially can leadto unacceptable solutions Nevertheless this analysis is farbeyond the scope of this paper which focuses on solving amathematical equation over unstructured grids
32 The 2D IAEA PWR Benchmark Problem This is aclassical two-group neutron diffusion problem first designedin the early 1970s and taken as a reference benchmark forcomputational codes A number of codes were used to solveeither this problem or its three-dimensional formulation[22 23] including milonga using a structured grid [24] Theoriginal formulation can be found in the reference [10] andthere is a reproduction that may be easily found online inreference [24] The geometry consists of a one-quarter ofa PWR core depicted in Figure 9 and the homogeneousmacroscopic cross sections are listed in Table 1 An axialbuckling term 119861
2
119911119892= 08times10
minus4 should be taken into accountThe external surface should be subject to a zero incomingcurrent condition which may be written as
120597120601119892
120597119899= minus
04692
119863119892
sdot 120601119892 (8)
The expected results are as follows
(1) Maximum eigenvalue(2) Fundamental flux distributions
(a) Radial flux traverses 120601119892(119909 0) and 120601119892(119909 119909)
Science and Technology of Nuclear Installations 19
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
1
3
10
30
100
300Ti
me c
onsu
med
to m
esh
the g
eom
etry
(sec
onds
)
Figure 21 Time needed to mesh the geometry versus number of unknowns
Table 1 Macroscopic cross-sections (units are not stated in the original reference but they are assumed to be in cmminus1 or cm as appropriate)
1198631
1198632
Σ1rarr2
Σ1198861
Σ1198862
]Σ1198912
Material1 15 04 002 001 0080 0135 Fuel 12 15 04 002 001 0085 0135 Fuel 23 15 04 002 001 0130 0135 Fuel 2 + rod4 20 03 004 0 0010 0 Reflector
Note the fluxes shall be normalized such that
1
119881coreint119881core
sum
119892
]Σ119891119892 sdot 120601119892119889119881 = 1 (9)
(b) Value and location of maximum power densityThis corresponds to maximum of 1206012 in thecore It is recommended that the maximumvalues of 1206012 both in the inner core and at thecorereflector interface be given
(3) Average subassembly powers 119875119896
119875119896 =1
119881119896
int119881119896
sum
119892
]Σ119891119892 sdot 120601119892119889119881 (10)
where 119881119896 is the volume of the 119896th subassembly and119896 designates the fuel subassemblies as shown inFigure 9
(4) Number of unknowns in the problem number ofiterations and total and outer
(5) Total computing time iteration time IO-time andcomputer used
(6) Type and numerical values of convergence criteria(7) Table of average group fluxes for a square mesh grid
of 20 times 20 cm(8) Dependence of results on mesh spacing
Even though the original problem is based on a quarter-core situation the problem has an eighth-core symmetry
20 Science and Technology of Nuclear Installations
Quarter Delaunay triangles elements
Tim
e con
sum
ed to
read
the m
esh
(sec
onds
)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumes
Quarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
1
3
10
30
100
300
Figure 22 Time needed to read the mesh versus number of unknowns
which cannot be taken into account by structured gridswhich are the main target of the benchmark Howevernonstructured grids can take into consideration any kind ofsymmetry almost without loss of accuracy and at the sametime reducing roughly the number of unknowns by a halfand the associated computational effort needed to solve theproblem by a factor of four Answers to items (1)ndash(7) canbe given completely by milonga using a single input file(see Supplementary data in SupplementaryMaterial availableonline at httpdxdoiorg1011552013641863) As asked initem (8) how results depend not only on the mesh spacingbut also on the meshing algorithm on the gridrsquos elementarygeometric shape and on the discretization scheme may shedlights on the subject whichmay be evenmore interesting thanthe numerical results themselves
Taking advantage of milongarsquos capability of reading andparsing command-line arguments the selection of the coregeometry (quarter or eighth) the meshing algorithm (delau-nay [16] or delquad [25]) the shape of the elementaryentities (triangles or quadrangles) the discretization scheme(volumes or elements) and the characteristic length ℓ119888 ofthe mesh can be provided at run time Fixing five values forℓ119888 = 4 3 2 1 05 gives 2 times 2 times 2 times 2 times 5 = 80 possiblecombinations which we solve with successive invocations to
milonga with the same input file but with different argumentsfrom a simple script Figures 10 11 12 13 14 15 16 17 and18 show the results corresponding to items (1)ndash(7) for someillustrative cases The complete set of figures and the codeused to generate themmay be provided upon request Table 2compiles the answers to the problem for every case studied
As the milonga code is still under development itsnumerical routines are not yet fully optimized nor designedfor parallel computation Therefore the reported times areonly rough estimates and should be taken with care Thesolution comprises five steps
(1) generate the grid with the requested geometry mesh-ing algorithm basic shape and characteristic lengthby calling to gmsh
(2) read the generated mesh(3) build the matrices(4) solve the eigenvalue problem(5) compute the requested results
The CPU time reported in Figures 10ndash18 is thus thesum of all of these five steps but not the time needed togenerate the figuresmdashwhich in some caseswith coarsemeshes
Science and Technology of Nuclear Installations 21
Table2
Resultsofthe2
DIAEA
PWRBe
nchm
arkp
roblem
obtained
with
them
ilong
acod
efor
thee
ightyp
ropo
sedcombinatio
nsofsymmetrym
eshing
algorithm
basicshapediscretization
schemeandcharacteris
ticelem
entc
elllengthTh
ereference
solutio
nis120588=28749times10minus5w
hich
correspo
ndsto119896eff=10296
Case
no
Symm-
etry
Mesh
algorithm
Basic
shape
Solutio
nmetho
dℓ119888
(cm)
Δ120588
(pcm
)max1206012
(mdash)
Total
unkn
owns
Outer
iter
Linear
iter
Inner
iter
Resid
ual
norm
Relativ
eerror
Error
estim
ate
Mem
ory
(Mb)
Page
faults
114
Delaunay
998779Vo
lumes
40
minus120
1145
8264
332
1033
6119890minus09
3119890minus09
1119890minus09
349470
214
Delaunay
998779Vo
lumes
30
85
1135
15740
332
1967
2119890minus08
1119890minus08
4119890minus09
4813401
314
Delaunay
998779Vo
lumes
20
197
1124
31188
332
3898
1119890minus08
6119890minus09
3119890minus09
7621932
414
Delaunay
998779Vo
lumes
10153
1123
127476
332
15934
7119890minus09
3119890minus09
3119890minus09
273
81886
514
Delaunay
998779Vo
lumes
05
187
1119
510676
332
63834
2119890minus09
9119890minus10
1119890minus09
1148
352261
614
Delaunay
998779Elem
ents
40
173
1100
4308
332
538
2119890minus08
8119890minus09
4119890minus09
318794
714
Delaunay
998779Elem
ents
30
117
1107
8108
332
1013
5119890minus09
2119890minus09
2119890minus09
4712946
814
Delaunay
998779Elem
ents
20
84
1112
15936
332
1992
6119890minus09
3119890minus09
2119890minus09
7321347
914
Delaunay
998779Elem
ents
1062
1116
64478
332
8059
6119890minus09
3119890minus09
4119890minus09
262
77105
1014
Delaunay
998779Elem
ents
05
58
1117
256700
332
32087
1119890minus09
5119890minus10
1119890minus09
1091
331969
1114
Delaunay
◻Vo
lumes
40
53
1149
5042
332
630
2119890minus08
1119890minus08
5119890minus09
288007
1214
Delaunay
◻Vo
lumes
30
346
1133
8560
332
1070
7119890minus09
3119890minus09
2119890minus09
3910895
1314
Delaunay
◻Vo
lumes
20
308
1131
15576
332
1947
1119890minus08
7119890minus09
3119890minus09
541564
514
14Delaunay
◻Vo
lumes
10177
1122
60774
332
7596
8119890minus09
4119890minus09
3119890minus09
167
50115
1514
Delaunay
◻Vo
lumes
05
199
1120
244936
332
30617
4119890minus09
2119890minus09
2119890minus09
698
215678
1614
Delaunay
◻Elem
ents
40
196
1100
5222
332
652
3119890minus08
1119890minus08
8119890minus09
4713098
1714
Delaunay
◻Elem
ents
30
123
1107
8810
332
1101
2119890minus08
9119890minus09
7119890minus09
6918598
1814
Delaunay
◻Elem
ents
20
901112
15922
332
1990
1119890minus08
5119890minus09
5119890minus09
107
30591
1914
Delaunay
◻Elem
ents
1063
1116
61456
332
7682
5119890minus09
2119890minus09
4119890minus09
389
113237
2014
Delaunay
◻Elem
ents
05
58
1117
246332
332
30791
8119890minus10
4119890minus10
1119890minus09
1676
498192
2114
Delq
uad
998779Vo
lumes
40
minus45
1144
6228
332
778
4119890minus08
2119890minus08
7119890minus09
308495
2214
Delq
uad
998779Vo
lumes
30
570
1153
11820
332
1477
3119890minus08
1119890minus08
4119890minus09
4010879
2314
Delq
uad
998779Vo
lumes
20
459
1103
24100
332
3012
2119890minus08
1119890minus08
5119890minus09
6217715
2414
Delq
uad
998779Vo
lumes
10512
1104
9640
03
3212050
2119890minus08
1119890minus08
9119890minus09
207
61714
2514
Delq
uad
998779Vo
lumes
05
528
1106
385600
332
48200
2119890minus09
1119890minus09
2119890minus09
838
255735
2614
Delqu
ad998779
Elem
ents
40
220
1093
3288
332
411
3119890minus08
2119890minus08
6119890minus09
308495
2714
Delq
uad
998779Elem
ents
30
140
1104
6148
332
768
4119890minus08
2119890minus08
8119890minus09
39110
0028
14Delq
uad
998779Elem
ents
20
82
1110
12392
332
1549
2119890minus08
9119890minus09
6119890minus09
6117067
2914
Delqu
ad998779
Elem
ents
1063
1115
48882
332
6110
2119890minus09
1119890minus09
1119890minus09
197
5804
630
14Delq
uad
998779Elem
ents
05
58
1116
194162
332
24270
2119890minus09
9119890minus10
2119890minus09
790
238769
3114
Delq
uad
◻Vo
lumes
40
minus416
1174
3132
332
391
5119890minus08
3119890minus08
8119890minus09
267322
3214
Delq
uad
◻Vo
lumes
30
minus248
1151
5922
332
740
9119890minus09
4119890minus09
2119890minus09
318779
3314
Delq
uad
◻Vo
lumes
20
minus132
1139
12050
332
1506
1119890minus08
7119890minus09
4119890minus09
4412239
3414
Delq
uad
◻Vo
lumes
10minus49
1126
48200
332
6025
6119890minus09
3119890minus09
2119890minus09
122
36094
3514
Delq
uad
◻Vo
lumes
05
minus23
1123
192800
332
24100
5119890minus09
2119890minus09
3119890minus09
464
140228
3614
Delq
uad
◻Elem
ents
40
224
1093
3294
332
411
3119890minus08
1119890minus08
7119890minus09
359852
3714
Delq
uad
◻Elem
ents
30
141
1104
6144
332
768
3119890minus08
2119890minus08
9119890minus09
5114018
3814
Delq
uad
◻Elem
ents
20
921111
12392
332
1549
1119890minus08
6119890minus09
5119890minus09
8122526
3914
Delq
uad
◻Elem
ents
1065
1115
48882
332
6110
5119890minus09
3119890minus09
4119890minus09
276
78431
22 Science and Technology of Nuclear Installations
Table2Con
tinued
Case
no
Symm-
etry
Mesh
algorithm
Basic
shape
Solutio
nmetho
dℓ119888
(cm)
Δ120588
(pcm
)max1206012
(mdash)
Total
unkn
owns
Outer
iter
Linear
iter
Inner
iter
Resid
ual
norm
Relativ
eerror
Error
estim
ate
Mem
ory
(Mb)
Page
faults
4014
Delq
uad
◻Elem
ents
05
59
1117
194162
332
24270
1119890minus09
5119890minus10
1119890minus09
1104
318967
4118
Delaunay
998779Vo
lumes
40
minus99
1139
4228
332
528
4119890minus12
2119890minus12
1119890minus12
359630
4218
Delaunay
998779Vo
lumes
30
50
1132
7712
332
964
1119890minus11
6119890minus12
2119890minus12
4211578
4318
Delaunay
998779Vo
lumes
20
177
1120
15776
332
1972
2119890minus12
1119890minus12
7119890minus13
5515436
4418
Delaunay
998779Vo
lumes
10142
1118
63902
332
7987
3119890minus12
1119890minus12
1119890minus12
160
46691
4518
Delaunay
998779Vo
lumes
05
176
1115
254612
332
31826
2119890minus12
8119890minus13
1119890minus12
555
168526
4618
Delaunay
998779Elem
ents
40
183
1096
2252
332
281
6119890minus12
3119890minus12
2119890minus12
359589
4718
Delaunay
998779Elem
ents
30
119
1104
4040
224
505
2119890minus08
1119890minus08
7119890minus09
41114
3748
18Delaunay
998779Elem
ents
20
85
1108
8156
332
1019
2119890minus12
8119890minus13
6119890minus13
5515053
4918
Delaunay
998779Elem
ents
1063
1112
32480
332
4060
1119890minus12
6119890minus13
8119890minus13
151
43671
5018
Delaunay
998779Elem
ents
05
58
1113
128358
332
1604
47119890minus13
3119890minus13
7119890minus13
527
157581
5118
Delaunay
◻Vo
lumes
40
37
1144
2380
224
297
5119890minus08
2119890minus08
1119890minus08
5314173
5218
Delaunay
◻Vo
lumes
30
218
1120
4194
332
524
7119890minus12
4119890minus12
2119890minus12
3610034
5318
Delaunay
◻Vo
lumes
20
345
1110
7960
332
995
5119890minus12
3119890minus12
1119890minus12
4712879
5418
Delaunay
◻Vo
lumes
10165
1117
30652
332
3831
2119890minus12
1119890minus12
7119890minus13
100
28851
5518
Delaunay
◻Vo
lumes
05
191
1117
122066
224
15258
2119890minus08
8119890minus09
9119890minus09
343
103825
5618
Delaunay
◻Elem
ents
40
175
1097
2526
332
315
4119890minus12
2119890minus12
1119890minus12
6016159
5718
Delaunay
◻Elem
ents
30
114
1104
4386
332
548
4119890minus12
2119890minus12
2119890minus12
5013768
5818
Delaunay
◻Elem
ents
20
901108
8234
332
1029
2119890minus12
1119890minus12
1119890minus12
7320094
5918
Delaunay
◻Elem
ents
1062
1112
31186
224
3898
2119890minus08
7119890minus09
9119890minus09
208
59589
6018
Delaunay
◻Elem
ents
05
58
1113
123122
224
15390
9119890minus09
5119890minus09
1119890minus08
807
2366
7061
18Delq
uad
998779Vo
lumes
40
minus210
1168
3224
332
403
1119890minus11
5119890minus12
2119890minus12
43117
8462
18Delq
uad
998779Vo
lumes
30
369
1150
6000
332
750
1119890minus11
6119890minus12
3119890minus12
5013502
6318
Delq
uad
998779Vo
lumes
20
312
1104
12316
332
1539
2119890minus11
8119890minus12
5119890minus12
6016395
6418
Delq
uad
998779Vo
lumes
1044
01094
48714
332
6089
9119890minus12
4119890minus12
4119890minus12
1183444
165
18Delq
uad
998779Vo
lumes
05
534
1081
193980
332
24247
5119890minus12
2119890minus12
4119890minus12
423
126768
6618
Delq
uad
998779Elem
ents
40
240
1090
1750
332
218
9119890minus12
4119890minus12
2119890minus12
43117
8667
18Delqu
ad998779
Elem
ents
30
150
1100
3184
332
398
1119890minus11
5119890minus12
2119890minus12
5013509
6818
Delq
uad
998779Elem
ents
20
89
1106
6426
332
803
5119890minus12
3119890minus12
2119890minus12
5916047
6918
Delq
uad
998779Elem
ents
1064
1111
24886
332
3110
2119890minus12
1119890minus12
1119890minus12
113
32788
7018
Delq
uad
998779Elem
ents
05
59
1113
98052
224
12256
1119890minus08
5119890minus09
8119890minus09
398
118153
7118
Delq
uad
◻Vo
lumes
40
minus398
1172
1650
332
206
1119890minus11
5119890minus12
2119890minus12
369907
7218
Delq
uad
◻Vo
lumes
30
minus249
1146
3058
332
382
4119890minus12
2119890minus12
1119890minus12
349349
7318
Delq
uad
◻Vo
lumes
20
minus131
1136
6228
224
778
3119890minus08
1119890minus08
5119890minus09
4010930
7418
Delq
uad
◻Vo
lumes
10minus51
1123
24536
224
3067
3119890minus08
2119890minus08
1119890minus08
7822716
7618
Delq
uad
◻Elem
ents
40
237
1090
1752
332
219
6119890minus12
3119890minus12
2119890minus12
4111213
7718
Delq
uad
◻Elem
ents
30
145
1101
3184
332
398
6119890minus12
3119890minus12
2119890minus12
43118
1178
18Delq
uad
◻Elem
ents
20
941107
6426
332
803
3119890minus12
1119890minus12
1119890minus12
6016220
7918
Delqu
ad◻
Elem
ents
1065
1111
24874
332
3109
1119890minus12
6119890minus13
8119890minus13
156
4400
080
18Delq
uad
◻Elem
ents
05
59
1113
9804
22
2412255
8119890minus09
4119890minus09
8119890minus09
567
162168
Science and Technology of Nuclear Installations 23
Tim
e con
sum
ed to
bui
ld th
e mat
rices
(sec
onds
)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay trianglesvolumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
10
30
100
300
1
3
Figure 23 Time needed to build the matrices versus number of unknowns
was considerably larger than the solution time itself Theeigenvalue problem was solved using a multilayer iterativeKrylov-Schur method [26] with a tolerance relative to thematrices norm
1003817100381710038171003817119877 sdot 120601 minus (1119896eff) sdot 119865 sdot 1206011003817100381710038171003817
119877 + (1119896eff) sdot 119865lt 10minus8 (11)
The reported residual norm and relative error are10038171003817100381710038171003817100381710038171003817119877 sdot 120601 minus
1
119896effsdot 119865 sdot 120601
10038171003817100381710038171003817100381710038171003817
1003817100381710038171003817119877 sdot 120601 minus (1119896eff) sdot 119865 sdot 1206011003817100381710038171003817
1003817100381710038171003817(1119896eff) sdot 1206011003817100381710038171003817
(12)
respectively The fields marked as outer linear and inneriterations refer to the number of steps needed to attainthe requested tolerance in each layer of the Krylov-Schuralgorithm The computer used to solve the problem has anIntel i7 920 267GHz processor with 4Gb of RAM runningDebian GNULinux Wheezy
When using a finite volumes-based scheme over anunstructured mesh the solver has to gather information
about which cells are neighbors and which are not Currentlygmsh does not write this kind of lists in its output filesso milonga has to explicitly solve the neighbors problemPerforming a linear search is an 119874(119873
2) task which is
unacceptable for problem sizes 119873 of interest The code usesa search based on a 119896-dimensional tree [27] which can inprinciple be performed in119874(119873) steps Still for large values of119873 the time needed to read and parse the mesh (number twopreviously mentioned) is the bottleneck of the solution Thisstep is not needed in finite elements although the construc-tion of the elementary matrices involves the computationof the multidimensional Jacobians and integrals which thenhave to be assembled Again for large 119873 this step (numberthree) takes up most of the time needed to solve the problem
The Delaunay algorithm is a standard method for gen-erating two-dimensional grids [16] by tessellating a planewith triangles If the mesh needs to be based on quadranglesinstead a recombination algorithm can be used to transformtwo adjacent triangles in one quadrangle whenever is possi-ble However for geometries which are based on rectangularshapes there exist other algorithms [25] both for meshingand for recombining the triangles that give rise to elementaryentities with right angles almost everywhere which may be a
24 Science and Technology of Nuclear Installations
Tim
e con
sum
ed to
solv
e the
eige
nval
ue p
robl
em (s
econ
ds)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
10
30
100
300
1
3
Figure 24 Time needed to solve the eigenvalue problem versus number of unknowns
desired property of the resulting grid For finite elements thesteps needed to build the matrices depend on the selectionof triangles or quadrangles as the basic elementary geometrybecause the shape functions change However the number ofunknowns is the same as the number of nodes that does notchange after a recombination procedure On the other handthe number of unknowns in the finite volumes schemes withtriangles is roughly twice as the number of unknowns withquadrangles for the same grid
Figure 19 shows how the computed static reactivity (ie1 minus 1119896eff) depends on the number of unknowns for eachof the sixteen combinations of geometry algorithm shapescheme The four accepted results published in the originalreference [10] are included for reference although it shouldbe taken into account that said reactivities were computedalmost forty years ago Figure 20 shows the total wall timeneeded to solve the problem as a function of 119873119866 whileFigures 21 22 23 and 24 show the times needed for eachof the first four steps involved in the solution maintainingthe same logarithmic scale for both the abscissas and theordinates Green data represent finite volumes whilst bluebullets correspond to finite elements Solid lines indicatequarter-core and dashed lines eighth-symmetry Fillet bullets
are results obtained by theDelaunay triangulation and emptybullets were computed with the delquad algorithm Finallytriangle-shaped data correspond to triangles and squares anddiamonds denote quadrangles as the basic geometry of thegrid
It can be seen that finite elements produce amuch smallerdispersion of eigenvalues 119896eff than finite volumes with therefinement of the mesh This can be explained because finitevolumes methods rely on a geometric condition of the meshwhich may change abruptly if the meshing algorithm decidesto allocate the cells in a rather different form for small changesin ℓ119888 Finite elements methods are less influenced by thesediscontinuous lay-out changes of the elements As expectedeighth-core symmetries give almost the same results as thequarter-core geometries with roughly half the unknownsFor small problems finite volumes run faster that finiteelements because the time needed to solve the neighborproblem is negligible When the problem size grows thistime increases and exceeds the overhead implied in the con-struction and assembly of the finite elements matrices Alsoat least for this configuration it is seen that the eigenvalueproblem is solved faster for finite volumes than for finiteelements
Science and Technology of Nuclear Installations 25
4 Conclusions
Unstructured grids provide the cognizant engineer witha wide variety of possibilities to deal with the design oranalysis of nuclear reactor cores These kinds of grids cansuccessfully reproduce continuous geometries commonlyfound in reactor cores such as cylinders and thereforenot only can the diffusion equation be better approximatedinside the domain of definition but also the fulfillment ofboundary conditions is improved A free computer codewas written from scratch that is able to completely solvethe 2D IAEA PWR Benchmark using unstructured gridsfor sixteen combinations of geometry meshing algorithmbasic shape and discretization scheme plus any value ofthe gridrsquos characteristic length by using a single input fileThe complete set of input files and codemdashexecutable andsourcemdashis available either online or upon request withcomments experiences suggestions and corrections beingmore than welcome Further development should includetackling full three-dimensional geometries with completethermal hydraulic feedback in order to analyze how thesolutions of the coupled neutronic-thermal problem dependon the spatial discretization scheme of the neutron leakageterm Parallelization of the computation and assembly of thematrices and of the solution of the eigenvalue problem andits implementation using GPUs are also desired features toimplement A problemwith direct applications that the futureversions of milonga ought to solve is the analysis of how thegeometry of the absorbing materials should be taken intoaccount in structured coarse grids in order to mitigate effectssuch as the rod-cusp problem
Suitable schemes for approximating the continuous dif-ferential operators by discrete matrix expressions includefinite volumes and finite elements families Finite volumesmethods compute cell mean values whilst finite elementsgive functional values at the gridrsquos nodes In general finiteelements are less sensitive to changes in the mesh so theresults they provide do not change significantly for differentmeshing algorithms or elementary shapes Small problemsare best solved by finite volumes as the neighbor-findingproblem is faster than the process of building and assemblingthe eigenvalue-problem matrices For a large number ofunknowns the process of finding which cell is neighbor ofwhichmdasheven based on a 119896-dimensional treemdashoverwhelmsthe computation and assembly of elementary matrices andfinite-element methods perform better
References
[1] G Glasstone and S Bell Nuclear Reactor Theory KriegerPublishing Company 1970
[2] J J Duderstadt and L J Hamilton Nuclear Reactor AnalysisJohn Wiley amp Sons New York NY USA 1976
[3] C Gho Reactor Physics Undergraduate Course Lecture NotesInstituto Balseiro 2006
[4] G G Theler and F J Bonetto ldquoOn the stability of the pointreactor kinetics equationsrdquoNuclear Engineering andDesign vol240 no 6 pp 1443ndash1449 2010
[5] E E Lewis andW F Miller Computational Methods of NeutronTransport John Wiley amp Sons 1984
[6] E Masiello ldquoAnalytical stability analysis of Coarse-Mesh finitedifference methodrdquo in Proceedings of the International Confer-ence on the Physics of Reactors (PHYSOR rsquo08) Nuclear Power ASustainable Resource pp 456ndash464 September 2008
[7] M L Zerkle Development of a polynomial nodal methodwith flux and current discontinuity factors [PhD thesis] Mas-sachusets Institute of Technology 1992
[8] J I Yoon andHG Joo ldquoTwo-level coarsemesh finite differenceformulation with multigroup source expansion nodal kernelsrdquoJournal of Nuclear Science andTechnology vol 45 no 7 pp 668ndash682 2008
[9] O Mazzantini M Schivo J Di Cesare R Garbero M Riveroand G Theler ldquoA coupled calculation suite for Atucha II oper-ational transients analysisrdquo Science and Technology of NuclearInstallations vol 2011 Article ID 785304 12 pages 2011
[10] Computational Benchmark Problem Comitee for the Mathe-matics and Computation Division of the American NuclearSociety ldquoArgonne Code Center Benchmark problem bookrdquoTech Rep ANL-7416 Supplement 2 Argonne National Labo-ratory 1977
[11] A F Henry Nuclear Reactor Analysis MIT Cambridge UK1975
[12] O C Zienkiewicz R L Taylor and J Z ZhuTheFinite ElementMethod Its Basis and Fundamentals vol 1 Elsevier 6th edition2005
[13] G Theler Difusion de neutrones en mallas no estructuradascomparacion entre volumenes y elementos finitos AcademicMonograph Universidad de Buenos Aires 2013
[14] G Theler ldquoMilonga a free nuclear reactor core analysis coderdquoAvailable online 2011
[15] G Theler F J Bonetto and A Clausse ldquoOptimizacion deparametros en reactores de potencia base de diseno delcodigo neutronico milongardquo in Reunion Anual de la AsociacionArgentina de Tecnologia Nuclear XXXVII 2010
[16] C Geuzaine and J-F Remacle ldquoGmsh a 3-D finite elementmesh generator with built-in pre- and post-processing facili-tiesrdquo International Journal for Numerical Methods in Engineer-ing vol 79 no 11 pp 1309ndash1331 2009
[17] S Balay J Brown K Buschelman et al ldquoPETSc users manualrdquoTech Rep ANL-9511-Revision 34 Argonne National Labora-tory 2013
[18] S Balay W D Gropp L Curfman McInnes and B F SmithldquoEfficient management of parallelism in object oriented numer-ical software librariesrdquo in Modern Software Tools in ScientificComputing E Arge A M Bruaset and H P Langtangen Edspp 163ndash202 Birkhauser 1997
[19] V Hernandez J E Roman and V Vidal ldquoSLEPC a scalable andflexible toolkit for the solution of eigenvalue problemsrdquo ACMTransactions on Mathematical Software vol 31 no 3 pp 351ndash362 2005
[20] M Galassi J Davies J Theiler et al GNU Scientific LibraryReference Manual 3rd edition 2009
[21] O C Zienkiewicz and R L Taylor The Finite Element MethodFor Solid and Structural Mechanics vol 3 Elsevier 6th edition2005
[22] A Imelda and K Doddy Evaluation of the IAEA 3-D PWRbenchmark problem using NESTLE code 2004
[23] R Mosteller ldquoStatic benchmarking of the NESTLE advancednodal coderdquo in Proceedings of the Joint International Conference
26 Science and Technology of Nuclear Installations
on Mathematical Methods and Supercomputing for NuclearApplications vol 2 pp 1596ndash1605 1997
[24] GTheler F J Bonetto andAClausse ldquoSolution of the 2D IAEAPWR Benchmark with the neutronic code milongardquo in Actasde la Reunion Anual de la Asociacion Argentina de TecnologıaNuclear XXXVIII 2011
[25] J F Remacle F Henrotte T Carrier-Baudouin et al ldquoAfrontal delaunay quad mesh generator using the 119871
infin normrdquoInternational Journal For Numerical Methods in Engineeringvol 94 no 5 pp 494ndash512 2013
[26] G W Stewart ldquoA Krylov-Schur algorithm for large eigenprob-lemsrdquo SIAM Journal on Matrix Analysis and Applications vol23 no 3 pp 601ndash614 2002
[27] J L Bentley ldquoMultidimensional binary search trees used forassociative searchingrdquo Communications of the ACM vol 18 no9 pp 509ndash517 1975
TribologyAdvances in
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Renewable Energy
Submit your manuscripts athttpwwwhindawicom
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StructuresJournal of
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Wind EnergyJournal of
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Nuclear EnergyInternational Journal of
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High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Science and Technology of Nuclear Installations 15
Largest eigenvalue 1029462 (286185 pcm)1136 (3093 2953)1233 (13100 5100)
Number of unknowns 6228Outer iterations 2Linear iterations 24Inner iterations 778Residual normR l ti
Max 1206012(x y) coreMax 1206012(x y) reflector
26 times 10minus8
1281 times 10minus8
4996 times 10minus9
keff
Relative errorError estimateMemory used 41460 kBSoft page faults 10930Hard page faults 0Total CPU time 0604 seconds
1281 times 10 8
4996 times 10minus9
Milongarsquos 2D LWR IAEA benchmark problem case no 073eighth-symmetry core meshed using Delaunay (quads 985747c = 2) solved with finite volumes
(a)
1 075 3297 5552 134 4268 9923 148 4660 10944 123 3926 9145 059 2645 4406 094 2997 6977 093 2914 6858 072 2002 5349 mdash 301 773
10 146 4606 108011 150 4735 111112 133 4212 98813 108 3448 80214 104 3271 76715 094 2954 69616 070 1954 52117 mdash 288 73518 149 4691 110119 136 4289 100720 119 3748 88021 107 3359 79022 096 2880 71423 064 1636 47524 mdash 217 56325 121 3810 89426 098 3110 72427 090 2829 66628 081 2227 59929 mdash 536 125730 mdash 062 25031 045 2002 33632 068 2040 50033 055 1414 40934 mdash 215 57935 055 1359 40836 mdash 359 83437 mdash 050 19938 mdash 061 236
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 17 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
31 A Coarse Bare Homogeneous Circle Figures 5 and 6 showthe results of solving a bare two-dimensional circular homo-geneous reactor of radius 119886 over an unstructured grid whichis deliberately coarse using the finite volumes method andthe finite elements method respectively Two group energies
were used and a null-flux boundary conditionwas fixed at theexternal surface Figures 5(a) and 6(a) compare the obtainedfast flux distributions In the first case the numerical solutionprovidesmean values for each neutron flux group in each cellwhile in the latter the solution is computed at the nodes and
16 Science and Technology of Nuclear Installations
Largest eigenvalue 1029851 (289858 pcm)1090 (3182 3182)851 (13000 5000)
Number of unknowns 1752Outer iterations 3Linear iterations 32Inner iterations 219Residual norm
Max 1206012(x y) coreMax 1206012(x y) reflector
566 times 10minus12
2789 times 10minus12
1595 times 10minus12
keff
Residual normRelative errorError estimateMemory used 42564 kBSoft page faults 11213Hard page faults 0Total CPU time 09921 seconds
566 times 10
2789 times 10minus12
1595 times 10minus12
Milongarsquos 2D LWR IAEA benchmark problem case no 076eighth-symmetry core meshed using Delaunay (quads 985747c = 4) solved with finite volumes
(a)
1 073 3151 5402 127 4059 9393 141 4457 10464 118 3768 8715 061 2615 4506 093 2979 6887 094 2971 6998 067 2092 5829 mdash 375 885
10 139 4397 103111 144 4541 106612 128 4052 95013 105 3365 77814 103 3259 76415 096 3016 71216 066 2045 56817 mdash 359 84118 143 4520 106119 132 4155 97520 116 3672 86221 107 3366 79222 098 2951 73323 054 1728 54324 mdash 273 65125 117 3700 86826 095 3057 70827 091 2860 67428 074 2311 64829 mdash 642 130030 mdash 084 33231 048 2041 35332 069 2103 51333 046 1493 46734 mdash 269 66735 046 1444 45536 mdash 432 87137 mdash 067 26338 mdash 074 297
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 18 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
continuous functions are evaluated by means of the shapefunctions used in the formulation Figures 5(b) and 6(b)illustrate the fast fluxes unknowns and its relative position inspace It can be noted that the mesh coarseness gives resultsthat differ substantially in both cases Finally the structure ofthe sparse eigenvalue problemmatrices is shown for each case
with blue red and cyan representing positive negative andexplicitly inserted zero values In the finite volume case thereare 184 unknowns (92 cells times 2 groups) while in the secondcase there are 218 unknowns (109 nodes times 2 groups) Thevolumesrsquo fissionmatrix is almost diagonal it has a bandwidthequal to the number of energy groups which in this case is
Science and Technology of Nuclear Installations 17
500 1000 2000 5000 10000 20000 50000Number of unknowns
2840
2860
2880
2900
2920
ORNLRisoslashKWUOntarioQuarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumes
Quarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elementsEighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumes
Eighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
120588(p
cm)
105 2 times 105 5 times 105
Figure 19 Static reactivity versus number of unknowns The four original solutions as published in 1977 [10] are included as reference
twoThe off-diagonal values appear right next to the diagonalelements because of the chosen ordering of the unknownsin the flux vector 120601 isin R184 Other orderings may be usedbut the rate of convergence of the eigenvalue problem can bedeteriorated On the other hand the elementsrsquo fission matrixhas a nontrivial structure because the grid is unstructuredand the net fission rate at each element depends on the fluxesat the nodes whose location inside thematrix depends in turnon how the gridwas generatedThis effect is similar to the onethat appears in structural analysis where mass matrices needto be lumped in order to simplify transient computations [21]The diagonal block of elementrsquos 119877 matrix and the null blockin 119865 correspond to the discrete equations that set the null-flux boundary conditions on the nodes located at the externalsurface Other types of boundary conditions lead to differentkinds of structures within the problem matrices
In case a part of the domain contained a nonmultiplicativematerial such as a reflector then there would appear sectionsof the fission matrix with null values in both methodsrendering 119865 singular in both methods Care should be takenwhen dealing with the numerical schemes for the eigenvalueproblem solution It can be seen in the elementsrsquo matricesa particular structure that implements the boundary condi-tions at the external surfaces This structure does not appearin the volumesrsquo matrices because the boundary conditions
appear as flux terms which are summed up over all thesurfaces of each cell so they aremasked inside the volumetricdiscretization of the divergence and gradient operators
As the two-group neutron diffusion equation with uni-form cross sections over a circle subject to null-flux bound-ary conditions has an analytical solution it is adequate tocompare how the two proposed numerical schemes relate toit In the studied problem we ignored upscattering and fastfissions Then the analytical effective multiplication factor is
119896eff =]Σ1198912 sdot Σ1199041rarr2
[Σ1198861 + Σ1199041rarr2 + 1198631(]0119886)2] [Σ1198862 + 1198632(]0119886)
2]
(7)
being ]0 = 24048 the smallest root of Besselrsquos first-kindfunction of order zero 1198690(119903)
Figure 7 shows how the two numerical effective multi-plication factors compare to the analytical solution givenby (7) as a function of the mesh refinement indicated bythe quotient 119886ℓ119888 between the radius of the circle and thecharacteristic length of the cellelements We can see thatthe 119896eff computed by the finite volumes (elements) methodis always greater (less) than the analytical solution Indeedit can be proven that for a bare one-dimensional slab thisis always the case [13] However this result does not hold
18 Science and Technology of Nuclear Installations
2000 3000 6000 10000 20000 30000 60000
01
003
001
03
1
3
10
30
100
300To
tal w
all t
ime (
seco
nds)
Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
Figure 20 Total wall time versus number of unknowns
for problems with nonuniform cross sections even in simpleone-dimensional reflected reactors
We may draw two other conclusions from Figure 7 Firstthat the finite element method seems to provide a bettersolution than the volumes-based scheme and second thateven though the error committed tends to decrease with finergrids its behavior is not monotonic for finite volumes Infact Figure 8 shows the difference between the numericaland analytical solutions using a logarithmic vertical scalewhere both conclusions are even more evident We deferthe discussion of such differences between the discretizationscheme until the next section It is worth to note howeverthat the fact that a finite-element-based scheme throws betterresults for the particular bare homogeneous circular reactorunder study than the finite volumes does not imply thatfinite volumes ought to be discarded as a valid tool forsolving the neutron diffusion equation in general cases Thecombination of lattice and core-level computations is usuallyperformedusing cell-based resultswhichwhen fed into node-based methods to solve the few-group neutron diffusionequation may introduce errors which potentially can leadto unacceptable solutions Nevertheless this analysis is farbeyond the scope of this paper which focuses on solving amathematical equation over unstructured grids
32 The 2D IAEA PWR Benchmark Problem This is aclassical two-group neutron diffusion problem first designedin the early 1970s and taken as a reference benchmark forcomputational codes A number of codes were used to solveeither this problem or its three-dimensional formulation[22 23] including milonga using a structured grid [24] Theoriginal formulation can be found in the reference [10] andthere is a reproduction that may be easily found online inreference [24] The geometry consists of a one-quarter ofa PWR core depicted in Figure 9 and the homogeneousmacroscopic cross sections are listed in Table 1 An axialbuckling term 119861
2
119911119892= 08times10
minus4 should be taken into accountThe external surface should be subject to a zero incomingcurrent condition which may be written as
120597120601119892
120597119899= minus
04692
119863119892
sdot 120601119892 (8)
The expected results are as follows
(1) Maximum eigenvalue(2) Fundamental flux distributions
(a) Radial flux traverses 120601119892(119909 0) and 120601119892(119909 119909)
Science and Technology of Nuclear Installations 19
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
1
3
10
30
100
300Ti
me c
onsu
med
to m
esh
the g
eom
etry
(sec
onds
)
Figure 21 Time needed to mesh the geometry versus number of unknowns
Table 1 Macroscopic cross-sections (units are not stated in the original reference but they are assumed to be in cmminus1 or cm as appropriate)
1198631
1198632
Σ1rarr2
Σ1198861
Σ1198862
]Σ1198912
Material1 15 04 002 001 0080 0135 Fuel 12 15 04 002 001 0085 0135 Fuel 23 15 04 002 001 0130 0135 Fuel 2 + rod4 20 03 004 0 0010 0 Reflector
Note the fluxes shall be normalized such that
1
119881coreint119881core
sum
119892
]Σ119891119892 sdot 120601119892119889119881 = 1 (9)
(b) Value and location of maximum power densityThis corresponds to maximum of 1206012 in thecore It is recommended that the maximumvalues of 1206012 both in the inner core and at thecorereflector interface be given
(3) Average subassembly powers 119875119896
119875119896 =1
119881119896
int119881119896
sum
119892
]Σ119891119892 sdot 120601119892119889119881 (10)
where 119881119896 is the volume of the 119896th subassembly and119896 designates the fuel subassemblies as shown inFigure 9
(4) Number of unknowns in the problem number ofiterations and total and outer
(5) Total computing time iteration time IO-time andcomputer used
(6) Type and numerical values of convergence criteria(7) Table of average group fluxes for a square mesh grid
of 20 times 20 cm(8) Dependence of results on mesh spacing
Even though the original problem is based on a quarter-core situation the problem has an eighth-core symmetry
20 Science and Technology of Nuclear Installations
Quarter Delaunay triangles elements
Tim
e con
sum
ed to
read
the m
esh
(sec
onds
)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumes
Quarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
1
3
10
30
100
300
Figure 22 Time needed to read the mesh versus number of unknowns
which cannot be taken into account by structured gridswhich are the main target of the benchmark Howevernonstructured grids can take into consideration any kind ofsymmetry almost without loss of accuracy and at the sametime reducing roughly the number of unknowns by a halfand the associated computational effort needed to solve theproblem by a factor of four Answers to items (1)ndash(7) canbe given completely by milonga using a single input file(see Supplementary data in SupplementaryMaterial availableonline at httpdxdoiorg1011552013641863) As asked initem (8) how results depend not only on the mesh spacingbut also on the meshing algorithm on the gridrsquos elementarygeometric shape and on the discretization scheme may shedlights on the subject whichmay be evenmore interesting thanthe numerical results themselves
Taking advantage of milongarsquos capability of reading andparsing command-line arguments the selection of the coregeometry (quarter or eighth) the meshing algorithm (delau-nay [16] or delquad [25]) the shape of the elementaryentities (triangles or quadrangles) the discretization scheme(volumes or elements) and the characteristic length ℓ119888 ofthe mesh can be provided at run time Fixing five values forℓ119888 = 4 3 2 1 05 gives 2 times 2 times 2 times 2 times 5 = 80 possiblecombinations which we solve with successive invocations to
milonga with the same input file but with different argumentsfrom a simple script Figures 10 11 12 13 14 15 16 17 and18 show the results corresponding to items (1)ndash(7) for someillustrative cases The complete set of figures and the codeused to generate themmay be provided upon request Table 2compiles the answers to the problem for every case studied
As the milonga code is still under development itsnumerical routines are not yet fully optimized nor designedfor parallel computation Therefore the reported times areonly rough estimates and should be taken with care Thesolution comprises five steps
(1) generate the grid with the requested geometry mesh-ing algorithm basic shape and characteristic lengthby calling to gmsh
(2) read the generated mesh(3) build the matrices(4) solve the eigenvalue problem(5) compute the requested results
The CPU time reported in Figures 10ndash18 is thus thesum of all of these five steps but not the time needed togenerate the figuresmdashwhich in some caseswith coarsemeshes
Science and Technology of Nuclear Installations 21
Table2
Resultsofthe2
DIAEA
PWRBe
nchm
arkp
roblem
obtained
with
them
ilong
acod
efor
thee
ightyp
ropo
sedcombinatio
nsofsymmetrym
eshing
algorithm
basicshapediscretization
schemeandcharacteris
ticelem
entc
elllengthTh
ereference
solutio
nis120588=28749times10minus5w
hich
correspo
ndsto119896eff=10296
Case
no
Symm-
etry
Mesh
algorithm
Basic
shape
Solutio
nmetho
dℓ119888
(cm)
Δ120588
(pcm
)max1206012
(mdash)
Total
unkn
owns
Outer
iter
Linear
iter
Inner
iter
Resid
ual
norm
Relativ
eerror
Error
estim
ate
Mem
ory
(Mb)
Page
faults
114
Delaunay
998779Vo
lumes
40
minus120
1145
8264
332
1033
6119890minus09
3119890minus09
1119890minus09
349470
214
Delaunay
998779Vo
lumes
30
85
1135
15740
332
1967
2119890minus08
1119890minus08
4119890minus09
4813401
314
Delaunay
998779Vo
lumes
20
197
1124
31188
332
3898
1119890minus08
6119890minus09
3119890minus09
7621932
414
Delaunay
998779Vo
lumes
10153
1123
127476
332
15934
7119890minus09
3119890minus09
3119890minus09
273
81886
514
Delaunay
998779Vo
lumes
05
187
1119
510676
332
63834
2119890minus09
9119890minus10
1119890minus09
1148
352261
614
Delaunay
998779Elem
ents
40
173
1100
4308
332
538
2119890minus08
8119890minus09
4119890minus09
318794
714
Delaunay
998779Elem
ents
30
117
1107
8108
332
1013
5119890minus09
2119890minus09
2119890minus09
4712946
814
Delaunay
998779Elem
ents
20
84
1112
15936
332
1992
6119890minus09
3119890minus09
2119890minus09
7321347
914
Delaunay
998779Elem
ents
1062
1116
64478
332
8059
6119890minus09
3119890minus09
4119890minus09
262
77105
1014
Delaunay
998779Elem
ents
05
58
1117
256700
332
32087
1119890minus09
5119890minus10
1119890minus09
1091
331969
1114
Delaunay
◻Vo
lumes
40
53
1149
5042
332
630
2119890minus08
1119890minus08
5119890minus09
288007
1214
Delaunay
◻Vo
lumes
30
346
1133
8560
332
1070
7119890minus09
3119890minus09
2119890minus09
3910895
1314
Delaunay
◻Vo
lumes
20
308
1131
15576
332
1947
1119890minus08
7119890minus09
3119890minus09
541564
514
14Delaunay
◻Vo
lumes
10177
1122
60774
332
7596
8119890minus09
4119890minus09
3119890minus09
167
50115
1514
Delaunay
◻Vo
lumes
05
199
1120
244936
332
30617
4119890minus09
2119890minus09
2119890minus09
698
215678
1614
Delaunay
◻Elem
ents
40
196
1100
5222
332
652
3119890minus08
1119890minus08
8119890minus09
4713098
1714
Delaunay
◻Elem
ents
30
123
1107
8810
332
1101
2119890minus08
9119890minus09
7119890minus09
6918598
1814
Delaunay
◻Elem
ents
20
901112
15922
332
1990
1119890minus08
5119890minus09
5119890minus09
107
30591
1914
Delaunay
◻Elem
ents
1063
1116
61456
332
7682
5119890minus09
2119890minus09
4119890minus09
389
113237
2014
Delaunay
◻Elem
ents
05
58
1117
246332
332
30791
8119890minus10
4119890minus10
1119890minus09
1676
498192
2114
Delq
uad
998779Vo
lumes
40
minus45
1144
6228
332
778
4119890minus08
2119890minus08
7119890minus09
308495
2214
Delq
uad
998779Vo
lumes
30
570
1153
11820
332
1477
3119890minus08
1119890minus08
4119890minus09
4010879
2314
Delq
uad
998779Vo
lumes
20
459
1103
24100
332
3012
2119890minus08
1119890minus08
5119890minus09
6217715
2414
Delq
uad
998779Vo
lumes
10512
1104
9640
03
3212050
2119890minus08
1119890minus08
9119890minus09
207
61714
2514
Delq
uad
998779Vo
lumes
05
528
1106
385600
332
48200
2119890minus09
1119890minus09
2119890minus09
838
255735
2614
Delqu
ad998779
Elem
ents
40
220
1093
3288
332
411
3119890minus08
2119890minus08
6119890minus09
308495
2714
Delq
uad
998779Elem
ents
30
140
1104
6148
332
768
4119890minus08
2119890minus08
8119890minus09
39110
0028
14Delq
uad
998779Elem
ents
20
82
1110
12392
332
1549
2119890minus08
9119890minus09
6119890minus09
6117067
2914
Delqu
ad998779
Elem
ents
1063
1115
48882
332
6110
2119890minus09
1119890minus09
1119890minus09
197
5804
630
14Delq
uad
998779Elem
ents
05
58
1116
194162
332
24270
2119890minus09
9119890minus10
2119890minus09
790
238769
3114
Delq
uad
◻Vo
lumes
40
minus416
1174
3132
332
391
5119890minus08
3119890minus08
8119890minus09
267322
3214
Delq
uad
◻Vo
lumes
30
minus248
1151
5922
332
740
9119890minus09
4119890minus09
2119890minus09
318779
3314
Delq
uad
◻Vo
lumes
20
minus132
1139
12050
332
1506
1119890minus08
7119890minus09
4119890minus09
4412239
3414
Delq
uad
◻Vo
lumes
10minus49
1126
48200
332
6025
6119890minus09
3119890minus09
2119890minus09
122
36094
3514
Delq
uad
◻Vo
lumes
05
minus23
1123
192800
332
24100
5119890minus09
2119890minus09
3119890minus09
464
140228
3614
Delq
uad
◻Elem
ents
40
224
1093
3294
332
411
3119890minus08
1119890minus08
7119890minus09
359852
3714
Delq
uad
◻Elem
ents
30
141
1104
6144
332
768
3119890minus08
2119890minus08
9119890minus09
5114018
3814
Delq
uad
◻Elem
ents
20
921111
12392
332
1549
1119890minus08
6119890minus09
5119890minus09
8122526
3914
Delq
uad
◻Elem
ents
1065
1115
48882
332
6110
5119890minus09
3119890minus09
4119890minus09
276
78431
22 Science and Technology of Nuclear Installations
Table2Con
tinued
Case
no
Symm-
etry
Mesh
algorithm
Basic
shape
Solutio
nmetho
dℓ119888
(cm)
Δ120588
(pcm
)max1206012
(mdash)
Total
unkn
owns
Outer
iter
Linear
iter
Inner
iter
Resid
ual
norm
Relativ
eerror
Error
estim
ate
Mem
ory
(Mb)
Page
faults
4014
Delq
uad
◻Elem
ents
05
59
1117
194162
332
24270
1119890minus09
5119890minus10
1119890minus09
1104
318967
4118
Delaunay
998779Vo
lumes
40
minus99
1139
4228
332
528
4119890minus12
2119890minus12
1119890minus12
359630
4218
Delaunay
998779Vo
lumes
30
50
1132
7712
332
964
1119890minus11
6119890minus12
2119890minus12
4211578
4318
Delaunay
998779Vo
lumes
20
177
1120
15776
332
1972
2119890minus12
1119890minus12
7119890minus13
5515436
4418
Delaunay
998779Vo
lumes
10142
1118
63902
332
7987
3119890minus12
1119890minus12
1119890minus12
160
46691
4518
Delaunay
998779Vo
lumes
05
176
1115
254612
332
31826
2119890minus12
8119890minus13
1119890minus12
555
168526
4618
Delaunay
998779Elem
ents
40
183
1096
2252
332
281
6119890minus12
3119890minus12
2119890minus12
359589
4718
Delaunay
998779Elem
ents
30
119
1104
4040
224
505
2119890minus08
1119890minus08
7119890minus09
41114
3748
18Delaunay
998779Elem
ents
20
85
1108
8156
332
1019
2119890minus12
8119890minus13
6119890minus13
5515053
4918
Delaunay
998779Elem
ents
1063
1112
32480
332
4060
1119890minus12
6119890minus13
8119890minus13
151
43671
5018
Delaunay
998779Elem
ents
05
58
1113
128358
332
1604
47119890minus13
3119890minus13
7119890minus13
527
157581
5118
Delaunay
◻Vo
lumes
40
37
1144
2380
224
297
5119890minus08
2119890minus08
1119890minus08
5314173
5218
Delaunay
◻Vo
lumes
30
218
1120
4194
332
524
7119890minus12
4119890minus12
2119890minus12
3610034
5318
Delaunay
◻Vo
lumes
20
345
1110
7960
332
995
5119890minus12
3119890minus12
1119890minus12
4712879
5418
Delaunay
◻Vo
lumes
10165
1117
30652
332
3831
2119890minus12
1119890minus12
7119890minus13
100
28851
5518
Delaunay
◻Vo
lumes
05
191
1117
122066
224
15258
2119890minus08
8119890minus09
9119890minus09
343
103825
5618
Delaunay
◻Elem
ents
40
175
1097
2526
332
315
4119890minus12
2119890minus12
1119890minus12
6016159
5718
Delaunay
◻Elem
ents
30
114
1104
4386
332
548
4119890minus12
2119890minus12
2119890minus12
5013768
5818
Delaunay
◻Elem
ents
20
901108
8234
332
1029
2119890minus12
1119890minus12
1119890minus12
7320094
5918
Delaunay
◻Elem
ents
1062
1112
31186
224
3898
2119890minus08
7119890minus09
9119890minus09
208
59589
6018
Delaunay
◻Elem
ents
05
58
1113
123122
224
15390
9119890minus09
5119890minus09
1119890minus08
807
2366
7061
18Delq
uad
998779Vo
lumes
40
minus210
1168
3224
332
403
1119890minus11
5119890minus12
2119890minus12
43117
8462
18Delq
uad
998779Vo
lumes
30
369
1150
6000
332
750
1119890minus11
6119890minus12
3119890minus12
5013502
6318
Delq
uad
998779Vo
lumes
20
312
1104
12316
332
1539
2119890minus11
8119890minus12
5119890minus12
6016395
6418
Delq
uad
998779Vo
lumes
1044
01094
48714
332
6089
9119890minus12
4119890minus12
4119890minus12
1183444
165
18Delq
uad
998779Vo
lumes
05
534
1081
193980
332
24247
5119890minus12
2119890minus12
4119890minus12
423
126768
6618
Delq
uad
998779Elem
ents
40
240
1090
1750
332
218
9119890minus12
4119890minus12
2119890minus12
43117
8667
18Delqu
ad998779
Elem
ents
30
150
1100
3184
332
398
1119890minus11
5119890minus12
2119890minus12
5013509
6818
Delq
uad
998779Elem
ents
20
89
1106
6426
332
803
5119890minus12
3119890minus12
2119890minus12
5916047
6918
Delq
uad
998779Elem
ents
1064
1111
24886
332
3110
2119890minus12
1119890minus12
1119890minus12
113
32788
7018
Delq
uad
998779Elem
ents
05
59
1113
98052
224
12256
1119890minus08
5119890minus09
8119890minus09
398
118153
7118
Delq
uad
◻Vo
lumes
40
minus398
1172
1650
332
206
1119890minus11
5119890minus12
2119890minus12
369907
7218
Delq
uad
◻Vo
lumes
30
minus249
1146
3058
332
382
4119890minus12
2119890minus12
1119890minus12
349349
7318
Delq
uad
◻Vo
lumes
20
minus131
1136
6228
224
778
3119890minus08
1119890minus08
5119890minus09
4010930
7418
Delq
uad
◻Vo
lumes
10minus51
1123
24536
224
3067
3119890minus08
2119890minus08
1119890minus08
7822716
7618
Delq
uad
◻Elem
ents
40
237
1090
1752
332
219
6119890minus12
3119890minus12
2119890minus12
4111213
7718
Delq
uad
◻Elem
ents
30
145
1101
3184
332
398
6119890minus12
3119890minus12
2119890minus12
43118
1178
18Delq
uad
◻Elem
ents
20
941107
6426
332
803
3119890minus12
1119890minus12
1119890minus12
6016220
7918
Delqu
ad◻
Elem
ents
1065
1111
24874
332
3109
1119890minus12
6119890minus13
8119890minus13
156
4400
080
18Delq
uad
◻Elem
ents
05
59
1113
9804
22
2412255
8119890minus09
4119890minus09
8119890minus09
567
162168
Science and Technology of Nuclear Installations 23
Tim
e con
sum
ed to
bui
ld th
e mat
rices
(sec
onds
)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay trianglesvolumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
10
30
100
300
1
3
Figure 23 Time needed to build the matrices versus number of unknowns
was considerably larger than the solution time itself Theeigenvalue problem was solved using a multilayer iterativeKrylov-Schur method [26] with a tolerance relative to thematrices norm
1003817100381710038171003817119877 sdot 120601 minus (1119896eff) sdot 119865 sdot 1206011003817100381710038171003817
119877 + (1119896eff) sdot 119865lt 10minus8 (11)
The reported residual norm and relative error are10038171003817100381710038171003817100381710038171003817119877 sdot 120601 minus
1
119896effsdot 119865 sdot 120601
10038171003817100381710038171003817100381710038171003817
1003817100381710038171003817119877 sdot 120601 minus (1119896eff) sdot 119865 sdot 1206011003817100381710038171003817
1003817100381710038171003817(1119896eff) sdot 1206011003817100381710038171003817
(12)
respectively The fields marked as outer linear and inneriterations refer to the number of steps needed to attainthe requested tolerance in each layer of the Krylov-Schuralgorithm The computer used to solve the problem has anIntel i7 920 267GHz processor with 4Gb of RAM runningDebian GNULinux Wheezy
When using a finite volumes-based scheme over anunstructured mesh the solver has to gather information
about which cells are neighbors and which are not Currentlygmsh does not write this kind of lists in its output filesso milonga has to explicitly solve the neighbors problemPerforming a linear search is an 119874(119873
2) task which is
unacceptable for problem sizes 119873 of interest The code usesa search based on a 119896-dimensional tree [27] which can inprinciple be performed in119874(119873) steps Still for large values of119873 the time needed to read and parse the mesh (number twopreviously mentioned) is the bottleneck of the solution Thisstep is not needed in finite elements although the construc-tion of the elementary matrices involves the computationof the multidimensional Jacobians and integrals which thenhave to be assembled Again for large 119873 this step (numberthree) takes up most of the time needed to solve the problem
The Delaunay algorithm is a standard method for gen-erating two-dimensional grids [16] by tessellating a planewith triangles If the mesh needs to be based on quadranglesinstead a recombination algorithm can be used to transformtwo adjacent triangles in one quadrangle whenever is possi-ble However for geometries which are based on rectangularshapes there exist other algorithms [25] both for meshingand for recombining the triangles that give rise to elementaryentities with right angles almost everywhere which may be a
24 Science and Technology of Nuclear Installations
Tim
e con
sum
ed to
solv
e the
eige
nval
ue p
robl
em (s
econ
ds)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
10
30
100
300
1
3
Figure 24 Time needed to solve the eigenvalue problem versus number of unknowns
desired property of the resulting grid For finite elements thesteps needed to build the matrices depend on the selectionof triangles or quadrangles as the basic elementary geometrybecause the shape functions change However the number ofunknowns is the same as the number of nodes that does notchange after a recombination procedure On the other handthe number of unknowns in the finite volumes schemes withtriangles is roughly twice as the number of unknowns withquadrangles for the same grid
Figure 19 shows how the computed static reactivity (ie1 minus 1119896eff) depends on the number of unknowns for eachof the sixteen combinations of geometry algorithm shapescheme The four accepted results published in the originalreference [10] are included for reference although it shouldbe taken into account that said reactivities were computedalmost forty years ago Figure 20 shows the total wall timeneeded to solve the problem as a function of 119873119866 whileFigures 21 22 23 and 24 show the times needed for eachof the first four steps involved in the solution maintainingthe same logarithmic scale for both the abscissas and theordinates Green data represent finite volumes whilst bluebullets correspond to finite elements Solid lines indicatequarter-core and dashed lines eighth-symmetry Fillet bullets
are results obtained by theDelaunay triangulation and emptybullets were computed with the delquad algorithm Finallytriangle-shaped data correspond to triangles and squares anddiamonds denote quadrangles as the basic geometry of thegrid
It can be seen that finite elements produce amuch smallerdispersion of eigenvalues 119896eff than finite volumes with therefinement of the mesh This can be explained because finitevolumes methods rely on a geometric condition of the meshwhich may change abruptly if the meshing algorithm decidesto allocate the cells in a rather different form for small changesin ℓ119888 Finite elements methods are less influenced by thesediscontinuous lay-out changes of the elements As expectedeighth-core symmetries give almost the same results as thequarter-core geometries with roughly half the unknownsFor small problems finite volumes run faster that finiteelements because the time needed to solve the neighborproblem is negligible When the problem size grows thistime increases and exceeds the overhead implied in the con-struction and assembly of the finite elements matrices Alsoat least for this configuration it is seen that the eigenvalueproblem is solved faster for finite volumes than for finiteelements
Science and Technology of Nuclear Installations 25
4 Conclusions
Unstructured grids provide the cognizant engineer witha wide variety of possibilities to deal with the design oranalysis of nuclear reactor cores These kinds of grids cansuccessfully reproduce continuous geometries commonlyfound in reactor cores such as cylinders and thereforenot only can the diffusion equation be better approximatedinside the domain of definition but also the fulfillment ofboundary conditions is improved A free computer codewas written from scratch that is able to completely solvethe 2D IAEA PWR Benchmark using unstructured gridsfor sixteen combinations of geometry meshing algorithmbasic shape and discretization scheme plus any value ofthe gridrsquos characteristic length by using a single input fileThe complete set of input files and codemdashexecutable andsourcemdashis available either online or upon request withcomments experiences suggestions and corrections beingmore than welcome Further development should includetackling full three-dimensional geometries with completethermal hydraulic feedback in order to analyze how thesolutions of the coupled neutronic-thermal problem dependon the spatial discretization scheme of the neutron leakageterm Parallelization of the computation and assembly of thematrices and of the solution of the eigenvalue problem andits implementation using GPUs are also desired features toimplement A problemwith direct applications that the futureversions of milonga ought to solve is the analysis of how thegeometry of the absorbing materials should be taken intoaccount in structured coarse grids in order to mitigate effectssuch as the rod-cusp problem
Suitable schemes for approximating the continuous dif-ferential operators by discrete matrix expressions includefinite volumes and finite elements families Finite volumesmethods compute cell mean values whilst finite elementsgive functional values at the gridrsquos nodes In general finiteelements are less sensitive to changes in the mesh so theresults they provide do not change significantly for differentmeshing algorithms or elementary shapes Small problemsare best solved by finite volumes as the neighbor-findingproblem is faster than the process of building and assemblingthe eigenvalue-problem matrices For a large number ofunknowns the process of finding which cell is neighbor ofwhichmdasheven based on a 119896-dimensional treemdashoverwhelmsthe computation and assembly of elementary matrices andfinite-element methods perform better
References
[1] G Glasstone and S Bell Nuclear Reactor Theory KriegerPublishing Company 1970
[2] J J Duderstadt and L J Hamilton Nuclear Reactor AnalysisJohn Wiley amp Sons New York NY USA 1976
[3] C Gho Reactor Physics Undergraduate Course Lecture NotesInstituto Balseiro 2006
[4] G G Theler and F J Bonetto ldquoOn the stability of the pointreactor kinetics equationsrdquoNuclear Engineering andDesign vol240 no 6 pp 1443ndash1449 2010
[5] E E Lewis andW F Miller Computational Methods of NeutronTransport John Wiley amp Sons 1984
[6] E Masiello ldquoAnalytical stability analysis of Coarse-Mesh finitedifference methodrdquo in Proceedings of the International Confer-ence on the Physics of Reactors (PHYSOR rsquo08) Nuclear Power ASustainable Resource pp 456ndash464 September 2008
[7] M L Zerkle Development of a polynomial nodal methodwith flux and current discontinuity factors [PhD thesis] Mas-sachusets Institute of Technology 1992
[8] J I Yoon andHG Joo ldquoTwo-level coarsemesh finite differenceformulation with multigroup source expansion nodal kernelsrdquoJournal of Nuclear Science andTechnology vol 45 no 7 pp 668ndash682 2008
[9] O Mazzantini M Schivo J Di Cesare R Garbero M Riveroand G Theler ldquoA coupled calculation suite for Atucha II oper-ational transients analysisrdquo Science and Technology of NuclearInstallations vol 2011 Article ID 785304 12 pages 2011
[10] Computational Benchmark Problem Comitee for the Mathe-matics and Computation Division of the American NuclearSociety ldquoArgonne Code Center Benchmark problem bookrdquoTech Rep ANL-7416 Supplement 2 Argonne National Labo-ratory 1977
[11] A F Henry Nuclear Reactor Analysis MIT Cambridge UK1975
[12] O C Zienkiewicz R L Taylor and J Z ZhuTheFinite ElementMethod Its Basis and Fundamentals vol 1 Elsevier 6th edition2005
[13] G Theler Difusion de neutrones en mallas no estructuradascomparacion entre volumenes y elementos finitos AcademicMonograph Universidad de Buenos Aires 2013
[14] G Theler ldquoMilonga a free nuclear reactor core analysis coderdquoAvailable online 2011
[15] G Theler F J Bonetto and A Clausse ldquoOptimizacion deparametros en reactores de potencia base de diseno delcodigo neutronico milongardquo in Reunion Anual de la AsociacionArgentina de Tecnologia Nuclear XXXVII 2010
[16] C Geuzaine and J-F Remacle ldquoGmsh a 3-D finite elementmesh generator with built-in pre- and post-processing facili-tiesrdquo International Journal for Numerical Methods in Engineer-ing vol 79 no 11 pp 1309ndash1331 2009
[17] S Balay J Brown K Buschelman et al ldquoPETSc users manualrdquoTech Rep ANL-9511-Revision 34 Argonne National Labora-tory 2013
[18] S Balay W D Gropp L Curfman McInnes and B F SmithldquoEfficient management of parallelism in object oriented numer-ical software librariesrdquo in Modern Software Tools in ScientificComputing E Arge A M Bruaset and H P Langtangen Edspp 163ndash202 Birkhauser 1997
[19] V Hernandez J E Roman and V Vidal ldquoSLEPC a scalable andflexible toolkit for the solution of eigenvalue problemsrdquo ACMTransactions on Mathematical Software vol 31 no 3 pp 351ndash362 2005
[20] M Galassi J Davies J Theiler et al GNU Scientific LibraryReference Manual 3rd edition 2009
[21] O C Zienkiewicz and R L Taylor The Finite Element MethodFor Solid and Structural Mechanics vol 3 Elsevier 6th edition2005
[22] A Imelda and K Doddy Evaluation of the IAEA 3-D PWRbenchmark problem using NESTLE code 2004
[23] R Mosteller ldquoStatic benchmarking of the NESTLE advancednodal coderdquo in Proceedings of the Joint International Conference
26 Science and Technology of Nuclear Installations
on Mathematical Methods and Supercomputing for NuclearApplications vol 2 pp 1596ndash1605 1997
[24] GTheler F J Bonetto andAClausse ldquoSolution of the 2D IAEAPWR Benchmark with the neutronic code milongardquo in Actasde la Reunion Anual de la Asociacion Argentina de TecnologıaNuclear XXXVIII 2011
[25] J F Remacle F Henrotte T Carrier-Baudouin et al ldquoAfrontal delaunay quad mesh generator using the 119871
infin normrdquoInternational Journal For Numerical Methods in Engineeringvol 94 no 5 pp 494ndash512 2013
[26] G W Stewart ldquoA Krylov-Schur algorithm for large eigenprob-lemsrdquo SIAM Journal on Matrix Analysis and Applications vol23 no 3 pp 601ndash614 2002
[27] J L Bentley ldquoMultidimensional binary search trees used forassociative searchingrdquo Communications of the ACM vol 18 no9 pp 509ndash517 1975
TribologyAdvances in
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Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Journal of
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Renewable Energy
Submit your manuscripts athttpwwwhindawicom
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International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
EnergyJournal of
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Journal ofEngineeringVolume 2014
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Nuclear InstallationsScience and Technology of
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Wind EnergyJournal of
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Nuclear EnergyInternational Journal of
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High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
16 Science and Technology of Nuclear Installations
Largest eigenvalue 1029851 (289858 pcm)1090 (3182 3182)851 (13000 5000)
Number of unknowns 1752Outer iterations 3Linear iterations 32Inner iterations 219Residual norm
Max 1206012(x y) coreMax 1206012(x y) reflector
566 times 10minus12
2789 times 10minus12
1595 times 10minus12
keff
Residual normRelative errorError estimateMemory used 42564 kBSoft page faults 11213Hard page faults 0Total CPU time 09921 seconds
566 times 10
2789 times 10minus12
1595 times 10minus12
Milongarsquos 2D LWR IAEA benchmark problem case no 076eighth-symmetry core meshed using Delaunay (quads 985747c = 4) solved with finite volumes
(a)
1 073 3151 5402 127 4059 9393 141 4457 10464 118 3768 8715 061 2615 4506 093 2979 6887 094 2971 6998 067 2092 5829 mdash 375 885
10 139 4397 103111 144 4541 106612 128 4052 95013 105 3365 77814 103 3259 76415 096 3016 71216 066 2045 56817 mdash 359 84118 143 4520 106119 132 4155 97520 116 3672 86221 107 3366 79222 098 2951 73323 054 1728 54324 mdash 273 65125 117 3700 86826 095 3057 70827 091 2860 67428 074 2311 64829 mdash 642 130030 mdash 084 33231 048 2041 35332 069 2103 51333 046 1493 46734 mdash 269 66735 046 1444 45536 mdash 432 87137 mdash 067 26338 mdash 074 297
k Pk 1206011k 1206012k
(b)
0 40 80 120 1600
15
30
45
1206012(x 0)1206011(x 0)
(c)
0 40 80 120 1600
15
30
45
1206012(x x)1206011(x x)
(d)
Figure 18 (a) Mesh and thermal flux distribution (b) Power and fluxes (c) Flux distribution 120601119892(119909 0) along the 119909-axis (d) Flux distribution
120601119892(119909 119909) along the diagonal
continuous functions are evaluated by means of the shapefunctions used in the formulation Figures 5(b) and 6(b)illustrate the fast fluxes unknowns and its relative position inspace It can be noted that the mesh coarseness gives resultsthat differ substantially in both cases Finally the structure ofthe sparse eigenvalue problemmatrices is shown for each case
with blue red and cyan representing positive negative andexplicitly inserted zero values In the finite volume case thereare 184 unknowns (92 cells times 2 groups) while in the secondcase there are 218 unknowns (109 nodes times 2 groups) Thevolumesrsquo fissionmatrix is almost diagonal it has a bandwidthequal to the number of energy groups which in this case is
Science and Technology of Nuclear Installations 17
500 1000 2000 5000 10000 20000 50000Number of unknowns
2840
2860
2880
2900
2920
ORNLRisoslashKWUOntarioQuarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumes
Quarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elementsEighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumes
Eighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
120588(p
cm)
105 2 times 105 5 times 105
Figure 19 Static reactivity versus number of unknowns The four original solutions as published in 1977 [10] are included as reference
twoThe off-diagonal values appear right next to the diagonalelements because of the chosen ordering of the unknownsin the flux vector 120601 isin R184 Other orderings may be usedbut the rate of convergence of the eigenvalue problem can bedeteriorated On the other hand the elementsrsquo fission matrixhas a nontrivial structure because the grid is unstructuredand the net fission rate at each element depends on the fluxesat the nodes whose location inside thematrix depends in turnon how the gridwas generatedThis effect is similar to the onethat appears in structural analysis where mass matrices needto be lumped in order to simplify transient computations [21]The diagonal block of elementrsquos 119877 matrix and the null blockin 119865 correspond to the discrete equations that set the null-flux boundary conditions on the nodes located at the externalsurface Other types of boundary conditions lead to differentkinds of structures within the problem matrices
In case a part of the domain contained a nonmultiplicativematerial such as a reflector then there would appear sectionsof the fission matrix with null values in both methodsrendering 119865 singular in both methods Care should be takenwhen dealing with the numerical schemes for the eigenvalueproblem solution It can be seen in the elementsrsquo matricesa particular structure that implements the boundary condi-tions at the external surfaces This structure does not appearin the volumesrsquo matrices because the boundary conditions
appear as flux terms which are summed up over all thesurfaces of each cell so they aremasked inside the volumetricdiscretization of the divergence and gradient operators
As the two-group neutron diffusion equation with uni-form cross sections over a circle subject to null-flux bound-ary conditions has an analytical solution it is adequate tocompare how the two proposed numerical schemes relate toit In the studied problem we ignored upscattering and fastfissions Then the analytical effective multiplication factor is
119896eff =]Σ1198912 sdot Σ1199041rarr2
[Σ1198861 + Σ1199041rarr2 + 1198631(]0119886)2] [Σ1198862 + 1198632(]0119886)
2]
(7)
being ]0 = 24048 the smallest root of Besselrsquos first-kindfunction of order zero 1198690(119903)
Figure 7 shows how the two numerical effective multi-plication factors compare to the analytical solution givenby (7) as a function of the mesh refinement indicated bythe quotient 119886ℓ119888 between the radius of the circle and thecharacteristic length of the cellelements We can see thatthe 119896eff computed by the finite volumes (elements) methodis always greater (less) than the analytical solution Indeedit can be proven that for a bare one-dimensional slab thisis always the case [13] However this result does not hold
18 Science and Technology of Nuclear Installations
2000 3000 6000 10000 20000 30000 60000
01
003
001
03
1
3
10
30
100
300To
tal w
all t
ime (
seco
nds)
Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
Figure 20 Total wall time versus number of unknowns
for problems with nonuniform cross sections even in simpleone-dimensional reflected reactors
We may draw two other conclusions from Figure 7 Firstthat the finite element method seems to provide a bettersolution than the volumes-based scheme and second thateven though the error committed tends to decrease with finergrids its behavior is not monotonic for finite volumes Infact Figure 8 shows the difference between the numericaland analytical solutions using a logarithmic vertical scalewhere both conclusions are even more evident We deferthe discussion of such differences between the discretizationscheme until the next section It is worth to note howeverthat the fact that a finite-element-based scheme throws betterresults for the particular bare homogeneous circular reactorunder study than the finite volumes does not imply thatfinite volumes ought to be discarded as a valid tool forsolving the neutron diffusion equation in general cases Thecombination of lattice and core-level computations is usuallyperformedusing cell-based resultswhichwhen fed into node-based methods to solve the few-group neutron diffusionequation may introduce errors which potentially can leadto unacceptable solutions Nevertheless this analysis is farbeyond the scope of this paper which focuses on solving amathematical equation over unstructured grids
32 The 2D IAEA PWR Benchmark Problem This is aclassical two-group neutron diffusion problem first designedin the early 1970s and taken as a reference benchmark forcomputational codes A number of codes were used to solveeither this problem or its three-dimensional formulation[22 23] including milonga using a structured grid [24] Theoriginal formulation can be found in the reference [10] andthere is a reproduction that may be easily found online inreference [24] The geometry consists of a one-quarter ofa PWR core depicted in Figure 9 and the homogeneousmacroscopic cross sections are listed in Table 1 An axialbuckling term 119861
2
119911119892= 08times10
minus4 should be taken into accountThe external surface should be subject to a zero incomingcurrent condition which may be written as
120597120601119892
120597119899= minus
04692
119863119892
sdot 120601119892 (8)
The expected results are as follows
(1) Maximum eigenvalue(2) Fundamental flux distributions
(a) Radial flux traverses 120601119892(119909 0) and 120601119892(119909 119909)
Science and Technology of Nuclear Installations 19
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
1
3
10
30
100
300Ti
me c
onsu
med
to m
esh
the g
eom
etry
(sec
onds
)
Figure 21 Time needed to mesh the geometry versus number of unknowns
Table 1 Macroscopic cross-sections (units are not stated in the original reference but they are assumed to be in cmminus1 or cm as appropriate)
1198631
1198632
Σ1rarr2
Σ1198861
Σ1198862
]Σ1198912
Material1 15 04 002 001 0080 0135 Fuel 12 15 04 002 001 0085 0135 Fuel 23 15 04 002 001 0130 0135 Fuel 2 + rod4 20 03 004 0 0010 0 Reflector
Note the fluxes shall be normalized such that
1
119881coreint119881core
sum
119892
]Σ119891119892 sdot 120601119892119889119881 = 1 (9)
(b) Value and location of maximum power densityThis corresponds to maximum of 1206012 in thecore It is recommended that the maximumvalues of 1206012 both in the inner core and at thecorereflector interface be given
(3) Average subassembly powers 119875119896
119875119896 =1
119881119896
int119881119896
sum
119892
]Σ119891119892 sdot 120601119892119889119881 (10)
where 119881119896 is the volume of the 119896th subassembly and119896 designates the fuel subassemblies as shown inFigure 9
(4) Number of unknowns in the problem number ofiterations and total and outer
(5) Total computing time iteration time IO-time andcomputer used
(6) Type and numerical values of convergence criteria(7) Table of average group fluxes for a square mesh grid
of 20 times 20 cm(8) Dependence of results on mesh spacing
Even though the original problem is based on a quarter-core situation the problem has an eighth-core symmetry
20 Science and Technology of Nuclear Installations
Quarter Delaunay triangles elements
Tim
e con
sum
ed to
read
the m
esh
(sec
onds
)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumes
Quarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
1
3
10
30
100
300
Figure 22 Time needed to read the mesh versus number of unknowns
which cannot be taken into account by structured gridswhich are the main target of the benchmark Howevernonstructured grids can take into consideration any kind ofsymmetry almost without loss of accuracy and at the sametime reducing roughly the number of unknowns by a halfand the associated computational effort needed to solve theproblem by a factor of four Answers to items (1)ndash(7) canbe given completely by milonga using a single input file(see Supplementary data in SupplementaryMaterial availableonline at httpdxdoiorg1011552013641863) As asked initem (8) how results depend not only on the mesh spacingbut also on the meshing algorithm on the gridrsquos elementarygeometric shape and on the discretization scheme may shedlights on the subject whichmay be evenmore interesting thanthe numerical results themselves
Taking advantage of milongarsquos capability of reading andparsing command-line arguments the selection of the coregeometry (quarter or eighth) the meshing algorithm (delau-nay [16] or delquad [25]) the shape of the elementaryentities (triangles or quadrangles) the discretization scheme(volumes or elements) and the characteristic length ℓ119888 ofthe mesh can be provided at run time Fixing five values forℓ119888 = 4 3 2 1 05 gives 2 times 2 times 2 times 2 times 5 = 80 possiblecombinations which we solve with successive invocations to
milonga with the same input file but with different argumentsfrom a simple script Figures 10 11 12 13 14 15 16 17 and18 show the results corresponding to items (1)ndash(7) for someillustrative cases The complete set of figures and the codeused to generate themmay be provided upon request Table 2compiles the answers to the problem for every case studied
As the milonga code is still under development itsnumerical routines are not yet fully optimized nor designedfor parallel computation Therefore the reported times areonly rough estimates and should be taken with care Thesolution comprises five steps
(1) generate the grid with the requested geometry mesh-ing algorithm basic shape and characteristic lengthby calling to gmsh
(2) read the generated mesh(3) build the matrices(4) solve the eigenvalue problem(5) compute the requested results
The CPU time reported in Figures 10ndash18 is thus thesum of all of these five steps but not the time needed togenerate the figuresmdashwhich in some caseswith coarsemeshes
Science and Technology of Nuclear Installations 21
Table2
Resultsofthe2
DIAEA
PWRBe
nchm
arkp
roblem
obtained
with
them
ilong
acod
efor
thee
ightyp
ropo
sedcombinatio
nsofsymmetrym
eshing
algorithm
basicshapediscretization
schemeandcharacteris
ticelem
entc
elllengthTh
ereference
solutio
nis120588=28749times10minus5w
hich
correspo
ndsto119896eff=10296
Case
no
Symm-
etry
Mesh
algorithm
Basic
shape
Solutio
nmetho
dℓ119888
(cm)
Δ120588
(pcm
)max1206012
(mdash)
Total
unkn
owns
Outer
iter
Linear
iter
Inner
iter
Resid
ual
norm
Relativ
eerror
Error
estim
ate
Mem
ory
(Mb)
Page
faults
114
Delaunay
998779Vo
lumes
40
minus120
1145
8264
332
1033
6119890minus09
3119890minus09
1119890minus09
349470
214
Delaunay
998779Vo
lumes
30
85
1135
15740
332
1967
2119890minus08
1119890minus08
4119890minus09
4813401
314
Delaunay
998779Vo
lumes
20
197
1124
31188
332
3898
1119890minus08
6119890minus09
3119890minus09
7621932
414
Delaunay
998779Vo
lumes
10153
1123
127476
332
15934
7119890minus09
3119890minus09
3119890minus09
273
81886
514
Delaunay
998779Vo
lumes
05
187
1119
510676
332
63834
2119890minus09
9119890minus10
1119890minus09
1148
352261
614
Delaunay
998779Elem
ents
40
173
1100
4308
332
538
2119890minus08
8119890minus09
4119890minus09
318794
714
Delaunay
998779Elem
ents
30
117
1107
8108
332
1013
5119890minus09
2119890minus09
2119890minus09
4712946
814
Delaunay
998779Elem
ents
20
84
1112
15936
332
1992
6119890minus09
3119890minus09
2119890minus09
7321347
914
Delaunay
998779Elem
ents
1062
1116
64478
332
8059
6119890minus09
3119890minus09
4119890minus09
262
77105
1014
Delaunay
998779Elem
ents
05
58
1117
256700
332
32087
1119890minus09
5119890minus10
1119890minus09
1091
331969
1114
Delaunay
◻Vo
lumes
40
53
1149
5042
332
630
2119890minus08
1119890minus08
5119890minus09
288007
1214
Delaunay
◻Vo
lumes
30
346
1133
8560
332
1070
7119890minus09
3119890minus09
2119890minus09
3910895
1314
Delaunay
◻Vo
lumes
20
308
1131
15576
332
1947
1119890minus08
7119890minus09
3119890minus09
541564
514
14Delaunay
◻Vo
lumes
10177
1122
60774
332
7596
8119890minus09
4119890minus09
3119890minus09
167
50115
1514
Delaunay
◻Vo
lumes
05
199
1120
244936
332
30617
4119890minus09
2119890minus09
2119890minus09
698
215678
1614
Delaunay
◻Elem
ents
40
196
1100
5222
332
652
3119890minus08
1119890minus08
8119890minus09
4713098
1714
Delaunay
◻Elem
ents
30
123
1107
8810
332
1101
2119890minus08
9119890minus09
7119890minus09
6918598
1814
Delaunay
◻Elem
ents
20
901112
15922
332
1990
1119890minus08
5119890minus09
5119890minus09
107
30591
1914
Delaunay
◻Elem
ents
1063
1116
61456
332
7682
5119890minus09
2119890minus09
4119890minus09
389
113237
2014
Delaunay
◻Elem
ents
05
58
1117
246332
332
30791
8119890minus10
4119890minus10
1119890minus09
1676
498192
2114
Delq
uad
998779Vo
lumes
40
minus45
1144
6228
332
778
4119890minus08
2119890minus08
7119890minus09
308495
2214
Delq
uad
998779Vo
lumes
30
570
1153
11820
332
1477
3119890minus08
1119890minus08
4119890minus09
4010879
2314
Delq
uad
998779Vo
lumes
20
459
1103
24100
332
3012
2119890minus08
1119890minus08
5119890minus09
6217715
2414
Delq
uad
998779Vo
lumes
10512
1104
9640
03
3212050
2119890minus08
1119890minus08
9119890minus09
207
61714
2514
Delq
uad
998779Vo
lumes
05
528
1106
385600
332
48200
2119890minus09
1119890minus09
2119890minus09
838
255735
2614
Delqu
ad998779
Elem
ents
40
220
1093
3288
332
411
3119890minus08
2119890minus08
6119890minus09
308495
2714
Delq
uad
998779Elem
ents
30
140
1104
6148
332
768
4119890minus08
2119890minus08
8119890minus09
39110
0028
14Delq
uad
998779Elem
ents
20
82
1110
12392
332
1549
2119890minus08
9119890minus09
6119890minus09
6117067
2914
Delqu
ad998779
Elem
ents
1063
1115
48882
332
6110
2119890minus09
1119890minus09
1119890minus09
197
5804
630
14Delq
uad
998779Elem
ents
05
58
1116
194162
332
24270
2119890minus09
9119890minus10
2119890minus09
790
238769
3114
Delq
uad
◻Vo
lumes
40
minus416
1174
3132
332
391
5119890minus08
3119890minus08
8119890minus09
267322
3214
Delq
uad
◻Vo
lumes
30
minus248
1151
5922
332
740
9119890minus09
4119890minus09
2119890minus09
318779
3314
Delq
uad
◻Vo
lumes
20
minus132
1139
12050
332
1506
1119890minus08
7119890minus09
4119890minus09
4412239
3414
Delq
uad
◻Vo
lumes
10minus49
1126
48200
332
6025
6119890minus09
3119890minus09
2119890minus09
122
36094
3514
Delq
uad
◻Vo
lumes
05
minus23
1123
192800
332
24100
5119890minus09
2119890minus09
3119890minus09
464
140228
3614
Delq
uad
◻Elem
ents
40
224
1093
3294
332
411
3119890minus08
1119890minus08
7119890minus09
359852
3714
Delq
uad
◻Elem
ents
30
141
1104
6144
332
768
3119890minus08
2119890minus08
9119890minus09
5114018
3814
Delq
uad
◻Elem
ents
20
921111
12392
332
1549
1119890minus08
6119890minus09
5119890minus09
8122526
3914
Delq
uad
◻Elem
ents
1065
1115
48882
332
6110
5119890minus09
3119890minus09
4119890minus09
276
78431
22 Science and Technology of Nuclear Installations
Table2Con
tinued
Case
no
Symm-
etry
Mesh
algorithm
Basic
shape
Solutio
nmetho
dℓ119888
(cm)
Δ120588
(pcm
)max1206012
(mdash)
Total
unkn
owns
Outer
iter
Linear
iter
Inner
iter
Resid
ual
norm
Relativ
eerror
Error
estim
ate
Mem
ory
(Mb)
Page
faults
4014
Delq
uad
◻Elem
ents
05
59
1117
194162
332
24270
1119890minus09
5119890minus10
1119890minus09
1104
318967
4118
Delaunay
998779Vo
lumes
40
minus99
1139
4228
332
528
4119890minus12
2119890minus12
1119890minus12
359630
4218
Delaunay
998779Vo
lumes
30
50
1132
7712
332
964
1119890minus11
6119890minus12
2119890minus12
4211578
4318
Delaunay
998779Vo
lumes
20
177
1120
15776
332
1972
2119890minus12
1119890minus12
7119890minus13
5515436
4418
Delaunay
998779Vo
lumes
10142
1118
63902
332
7987
3119890minus12
1119890minus12
1119890minus12
160
46691
4518
Delaunay
998779Vo
lumes
05
176
1115
254612
332
31826
2119890minus12
8119890minus13
1119890minus12
555
168526
4618
Delaunay
998779Elem
ents
40
183
1096
2252
332
281
6119890minus12
3119890minus12
2119890minus12
359589
4718
Delaunay
998779Elem
ents
30
119
1104
4040
224
505
2119890minus08
1119890minus08
7119890minus09
41114
3748
18Delaunay
998779Elem
ents
20
85
1108
8156
332
1019
2119890minus12
8119890minus13
6119890minus13
5515053
4918
Delaunay
998779Elem
ents
1063
1112
32480
332
4060
1119890minus12
6119890minus13
8119890minus13
151
43671
5018
Delaunay
998779Elem
ents
05
58
1113
128358
332
1604
47119890minus13
3119890minus13
7119890minus13
527
157581
5118
Delaunay
◻Vo
lumes
40
37
1144
2380
224
297
5119890minus08
2119890minus08
1119890minus08
5314173
5218
Delaunay
◻Vo
lumes
30
218
1120
4194
332
524
7119890minus12
4119890minus12
2119890minus12
3610034
5318
Delaunay
◻Vo
lumes
20
345
1110
7960
332
995
5119890minus12
3119890minus12
1119890minus12
4712879
5418
Delaunay
◻Vo
lumes
10165
1117
30652
332
3831
2119890minus12
1119890minus12
7119890minus13
100
28851
5518
Delaunay
◻Vo
lumes
05
191
1117
122066
224
15258
2119890minus08
8119890minus09
9119890minus09
343
103825
5618
Delaunay
◻Elem
ents
40
175
1097
2526
332
315
4119890minus12
2119890minus12
1119890minus12
6016159
5718
Delaunay
◻Elem
ents
30
114
1104
4386
332
548
4119890minus12
2119890minus12
2119890minus12
5013768
5818
Delaunay
◻Elem
ents
20
901108
8234
332
1029
2119890minus12
1119890minus12
1119890minus12
7320094
5918
Delaunay
◻Elem
ents
1062
1112
31186
224
3898
2119890minus08
7119890minus09
9119890minus09
208
59589
6018
Delaunay
◻Elem
ents
05
58
1113
123122
224
15390
9119890minus09
5119890minus09
1119890minus08
807
2366
7061
18Delq
uad
998779Vo
lumes
40
minus210
1168
3224
332
403
1119890minus11
5119890minus12
2119890minus12
43117
8462
18Delq
uad
998779Vo
lumes
30
369
1150
6000
332
750
1119890minus11
6119890minus12
3119890minus12
5013502
6318
Delq
uad
998779Vo
lumes
20
312
1104
12316
332
1539
2119890minus11
8119890minus12
5119890minus12
6016395
6418
Delq
uad
998779Vo
lumes
1044
01094
48714
332
6089
9119890minus12
4119890minus12
4119890minus12
1183444
165
18Delq
uad
998779Vo
lumes
05
534
1081
193980
332
24247
5119890minus12
2119890minus12
4119890minus12
423
126768
6618
Delq
uad
998779Elem
ents
40
240
1090
1750
332
218
9119890minus12
4119890minus12
2119890minus12
43117
8667
18Delqu
ad998779
Elem
ents
30
150
1100
3184
332
398
1119890minus11
5119890minus12
2119890minus12
5013509
6818
Delq
uad
998779Elem
ents
20
89
1106
6426
332
803
5119890minus12
3119890minus12
2119890minus12
5916047
6918
Delq
uad
998779Elem
ents
1064
1111
24886
332
3110
2119890minus12
1119890minus12
1119890minus12
113
32788
7018
Delq
uad
998779Elem
ents
05
59
1113
98052
224
12256
1119890minus08
5119890minus09
8119890minus09
398
118153
7118
Delq
uad
◻Vo
lumes
40
minus398
1172
1650
332
206
1119890minus11
5119890minus12
2119890minus12
369907
7218
Delq
uad
◻Vo
lumes
30
minus249
1146
3058
332
382
4119890minus12
2119890minus12
1119890minus12
349349
7318
Delq
uad
◻Vo
lumes
20
minus131
1136
6228
224
778
3119890minus08
1119890minus08
5119890minus09
4010930
7418
Delq
uad
◻Vo
lumes
10minus51
1123
24536
224
3067
3119890minus08
2119890minus08
1119890minus08
7822716
7618
Delq
uad
◻Elem
ents
40
237
1090
1752
332
219
6119890minus12
3119890minus12
2119890minus12
4111213
7718
Delq
uad
◻Elem
ents
30
145
1101
3184
332
398
6119890minus12
3119890minus12
2119890minus12
43118
1178
18Delq
uad
◻Elem
ents
20
941107
6426
332
803
3119890minus12
1119890minus12
1119890minus12
6016220
7918
Delqu
ad◻
Elem
ents
1065
1111
24874
332
3109
1119890minus12
6119890minus13
8119890minus13
156
4400
080
18Delq
uad
◻Elem
ents
05
59
1113
9804
22
2412255
8119890minus09
4119890minus09
8119890minus09
567
162168
Science and Technology of Nuclear Installations 23
Tim
e con
sum
ed to
bui
ld th
e mat
rices
(sec
onds
)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay trianglesvolumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
10
30
100
300
1
3
Figure 23 Time needed to build the matrices versus number of unknowns
was considerably larger than the solution time itself Theeigenvalue problem was solved using a multilayer iterativeKrylov-Schur method [26] with a tolerance relative to thematrices norm
1003817100381710038171003817119877 sdot 120601 minus (1119896eff) sdot 119865 sdot 1206011003817100381710038171003817
119877 + (1119896eff) sdot 119865lt 10minus8 (11)
The reported residual norm and relative error are10038171003817100381710038171003817100381710038171003817119877 sdot 120601 minus
1
119896effsdot 119865 sdot 120601
10038171003817100381710038171003817100381710038171003817
1003817100381710038171003817119877 sdot 120601 minus (1119896eff) sdot 119865 sdot 1206011003817100381710038171003817
1003817100381710038171003817(1119896eff) sdot 1206011003817100381710038171003817
(12)
respectively The fields marked as outer linear and inneriterations refer to the number of steps needed to attainthe requested tolerance in each layer of the Krylov-Schuralgorithm The computer used to solve the problem has anIntel i7 920 267GHz processor with 4Gb of RAM runningDebian GNULinux Wheezy
When using a finite volumes-based scheme over anunstructured mesh the solver has to gather information
about which cells are neighbors and which are not Currentlygmsh does not write this kind of lists in its output filesso milonga has to explicitly solve the neighbors problemPerforming a linear search is an 119874(119873
2) task which is
unacceptable for problem sizes 119873 of interest The code usesa search based on a 119896-dimensional tree [27] which can inprinciple be performed in119874(119873) steps Still for large values of119873 the time needed to read and parse the mesh (number twopreviously mentioned) is the bottleneck of the solution Thisstep is not needed in finite elements although the construc-tion of the elementary matrices involves the computationof the multidimensional Jacobians and integrals which thenhave to be assembled Again for large 119873 this step (numberthree) takes up most of the time needed to solve the problem
The Delaunay algorithm is a standard method for gen-erating two-dimensional grids [16] by tessellating a planewith triangles If the mesh needs to be based on quadranglesinstead a recombination algorithm can be used to transformtwo adjacent triangles in one quadrangle whenever is possi-ble However for geometries which are based on rectangularshapes there exist other algorithms [25] both for meshingand for recombining the triangles that give rise to elementaryentities with right angles almost everywhere which may be a
24 Science and Technology of Nuclear Installations
Tim
e con
sum
ed to
solv
e the
eige
nval
ue p
robl
em (s
econ
ds)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
10
30
100
300
1
3
Figure 24 Time needed to solve the eigenvalue problem versus number of unknowns
desired property of the resulting grid For finite elements thesteps needed to build the matrices depend on the selectionof triangles or quadrangles as the basic elementary geometrybecause the shape functions change However the number ofunknowns is the same as the number of nodes that does notchange after a recombination procedure On the other handthe number of unknowns in the finite volumes schemes withtriangles is roughly twice as the number of unknowns withquadrangles for the same grid
Figure 19 shows how the computed static reactivity (ie1 minus 1119896eff) depends on the number of unknowns for eachof the sixteen combinations of geometry algorithm shapescheme The four accepted results published in the originalreference [10] are included for reference although it shouldbe taken into account that said reactivities were computedalmost forty years ago Figure 20 shows the total wall timeneeded to solve the problem as a function of 119873119866 whileFigures 21 22 23 and 24 show the times needed for eachof the first four steps involved in the solution maintainingthe same logarithmic scale for both the abscissas and theordinates Green data represent finite volumes whilst bluebullets correspond to finite elements Solid lines indicatequarter-core and dashed lines eighth-symmetry Fillet bullets
are results obtained by theDelaunay triangulation and emptybullets were computed with the delquad algorithm Finallytriangle-shaped data correspond to triangles and squares anddiamonds denote quadrangles as the basic geometry of thegrid
It can be seen that finite elements produce amuch smallerdispersion of eigenvalues 119896eff than finite volumes with therefinement of the mesh This can be explained because finitevolumes methods rely on a geometric condition of the meshwhich may change abruptly if the meshing algorithm decidesto allocate the cells in a rather different form for small changesin ℓ119888 Finite elements methods are less influenced by thesediscontinuous lay-out changes of the elements As expectedeighth-core symmetries give almost the same results as thequarter-core geometries with roughly half the unknownsFor small problems finite volumes run faster that finiteelements because the time needed to solve the neighborproblem is negligible When the problem size grows thistime increases and exceeds the overhead implied in the con-struction and assembly of the finite elements matrices Alsoat least for this configuration it is seen that the eigenvalueproblem is solved faster for finite volumes than for finiteelements
Science and Technology of Nuclear Installations 25
4 Conclusions
Unstructured grids provide the cognizant engineer witha wide variety of possibilities to deal with the design oranalysis of nuclear reactor cores These kinds of grids cansuccessfully reproduce continuous geometries commonlyfound in reactor cores such as cylinders and thereforenot only can the diffusion equation be better approximatedinside the domain of definition but also the fulfillment ofboundary conditions is improved A free computer codewas written from scratch that is able to completely solvethe 2D IAEA PWR Benchmark using unstructured gridsfor sixteen combinations of geometry meshing algorithmbasic shape and discretization scheme plus any value ofthe gridrsquos characteristic length by using a single input fileThe complete set of input files and codemdashexecutable andsourcemdashis available either online or upon request withcomments experiences suggestions and corrections beingmore than welcome Further development should includetackling full three-dimensional geometries with completethermal hydraulic feedback in order to analyze how thesolutions of the coupled neutronic-thermal problem dependon the spatial discretization scheme of the neutron leakageterm Parallelization of the computation and assembly of thematrices and of the solution of the eigenvalue problem andits implementation using GPUs are also desired features toimplement A problemwith direct applications that the futureversions of milonga ought to solve is the analysis of how thegeometry of the absorbing materials should be taken intoaccount in structured coarse grids in order to mitigate effectssuch as the rod-cusp problem
Suitable schemes for approximating the continuous dif-ferential operators by discrete matrix expressions includefinite volumes and finite elements families Finite volumesmethods compute cell mean values whilst finite elementsgive functional values at the gridrsquos nodes In general finiteelements are less sensitive to changes in the mesh so theresults they provide do not change significantly for differentmeshing algorithms or elementary shapes Small problemsare best solved by finite volumes as the neighbor-findingproblem is faster than the process of building and assemblingthe eigenvalue-problem matrices For a large number ofunknowns the process of finding which cell is neighbor ofwhichmdasheven based on a 119896-dimensional treemdashoverwhelmsthe computation and assembly of elementary matrices andfinite-element methods perform better
References
[1] G Glasstone and S Bell Nuclear Reactor Theory KriegerPublishing Company 1970
[2] J J Duderstadt and L J Hamilton Nuclear Reactor AnalysisJohn Wiley amp Sons New York NY USA 1976
[3] C Gho Reactor Physics Undergraduate Course Lecture NotesInstituto Balseiro 2006
[4] G G Theler and F J Bonetto ldquoOn the stability of the pointreactor kinetics equationsrdquoNuclear Engineering andDesign vol240 no 6 pp 1443ndash1449 2010
[5] E E Lewis andW F Miller Computational Methods of NeutronTransport John Wiley amp Sons 1984
[6] E Masiello ldquoAnalytical stability analysis of Coarse-Mesh finitedifference methodrdquo in Proceedings of the International Confer-ence on the Physics of Reactors (PHYSOR rsquo08) Nuclear Power ASustainable Resource pp 456ndash464 September 2008
[7] M L Zerkle Development of a polynomial nodal methodwith flux and current discontinuity factors [PhD thesis] Mas-sachusets Institute of Technology 1992
[8] J I Yoon andHG Joo ldquoTwo-level coarsemesh finite differenceformulation with multigroup source expansion nodal kernelsrdquoJournal of Nuclear Science andTechnology vol 45 no 7 pp 668ndash682 2008
[9] O Mazzantini M Schivo J Di Cesare R Garbero M Riveroand G Theler ldquoA coupled calculation suite for Atucha II oper-ational transients analysisrdquo Science and Technology of NuclearInstallations vol 2011 Article ID 785304 12 pages 2011
[10] Computational Benchmark Problem Comitee for the Mathe-matics and Computation Division of the American NuclearSociety ldquoArgonne Code Center Benchmark problem bookrdquoTech Rep ANL-7416 Supplement 2 Argonne National Labo-ratory 1977
[11] A F Henry Nuclear Reactor Analysis MIT Cambridge UK1975
[12] O C Zienkiewicz R L Taylor and J Z ZhuTheFinite ElementMethod Its Basis and Fundamentals vol 1 Elsevier 6th edition2005
[13] G Theler Difusion de neutrones en mallas no estructuradascomparacion entre volumenes y elementos finitos AcademicMonograph Universidad de Buenos Aires 2013
[14] G Theler ldquoMilonga a free nuclear reactor core analysis coderdquoAvailable online 2011
[15] G Theler F J Bonetto and A Clausse ldquoOptimizacion deparametros en reactores de potencia base de diseno delcodigo neutronico milongardquo in Reunion Anual de la AsociacionArgentina de Tecnologia Nuclear XXXVII 2010
[16] C Geuzaine and J-F Remacle ldquoGmsh a 3-D finite elementmesh generator with built-in pre- and post-processing facili-tiesrdquo International Journal for Numerical Methods in Engineer-ing vol 79 no 11 pp 1309ndash1331 2009
[17] S Balay J Brown K Buschelman et al ldquoPETSc users manualrdquoTech Rep ANL-9511-Revision 34 Argonne National Labora-tory 2013
[18] S Balay W D Gropp L Curfman McInnes and B F SmithldquoEfficient management of parallelism in object oriented numer-ical software librariesrdquo in Modern Software Tools in ScientificComputing E Arge A M Bruaset and H P Langtangen Edspp 163ndash202 Birkhauser 1997
[19] V Hernandez J E Roman and V Vidal ldquoSLEPC a scalable andflexible toolkit for the solution of eigenvalue problemsrdquo ACMTransactions on Mathematical Software vol 31 no 3 pp 351ndash362 2005
[20] M Galassi J Davies J Theiler et al GNU Scientific LibraryReference Manual 3rd edition 2009
[21] O C Zienkiewicz and R L Taylor The Finite Element MethodFor Solid and Structural Mechanics vol 3 Elsevier 6th edition2005
[22] A Imelda and K Doddy Evaluation of the IAEA 3-D PWRbenchmark problem using NESTLE code 2004
[23] R Mosteller ldquoStatic benchmarking of the NESTLE advancednodal coderdquo in Proceedings of the Joint International Conference
26 Science and Technology of Nuclear Installations
on Mathematical Methods and Supercomputing for NuclearApplications vol 2 pp 1596ndash1605 1997
[24] GTheler F J Bonetto andAClausse ldquoSolution of the 2D IAEAPWR Benchmark with the neutronic code milongardquo in Actasde la Reunion Anual de la Asociacion Argentina de TecnologıaNuclear XXXVIII 2011
[25] J F Remacle F Henrotte T Carrier-Baudouin et al ldquoAfrontal delaunay quad mesh generator using the 119871
infin normrdquoInternational Journal For Numerical Methods in Engineeringvol 94 no 5 pp 494ndash512 2013
[26] G W Stewart ldquoA Krylov-Schur algorithm for large eigenprob-lemsrdquo SIAM Journal on Matrix Analysis and Applications vol23 no 3 pp 601ndash614 2002
[27] J L Bentley ldquoMultidimensional binary search trees used forassociative searchingrdquo Communications of the ACM vol 18 no9 pp 509ndash517 1975
TribologyAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FuelsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofPetroleum Engineering
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
CombustionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Renewable Energy
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
StructuresJournal of
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear InstallationsScience and Technology of
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Solar EnergyJournal of
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Wind EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear EnergyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Science and Technology of Nuclear Installations 17
500 1000 2000 5000 10000 20000 50000Number of unknowns
2840
2860
2880
2900
2920
ORNLRisoslashKWUOntarioQuarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumes
Quarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elementsEighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumes
Eighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
120588(p
cm)
105 2 times 105 5 times 105
Figure 19 Static reactivity versus number of unknowns The four original solutions as published in 1977 [10] are included as reference
twoThe off-diagonal values appear right next to the diagonalelements because of the chosen ordering of the unknownsin the flux vector 120601 isin R184 Other orderings may be usedbut the rate of convergence of the eigenvalue problem can bedeteriorated On the other hand the elementsrsquo fission matrixhas a nontrivial structure because the grid is unstructuredand the net fission rate at each element depends on the fluxesat the nodes whose location inside thematrix depends in turnon how the gridwas generatedThis effect is similar to the onethat appears in structural analysis where mass matrices needto be lumped in order to simplify transient computations [21]The diagonal block of elementrsquos 119877 matrix and the null blockin 119865 correspond to the discrete equations that set the null-flux boundary conditions on the nodes located at the externalsurface Other types of boundary conditions lead to differentkinds of structures within the problem matrices
In case a part of the domain contained a nonmultiplicativematerial such as a reflector then there would appear sectionsof the fission matrix with null values in both methodsrendering 119865 singular in both methods Care should be takenwhen dealing with the numerical schemes for the eigenvalueproblem solution It can be seen in the elementsrsquo matricesa particular structure that implements the boundary condi-tions at the external surfaces This structure does not appearin the volumesrsquo matrices because the boundary conditions
appear as flux terms which are summed up over all thesurfaces of each cell so they aremasked inside the volumetricdiscretization of the divergence and gradient operators
As the two-group neutron diffusion equation with uni-form cross sections over a circle subject to null-flux bound-ary conditions has an analytical solution it is adequate tocompare how the two proposed numerical schemes relate toit In the studied problem we ignored upscattering and fastfissions Then the analytical effective multiplication factor is
119896eff =]Σ1198912 sdot Σ1199041rarr2
[Σ1198861 + Σ1199041rarr2 + 1198631(]0119886)2] [Σ1198862 + 1198632(]0119886)
2]
(7)
being ]0 = 24048 the smallest root of Besselrsquos first-kindfunction of order zero 1198690(119903)
Figure 7 shows how the two numerical effective multi-plication factors compare to the analytical solution givenby (7) as a function of the mesh refinement indicated bythe quotient 119886ℓ119888 between the radius of the circle and thecharacteristic length of the cellelements We can see thatthe 119896eff computed by the finite volumes (elements) methodis always greater (less) than the analytical solution Indeedit can be proven that for a bare one-dimensional slab thisis always the case [13] However this result does not hold
18 Science and Technology of Nuclear Installations
2000 3000 6000 10000 20000 30000 60000
01
003
001
03
1
3
10
30
100
300To
tal w
all t
ime (
seco
nds)
Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
Figure 20 Total wall time versus number of unknowns
for problems with nonuniform cross sections even in simpleone-dimensional reflected reactors
We may draw two other conclusions from Figure 7 Firstthat the finite element method seems to provide a bettersolution than the volumes-based scheme and second thateven though the error committed tends to decrease with finergrids its behavior is not monotonic for finite volumes Infact Figure 8 shows the difference between the numericaland analytical solutions using a logarithmic vertical scalewhere both conclusions are even more evident We deferthe discussion of such differences between the discretizationscheme until the next section It is worth to note howeverthat the fact that a finite-element-based scheme throws betterresults for the particular bare homogeneous circular reactorunder study than the finite volumes does not imply thatfinite volumes ought to be discarded as a valid tool forsolving the neutron diffusion equation in general cases Thecombination of lattice and core-level computations is usuallyperformedusing cell-based resultswhichwhen fed into node-based methods to solve the few-group neutron diffusionequation may introduce errors which potentially can leadto unacceptable solutions Nevertheless this analysis is farbeyond the scope of this paper which focuses on solving amathematical equation over unstructured grids
32 The 2D IAEA PWR Benchmark Problem This is aclassical two-group neutron diffusion problem first designedin the early 1970s and taken as a reference benchmark forcomputational codes A number of codes were used to solveeither this problem or its three-dimensional formulation[22 23] including milonga using a structured grid [24] Theoriginal formulation can be found in the reference [10] andthere is a reproduction that may be easily found online inreference [24] The geometry consists of a one-quarter ofa PWR core depicted in Figure 9 and the homogeneousmacroscopic cross sections are listed in Table 1 An axialbuckling term 119861
2
119911119892= 08times10
minus4 should be taken into accountThe external surface should be subject to a zero incomingcurrent condition which may be written as
120597120601119892
120597119899= minus
04692
119863119892
sdot 120601119892 (8)
The expected results are as follows
(1) Maximum eigenvalue(2) Fundamental flux distributions
(a) Radial flux traverses 120601119892(119909 0) and 120601119892(119909 119909)
Science and Technology of Nuclear Installations 19
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
1
3
10
30
100
300Ti
me c
onsu
med
to m
esh
the g
eom
etry
(sec
onds
)
Figure 21 Time needed to mesh the geometry versus number of unknowns
Table 1 Macroscopic cross-sections (units are not stated in the original reference but they are assumed to be in cmminus1 or cm as appropriate)
1198631
1198632
Σ1rarr2
Σ1198861
Σ1198862
]Σ1198912
Material1 15 04 002 001 0080 0135 Fuel 12 15 04 002 001 0085 0135 Fuel 23 15 04 002 001 0130 0135 Fuel 2 + rod4 20 03 004 0 0010 0 Reflector
Note the fluxes shall be normalized such that
1
119881coreint119881core
sum
119892
]Σ119891119892 sdot 120601119892119889119881 = 1 (9)
(b) Value and location of maximum power densityThis corresponds to maximum of 1206012 in thecore It is recommended that the maximumvalues of 1206012 both in the inner core and at thecorereflector interface be given
(3) Average subassembly powers 119875119896
119875119896 =1
119881119896
int119881119896
sum
119892
]Σ119891119892 sdot 120601119892119889119881 (10)
where 119881119896 is the volume of the 119896th subassembly and119896 designates the fuel subassemblies as shown inFigure 9
(4) Number of unknowns in the problem number ofiterations and total and outer
(5) Total computing time iteration time IO-time andcomputer used
(6) Type and numerical values of convergence criteria(7) Table of average group fluxes for a square mesh grid
of 20 times 20 cm(8) Dependence of results on mesh spacing
Even though the original problem is based on a quarter-core situation the problem has an eighth-core symmetry
20 Science and Technology of Nuclear Installations
Quarter Delaunay triangles elements
Tim
e con
sum
ed to
read
the m
esh
(sec
onds
)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumes
Quarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
1
3
10
30
100
300
Figure 22 Time needed to read the mesh versus number of unknowns
which cannot be taken into account by structured gridswhich are the main target of the benchmark Howevernonstructured grids can take into consideration any kind ofsymmetry almost without loss of accuracy and at the sametime reducing roughly the number of unknowns by a halfand the associated computational effort needed to solve theproblem by a factor of four Answers to items (1)ndash(7) canbe given completely by milonga using a single input file(see Supplementary data in SupplementaryMaterial availableonline at httpdxdoiorg1011552013641863) As asked initem (8) how results depend not only on the mesh spacingbut also on the meshing algorithm on the gridrsquos elementarygeometric shape and on the discretization scheme may shedlights on the subject whichmay be evenmore interesting thanthe numerical results themselves
Taking advantage of milongarsquos capability of reading andparsing command-line arguments the selection of the coregeometry (quarter or eighth) the meshing algorithm (delau-nay [16] or delquad [25]) the shape of the elementaryentities (triangles or quadrangles) the discretization scheme(volumes or elements) and the characteristic length ℓ119888 ofthe mesh can be provided at run time Fixing five values forℓ119888 = 4 3 2 1 05 gives 2 times 2 times 2 times 2 times 5 = 80 possiblecombinations which we solve with successive invocations to
milonga with the same input file but with different argumentsfrom a simple script Figures 10 11 12 13 14 15 16 17 and18 show the results corresponding to items (1)ndash(7) for someillustrative cases The complete set of figures and the codeused to generate themmay be provided upon request Table 2compiles the answers to the problem for every case studied
As the milonga code is still under development itsnumerical routines are not yet fully optimized nor designedfor parallel computation Therefore the reported times areonly rough estimates and should be taken with care Thesolution comprises five steps
(1) generate the grid with the requested geometry mesh-ing algorithm basic shape and characteristic lengthby calling to gmsh
(2) read the generated mesh(3) build the matrices(4) solve the eigenvalue problem(5) compute the requested results
The CPU time reported in Figures 10ndash18 is thus thesum of all of these five steps but not the time needed togenerate the figuresmdashwhich in some caseswith coarsemeshes
Science and Technology of Nuclear Installations 21
Table2
Resultsofthe2
DIAEA
PWRBe
nchm
arkp
roblem
obtained
with
them
ilong
acod
efor
thee
ightyp
ropo
sedcombinatio
nsofsymmetrym
eshing
algorithm
basicshapediscretization
schemeandcharacteris
ticelem
entc
elllengthTh
ereference
solutio
nis120588=28749times10minus5w
hich
correspo
ndsto119896eff=10296
Case
no
Symm-
etry
Mesh
algorithm
Basic
shape
Solutio
nmetho
dℓ119888
(cm)
Δ120588
(pcm
)max1206012
(mdash)
Total
unkn
owns
Outer
iter
Linear
iter
Inner
iter
Resid
ual
norm
Relativ
eerror
Error
estim
ate
Mem
ory
(Mb)
Page
faults
114
Delaunay
998779Vo
lumes
40
minus120
1145
8264
332
1033
6119890minus09
3119890minus09
1119890minus09
349470
214
Delaunay
998779Vo
lumes
30
85
1135
15740
332
1967
2119890minus08
1119890minus08
4119890minus09
4813401
314
Delaunay
998779Vo
lumes
20
197
1124
31188
332
3898
1119890minus08
6119890minus09
3119890minus09
7621932
414
Delaunay
998779Vo
lumes
10153
1123
127476
332
15934
7119890minus09
3119890minus09
3119890minus09
273
81886
514
Delaunay
998779Vo
lumes
05
187
1119
510676
332
63834
2119890minus09
9119890minus10
1119890minus09
1148
352261
614
Delaunay
998779Elem
ents
40
173
1100
4308
332
538
2119890minus08
8119890minus09
4119890minus09
318794
714
Delaunay
998779Elem
ents
30
117
1107
8108
332
1013
5119890minus09
2119890minus09
2119890minus09
4712946
814
Delaunay
998779Elem
ents
20
84
1112
15936
332
1992
6119890minus09
3119890minus09
2119890minus09
7321347
914
Delaunay
998779Elem
ents
1062
1116
64478
332
8059
6119890minus09
3119890minus09
4119890minus09
262
77105
1014
Delaunay
998779Elem
ents
05
58
1117
256700
332
32087
1119890minus09
5119890minus10
1119890minus09
1091
331969
1114
Delaunay
◻Vo
lumes
40
53
1149
5042
332
630
2119890minus08
1119890minus08
5119890minus09
288007
1214
Delaunay
◻Vo
lumes
30
346
1133
8560
332
1070
7119890minus09
3119890minus09
2119890minus09
3910895
1314
Delaunay
◻Vo
lumes
20
308
1131
15576
332
1947
1119890minus08
7119890minus09
3119890minus09
541564
514
14Delaunay
◻Vo
lumes
10177
1122
60774
332
7596
8119890minus09
4119890minus09
3119890minus09
167
50115
1514
Delaunay
◻Vo
lumes
05
199
1120
244936
332
30617
4119890minus09
2119890minus09
2119890minus09
698
215678
1614
Delaunay
◻Elem
ents
40
196
1100
5222
332
652
3119890minus08
1119890minus08
8119890minus09
4713098
1714
Delaunay
◻Elem
ents
30
123
1107
8810
332
1101
2119890minus08
9119890minus09
7119890minus09
6918598
1814
Delaunay
◻Elem
ents
20
901112
15922
332
1990
1119890minus08
5119890minus09
5119890minus09
107
30591
1914
Delaunay
◻Elem
ents
1063
1116
61456
332
7682
5119890minus09
2119890minus09
4119890minus09
389
113237
2014
Delaunay
◻Elem
ents
05
58
1117
246332
332
30791
8119890minus10
4119890minus10
1119890minus09
1676
498192
2114
Delq
uad
998779Vo
lumes
40
minus45
1144
6228
332
778
4119890minus08
2119890minus08
7119890minus09
308495
2214
Delq
uad
998779Vo
lumes
30
570
1153
11820
332
1477
3119890minus08
1119890minus08
4119890minus09
4010879
2314
Delq
uad
998779Vo
lumes
20
459
1103
24100
332
3012
2119890minus08
1119890minus08
5119890minus09
6217715
2414
Delq
uad
998779Vo
lumes
10512
1104
9640
03
3212050
2119890minus08
1119890minus08
9119890minus09
207
61714
2514
Delq
uad
998779Vo
lumes
05
528
1106
385600
332
48200
2119890minus09
1119890minus09
2119890minus09
838
255735
2614
Delqu
ad998779
Elem
ents
40
220
1093
3288
332
411
3119890minus08
2119890minus08
6119890minus09
308495
2714
Delq
uad
998779Elem
ents
30
140
1104
6148
332
768
4119890minus08
2119890minus08
8119890minus09
39110
0028
14Delq
uad
998779Elem
ents
20
82
1110
12392
332
1549
2119890minus08
9119890minus09
6119890minus09
6117067
2914
Delqu
ad998779
Elem
ents
1063
1115
48882
332
6110
2119890minus09
1119890minus09
1119890minus09
197
5804
630
14Delq
uad
998779Elem
ents
05
58
1116
194162
332
24270
2119890minus09
9119890minus10
2119890minus09
790
238769
3114
Delq
uad
◻Vo
lumes
40
minus416
1174
3132
332
391
5119890minus08
3119890minus08
8119890minus09
267322
3214
Delq
uad
◻Vo
lumes
30
minus248
1151
5922
332
740
9119890minus09
4119890minus09
2119890minus09
318779
3314
Delq
uad
◻Vo
lumes
20
minus132
1139
12050
332
1506
1119890minus08
7119890minus09
4119890minus09
4412239
3414
Delq
uad
◻Vo
lumes
10minus49
1126
48200
332
6025
6119890minus09
3119890minus09
2119890minus09
122
36094
3514
Delq
uad
◻Vo
lumes
05
minus23
1123
192800
332
24100
5119890minus09
2119890minus09
3119890minus09
464
140228
3614
Delq
uad
◻Elem
ents
40
224
1093
3294
332
411
3119890minus08
1119890minus08
7119890minus09
359852
3714
Delq
uad
◻Elem
ents
30
141
1104
6144
332
768
3119890minus08
2119890minus08
9119890minus09
5114018
3814
Delq
uad
◻Elem
ents
20
921111
12392
332
1549
1119890minus08
6119890minus09
5119890minus09
8122526
3914
Delq
uad
◻Elem
ents
1065
1115
48882
332
6110
5119890minus09
3119890minus09
4119890minus09
276
78431
22 Science and Technology of Nuclear Installations
Table2Con
tinued
Case
no
Symm-
etry
Mesh
algorithm
Basic
shape
Solutio
nmetho
dℓ119888
(cm)
Δ120588
(pcm
)max1206012
(mdash)
Total
unkn
owns
Outer
iter
Linear
iter
Inner
iter
Resid
ual
norm
Relativ
eerror
Error
estim
ate
Mem
ory
(Mb)
Page
faults
4014
Delq
uad
◻Elem
ents
05
59
1117
194162
332
24270
1119890minus09
5119890minus10
1119890minus09
1104
318967
4118
Delaunay
998779Vo
lumes
40
minus99
1139
4228
332
528
4119890minus12
2119890minus12
1119890minus12
359630
4218
Delaunay
998779Vo
lumes
30
50
1132
7712
332
964
1119890minus11
6119890minus12
2119890minus12
4211578
4318
Delaunay
998779Vo
lumes
20
177
1120
15776
332
1972
2119890minus12
1119890minus12
7119890minus13
5515436
4418
Delaunay
998779Vo
lumes
10142
1118
63902
332
7987
3119890minus12
1119890minus12
1119890minus12
160
46691
4518
Delaunay
998779Vo
lumes
05
176
1115
254612
332
31826
2119890minus12
8119890minus13
1119890minus12
555
168526
4618
Delaunay
998779Elem
ents
40
183
1096
2252
332
281
6119890minus12
3119890minus12
2119890minus12
359589
4718
Delaunay
998779Elem
ents
30
119
1104
4040
224
505
2119890minus08
1119890minus08
7119890minus09
41114
3748
18Delaunay
998779Elem
ents
20
85
1108
8156
332
1019
2119890minus12
8119890minus13
6119890minus13
5515053
4918
Delaunay
998779Elem
ents
1063
1112
32480
332
4060
1119890minus12
6119890minus13
8119890minus13
151
43671
5018
Delaunay
998779Elem
ents
05
58
1113
128358
332
1604
47119890minus13
3119890minus13
7119890minus13
527
157581
5118
Delaunay
◻Vo
lumes
40
37
1144
2380
224
297
5119890minus08
2119890minus08
1119890minus08
5314173
5218
Delaunay
◻Vo
lumes
30
218
1120
4194
332
524
7119890minus12
4119890minus12
2119890minus12
3610034
5318
Delaunay
◻Vo
lumes
20
345
1110
7960
332
995
5119890minus12
3119890minus12
1119890minus12
4712879
5418
Delaunay
◻Vo
lumes
10165
1117
30652
332
3831
2119890minus12
1119890minus12
7119890minus13
100
28851
5518
Delaunay
◻Vo
lumes
05
191
1117
122066
224
15258
2119890minus08
8119890minus09
9119890minus09
343
103825
5618
Delaunay
◻Elem
ents
40
175
1097
2526
332
315
4119890minus12
2119890minus12
1119890minus12
6016159
5718
Delaunay
◻Elem
ents
30
114
1104
4386
332
548
4119890minus12
2119890minus12
2119890minus12
5013768
5818
Delaunay
◻Elem
ents
20
901108
8234
332
1029
2119890minus12
1119890minus12
1119890minus12
7320094
5918
Delaunay
◻Elem
ents
1062
1112
31186
224
3898
2119890minus08
7119890minus09
9119890minus09
208
59589
6018
Delaunay
◻Elem
ents
05
58
1113
123122
224
15390
9119890minus09
5119890minus09
1119890minus08
807
2366
7061
18Delq
uad
998779Vo
lumes
40
minus210
1168
3224
332
403
1119890minus11
5119890minus12
2119890minus12
43117
8462
18Delq
uad
998779Vo
lumes
30
369
1150
6000
332
750
1119890minus11
6119890minus12
3119890minus12
5013502
6318
Delq
uad
998779Vo
lumes
20
312
1104
12316
332
1539
2119890minus11
8119890minus12
5119890minus12
6016395
6418
Delq
uad
998779Vo
lumes
1044
01094
48714
332
6089
9119890minus12
4119890minus12
4119890minus12
1183444
165
18Delq
uad
998779Vo
lumes
05
534
1081
193980
332
24247
5119890minus12
2119890minus12
4119890minus12
423
126768
6618
Delq
uad
998779Elem
ents
40
240
1090
1750
332
218
9119890minus12
4119890minus12
2119890minus12
43117
8667
18Delqu
ad998779
Elem
ents
30
150
1100
3184
332
398
1119890minus11
5119890minus12
2119890minus12
5013509
6818
Delq
uad
998779Elem
ents
20
89
1106
6426
332
803
5119890minus12
3119890minus12
2119890minus12
5916047
6918
Delq
uad
998779Elem
ents
1064
1111
24886
332
3110
2119890minus12
1119890minus12
1119890minus12
113
32788
7018
Delq
uad
998779Elem
ents
05
59
1113
98052
224
12256
1119890minus08
5119890minus09
8119890minus09
398
118153
7118
Delq
uad
◻Vo
lumes
40
minus398
1172
1650
332
206
1119890minus11
5119890minus12
2119890minus12
369907
7218
Delq
uad
◻Vo
lumes
30
minus249
1146
3058
332
382
4119890minus12
2119890minus12
1119890minus12
349349
7318
Delq
uad
◻Vo
lumes
20
minus131
1136
6228
224
778
3119890minus08
1119890minus08
5119890minus09
4010930
7418
Delq
uad
◻Vo
lumes
10minus51
1123
24536
224
3067
3119890minus08
2119890minus08
1119890minus08
7822716
7618
Delq
uad
◻Elem
ents
40
237
1090
1752
332
219
6119890minus12
3119890minus12
2119890minus12
4111213
7718
Delq
uad
◻Elem
ents
30
145
1101
3184
332
398
6119890minus12
3119890minus12
2119890minus12
43118
1178
18Delq
uad
◻Elem
ents
20
941107
6426
332
803
3119890minus12
1119890minus12
1119890minus12
6016220
7918
Delqu
ad◻
Elem
ents
1065
1111
24874
332
3109
1119890minus12
6119890minus13
8119890minus13
156
4400
080
18Delq
uad
◻Elem
ents
05
59
1113
9804
22
2412255
8119890minus09
4119890minus09
8119890minus09
567
162168
Science and Technology of Nuclear Installations 23
Tim
e con
sum
ed to
bui
ld th
e mat
rices
(sec
onds
)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay trianglesvolumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
10
30
100
300
1
3
Figure 23 Time needed to build the matrices versus number of unknowns
was considerably larger than the solution time itself Theeigenvalue problem was solved using a multilayer iterativeKrylov-Schur method [26] with a tolerance relative to thematrices norm
1003817100381710038171003817119877 sdot 120601 minus (1119896eff) sdot 119865 sdot 1206011003817100381710038171003817
119877 + (1119896eff) sdot 119865lt 10minus8 (11)
The reported residual norm and relative error are10038171003817100381710038171003817100381710038171003817119877 sdot 120601 minus
1
119896effsdot 119865 sdot 120601
10038171003817100381710038171003817100381710038171003817
1003817100381710038171003817119877 sdot 120601 minus (1119896eff) sdot 119865 sdot 1206011003817100381710038171003817
1003817100381710038171003817(1119896eff) sdot 1206011003817100381710038171003817
(12)
respectively The fields marked as outer linear and inneriterations refer to the number of steps needed to attainthe requested tolerance in each layer of the Krylov-Schuralgorithm The computer used to solve the problem has anIntel i7 920 267GHz processor with 4Gb of RAM runningDebian GNULinux Wheezy
When using a finite volumes-based scheme over anunstructured mesh the solver has to gather information
about which cells are neighbors and which are not Currentlygmsh does not write this kind of lists in its output filesso milonga has to explicitly solve the neighbors problemPerforming a linear search is an 119874(119873
2) task which is
unacceptable for problem sizes 119873 of interest The code usesa search based on a 119896-dimensional tree [27] which can inprinciple be performed in119874(119873) steps Still for large values of119873 the time needed to read and parse the mesh (number twopreviously mentioned) is the bottleneck of the solution Thisstep is not needed in finite elements although the construc-tion of the elementary matrices involves the computationof the multidimensional Jacobians and integrals which thenhave to be assembled Again for large 119873 this step (numberthree) takes up most of the time needed to solve the problem
The Delaunay algorithm is a standard method for gen-erating two-dimensional grids [16] by tessellating a planewith triangles If the mesh needs to be based on quadranglesinstead a recombination algorithm can be used to transformtwo adjacent triangles in one quadrangle whenever is possi-ble However for geometries which are based on rectangularshapes there exist other algorithms [25] both for meshingand for recombining the triangles that give rise to elementaryentities with right angles almost everywhere which may be a
24 Science and Technology of Nuclear Installations
Tim
e con
sum
ed to
solv
e the
eige
nval
ue p
robl
em (s
econ
ds)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
10
30
100
300
1
3
Figure 24 Time needed to solve the eigenvalue problem versus number of unknowns
desired property of the resulting grid For finite elements thesteps needed to build the matrices depend on the selectionof triangles or quadrangles as the basic elementary geometrybecause the shape functions change However the number ofunknowns is the same as the number of nodes that does notchange after a recombination procedure On the other handthe number of unknowns in the finite volumes schemes withtriangles is roughly twice as the number of unknowns withquadrangles for the same grid
Figure 19 shows how the computed static reactivity (ie1 minus 1119896eff) depends on the number of unknowns for eachof the sixteen combinations of geometry algorithm shapescheme The four accepted results published in the originalreference [10] are included for reference although it shouldbe taken into account that said reactivities were computedalmost forty years ago Figure 20 shows the total wall timeneeded to solve the problem as a function of 119873119866 whileFigures 21 22 23 and 24 show the times needed for eachof the first four steps involved in the solution maintainingthe same logarithmic scale for both the abscissas and theordinates Green data represent finite volumes whilst bluebullets correspond to finite elements Solid lines indicatequarter-core and dashed lines eighth-symmetry Fillet bullets
are results obtained by theDelaunay triangulation and emptybullets were computed with the delquad algorithm Finallytriangle-shaped data correspond to triangles and squares anddiamonds denote quadrangles as the basic geometry of thegrid
It can be seen that finite elements produce amuch smallerdispersion of eigenvalues 119896eff than finite volumes with therefinement of the mesh This can be explained because finitevolumes methods rely on a geometric condition of the meshwhich may change abruptly if the meshing algorithm decidesto allocate the cells in a rather different form for small changesin ℓ119888 Finite elements methods are less influenced by thesediscontinuous lay-out changes of the elements As expectedeighth-core symmetries give almost the same results as thequarter-core geometries with roughly half the unknownsFor small problems finite volumes run faster that finiteelements because the time needed to solve the neighborproblem is negligible When the problem size grows thistime increases and exceeds the overhead implied in the con-struction and assembly of the finite elements matrices Alsoat least for this configuration it is seen that the eigenvalueproblem is solved faster for finite volumes than for finiteelements
Science and Technology of Nuclear Installations 25
4 Conclusions
Unstructured grids provide the cognizant engineer witha wide variety of possibilities to deal with the design oranalysis of nuclear reactor cores These kinds of grids cansuccessfully reproduce continuous geometries commonlyfound in reactor cores such as cylinders and thereforenot only can the diffusion equation be better approximatedinside the domain of definition but also the fulfillment ofboundary conditions is improved A free computer codewas written from scratch that is able to completely solvethe 2D IAEA PWR Benchmark using unstructured gridsfor sixteen combinations of geometry meshing algorithmbasic shape and discretization scheme plus any value ofthe gridrsquos characteristic length by using a single input fileThe complete set of input files and codemdashexecutable andsourcemdashis available either online or upon request withcomments experiences suggestions and corrections beingmore than welcome Further development should includetackling full three-dimensional geometries with completethermal hydraulic feedback in order to analyze how thesolutions of the coupled neutronic-thermal problem dependon the spatial discretization scheme of the neutron leakageterm Parallelization of the computation and assembly of thematrices and of the solution of the eigenvalue problem andits implementation using GPUs are also desired features toimplement A problemwith direct applications that the futureversions of milonga ought to solve is the analysis of how thegeometry of the absorbing materials should be taken intoaccount in structured coarse grids in order to mitigate effectssuch as the rod-cusp problem
Suitable schemes for approximating the continuous dif-ferential operators by discrete matrix expressions includefinite volumes and finite elements families Finite volumesmethods compute cell mean values whilst finite elementsgive functional values at the gridrsquos nodes In general finiteelements are less sensitive to changes in the mesh so theresults they provide do not change significantly for differentmeshing algorithms or elementary shapes Small problemsare best solved by finite volumes as the neighbor-findingproblem is faster than the process of building and assemblingthe eigenvalue-problem matrices For a large number ofunknowns the process of finding which cell is neighbor ofwhichmdasheven based on a 119896-dimensional treemdashoverwhelmsthe computation and assembly of elementary matrices andfinite-element methods perform better
References
[1] G Glasstone and S Bell Nuclear Reactor Theory KriegerPublishing Company 1970
[2] J J Duderstadt and L J Hamilton Nuclear Reactor AnalysisJohn Wiley amp Sons New York NY USA 1976
[3] C Gho Reactor Physics Undergraduate Course Lecture NotesInstituto Balseiro 2006
[4] G G Theler and F J Bonetto ldquoOn the stability of the pointreactor kinetics equationsrdquoNuclear Engineering andDesign vol240 no 6 pp 1443ndash1449 2010
[5] E E Lewis andW F Miller Computational Methods of NeutronTransport John Wiley amp Sons 1984
[6] E Masiello ldquoAnalytical stability analysis of Coarse-Mesh finitedifference methodrdquo in Proceedings of the International Confer-ence on the Physics of Reactors (PHYSOR rsquo08) Nuclear Power ASustainable Resource pp 456ndash464 September 2008
[7] M L Zerkle Development of a polynomial nodal methodwith flux and current discontinuity factors [PhD thesis] Mas-sachusets Institute of Technology 1992
[8] J I Yoon andHG Joo ldquoTwo-level coarsemesh finite differenceformulation with multigroup source expansion nodal kernelsrdquoJournal of Nuclear Science andTechnology vol 45 no 7 pp 668ndash682 2008
[9] O Mazzantini M Schivo J Di Cesare R Garbero M Riveroand G Theler ldquoA coupled calculation suite for Atucha II oper-ational transients analysisrdquo Science and Technology of NuclearInstallations vol 2011 Article ID 785304 12 pages 2011
[10] Computational Benchmark Problem Comitee for the Mathe-matics and Computation Division of the American NuclearSociety ldquoArgonne Code Center Benchmark problem bookrdquoTech Rep ANL-7416 Supplement 2 Argonne National Labo-ratory 1977
[11] A F Henry Nuclear Reactor Analysis MIT Cambridge UK1975
[12] O C Zienkiewicz R L Taylor and J Z ZhuTheFinite ElementMethod Its Basis and Fundamentals vol 1 Elsevier 6th edition2005
[13] G Theler Difusion de neutrones en mallas no estructuradascomparacion entre volumenes y elementos finitos AcademicMonograph Universidad de Buenos Aires 2013
[14] G Theler ldquoMilonga a free nuclear reactor core analysis coderdquoAvailable online 2011
[15] G Theler F J Bonetto and A Clausse ldquoOptimizacion deparametros en reactores de potencia base de diseno delcodigo neutronico milongardquo in Reunion Anual de la AsociacionArgentina de Tecnologia Nuclear XXXVII 2010
[16] C Geuzaine and J-F Remacle ldquoGmsh a 3-D finite elementmesh generator with built-in pre- and post-processing facili-tiesrdquo International Journal for Numerical Methods in Engineer-ing vol 79 no 11 pp 1309ndash1331 2009
[17] S Balay J Brown K Buschelman et al ldquoPETSc users manualrdquoTech Rep ANL-9511-Revision 34 Argonne National Labora-tory 2013
[18] S Balay W D Gropp L Curfman McInnes and B F SmithldquoEfficient management of parallelism in object oriented numer-ical software librariesrdquo in Modern Software Tools in ScientificComputing E Arge A M Bruaset and H P Langtangen Edspp 163ndash202 Birkhauser 1997
[19] V Hernandez J E Roman and V Vidal ldquoSLEPC a scalable andflexible toolkit for the solution of eigenvalue problemsrdquo ACMTransactions on Mathematical Software vol 31 no 3 pp 351ndash362 2005
[20] M Galassi J Davies J Theiler et al GNU Scientific LibraryReference Manual 3rd edition 2009
[21] O C Zienkiewicz and R L Taylor The Finite Element MethodFor Solid and Structural Mechanics vol 3 Elsevier 6th edition2005
[22] A Imelda and K Doddy Evaluation of the IAEA 3-D PWRbenchmark problem using NESTLE code 2004
[23] R Mosteller ldquoStatic benchmarking of the NESTLE advancednodal coderdquo in Proceedings of the Joint International Conference
26 Science and Technology of Nuclear Installations
on Mathematical Methods and Supercomputing for NuclearApplications vol 2 pp 1596ndash1605 1997
[24] GTheler F J Bonetto andAClausse ldquoSolution of the 2D IAEAPWR Benchmark with the neutronic code milongardquo in Actasde la Reunion Anual de la Asociacion Argentina de TecnologıaNuclear XXXVIII 2011
[25] J F Remacle F Henrotte T Carrier-Baudouin et al ldquoAfrontal delaunay quad mesh generator using the 119871
infin normrdquoInternational Journal For Numerical Methods in Engineeringvol 94 no 5 pp 494ndash512 2013
[26] G W Stewart ldquoA Krylov-Schur algorithm for large eigenprob-lemsrdquo SIAM Journal on Matrix Analysis and Applications vol23 no 3 pp 601ndash614 2002
[27] J L Bentley ldquoMultidimensional binary search trees used forassociative searchingrdquo Communications of the ACM vol 18 no9 pp 509ndash517 1975
TribologyAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FuelsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofPetroleum Engineering
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Industrial EngineeringJournal of
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18 Science and Technology of Nuclear Installations
2000 3000 6000 10000 20000 30000 60000
01
003
001
03
1
3
10
30
100
300To
tal w
all t
ime (
seco
nds)
Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
Figure 20 Total wall time versus number of unknowns
for problems with nonuniform cross sections even in simpleone-dimensional reflected reactors
We may draw two other conclusions from Figure 7 Firstthat the finite element method seems to provide a bettersolution than the volumes-based scheme and second thateven though the error committed tends to decrease with finergrids its behavior is not monotonic for finite volumes Infact Figure 8 shows the difference between the numericaland analytical solutions using a logarithmic vertical scalewhere both conclusions are even more evident We deferthe discussion of such differences between the discretizationscheme until the next section It is worth to note howeverthat the fact that a finite-element-based scheme throws betterresults for the particular bare homogeneous circular reactorunder study than the finite volumes does not imply thatfinite volumes ought to be discarded as a valid tool forsolving the neutron diffusion equation in general cases Thecombination of lattice and core-level computations is usuallyperformedusing cell-based resultswhichwhen fed into node-based methods to solve the few-group neutron diffusionequation may introduce errors which potentially can leadto unacceptable solutions Nevertheless this analysis is farbeyond the scope of this paper which focuses on solving amathematical equation over unstructured grids
32 The 2D IAEA PWR Benchmark Problem This is aclassical two-group neutron diffusion problem first designedin the early 1970s and taken as a reference benchmark forcomputational codes A number of codes were used to solveeither this problem or its three-dimensional formulation[22 23] including milonga using a structured grid [24] Theoriginal formulation can be found in the reference [10] andthere is a reproduction that may be easily found online inreference [24] The geometry consists of a one-quarter ofa PWR core depicted in Figure 9 and the homogeneousmacroscopic cross sections are listed in Table 1 An axialbuckling term 119861
2
119911119892= 08times10
minus4 should be taken into accountThe external surface should be subject to a zero incomingcurrent condition which may be written as
120597120601119892
120597119899= minus
04692
119863119892
sdot 120601119892 (8)
The expected results are as follows
(1) Maximum eigenvalue(2) Fundamental flux distributions
(a) Radial flux traverses 120601119892(119909 0) and 120601119892(119909 119909)
Science and Technology of Nuclear Installations 19
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
1
3
10
30
100
300Ti
me c
onsu
med
to m
esh
the g
eom
etry
(sec
onds
)
Figure 21 Time needed to mesh the geometry versus number of unknowns
Table 1 Macroscopic cross-sections (units are not stated in the original reference but they are assumed to be in cmminus1 or cm as appropriate)
1198631
1198632
Σ1rarr2
Σ1198861
Σ1198862
]Σ1198912
Material1 15 04 002 001 0080 0135 Fuel 12 15 04 002 001 0085 0135 Fuel 23 15 04 002 001 0130 0135 Fuel 2 + rod4 20 03 004 0 0010 0 Reflector
Note the fluxes shall be normalized such that
1
119881coreint119881core
sum
119892
]Σ119891119892 sdot 120601119892119889119881 = 1 (9)
(b) Value and location of maximum power densityThis corresponds to maximum of 1206012 in thecore It is recommended that the maximumvalues of 1206012 both in the inner core and at thecorereflector interface be given
(3) Average subassembly powers 119875119896
119875119896 =1
119881119896
int119881119896
sum
119892
]Σ119891119892 sdot 120601119892119889119881 (10)
where 119881119896 is the volume of the 119896th subassembly and119896 designates the fuel subassemblies as shown inFigure 9
(4) Number of unknowns in the problem number ofiterations and total and outer
(5) Total computing time iteration time IO-time andcomputer used
(6) Type and numerical values of convergence criteria(7) Table of average group fluxes for a square mesh grid
of 20 times 20 cm(8) Dependence of results on mesh spacing
Even though the original problem is based on a quarter-core situation the problem has an eighth-core symmetry
20 Science and Technology of Nuclear Installations
Quarter Delaunay triangles elements
Tim
e con
sum
ed to
read
the m
esh
(sec
onds
)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumes
Quarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
1
3
10
30
100
300
Figure 22 Time needed to read the mesh versus number of unknowns
which cannot be taken into account by structured gridswhich are the main target of the benchmark Howevernonstructured grids can take into consideration any kind ofsymmetry almost without loss of accuracy and at the sametime reducing roughly the number of unknowns by a halfand the associated computational effort needed to solve theproblem by a factor of four Answers to items (1)ndash(7) canbe given completely by milonga using a single input file(see Supplementary data in SupplementaryMaterial availableonline at httpdxdoiorg1011552013641863) As asked initem (8) how results depend not only on the mesh spacingbut also on the meshing algorithm on the gridrsquos elementarygeometric shape and on the discretization scheme may shedlights on the subject whichmay be evenmore interesting thanthe numerical results themselves
Taking advantage of milongarsquos capability of reading andparsing command-line arguments the selection of the coregeometry (quarter or eighth) the meshing algorithm (delau-nay [16] or delquad [25]) the shape of the elementaryentities (triangles or quadrangles) the discretization scheme(volumes or elements) and the characteristic length ℓ119888 ofthe mesh can be provided at run time Fixing five values forℓ119888 = 4 3 2 1 05 gives 2 times 2 times 2 times 2 times 5 = 80 possiblecombinations which we solve with successive invocations to
milonga with the same input file but with different argumentsfrom a simple script Figures 10 11 12 13 14 15 16 17 and18 show the results corresponding to items (1)ndash(7) for someillustrative cases The complete set of figures and the codeused to generate themmay be provided upon request Table 2compiles the answers to the problem for every case studied
As the milonga code is still under development itsnumerical routines are not yet fully optimized nor designedfor parallel computation Therefore the reported times areonly rough estimates and should be taken with care Thesolution comprises five steps
(1) generate the grid with the requested geometry mesh-ing algorithm basic shape and characteristic lengthby calling to gmsh
(2) read the generated mesh(3) build the matrices(4) solve the eigenvalue problem(5) compute the requested results
The CPU time reported in Figures 10ndash18 is thus thesum of all of these five steps but not the time needed togenerate the figuresmdashwhich in some caseswith coarsemeshes
Science and Technology of Nuclear Installations 21
Table2
Resultsofthe2
DIAEA
PWRBe
nchm
arkp
roblem
obtained
with
them
ilong
acod
efor
thee
ightyp
ropo
sedcombinatio
nsofsymmetrym
eshing
algorithm
basicshapediscretization
schemeandcharacteris
ticelem
entc
elllengthTh
ereference
solutio
nis120588=28749times10minus5w
hich
correspo
ndsto119896eff=10296
Case
no
Symm-
etry
Mesh
algorithm
Basic
shape
Solutio
nmetho
dℓ119888
(cm)
Δ120588
(pcm
)max1206012
(mdash)
Total
unkn
owns
Outer
iter
Linear
iter
Inner
iter
Resid
ual
norm
Relativ
eerror
Error
estim
ate
Mem
ory
(Mb)
Page
faults
114
Delaunay
998779Vo
lumes
40
minus120
1145
8264
332
1033
6119890minus09
3119890minus09
1119890minus09
349470
214
Delaunay
998779Vo
lumes
30
85
1135
15740
332
1967
2119890minus08
1119890minus08
4119890minus09
4813401
314
Delaunay
998779Vo
lumes
20
197
1124
31188
332
3898
1119890minus08
6119890minus09
3119890minus09
7621932
414
Delaunay
998779Vo
lumes
10153
1123
127476
332
15934
7119890minus09
3119890minus09
3119890minus09
273
81886
514
Delaunay
998779Vo
lumes
05
187
1119
510676
332
63834
2119890minus09
9119890minus10
1119890minus09
1148
352261
614
Delaunay
998779Elem
ents
40
173
1100
4308
332
538
2119890minus08
8119890minus09
4119890minus09
318794
714
Delaunay
998779Elem
ents
30
117
1107
8108
332
1013
5119890minus09
2119890minus09
2119890minus09
4712946
814
Delaunay
998779Elem
ents
20
84
1112
15936
332
1992
6119890minus09
3119890minus09
2119890minus09
7321347
914
Delaunay
998779Elem
ents
1062
1116
64478
332
8059
6119890minus09
3119890minus09
4119890minus09
262
77105
1014
Delaunay
998779Elem
ents
05
58
1117
256700
332
32087
1119890minus09
5119890minus10
1119890minus09
1091
331969
1114
Delaunay
◻Vo
lumes
40
53
1149
5042
332
630
2119890minus08
1119890minus08
5119890minus09
288007
1214
Delaunay
◻Vo
lumes
30
346
1133
8560
332
1070
7119890minus09
3119890minus09
2119890minus09
3910895
1314
Delaunay
◻Vo
lumes
20
308
1131
15576
332
1947
1119890minus08
7119890minus09
3119890minus09
541564
514
14Delaunay
◻Vo
lumes
10177
1122
60774
332
7596
8119890minus09
4119890minus09
3119890minus09
167
50115
1514
Delaunay
◻Vo
lumes
05
199
1120
244936
332
30617
4119890minus09
2119890minus09
2119890minus09
698
215678
1614
Delaunay
◻Elem
ents
40
196
1100
5222
332
652
3119890minus08
1119890minus08
8119890minus09
4713098
1714
Delaunay
◻Elem
ents
30
123
1107
8810
332
1101
2119890minus08
9119890minus09
7119890minus09
6918598
1814
Delaunay
◻Elem
ents
20
901112
15922
332
1990
1119890minus08
5119890minus09
5119890minus09
107
30591
1914
Delaunay
◻Elem
ents
1063
1116
61456
332
7682
5119890minus09
2119890minus09
4119890minus09
389
113237
2014
Delaunay
◻Elem
ents
05
58
1117
246332
332
30791
8119890minus10
4119890minus10
1119890minus09
1676
498192
2114
Delq
uad
998779Vo
lumes
40
minus45
1144
6228
332
778
4119890minus08
2119890minus08
7119890minus09
308495
2214
Delq
uad
998779Vo
lumes
30
570
1153
11820
332
1477
3119890minus08
1119890minus08
4119890minus09
4010879
2314
Delq
uad
998779Vo
lumes
20
459
1103
24100
332
3012
2119890minus08
1119890minus08
5119890minus09
6217715
2414
Delq
uad
998779Vo
lumes
10512
1104
9640
03
3212050
2119890minus08
1119890minus08
9119890minus09
207
61714
2514
Delq
uad
998779Vo
lumes
05
528
1106
385600
332
48200
2119890minus09
1119890minus09
2119890minus09
838
255735
2614
Delqu
ad998779
Elem
ents
40
220
1093
3288
332
411
3119890minus08
2119890minus08
6119890minus09
308495
2714
Delq
uad
998779Elem
ents
30
140
1104
6148
332
768
4119890minus08
2119890minus08
8119890minus09
39110
0028
14Delq
uad
998779Elem
ents
20
82
1110
12392
332
1549
2119890minus08
9119890minus09
6119890minus09
6117067
2914
Delqu
ad998779
Elem
ents
1063
1115
48882
332
6110
2119890minus09
1119890minus09
1119890minus09
197
5804
630
14Delq
uad
998779Elem
ents
05
58
1116
194162
332
24270
2119890minus09
9119890minus10
2119890minus09
790
238769
3114
Delq
uad
◻Vo
lumes
40
minus416
1174
3132
332
391
5119890minus08
3119890minus08
8119890minus09
267322
3214
Delq
uad
◻Vo
lumes
30
minus248
1151
5922
332
740
9119890minus09
4119890minus09
2119890minus09
318779
3314
Delq
uad
◻Vo
lumes
20
minus132
1139
12050
332
1506
1119890minus08
7119890minus09
4119890minus09
4412239
3414
Delq
uad
◻Vo
lumes
10minus49
1126
48200
332
6025
6119890minus09
3119890minus09
2119890minus09
122
36094
3514
Delq
uad
◻Vo
lumes
05
minus23
1123
192800
332
24100
5119890minus09
2119890minus09
3119890minus09
464
140228
3614
Delq
uad
◻Elem
ents
40
224
1093
3294
332
411
3119890minus08
1119890minus08
7119890minus09
359852
3714
Delq
uad
◻Elem
ents
30
141
1104
6144
332
768
3119890minus08
2119890minus08
9119890minus09
5114018
3814
Delq
uad
◻Elem
ents
20
921111
12392
332
1549
1119890minus08
6119890minus09
5119890minus09
8122526
3914
Delq
uad
◻Elem
ents
1065
1115
48882
332
6110
5119890minus09
3119890minus09
4119890minus09
276
78431
22 Science and Technology of Nuclear Installations
Table2Con
tinued
Case
no
Symm-
etry
Mesh
algorithm
Basic
shape
Solutio
nmetho
dℓ119888
(cm)
Δ120588
(pcm
)max1206012
(mdash)
Total
unkn
owns
Outer
iter
Linear
iter
Inner
iter
Resid
ual
norm
Relativ
eerror
Error
estim
ate
Mem
ory
(Mb)
Page
faults
4014
Delq
uad
◻Elem
ents
05
59
1117
194162
332
24270
1119890minus09
5119890minus10
1119890minus09
1104
318967
4118
Delaunay
998779Vo
lumes
40
minus99
1139
4228
332
528
4119890minus12
2119890minus12
1119890minus12
359630
4218
Delaunay
998779Vo
lumes
30
50
1132
7712
332
964
1119890minus11
6119890minus12
2119890minus12
4211578
4318
Delaunay
998779Vo
lumes
20
177
1120
15776
332
1972
2119890minus12
1119890minus12
7119890minus13
5515436
4418
Delaunay
998779Vo
lumes
10142
1118
63902
332
7987
3119890minus12
1119890minus12
1119890minus12
160
46691
4518
Delaunay
998779Vo
lumes
05
176
1115
254612
332
31826
2119890minus12
8119890minus13
1119890minus12
555
168526
4618
Delaunay
998779Elem
ents
40
183
1096
2252
332
281
6119890minus12
3119890minus12
2119890minus12
359589
4718
Delaunay
998779Elem
ents
30
119
1104
4040
224
505
2119890minus08
1119890minus08
7119890minus09
41114
3748
18Delaunay
998779Elem
ents
20
85
1108
8156
332
1019
2119890minus12
8119890minus13
6119890minus13
5515053
4918
Delaunay
998779Elem
ents
1063
1112
32480
332
4060
1119890minus12
6119890minus13
8119890minus13
151
43671
5018
Delaunay
998779Elem
ents
05
58
1113
128358
332
1604
47119890minus13
3119890minus13
7119890minus13
527
157581
5118
Delaunay
◻Vo
lumes
40
37
1144
2380
224
297
5119890minus08
2119890minus08
1119890minus08
5314173
5218
Delaunay
◻Vo
lumes
30
218
1120
4194
332
524
7119890minus12
4119890minus12
2119890minus12
3610034
5318
Delaunay
◻Vo
lumes
20
345
1110
7960
332
995
5119890minus12
3119890minus12
1119890minus12
4712879
5418
Delaunay
◻Vo
lumes
10165
1117
30652
332
3831
2119890minus12
1119890minus12
7119890minus13
100
28851
5518
Delaunay
◻Vo
lumes
05
191
1117
122066
224
15258
2119890minus08
8119890minus09
9119890minus09
343
103825
5618
Delaunay
◻Elem
ents
40
175
1097
2526
332
315
4119890minus12
2119890minus12
1119890minus12
6016159
5718
Delaunay
◻Elem
ents
30
114
1104
4386
332
548
4119890minus12
2119890minus12
2119890minus12
5013768
5818
Delaunay
◻Elem
ents
20
901108
8234
332
1029
2119890minus12
1119890minus12
1119890minus12
7320094
5918
Delaunay
◻Elem
ents
1062
1112
31186
224
3898
2119890minus08
7119890minus09
9119890minus09
208
59589
6018
Delaunay
◻Elem
ents
05
58
1113
123122
224
15390
9119890minus09
5119890minus09
1119890minus08
807
2366
7061
18Delq
uad
998779Vo
lumes
40
minus210
1168
3224
332
403
1119890minus11
5119890minus12
2119890minus12
43117
8462
18Delq
uad
998779Vo
lumes
30
369
1150
6000
332
750
1119890minus11
6119890minus12
3119890minus12
5013502
6318
Delq
uad
998779Vo
lumes
20
312
1104
12316
332
1539
2119890minus11
8119890minus12
5119890minus12
6016395
6418
Delq
uad
998779Vo
lumes
1044
01094
48714
332
6089
9119890minus12
4119890minus12
4119890minus12
1183444
165
18Delq
uad
998779Vo
lumes
05
534
1081
193980
332
24247
5119890minus12
2119890minus12
4119890minus12
423
126768
6618
Delq
uad
998779Elem
ents
40
240
1090
1750
332
218
9119890minus12
4119890minus12
2119890minus12
43117
8667
18Delqu
ad998779
Elem
ents
30
150
1100
3184
332
398
1119890minus11
5119890minus12
2119890minus12
5013509
6818
Delq
uad
998779Elem
ents
20
89
1106
6426
332
803
5119890minus12
3119890minus12
2119890minus12
5916047
6918
Delq
uad
998779Elem
ents
1064
1111
24886
332
3110
2119890minus12
1119890minus12
1119890minus12
113
32788
7018
Delq
uad
998779Elem
ents
05
59
1113
98052
224
12256
1119890minus08
5119890minus09
8119890minus09
398
118153
7118
Delq
uad
◻Vo
lumes
40
minus398
1172
1650
332
206
1119890minus11
5119890minus12
2119890minus12
369907
7218
Delq
uad
◻Vo
lumes
30
minus249
1146
3058
332
382
4119890minus12
2119890minus12
1119890minus12
349349
7318
Delq
uad
◻Vo
lumes
20
minus131
1136
6228
224
778
3119890minus08
1119890minus08
5119890minus09
4010930
7418
Delq
uad
◻Vo
lumes
10minus51
1123
24536
224
3067
3119890minus08
2119890minus08
1119890minus08
7822716
7618
Delq
uad
◻Elem
ents
40
237
1090
1752
332
219
6119890minus12
3119890minus12
2119890minus12
4111213
7718
Delq
uad
◻Elem
ents
30
145
1101
3184
332
398
6119890minus12
3119890minus12
2119890minus12
43118
1178
18Delq
uad
◻Elem
ents
20
941107
6426
332
803
3119890minus12
1119890minus12
1119890minus12
6016220
7918
Delqu
ad◻
Elem
ents
1065
1111
24874
332
3109
1119890minus12
6119890minus13
8119890minus13
156
4400
080
18Delq
uad
◻Elem
ents
05
59
1113
9804
22
2412255
8119890minus09
4119890minus09
8119890minus09
567
162168
Science and Technology of Nuclear Installations 23
Tim
e con
sum
ed to
bui
ld th
e mat
rices
(sec
onds
)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay trianglesvolumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
10
30
100
300
1
3
Figure 23 Time needed to build the matrices versus number of unknowns
was considerably larger than the solution time itself Theeigenvalue problem was solved using a multilayer iterativeKrylov-Schur method [26] with a tolerance relative to thematrices norm
1003817100381710038171003817119877 sdot 120601 minus (1119896eff) sdot 119865 sdot 1206011003817100381710038171003817
119877 + (1119896eff) sdot 119865lt 10minus8 (11)
The reported residual norm and relative error are10038171003817100381710038171003817100381710038171003817119877 sdot 120601 minus
1
119896effsdot 119865 sdot 120601
10038171003817100381710038171003817100381710038171003817
1003817100381710038171003817119877 sdot 120601 minus (1119896eff) sdot 119865 sdot 1206011003817100381710038171003817
1003817100381710038171003817(1119896eff) sdot 1206011003817100381710038171003817
(12)
respectively The fields marked as outer linear and inneriterations refer to the number of steps needed to attainthe requested tolerance in each layer of the Krylov-Schuralgorithm The computer used to solve the problem has anIntel i7 920 267GHz processor with 4Gb of RAM runningDebian GNULinux Wheezy
When using a finite volumes-based scheme over anunstructured mesh the solver has to gather information
about which cells are neighbors and which are not Currentlygmsh does not write this kind of lists in its output filesso milonga has to explicitly solve the neighbors problemPerforming a linear search is an 119874(119873
2) task which is
unacceptable for problem sizes 119873 of interest The code usesa search based on a 119896-dimensional tree [27] which can inprinciple be performed in119874(119873) steps Still for large values of119873 the time needed to read and parse the mesh (number twopreviously mentioned) is the bottleneck of the solution Thisstep is not needed in finite elements although the construc-tion of the elementary matrices involves the computationof the multidimensional Jacobians and integrals which thenhave to be assembled Again for large 119873 this step (numberthree) takes up most of the time needed to solve the problem
The Delaunay algorithm is a standard method for gen-erating two-dimensional grids [16] by tessellating a planewith triangles If the mesh needs to be based on quadranglesinstead a recombination algorithm can be used to transformtwo adjacent triangles in one quadrangle whenever is possi-ble However for geometries which are based on rectangularshapes there exist other algorithms [25] both for meshingand for recombining the triangles that give rise to elementaryentities with right angles almost everywhere which may be a
24 Science and Technology of Nuclear Installations
Tim
e con
sum
ed to
solv
e the
eige
nval
ue p
robl
em (s
econ
ds)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
10
30
100
300
1
3
Figure 24 Time needed to solve the eigenvalue problem versus number of unknowns
desired property of the resulting grid For finite elements thesteps needed to build the matrices depend on the selectionof triangles or quadrangles as the basic elementary geometrybecause the shape functions change However the number ofunknowns is the same as the number of nodes that does notchange after a recombination procedure On the other handthe number of unknowns in the finite volumes schemes withtriangles is roughly twice as the number of unknowns withquadrangles for the same grid
Figure 19 shows how the computed static reactivity (ie1 minus 1119896eff) depends on the number of unknowns for eachof the sixteen combinations of geometry algorithm shapescheme The four accepted results published in the originalreference [10] are included for reference although it shouldbe taken into account that said reactivities were computedalmost forty years ago Figure 20 shows the total wall timeneeded to solve the problem as a function of 119873119866 whileFigures 21 22 23 and 24 show the times needed for eachof the first four steps involved in the solution maintainingthe same logarithmic scale for both the abscissas and theordinates Green data represent finite volumes whilst bluebullets correspond to finite elements Solid lines indicatequarter-core and dashed lines eighth-symmetry Fillet bullets
are results obtained by theDelaunay triangulation and emptybullets were computed with the delquad algorithm Finallytriangle-shaped data correspond to triangles and squares anddiamonds denote quadrangles as the basic geometry of thegrid
It can be seen that finite elements produce amuch smallerdispersion of eigenvalues 119896eff than finite volumes with therefinement of the mesh This can be explained because finitevolumes methods rely on a geometric condition of the meshwhich may change abruptly if the meshing algorithm decidesto allocate the cells in a rather different form for small changesin ℓ119888 Finite elements methods are less influenced by thesediscontinuous lay-out changes of the elements As expectedeighth-core symmetries give almost the same results as thequarter-core geometries with roughly half the unknownsFor small problems finite volumes run faster that finiteelements because the time needed to solve the neighborproblem is negligible When the problem size grows thistime increases and exceeds the overhead implied in the con-struction and assembly of the finite elements matrices Alsoat least for this configuration it is seen that the eigenvalueproblem is solved faster for finite volumes than for finiteelements
Science and Technology of Nuclear Installations 25
4 Conclusions
Unstructured grids provide the cognizant engineer witha wide variety of possibilities to deal with the design oranalysis of nuclear reactor cores These kinds of grids cansuccessfully reproduce continuous geometries commonlyfound in reactor cores such as cylinders and thereforenot only can the diffusion equation be better approximatedinside the domain of definition but also the fulfillment ofboundary conditions is improved A free computer codewas written from scratch that is able to completely solvethe 2D IAEA PWR Benchmark using unstructured gridsfor sixteen combinations of geometry meshing algorithmbasic shape and discretization scheme plus any value ofthe gridrsquos characteristic length by using a single input fileThe complete set of input files and codemdashexecutable andsourcemdashis available either online or upon request withcomments experiences suggestions and corrections beingmore than welcome Further development should includetackling full three-dimensional geometries with completethermal hydraulic feedback in order to analyze how thesolutions of the coupled neutronic-thermal problem dependon the spatial discretization scheme of the neutron leakageterm Parallelization of the computation and assembly of thematrices and of the solution of the eigenvalue problem andits implementation using GPUs are also desired features toimplement A problemwith direct applications that the futureversions of milonga ought to solve is the analysis of how thegeometry of the absorbing materials should be taken intoaccount in structured coarse grids in order to mitigate effectssuch as the rod-cusp problem
Suitable schemes for approximating the continuous dif-ferential operators by discrete matrix expressions includefinite volumes and finite elements families Finite volumesmethods compute cell mean values whilst finite elementsgive functional values at the gridrsquos nodes In general finiteelements are less sensitive to changes in the mesh so theresults they provide do not change significantly for differentmeshing algorithms or elementary shapes Small problemsare best solved by finite volumes as the neighbor-findingproblem is faster than the process of building and assemblingthe eigenvalue-problem matrices For a large number ofunknowns the process of finding which cell is neighbor ofwhichmdasheven based on a 119896-dimensional treemdashoverwhelmsthe computation and assembly of elementary matrices andfinite-element methods perform better
References
[1] G Glasstone and S Bell Nuclear Reactor Theory KriegerPublishing Company 1970
[2] J J Duderstadt and L J Hamilton Nuclear Reactor AnalysisJohn Wiley amp Sons New York NY USA 1976
[3] C Gho Reactor Physics Undergraduate Course Lecture NotesInstituto Balseiro 2006
[4] G G Theler and F J Bonetto ldquoOn the stability of the pointreactor kinetics equationsrdquoNuclear Engineering andDesign vol240 no 6 pp 1443ndash1449 2010
[5] E E Lewis andW F Miller Computational Methods of NeutronTransport John Wiley amp Sons 1984
[6] E Masiello ldquoAnalytical stability analysis of Coarse-Mesh finitedifference methodrdquo in Proceedings of the International Confer-ence on the Physics of Reactors (PHYSOR rsquo08) Nuclear Power ASustainable Resource pp 456ndash464 September 2008
[7] M L Zerkle Development of a polynomial nodal methodwith flux and current discontinuity factors [PhD thesis] Mas-sachusets Institute of Technology 1992
[8] J I Yoon andHG Joo ldquoTwo-level coarsemesh finite differenceformulation with multigroup source expansion nodal kernelsrdquoJournal of Nuclear Science andTechnology vol 45 no 7 pp 668ndash682 2008
[9] O Mazzantini M Schivo J Di Cesare R Garbero M Riveroand G Theler ldquoA coupled calculation suite for Atucha II oper-ational transients analysisrdquo Science and Technology of NuclearInstallations vol 2011 Article ID 785304 12 pages 2011
[10] Computational Benchmark Problem Comitee for the Mathe-matics and Computation Division of the American NuclearSociety ldquoArgonne Code Center Benchmark problem bookrdquoTech Rep ANL-7416 Supplement 2 Argonne National Labo-ratory 1977
[11] A F Henry Nuclear Reactor Analysis MIT Cambridge UK1975
[12] O C Zienkiewicz R L Taylor and J Z ZhuTheFinite ElementMethod Its Basis and Fundamentals vol 1 Elsevier 6th edition2005
[13] G Theler Difusion de neutrones en mallas no estructuradascomparacion entre volumenes y elementos finitos AcademicMonograph Universidad de Buenos Aires 2013
[14] G Theler ldquoMilonga a free nuclear reactor core analysis coderdquoAvailable online 2011
[15] G Theler F J Bonetto and A Clausse ldquoOptimizacion deparametros en reactores de potencia base de diseno delcodigo neutronico milongardquo in Reunion Anual de la AsociacionArgentina de Tecnologia Nuclear XXXVII 2010
[16] C Geuzaine and J-F Remacle ldquoGmsh a 3-D finite elementmesh generator with built-in pre- and post-processing facili-tiesrdquo International Journal for Numerical Methods in Engineer-ing vol 79 no 11 pp 1309ndash1331 2009
[17] S Balay J Brown K Buschelman et al ldquoPETSc users manualrdquoTech Rep ANL-9511-Revision 34 Argonne National Labora-tory 2013
[18] S Balay W D Gropp L Curfman McInnes and B F SmithldquoEfficient management of parallelism in object oriented numer-ical software librariesrdquo in Modern Software Tools in ScientificComputing E Arge A M Bruaset and H P Langtangen Edspp 163ndash202 Birkhauser 1997
[19] V Hernandez J E Roman and V Vidal ldquoSLEPC a scalable andflexible toolkit for the solution of eigenvalue problemsrdquo ACMTransactions on Mathematical Software vol 31 no 3 pp 351ndash362 2005
[20] M Galassi J Davies J Theiler et al GNU Scientific LibraryReference Manual 3rd edition 2009
[21] O C Zienkiewicz and R L Taylor The Finite Element MethodFor Solid and Structural Mechanics vol 3 Elsevier 6th edition2005
[22] A Imelda and K Doddy Evaluation of the IAEA 3-D PWRbenchmark problem using NESTLE code 2004
[23] R Mosteller ldquoStatic benchmarking of the NESTLE advancednodal coderdquo in Proceedings of the Joint International Conference
26 Science and Technology of Nuclear Installations
on Mathematical Methods and Supercomputing for NuclearApplications vol 2 pp 1596ndash1605 1997
[24] GTheler F J Bonetto andAClausse ldquoSolution of the 2D IAEAPWR Benchmark with the neutronic code milongardquo in Actasde la Reunion Anual de la Asociacion Argentina de TecnologıaNuclear XXXVIII 2011
[25] J F Remacle F Henrotte T Carrier-Baudouin et al ldquoAfrontal delaunay quad mesh generator using the 119871
infin normrdquoInternational Journal For Numerical Methods in Engineeringvol 94 no 5 pp 494ndash512 2013
[26] G W Stewart ldquoA Krylov-Schur algorithm for large eigenprob-lemsrdquo SIAM Journal on Matrix Analysis and Applications vol23 no 3 pp 601ndash614 2002
[27] J L Bentley ldquoMultidimensional binary search trees used forassociative searchingrdquo Communications of the ACM vol 18 no9 pp 509ndash517 1975
TribologyAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FuelsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofPetroleum Engineering
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
CombustionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Renewable Energy
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
StructuresJournal of
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear InstallationsScience and Technology of
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Solar EnergyJournal of
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Wind EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear EnergyInternational Journal of
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High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Science and Technology of Nuclear Installations 19
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
1
3
10
30
100
300Ti
me c
onsu
med
to m
esh
the g
eom
etry
(sec
onds
)
Figure 21 Time needed to mesh the geometry versus number of unknowns
Table 1 Macroscopic cross-sections (units are not stated in the original reference but they are assumed to be in cmminus1 or cm as appropriate)
1198631
1198632
Σ1rarr2
Σ1198861
Σ1198862
]Σ1198912
Material1 15 04 002 001 0080 0135 Fuel 12 15 04 002 001 0085 0135 Fuel 23 15 04 002 001 0130 0135 Fuel 2 + rod4 20 03 004 0 0010 0 Reflector
Note the fluxes shall be normalized such that
1
119881coreint119881core
sum
119892
]Σ119891119892 sdot 120601119892119889119881 = 1 (9)
(b) Value and location of maximum power densityThis corresponds to maximum of 1206012 in thecore It is recommended that the maximumvalues of 1206012 both in the inner core and at thecorereflector interface be given
(3) Average subassembly powers 119875119896
119875119896 =1
119881119896
int119881119896
sum
119892
]Σ119891119892 sdot 120601119892119889119881 (10)
where 119881119896 is the volume of the 119896th subassembly and119896 designates the fuel subassemblies as shown inFigure 9
(4) Number of unknowns in the problem number ofiterations and total and outer
(5) Total computing time iteration time IO-time andcomputer used
(6) Type and numerical values of convergence criteria(7) Table of average group fluxes for a square mesh grid
of 20 times 20 cm(8) Dependence of results on mesh spacing
Even though the original problem is based on a quarter-core situation the problem has an eighth-core symmetry
20 Science and Technology of Nuclear Installations
Quarter Delaunay triangles elements
Tim
e con
sum
ed to
read
the m
esh
(sec
onds
)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumes
Quarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
1
3
10
30
100
300
Figure 22 Time needed to read the mesh versus number of unknowns
which cannot be taken into account by structured gridswhich are the main target of the benchmark Howevernonstructured grids can take into consideration any kind ofsymmetry almost without loss of accuracy and at the sametime reducing roughly the number of unknowns by a halfand the associated computational effort needed to solve theproblem by a factor of four Answers to items (1)ndash(7) canbe given completely by milonga using a single input file(see Supplementary data in SupplementaryMaterial availableonline at httpdxdoiorg1011552013641863) As asked initem (8) how results depend not only on the mesh spacingbut also on the meshing algorithm on the gridrsquos elementarygeometric shape and on the discretization scheme may shedlights on the subject whichmay be evenmore interesting thanthe numerical results themselves
Taking advantage of milongarsquos capability of reading andparsing command-line arguments the selection of the coregeometry (quarter or eighth) the meshing algorithm (delau-nay [16] or delquad [25]) the shape of the elementaryentities (triangles or quadrangles) the discretization scheme(volumes or elements) and the characteristic length ℓ119888 ofthe mesh can be provided at run time Fixing five values forℓ119888 = 4 3 2 1 05 gives 2 times 2 times 2 times 2 times 5 = 80 possiblecombinations which we solve with successive invocations to
milonga with the same input file but with different argumentsfrom a simple script Figures 10 11 12 13 14 15 16 17 and18 show the results corresponding to items (1)ndash(7) for someillustrative cases The complete set of figures and the codeused to generate themmay be provided upon request Table 2compiles the answers to the problem for every case studied
As the milonga code is still under development itsnumerical routines are not yet fully optimized nor designedfor parallel computation Therefore the reported times areonly rough estimates and should be taken with care Thesolution comprises five steps
(1) generate the grid with the requested geometry mesh-ing algorithm basic shape and characteristic lengthby calling to gmsh
(2) read the generated mesh(3) build the matrices(4) solve the eigenvalue problem(5) compute the requested results
The CPU time reported in Figures 10ndash18 is thus thesum of all of these five steps but not the time needed togenerate the figuresmdashwhich in some caseswith coarsemeshes
Science and Technology of Nuclear Installations 21
Table2
Resultsofthe2
DIAEA
PWRBe
nchm
arkp
roblem
obtained
with
them
ilong
acod
efor
thee
ightyp
ropo
sedcombinatio
nsofsymmetrym
eshing
algorithm
basicshapediscretization
schemeandcharacteris
ticelem
entc
elllengthTh
ereference
solutio
nis120588=28749times10minus5w
hich
correspo
ndsto119896eff=10296
Case
no
Symm-
etry
Mesh
algorithm
Basic
shape
Solutio
nmetho
dℓ119888
(cm)
Δ120588
(pcm
)max1206012
(mdash)
Total
unkn
owns
Outer
iter
Linear
iter
Inner
iter
Resid
ual
norm
Relativ
eerror
Error
estim
ate
Mem
ory
(Mb)
Page
faults
114
Delaunay
998779Vo
lumes
40
minus120
1145
8264
332
1033
6119890minus09
3119890minus09
1119890minus09
349470
214
Delaunay
998779Vo
lumes
30
85
1135
15740
332
1967
2119890minus08
1119890minus08
4119890minus09
4813401
314
Delaunay
998779Vo
lumes
20
197
1124
31188
332
3898
1119890minus08
6119890minus09
3119890minus09
7621932
414
Delaunay
998779Vo
lumes
10153
1123
127476
332
15934
7119890minus09
3119890minus09
3119890minus09
273
81886
514
Delaunay
998779Vo
lumes
05
187
1119
510676
332
63834
2119890minus09
9119890minus10
1119890minus09
1148
352261
614
Delaunay
998779Elem
ents
40
173
1100
4308
332
538
2119890minus08
8119890minus09
4119890minus09
318794
714
Delaunay
998779Elem
ents
30
117
1107
8108
332
1013
5119890minus09
2119890minus09
2119890minus09
4712946
814
Delaunay
998779Elem
ents
20
84
1112
15936
332
1992
6119890minus09
3119890minus09
2119890minus09
7321347
914
Delaunay
998779Elem
ents
1062
1116
64478
332
8059
6119890minus09
3119890minus09
4119890minus09
262
77105
1014
Delaunay
998779Elem
ents
05
58
1117
256700
332
32087
1119890minus09
5119890minus10
1119890minus09
1091
331969
1114
Delaunay
◻Vo
lumes
40
53
1149
5042
332
630
2119890minus08
1119890minus08
5119890minus09
288007
1214
Delaunay
◻Vo
lumes
30
346
1133
8560
332
1070
7119890minus09
3119890minus09
2119890minus09
3910895
1314
Delaunay
◻Vo
lumes
20
308
1131
15576
332
1947
1119890minus08
7119890minus09
3119890minus09
541564
514
14Delaunay
◻Vo
lumes
10177
1122
60774
332
7596
8119890minus09
4119890minus09
3119890minus09
167
50115
1514
Delaunay
◻Vo
lumes
05
199
1120
244936
332
30617
4119890minus09
2119890minus09
2119890minus09
698
215678
1614
Delaunay
◻Elem
ents
40
196
1100
5222
332
652
3119890minus08
1119890minus08
8119890minus09
4713098
1714
Delaunay
◻Elem
ents
30
123
1107
8810
332
1101
2119890minus08
9119890minus09
7119890minus09
6918598
1814
Delaunay
◻Elem
ents
20
901112
15922
332
1990
1119890minus08
5119890minus09
5119890minus09
107
30591
1914
Delaunay
◻Elem
ents
1063
1116
61456
332
7682
5119890minus09
2119890minus09
4119890minus09
389
113237
2014
Delaunay
◻Elem
ents
05
58
1117
246332
332
30791
8119890minus10
4119890minus10
1119890minus09
1676
498192
2114
Delq
uad
998779Vo
lumes
40
minus45
1144
6228
332
778
4119890minus08
2119890minus08
7119890minus09
308495
2214
Delq
uad
998779Vo
lumes
30
570
1153
11820
332
1477
3119890minus08
1119890minus08
4119890minus09
4010879
2314
Delq
uad
998779Vo
lumes
20
459
1103
24100
332
3012
2119890minus08
1119890minus08
5119890minus09
6217715
2414
Delq
uad
998779Vo
lumes
10512
1104
9640
03
3212050
2119890minus08
1119890minus08
9119890minus09
207
61714
2514
Delq
uad
998779Vo
lumes
05
528
1106
385600
332
48200
2119890minus09
1119890minus09
2119890minus09
838
255735
2614
Delqu
ad998779
Elem
ents
40
220
1093
3288
332
411
3119890minus08
2119890minus08
6119890minus09
308495
2714
Delq
uad
998779Elem
ents
30
140
1104
6148
332
768
4119890minus08
2119890minus08
8119890minus09
39110
0028
14Delq
uad
998779Elem
ents
20
82
1110
12392
332
1549
2119890minus08
9119890minus09
6119890minus09
6117067
2914
Delqu
ad998779
Elem
ents
1063
1115
48882
332
6110
2119890minus09
1119890minus09
1119890minus09
197
5804
630
14Delq
uad
998779Elem
ents
05
58
1116
194162
332
24270
2119890minus09
9119890minus10
2119890minus09
790
238769
3114
Delq
uad
◻Vo
lumes
40
minus416
1174
3132
332
391
5119890minus08
3119890minus08
8119890minus09
267322
3214
Delq
uad
◻Vo
lumes
30
minus248
1151
5922
332
740
9119890minus09
4119890minus09
2119890minus09
318779
3314
Delq
uad
◻Vo
lumes
20
minus132
1139
12050
332
1506
1119890minus08
7119890minus09
4119890minus09
4412239
3414
Delq
uad
◻Vo
lumes
10minus49
1126
48200
332
6025
6119890minus09
3119890minus09
2119890minus09
122
36094
3514
Delq
uad
◻Vo
lumes
05
minus23
1123
192800
332
24100
5119890minus09
2119890minus09
3119890minus09
464
140228
3614
Delq
uad
◻Elem
ents
40
224
1093
3294
332
411
3119890minus08
1119890minus08
7119890minus09
359852
3714
Delq
uad
◻Elem
ents
30
141
1104
6144
332
768
3119890minus08
2119890minus08
9119890minus09
5114018
3814
Delq
uad
◻Elem
ents
20
921111
12392
332
1549
1119890minus08
6119890minus09
5119890minus09
8122526
3914
Delq
uad
◻Elem
ents
1065
1115
48882
332
6110
5119890minus09
3119890minus09
4119890minus09
276
78431
22 Science and Technology of Nuclear Installations
Table2Con
tinued
Case
no
Symm-
etry
Mesh
algorithm
Basic
shape
Solutio
nmetho
dℓ119888
(cm)
Δ120588
(pcm
)max1206012
(mdash)
Total
unkn
owns
Outer
iter
Linear
iter
Inner
iter
Resid
ual
norm
Relativ
eerror
Error
estim
ate
Mem
ory
(Mb)
Page
faults
4014
Delq
uad
◻Elem
ents
05
59
1117
194162
332
24270
1119890minus09
5119890minus10
1119890minus09
1104
318967
4118
Delaunay
998779Vo
lumes
40
minus99
1139
4228
332
528
4119890minus12
2119890minus12
1119890minus12
359630
4218
Delaunay
998779Vo
lumes
30
50
1132
7712
332
964
1119890minus11
6119890minus12
2119890minus12
4211578
4318
Delaunay
998779Vo
lumes
20
177
1120
15776
332
1972
2119890minus12
1119890minus12
7119890minus13
5515436
4418
Delaunay
998779Vo
lumes
10142
1118
63902
332
7987
3119890minus12
1119890minus12
1119890minus12
160
46691
4518
Delaunay
998779Vo
lumes
05
176
1115
254612
332
31826
2119890minus12
8119890minus13
1119890minus12
555
168526
4618
Delaunay
998779Elem
ents
40
183
1096
2252
332
281
6119890minus12
3119890minus12
2119890minus12
359589
4718
Delaunay
998779Elem
ents
30
119
1104
4040
224
505
2119890minus08
1119890minus08
7119890minus09
41114
3748
18Delaunay
998779Elem
ents
20
85
1108
8156
332
1019
2119890minus12
8119890minus13
6119890minus13
5515053
4918
Delaunay
998779Elem
ents
1063
1112
32480
332
4060
1119890minus12
6119890minus13
8119890minus13
151
43671
5018
Delaunay
998779Elem
ents
05
58
1113
128358
332
1604
47119890minus13
3119890minus13
7119890minus13
527
157581
5118
Delaunay
◻Vo
lumes
40
37
1144
2380
224
297
5119890minus08
2119890minus08
1119890minus08
5314173
5218
Delaunay
◻Vo
lumes
30
218
1120
4194
332
524
7119890minus12
4119890minus12
2119890minus12
3610034
5318
Delaunay
◻Vo
lumes
20
345
1110
7960
332
995
5119890minus12
3119890minus12
1119890minus12
4712879
5418
Delaunay
◻Vo
lumes
10165
1117
30652
332
3831
2119890minus12
1119890minus12
7119890minus13
100
28851
5518
Delaunay
◻Vo
lumes
05
191
1117
122066
224
15258
2119890minus08
8119890minus09
9119890minus09
343
103825
5618
Delaunay
◻Elem
ents
40
175
1097
2526
332
315
4119890minus12
2119890minus12
1119890minus12
6016159
5718
Delaunay
◻Elem
ents
30
114
1104
4386
332
548
4119890minus12
2119890minus12
2119890minus12
5013768
5818
Delaunay
◻Elem
ents
20
901108
8234
332
1029
2119890minus12
1119890minus12
1119890minus12
7320094
5918
Delaunay
◻Elem
ents
1062
1112
31186
224
3898
2119890minus08
7119890minus09
9119890minus09
208
59589
6018
Delaunay
◻Elem
ents
05
58
1113
123122
224
15390
9119890minus09
5119890minus09
1119890minus08
807
2366
7061
18Delq
uad
998779Vo
lumes
40
minus210
1168
3224
332
403
1119890minus11
5119890minus12
2119890minus12
43117
8462
18Delq
uad
998779Vo
lumes
30
369
1150
6000
332
750
1119890minus11
6119890minus12
3119890minus12
5013502
6318
Delq
uad
998779Vo
lumes
20
312
1104
12316
332
1539
2119890minus11
8119890minus12
5119890minus12
6016395
6418
Delq
uad
998779Vo
lumes
1044
01094
48714
332
6089
9119890minus12
4119890minus12
4119890minus12
1183444
165
18Delq
uad
998779Vo
lumes
05
534
1081
193980
332
24247
5119890minus12
2119890minus12
4119890minus12
423
126768
6618
Delq
uad
998779Elem
ents
40
240
1090
1750
332
218
9119890minus12
4119890minus12
2119890minus12
43117
8667
18Delqu
ad998779
Elem
ents
30
150
1100
3184
332
398
1119890minus11
5119890minus12
2119890minus12
5013509
6818
Delq
uad
998779Elem
ents
20
89
1106
6426
332
803
5119890minus12
3119890minus12
2119890minus12
5916047
6918
Delq
uad
998779Elem
ents
1064
1111
24886
332
3110
2119890minus12
1119890minus12
1119890minus12
113
32788
7018
Delq
uad
998779Elem
ents
05
59
1113
98052
224
12256
1119890minus08
5119890minus09
8119890minus09
398
118153
7118
Delq
uad
◻Vo
lumes
40
minus398
1172
1650
332
206
1119890minus11
5119890minus12
2119890minus12
369907
7218
Delq
uad
◻Vo
lumes
30
minus249
1146
3058
332
382
4119890minus12
2119890minus12
1119890minus12
349349
7318
Delq
uad
◻Vo
lumes
20
minus131
1136
6228
224
778
3119890minus08
1119890minus08
5119890minus09
4010930
7418
Delq
uad
◻Vo
lumes
10minus51
1123
24536
224
3067
3119890minus08
2119890minus08
1119890minus08
7822716
7618
Delq
uad
◻Elem
ents
40
237
1090
1752
332
219
6119890minus12
3119890minus12
2119890minus12
4111213
7718
Delq
uad
◻Elem
ents
30
145
1101
3184
332
398
6119890minus12
3119890minus12
2119890minus12
43118
1178
18Delq
uad
◻Elem
ents
20
941107
6426
332
803
3119890minus12
1119890minus12
1119890minus12
6016220
7918
Delqu
ad◻
Elem
ents
1065
1111
24874
332
3109
1119890minus12
6119890minus13
8119890minus13
156
4400
080
18Delq
uad
◻Elem
ents
05
59
1113
9804
22
2412255
8119890minus09
4119890minus09
8119890minus09
567
162168
Science and Technology of Nuclear Installations 23
Tim
e con
sum
ed to
bui
ld th
e mat
rices
(sec
onds
)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay trianglesvolumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
10
30
100
300
1
3
Figure 23 Time needed to build the matrices versus number of unknowns
was considerably larger than the solution time itself Theeigenvalue problem was solved using a multilayer iterativeKrylov-Schur method [26] with a tolerance relative to thematrices norm
1003817100381710038171003817119877 sdot 120601 minus (1119896eff) sdot 119865 sdot 1206011003817100381710038171003817
119877 + (1119896eff) sdot 119865lt 10minus8 (11)
The reported residual norm and relative error are10038171003817100381710038171003817100381710038171003817119877 sdot 120601 minus
1
119896effsdot 119865 sdot 120601
10038171003817100381710038171003817100381710038171003817
1003817100381710038171003817119877 sdot 120601 minus (1119896eff) sdot 119865 sdot 1206011003817100381710038171003817
1003817100381710038171003817(1119896eff) sdot 1206011003817100381710038171003817
(12)
respectively The fields marked as outer linear and inneriterations refer to the number of steps needed to attainthe requested tolerance in each layer of the Krylov-Schuralgorithm The computer used to solve the problem has anIntel i7 920 267GHz processor with 4Gb of RAM runningDebian GNULinux Wheezy
When using a finite volumes-based scheme over anunstructured mesh the solver has to gather information
about which cells are neighbors and which are not Currentlygmsh does not write this kind of lists in its output filesso milonga has to explicitly solve the neighbors problemPerforming a linear search is an 119874(119873
2) task which is
unacceptable for problem sizes 119873 of interest The code usesa search based on a 119896-dimensional tree [27] which can inprinciple be performed in119874(119873) steps Still for large values of119873 the time needed to read and parse the mesh (number twopreviously mentioned) is the bottleneck of the solution Thisstep is not needed in finite elements although the construc-tion of the elementary matrices involves the computationof the multidimensional Jacobians and integrals which thenhave to be assembled Again for large 119873 this step (numberthree) takes up most of the time needed to solve the problem
The Delaunay algorithm is a standard method for gen-erating two-dimensional grids [16] by tessellating a planewith triangles If the mesh needs to be based on quadranglesinstead a recombination algorithm can be used to transformtwo adjacent triangles in one quadrangle whenever is possi-ble However for geometries which are based on rectangularshapes there exist other algorithms [25] both for meshingand for recombining the triangles that give rise to elementaryentities with right angles almost everywhere which may be a
24 Science and Technology of Nuclear Installations
Tim
e con
sum
ed to
solv
e the
eige
nval
ue p
robl
em (s
econ
ds)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
10
30
100
300
1
3
Figure 24 Time needed to solve the eigenvalue problem versus number of unknowns
desired property of the resulting grid For finite elements thesteps needed to build the matrices depend on the selectionof triangles or quadrangles as the basic elementary geometrybecause the shape functions change However the number ofunknowns is the same as the number of nodes that does notchange after a recombination procedure On the other handthe number of unknowns in the finite volumes schemes withtriangles is roughly twice as the number of unknowns withquadrangles for the same grid
Figure 19 shows how the computed static reactivity (ie1 minus 1119896eff) depends on the number of unknowns for eachof the sixteen combinations of geometry algorithm shapescheme The four accepted results published in the originalreference [10] are included for reference although it shouldbe taken into account that said reactivities were computedalmost forty years ago Figure 20 shows the total wall timeneeded to solve the problem as a function of 119873119866 whileFigures 21 22 23 and 24 show the times needed for eachof the first four steps involved in the solution maintainingthe same logarithmic scale for both the abscissas and theordinates Green data represent finite volumes whilst bluebullets correspond to finite elements Solid lines indicatequarter-core and dashed lines eighth-symmetry Fillet bullets
are results obtained by theDelaunay triangulation and emptybullets were computed with the delquad algorithm Finallytriangle-shaped data correspond to triangles and squares anddiamonds denote quadrangles as the basic geometry of thegrid
It can be seen that finite elements produce amuch smallerdispersion of eigenvalues 119896eff than finite volumes with therefinement of the mesh This can be explained because finitevolumes methods rely on a geometric condition of the meshwhich may change abruptly if the meshing algorithm decidesto allocate the cells in a rather different form for small changesin ℓ119888 Finite elements methods are less influenced by thesediscontinuous lay-out changes of the elements As expectedeighth-core symmetries give almost the same results as thequarter-core geometries with roughly half the unknownsFor small problems finite volumes run faster that finiteelements because the time needed to solve the neighborproblem is negligible When the problem size grows thistime increases and exceeds the overhead implied in the con-struction and assembly of the finite elements matrices Alsoat least for this configuration it is seen that the eigenvalueproblem is solved faster for finite volumes than for finiteelements
Science and Technology of Nuclear Installations 25
4 Conclusions
Unstructured grids provide the cognizant engineer witha wide variety of possibilities to deal with the design oranalysis of nuclear reactor cores These kinds of grids cansuccessfully reproduce continuous geometries commonlyfound in reactor cores such as cylinders and thereforenot only can the diffusion equation be better approximatedinside the domain of definition but also the fulfillment ofboundary conditions is improved A free computer codewas written from scratch that is able to completely solvethe 2D IAEA PWR Benchmark using unstructured gridsfor sixteen combinations of geometry meshing algorithmbasic shape and discretization scheme plus any value ofthe gridrsquos characteristic length by using a single input fileThe complete set of input files and codemdashexecutable andsourcemdashis available either online or upon request withcomments experiences suggestions and corrections beingmore than welcome Further development should includetackling full three-dimensional geometries with completethermal hydraulic feedback in order to analyze how thesolutions of the coupled neutronic-thermal problem dependon the spatial discretization scheme of the neutron leakageterm Parallelization of the computation and assembly of thematrices and of the solution of the eigenvalue problem andits implementation using GPUs are also desired features toimplement A problemwith direct applications that the futureversions of milonga ought to solve is the analysis of how thegeometry of the absorbing materials should be taken intoaccount in structured coarse grids in order to mitigate effectssuch as the rod-cusp problem
Suitable schemes for approximating the continuous dif-ferential operators by discrete matrix expressions includefinite volumes and finite elements families Finite volumesmethods compute cell mean values whilst finite elementsgive functional values at the gridrsquos nodes In general finiteelements are less sensitive to changes in the mesh so theresults they provide do not change significantly for differentmeshing algorithms or elementary shapes Small problemsare best solved by finite volumes as the neighbor-findingproblem is faster than the process of building and assemblingthe eigenvalue-problem matrices For a large number ofunknowns the process of finding which cell is neighbor ofwhichmdasheven based on a 119896-dimensional treemdashoverwhelmsthe computation and assembly of elementary matrices andfinite-element methods perform better
References
[1] G Glasstone and S Bell Nuclear Reactor Theory KriegerPublishing Company 1970
[2] J J Duderstadt and L J Hamilton Nuclear Reactor AnalysisJohn Wiley amp Sons New York NY USA 1976
[3] C Gho Reactor Physics Undergraduate Course Lecture NotesInstituto Balseiro 2006
[4] G G Theler and F J Bonetto ldquoOn the stability of the pointreactor kinetics equationsrdquoNuclear Engineering andDesign vol240 no 6 pp 1443ndash1449 2010
[5] E E Lewis andW F Miller Computational Methods of NeutronTransport John Wiley amp Sons 1984
[6] E Masiello ldquoAnalytical stability analysis of Coarse-Mesh finitedifference methodrdquo in Proceedings of the International Confer-ence on the Physics of Reactors (PHYSOR rsquo08) Nuclear Power ASustainable Resource pp 456ndash464 September 2008
[7] M L Zerkle Development of a polynomial nodal methodwith flux and current discontinuity factors [PhD thesis] Mas-sachusets Institute of Technology 1992
[8] J I Yoon andHG Joo ldquoTwo-level coarsemesh finite differenceformulation with multigroup source expansion nodal kernelsrdquoJournal of Nuclear Science andTechnology vol 45 no 7 pp 668ndash682 2008
[9] O Mazzantini M Schivo J Di Cesare R Garbero M Riveroand G Theler ldquoA coupled calculation suite for Atucha II oper-ational transients analysisrdquo Science and Technology of NuclearInstallations vol 2011 Article ID 785304 12 pages 2011
[10] Computational Benchmark Problem Comitee for the Mathe-matics and Computation Division of the American NuclearSociety ldquoArgonne Code Center Benchmark problem bookrdquoTech Rep ANL-7416 Supplement 2 Argonne National Labo-ratory 1977
[11] A F Henry Nuclear Reactor Analysis MIT Cambridge UK1975
[12] O C Zienkiewicz R L Taylor and J Z ZhuTheFinite ElementMethod Its Basis and Fundamentals vol 1 Elsevier 6th edition2005
[13] G Theler Difusion de neutrones en mallas no estructuradascomparacion entre volumenes y elementos finitos AcademicMonograph Universidad de Buenos Aires 2013
[14] G Theler ldquoMilonga a free nuclear reactor core analysis coderdquoAvailable online 2011
[15] G Theler F J Bonetto and A Clausse ldquoOptimizacion deparametros en reactores de potencia base de diseno delcodigo neutronico milongardquo in Reunion Anual de la AsociacionArgentina de Tecnologia Nuclear XXXVII 2010
[16] C Geuzaine and J-F Remacle ldquoGmsh a 3-D finite elementmesh generator with built-in pre- and post-processing facili-tiesrdquo International Journal for Numerical Methods in Engineer-ing vol 79 no 11 pp 1309ndash1331 2009
[17] S Balay J Brown K Buschelman et al ldquoPETSc users manualrdquoTech Rep ANL-9511-Revision 34 Argonne National Labora-tory 2013
[18] S Balay W D Gropp L Curfman McInnes and B F SmithldquoEfficient management of parallelism in object oriented numer-ical software librariesrdquo in Modern Software Tools in ScientificComputing E Arge A M Bruaset and H P Langtangen Edspp 163ndash202 Birkhauser 1997
[19] V Hernandez J E Roman and V Vidal ldquoSLEPC a scalable andflexible toolkit for the solution of eigenvalue problemsrdquo ACMTransactions on Mathematical Software vol 31 no 3 pp 351ndash362 2005
[20] M Galassi J Davies J Theiler et al GNU Scientific LibraryReference Manual 3rd edition 2009
[21] O C Zienkiewicz and R L Taylor The Finite Element MethodFor Solid and Structural Mechanics vol 3 Elsevier 6th edition2005
[22] A Imelda and K Doddy Evaluation of the IAEA 3-D PWRbenchmark problem using NESTLE code 2004
[23] R Mosteller ldquoStatic benchmarking of the NESTLE advancednodal coderdquo in Proceedings of the Joint International Conference
26 Science and Technology of Nuclear Installations
on Mathematical Methods and Supercomputing for NuclearApplications vol 2 pp 1596ndash1605 1997
[24] GTheler F J Bonetto andAClausse ldquoSolution of the 2D IAEAPWR Benchmark with the neutronic code milongardquo in Actasde la Reunion Anual de la Asociacion Argentina de TecnologıaNuclear XXXVIII 2011
[25] J F Remacle F Henrotte T Carrier-Baudouin et al ldquoAfrontal delaunay quad mesh generator using the 119871
infin normrdquoInternational Journal For Numerical Methods in Engineeringvol 94 no 5 pp 494ndash512 2013
[26] G W Stewart ldquoA Krylov-Schur algorithm for large eigenprob-lemsrdquo SIAM Journal on Matrix Analysis and Applications vol23 no 3 pp 601ndash614 2002
[27] J L Bentley ldquoMultidimensional binary search trees used forassociative searchingrdquo Communications of the ACM vol 18 no9 pp 509ndash517 1975
TribologyAdvances in
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International Journal of
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Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Renewable Energy
Submit your manuscripts athttpwwwhindawicom
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StructuresJournal of
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
EnergyJournal of
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Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear InstallationsScience and Technology of
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Wind EnergyJournal of
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Nuclear EnergyInternational Journal of
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High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
20 Science and Technology of Nuclear Installations
Quarter Delaunay triangles elements
Tim
e con
sum
ed to
read
the m
esh
(sec
onds
)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumes
Quarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
1
3
10
30
100
300
Figure 22 Time needed to read the mesh versus number of unknowns
which cannot be taken into account by structured gridswhich are the main target of the benchmark Howevernonstructured grids can take into consideration any kind ofsymmetry almost without loss of accuracy and at the sametime reducing roughly the number of unknowns by a halfand the associated computational effort needed to solve theproblem by a factor of four Answers to items (1)ndash(7) canbe given completely by milonga using a single input file(see Supplementary data in SupplementaryMaterial availableonline at httpdxdoiorg1011552013641863) As asked initem (8) how results depend not only on the mesh spacingbut also on the meshing algorithm on the gridrsquos elementarygeometric shape and on the discretization scheme may shedlights on the subject whichmay be evenmore interesting thanthe numerical results themselves
Taking advantage of milongarsquos capability of reading andparsing command-line arguments the selection of the coregeometry (quarter or eighth) the meshing algorithm (delau-nay [16] or delquad [25]) the shape of the elementaryentities (triangles or quadrangles) the discretization scheme(volumes or elements) and the characteristic length ℓ119888 ofthe mesh can be provided at run time Fixing five values forℓ119888 = 4 3 2 1 05 gives 2 times 2 times 2 times 2 times 5 = 80 possiblecombinations which we solve with successive invocations to
milonga with the same input file but with different argumentsfrom a simple script Figures 10 11 12 13 14 15 16 17 and18 show the results corresponding to items (1)ndash(7) for someillustrative cases The complete set of figures and the codeused to generate themmay be provided upon request Table 2compiles the answers to the problem for every case studied
As the milonga code is still under development itsnumerical routines are not yet fully optimized nor designedfor parallel computation Therefore the reported times areonly rough estimates and should be taken with care Thesolution comprises five steps
(1) generate the grid with the requested geometry mesh-ing algorithm basic shape and characteristic lengthby calling to gmsh
(2) read the generated mesh(3) build the matrices(4) solve the eigenvalue problem(5) compute the requested results
The CPU time reported in Figures 10ndash18 is thus thesum of all of these five steps but not the time needed togenerate the figuresmdashwhich in some caseswith coarsemeshes
Science and Technology of Nuclear Installations 21
Table2
Resultsofthe2
DIAEA
PWRBe
nchm
arkp
roblem
obtained
with
them
ilong
acod
efor
thee
ightyp
ropo
sedcombinatio
nsofsymmetrym
eshing
algorithm
basicshapediscretization
schemeandcharacteris
ticelem
entc
elllengthTh
ereference
solutio
nis120588=28749times10minus5w
hich
correspo
ndsto119896eff=10296
Case
no
Symm-
etry
Mesh
algorithm
Basic
shape
Solutio
nmetho
dℓ119888
(cm)
Δ120588
(pcm
)max1206012
(mdash)
Total
unkn
owns
Outer
iter
Linear
iter
Inner
iter
Resid
ual
norm
Relativ
eerror
Error
estim
ate
Mem
ory
(Mb)
Page
faults
114
Delaunay
998779Vo
lumes
40
minus120
1145
8264
332
1033
6119890minus09
3119890minus09
1119890minus09
349470
214
Delaunay
998779Vo
lumes
30
85
1135
15740
332
1967
2119890minus08
1119890minus08
4119890minus09
4813401
314
Delaunay
998779Vo
lumes
20
197
1124
31188
332
3898
1119890minus08
6119890minus09
3119890minus09
7621932
414
Delaunay
998779Vo
lumes
10153
1123
127476
332
15934
7119890minus09
3119890minus09
3119890minus09
273
81886
514
Delaunay
998779Vo
lumes
05
187
1119
510676
332
63834
2119890minus09
9119890minus10
1119890minus09
1148
352261
614
Delaunay
998779Elem
ents
40
173
1100
4308
332
538
2119890minus08
8119890minus09
4119890minus09
318794
714
Delaunay
998779Elem
ents
30
117
1107
8108
332
1013
5119890minus09
2119890minus09
2119890minus09
4712946
814
Delaunay
998779Elem
ents
20
84
1112
15936
332
1992
6119890minus09
3119890minus09
2119890minus09
7321347
914
Delaunay
998779Elem
ents
1062
1116
64478
332
8059
6119890minus09
3119890minus09
4119890minus09
262
77105
1014
Delaunay
998779Elem
ents
05
58
1117
256700
332
32087
1119890minus09
5119890minus10
1119890minus09
1091
331969
1114
Delaunay
◻Vo
lumes
40
53
1149
5042
332
630
2119890minus08
1119890minus08
5119890minus09
288007
1214
Delaunay
◻Vo
lumes
30
346
1133
8560
332
1070
7119890minus09
3119890minus09
2119890minus09
3910895
1314
Delaunay
◻Vo
lumes
20
308
1131
15576
332
1947
1119890minus08
7119890minus09
3119890minus09
541564
514
14Delaunay
◻Vo
lumes
10177
1122
60774
332
7596
8119890minus09
4119890minus09
3119890minus09
167
50115
1514
Delaunay
◻Vo
lumes
05
199
1120
244936
332
30617
4119890minus09
2119890minus09
2119890minus09
698
215678
1614
Delaunay
◻Elem
ents
40
196
1100
5222
332
652
3119890minus08
1119890minus08
8119890minus09
4713098
1714
Delaunay
◻Elem
ents
30
123
1107
8810
332
1101
2119890minus08
9119890minus09
7119890minus09
6918598
1814
Delaunay
◻Elem
ents
20
901112
15922
332
1990
1119890minus08
5119890minus09
5119890minus09
107
30591
1914
Delaunay
◻Elem
ents
1063
1116
61456
332
7682
5119890minus09
2119890minus09
4119890minus09
389
113237
2014
Delaunay
◻Elem
ents
05
58
1117
246332
332
30791
8119890minus10
4119890minus10
1119890minus09
1676
498192
2114
Delq
uad
998779Vo
lumes
40
minus45
1144
6228
332
778
4119890minus08
2119890minus08
7119890minus09
308495
2214
Delq
uad
998779Vo
lumes
30
570
1153
11820
332
1477
3119890minus08
1119890minus08
4119890minus09
4010879
2314
Delq
uad
998779Vo
lumes
20
459
1103
24100
332
3012
2119890minus08
1119890minus08
5119890minus09
6217715
2414
Delq
uad
998779Vo
lumes
10512
1104
9640
03
3212050
2119890minus08
1119890minus08
9119890minus09
207
61714
2514
Delq
uad
998779Vo
lumes
05
528
1106
385600
332
48200
2119890minus09
1119890minus09
2119890minus09
838
255735
2614
Delqu
ad998779
Elem
ents
40
220
1093
3288
332
411
3119890minus08
2119890minus08
6119890minus09
308495
2714
Delq
uad
998779Elem
ents
30
140
1104
6148
332
768
4119890minus08
2119890minus08
8119890minus09
39110
0028
14Delq
uad
998779Elem
ents
20
82
1110
12392
332
1549
2119890minus08
9119890minus09
6119890minus09
6117067
2914
Delqu
ad998779
Elem
ents
1063
1115
48882
332
6110
2119890minus09
1119890minus09
1119890minus09
197
5804
630
14Delq
uad
998779Elem
ents
05
58
1116
194162
332
24270
2119890minus09
9119890minus10
2119890minus09
790
238769
3114
Delq
uad
◻Vo
lumes
40
minus416
1174
3132
332
391
5119890minus08
3119890minus08
8119890minus09
267322
3214
Delq
uad
◻Vo
lumes
30
minus248
1151
5922
332
740
9119890minus09
4119890minus09
2119890minus09
318779
3314
Delq
uad
◻Vo
lumes
20
minus132
1139
12050
332
1506
1119890minus08
7119890minus09
4119890minus09
4412239
3414
Delq
uad
◻Vo
lumes
10minus49
1126
48200
332
6025
6119890minus09
3119890minus09
2119890minus09
122
36094
3514
Delq
uad
◻Vo
lumes
05
minus23
1123
192800
332
24100
5119890minus09
2119890minus09
3119890minus09
464
140228
3614
Delq
uad
◻Elem
ents
40
224
1093
3294
332
411
3119890minus08
1119890minus08
7119890minus09
359852
3714
Delq
uad
◻Elem
ents
30
141
1104
6144
332
768
3119890minus08
2119890minus08
9119890minus09
5114018
3814
Delq
uad
◻Elem
ents
20
921111
12392
332
1549
1119890minus08
6119890minus09
5119890minus09
8122526
3914
Delq
uad
◻Elem
ents
1065
1115
48882
332
6110
5119890minus09
3119890minus09
4119890minus09
276
78431
22 Science and Technology of Nuclear Installations
Table2Con
tinued
Case
no
Symm-
etry
Mesh
algorithm
Basic
shape
Solutio
nmetho
dℓ119888
(cm)
Δ120588
(pcm
)max1206012
(mdash)
Total
unkn
owns
Outer
iter
Linear
iter
Inner
iter
Resid
ual
norm
Relativ
eerror
Error
estim
ate
Mem
ory
(Mb)
Page
faults
4014
Delq
uad
◻Elem
ents
05
59
1117
194162
332
24270
1119890minus09
5119890minus10
1119890minus09
1104
318967
4118
Delaunay
998779Vo
lumes
40
minus99
1139
4228
332
528
4119890minus12
2119890minus12
1119890minus12
359630
4218
Delaunay
998779Vo
lumes
30
50
1132
7712
332
964
1119890minus11
6119890minus12
2119890minus12
4211578
4318
Delaunay
998779Vo
lumes
20
177
1120
15776
332
1972
2119890minus12
1119890minus12
7119890minus13
5515436
4418
Delaunay
998779Vo
lumes
10142
1118
63902
332
7987
3119890minus12
1119890minus12
1119890minus12
160
46691
4518
Delaunay
998779Vo
lumes
05
176
1115
254612
332
31826
2119890minus12
8119890minus13
1119890minus12
555
168526
4618
Delaunay
998779Elem
ents
40
183
1096
2252
332
281
6119890minus12
3119890minus12
2119890minus12
359589
4718
Delaunay
998779Elem
ents
30
119
1104
4040
224
505
2119890minus08
1119890minus08
7119890minus09
41114
3748
18Delaunay
998779Elem
ents
20
85
1108
8156
332
1019
2119890minus12
8119890minus13
6119890minus13
5515053
4918
Delaunay
998779Elem
ents
1063
1112
32480
332
4060
1119890minus12
6119890minus13
8119890minus13
151
43671
5018
Delaunay
998779Elem
ents
05
58
1113
128358
332
1604
47119890minus13
3119890minus13
7119890minus13
527
157581
5118
Delaunay
◻Vo
lumes
40
37
1144
2380
224
297
5119890minus08
2119890minus08
1119890minus08
5314173
5218
Delaunay
◻Vo
lumes
30
218
1120
4194
332
524
7119890minus12
4119890minus12
2119890minus12
3610034
5318
Delaunay
◻Vo
lumes
20
345
1110
7960
332
995
5119890minus12
3119890minus12
1119890minus12
4712879
5418
Delaunay
◻Vo
lumes
10165
1117
30652
332
3831
2119890minus12
1119890minus12
7119890minus13
100
28851
5518
Delaunay
◻Vo
lumes
05
191
1117
122066
224
15258
2119890minus08
8119890minus09
9119890minus09
343
103825
5618
Delaunay
◻Elem
ents
40
175
1097
2526
332
315
4119890minus12
2119890minus12
1119890minus12
6016159
5718
Delaunay
◻Elem
ents
30
114
1104
4386
332
548
4119890minus12
2119890minus12
2119890minus12
5013768
5818
Delaunay
◻Elem
ents
20
901108
8234
332
1029
2119890minus12
1119890minus12
1119890minus12
7320094
5918
Delaunay
◻Elem
ents
1062
1112
31186
224
3898
2119890minus08
7119890minus09
9119890minus09
208
59589
6018
Delaunay
◻Elem
ents
05
58
1113
123122
224
15390
9119890minus09
5119890minus09
1119890minus08
807
2366
7061
18Delq
uad
998779Vo
lumes
40
minus210
1168
3224
332
403
1119890minus11
5119890minus12
2119890minus12
43117
8462
18Delq
uad
998779Vo
lumes
30
369
1150
6000
332
750
1119890minus11
6119890minus12
3119890minus12
5013502
6318
Delq
uad
998779Vo
lumes
20
312
1104
12316
332
1539
2119890minus11
8119890minus12
5119890minus12
6016395
6418
Delq
uad
998779Vo
lumes
1044
01094
48714
332
6089
9119890minus12
4119890minus12
4119890minus12
1183444
165
18Delq
uad
998779Vo
lumes
05
534
1081
193980
332
24247
5119890minus12
2119890minus12
4119890minus12
423
126768
6618
Delq
uad
998779Elem
ents
40
240
1090
1750
332
218
9119890minus12
4119890minus12
2119890minus12
43117
8667
18Delqu
ad998779
Elem
ents
30
150
1100
3184
332
398
1119890minus11
5119890minus12
2119890minus12
5013509
6818
Delq
uad
998779Elem
ents
20
89
1106
6426
332
803
5119890minus12
3119890minus12
2119890minus12
5916047
6918
Delq
uad
998779Elem
ents
1064
1111
24886
332
3110
2119890minus12
1119890minus12
1119890minus12
113
32788
7018
Delq
uad
998779Elem
ents
05
59
1113
98052
224
12256
1119890minus08
5119890minus09
8119890minus09
398
118153
7118
Delq
uad
◻Vo
lumes
40
minus398
1172
1650
332
206
1119890minus11
5119890minus12
2119890minus12
369907
7218
Delq
uad
◻Vo
lumes
30
minus249
1146
3058
332
382
4119890minus12
2119890minus12
1119890minus12
349349
7318
Delq
uad
◻Vo
lumes
20
minus131
1136
6228
224
778
3119890minus08
1119890minus08
5119890minus09
4010930
7418
Delq
uad
◻Vo
lumes
10minus51
1123
24536
224
3067
3119890minus08
2119890minus08
1119890minus08
7822716
7618
Delq
uad
◻Elem
ents
40
237
1090
1752
332
219
6119890minus12
3119890minus12
2119890minus12
4111213
7718
Delq
uad
◻Elem
ents
30
145
1101
3184
332
398
6119890minus12
3119890minus12
2119890minus12
43118
1178
18Delq
uad
◻Elem
ents
20
941107
6426
332
803
3119890minus12
1119890minus12
1119890minus12
6016220
7918
Delqu
ad◻
Elem
ents
1065
1111
24874
332
3109
1119890minus12
6119890minus13
8119890minus13
156
4400
080
18Delq
uad
◻Elem
ents
05
59
1113
9804
22
2412255
8119890minus09
4119890minus09
8119890minus09
567
162168
Science and Technology of Nuclear Installations 23
Tim
e con
sum
ed to
bui
ld th
e mat
rices
(sec
onds
)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay trianglesvolumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
10
30
100
300
1
3
Figure 23 Time needed to build the matrices versus number of unknowns
was considerably larger than the solution time itself Theeigenvalue problem was solved using a multilayer iterativeKrylov-Schur method [26] with a tolerance relative to thematrices norm
1003817100381710038171003817119877 sdot 120601 minus (1119896eff) sdot 119865 sdot 1206011003817100381710038171003817
119877 + (1119896eff) sdot 119865lt 10minus8 (11)
The reported residual norm and relative error are10038171003817100381710038171003817100381710038171003817119877 sdot 120601 minus
1
119896effsdot 119865 sdot 120601
10038171003817100381710038171003817100381710038171003817
1003817100381710038171003817119877 sdot 120601 minus (1119896eff) sdot 119865 sdot 1206011003817100381710038171003817
1003817100381710038171003817(1119896eff) sdot 1206011003817100381710038171003817
(12)
respectively The fields marked as outer linear and inneriterations refer to the number of steps needed to attainthe requested tolerance in each layer of the Krylov-Schuralgorithm The computer used to solve the problem has anIntel i7 920 267GHz processor with 4Gb of RAM runningDebian GNULinux Wheezy
When using a finite volumes-based scheme over anunstructured mesh the solver has to gather information
about which cells are neighbors and which are not Currentlygmsh does not write this kind of lists in its output filesso milonga has to explicitly solve the neighbors problemPerforming a linear search is an 119874(119873
2) task which is
unacceptable for problem sizes 119873 of interest The code usesa search based on a 119896-dimensional tree [27] which can inprinciple be performed in119874(119873) steps Still for large values of119873 the time needed to read and parse the mesh (number twopreviously mentioned) is the bottleneck of the solution Thisstep is not needed in finite elements although the construc-tion of the elementary matrices involves the computationof the multidimensional Jacobians and integrals which thenhave to be assembled Again for large 119873 this step (numberthree) takes up most of the time needed to solve the problem
The Delaunay algorithm is a standard method for gen-erating two-dimensional grids [16] by tessellating a planewith triangles If the mesh needs to be based on quadranglesinstead a recombination algorithm can be used to transformtwo adjacent triangles in one quadrangle whenever is possi-ble However for geometries which are based on rectangularshapes there exist other algorithms [25] both for meshingand for recombining the triangles that give rise to elementaryentities with right angles almost everywhere which may be a
24 Science and Technology of Nuclear Installations
Tim
e con
sum
ed to
solv
e the
eige
nval
ue p
robl
em (s
econ
ds)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
10
30
100
300
1
3
Figure 24 Time needed to solve the eigenvalue problem versus number of unknowns
desired property of the resulting grid For finite elements thesteps needed to build the matrices depend on the selectionof triangles or quadrangles as the basic elementary geometrybecause the shape functions change However the number ofunknowns is the same as the number of nodes that does notchange after a recombination procedure On the other handthe number of unknowns in the finite volumes schemes withtriangles is roughly twice as the number of unknowns withquadrangles for the same grid
Figure 19 shows how the computed static reactivity (ie1 minus 1119896eff) depends on the number of unknowns for eachof the sixteen combinations of geometry algorithm shapescheme The four accepted results published in the originalreference [10] are included for reference although it shouldbe taken into account that said reactivities were computedalmost forty years ago Figure 20 shows the total wall timeneeded to solve the problem as a function of 119873119866 whileFigures 21 22 23 and 24 show the times needed for eachof the first four steps involved in the solution maintainingthe same logarithmic scale for both the abscissas and theordinates Green data represent finite volumes whilst bluebullets correspond to finite elements Solid lines indicatequarter-core and dashed lines eighth-symmetry Fillet bullets
are results obtained by theDelaunay triangulation and emptybullets were computed with the delquad algorithm Finallytriangle-shaped data correspond to triangles and squares anddiamonds denote quadrangles as the basic geometry of thegrid
It can be seen that finite elements produce amuch smallerdispersion of eigenvalues 119896eff than finite volumes with therefinement of the mesh This can be explained because finitevolumes methods rely on a geometric condition of the meshwhich may change abruptly if the meshing algorithm decidesto allocate the cells in a rather different form for small changesin ℓ119888 Finite elements methods are less influenced by thesediscontinuous lay-out changes of the elements As expectedeighth-core symmetries give almost the same results as thequarter-core geometries with roughly half the unknownsFor small problems finite volumes run faster that finiteelements because the time needed to solve the neighborproblem is negligible When the problem size grows thistime increases and exceeds the overhead implied in the con-struction and assembly of the finite elements matrices Alsoat least for this configuration it is seen that the eigenvalueproblem is solved faster for finite volumes than for finiteelements
Science and Technology of Nuclear Installations 25
4 Conclusions
Unstructured grids provide the cognizant engineer witha wide variety of possibilities to deal with the design oranalysis of nuclear reactor cores These kinds of grids cansuccessfully reproduce continuous geometries commonlyfound in reactor cores such as cylinders and thereforenot only can the diffusion equation be better approximatedinside the domain of definition but also the fulfillment ofboundary conditions is improved A free computer codewas written from scratch that is able to completely solvethe 2D IAEA PWR Benchmark using unstructured gridsfor sixteen combinations of geometry meshing algorithmbasic shape and discretization scheme plus any value ofthe gridrsquos characteristic length by using a single input fileThe complete set of input files and codemdashexecutable andsourcemdashis available either online or upon request withcomments experiences suggestions and corrections beingmore than welcome Further development should includetackling full three-dimensional geometries with completethermal hydraulic feedback in order to analyze how thesolutions of the coupled neutronic-thermal problem dependon the spatial discretization scheme of the neutron leakageterm Parallelization of the computation and assembly of thematrices and of the solution of the eigenvalue problem andits implementation using GPUs are also desired features toimplement A problemwith direct applications that the futureversions of milonga ought to solve is the analysis of how thegeometry of the absorbing materials should be taken intoaccount in structured coarse grids in order to mitigate effectssuch as the rod-cusp problem
Suitable schemes for approximating the continuous dif-ferential operators by discrete matrix expressions includefinite volumes and finite elements families Finite volumesmethods compute cell mean values whilst finite elementsgive functional values at the gridrsquos nodes In general finiteelements are less sensitive to changes in the mesh so theresults they provide do not change significantly for differentmeshing algorithms or elementary shapes Small problemsare best solved by finite volumes as the neighbor-findingproblem is faster than the process of building and assemblingthe eigenvalue-problem matrices For a large number ofunknowns the process of finding which cell is neighbor ofwhichmdasheven based on a 119896-dimensional treemdashoverwhelmsthe computation and assembly of elementary matrices andfinite-element methods perform better
References
[1] G Glasstone and S Bell Nuclear Reactor Theory KriegerPublishing Company 1970
[2] J J Duderstadt and L J Hamilton Nuclear Reactor AnalysisJohn Wiley amp Sons New York NY USA 1976
[3] C Gho Reactor Physics Undergraduate Course Lecture NotesInstituto Balseiro 2006
[4] G G Theler and F J Bonetto ldquoOn the stability of the pointreactor kinetics equationsrdquoNuclear Engineering andDesign vol240 no 6 pp 1443ndash1449 2010
[5] E E Lewis andW F Miller Computational Methods of NeutronTransport John Wiley amp Sons 1984
[6] E Masiello ldquoAnalytical stability analysis of Coarse-Mesh finitedifference methodrdquo in Proceedings of the International Confer-ence on the Physics of Reactors (PHYSOR rsquo08) Nuclear Power ASustainable Resource pp 456ndash464 September 2008
[7] M L Zerkle Development of a polynomial nodal methodwith flux and current discontinuity factors [PhD thesis] Mas-sachusets Institute of Technology 1992
[8] J I Yoon andHG Joo ldquoTwo-level coarsemesh finite differenceformulation with multigroup source expansion nodal kernelsrdquoJournal of Nuclear Science andTechnology vol 45 no 7 pp 668ndash682 2008
[9] O Mazzantini M Schivo J Di Cesare R Garbero M Riveroand G Theler ldquoA coupled calculation suite for Atucha II oper-ational transients analysisrdquo Science and Technology of NuclearInstallations vol 2011 Article ID 785304 12 pages 2011
[10] Computational Benchmark Problem Comitee for the Mathe-matics and Computation Division of the American NuclearSociety ldquoArgonne Code Center Benchmark problem bookrdquoTech Rep ANL-7416 Supplement 2 Argonne National Labo-ratory 1977
[11] A F Henry Nuclear Reactor Analysis MIT Cambridge UK1975
[12] O C Zienkiewicz R L Taylor and J Z ZhuTheFinite ElementMethod Its Basis and Fundamentals vol 1 Elsevier 6th edition2005
[13] G Theler Difusion de neutrones en mallas no estructuradascomparacion entre volumenes y elementos finitos AcademicMonograph Universidad de Buenos Aires 2013
[14] G Theler ldquoMilonga a free nuclear reactor core analysis coderdquoAvailable online 2011
[15] G Theler F J Bonetto and A Clausse ldquoOptimizacion deparametros en reactores de potencia base de diseno delcodigo neutronico milongardquo in Reunion Anual de la AsociacionArgentina de Tecnologia Nuclear XXXVII 2010
[16] C Geuzaine and J-F Remacle ldquoGmsh a 3-D finite elementmesh generator with built-in pre- and post-processing facili-tiesrdquo International Journal for Numerical Methods in Engineer-ing vol 79 no 11 pp 1309ndash1331 2009
[17] S Balay J Brown K Buschelman et al ldquoPETSc users manualrdquoTech Rep ANL-9511-Revision 34 Argonne National Labora-tory 2013
[18] S Balay W D Gropp L Curfman McInnes and B F SmithldquoEfficient management of parallelism in object oriented numer-ical software librariesrdquo in Modern Software Tools in ScientificComputing E Arge A M Bruaset and H P Langtangen Edspp 163ndash202 Birkhauser 1997
[19] V Hernandez J E Roman and V Vidal ldquoSLEPC a scalable andflexible toolkit for the solution of eigenvalue problemsrdquo ACMTransactions on Mathematical Software vol 31 no 3 pp 351ndash362 2005
[20] M Galassi J Davies J Theiler et al GNU Scientific LibraryReference Manual 3rd edition 2009
[21] O C Zienkiewicz and R L Taylor The Finite Element MethodFor Solid and Structural Mechanics vol 3 Elsevier 6th edition2005
[22] A Imelda and K Doddy Evaluation of the IAEA 3-D PWRbenchmark problem using NESTLE code 2004
[23] R Mosteller ldquoStatic benchmarking of the NESTLE advancednodal coderdquo in Proceedings of the Joint International Conference
26 Science and Technology of Nuclear Installations
on Mathematical Methods and Supercomputing for NuclearApplications vol 2 pp 1596ndash1605 1997
[24] GTheler F J Bonetto andAClausse ldquoSolution of the 2D IAEAPWR Benchmark with the neutronic code milongardquo in Actasde la Reunion Anual de la Asociacion Argentina de TecnologıaNuclear XXXVIII 2011
[25] J F Remacle F Henrotte T Carrier-Baudouin et al ldquoAfrontal delaunay quad mesh generator using the 119871
infin normrdquoInternational Journal For Numerical Methods in Engineeringvol 94 no 5 pp 494ndash512 2013
[26] G W Stewart ldquoA Krylov-Schur algorithm for large eigenprob-lemsrdquo SIAM Journal on Matrix Analysis and Applications vol23 no 3 pp 601ndash614 2002
[27] J L Bentley ldquoMultidimensional binary search trees used forassociative searchingrdquo Communications of the ACM vol 18 no9 pp 509ndash517 1975
TribologyAdvances in
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Submit your manuscripts athttpwwwhindawicom
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Science and Technology of Nuclear Installations 21
Table2
Resultsofthe2
DIAEA
PWRBe
nchm
arkp
roblem
obtained
with
them
ilong
acod
efor
thee
ightyp
ropo
sedcombinatio
nsofsymmetrym
eshing
algorithm
basicshapediscretization
schemeandcharacteris
ticelem
entc
elllengthTh
ereference
solutio
nis120588=28749times10minus5w
hich
correspo
ndsto119896eff=10296
Case
no
Symm-
etry
Mesh
algorithm
Basic
shape
Solutio
nmetho
dℓ119888
(cm)
Δ120588
(pcm
)max1206012
(mdash)
Total
unkn
owns
Outer
iter
Linear
iter
Inner
iter
Resid
ual
norm
Relativ
eerror
Error
estim
ate
Mem
ory
(Mb)
Page
faults
114
Delaunay
998779Vo
lumes
40
minus120
1145
8264
332
1033
6119890minus09
3119890minus09
1119890minus09
349470
214
Delaunay
998779Vo
lumes
30
85
1135
15740
332
1967
2119890minus08
1119890minus08
4119890minus09
4813401
314
Delaunay
998779Vo
lumes
20
197
1124
31188
332
3898
1119890minus08
6119890minus09
3119890minus09
7621932
414
Delaunay
998779Vo
lumes
10153
1123
127476
332
15934
7119890minus09
3119890minus09
3119890minus09
273
81886
514
Delaunay
998779Vo
lumes
05
187
1119
510676
332
63834
2119890minus09
9119890minus10
1119890minus09
1148
352261
614
Delaunay
998779Elem
ents
40
173
1100
4308
332
538
2119890minus08
8119890minus09
4119890minus09
318794
714
Delaunay
998779Elem
ents
30
117
1107
8108
332
1013
5119890minus09
2119890minus09
2119890minus09
4712946
814
Delaunay
998779Elem
ents
20
84
1112
15936
332
1992
6119890minus09
3119890minus09
2119890minus09
7321347
914
Delaunay
998779Elem
ents
1062
1116
64478
332
8059
6119890minus09
3119890minus09
4119890minus09
262
77105
1014
Delaunay
998779Elem
ents
05
58
1117
256700
332
32087
1119890minus09
5119890minus10
1119890minus09
1091
331969
1114
Delaunay
◻Vo
lumes
40
53
1149
5042
332
630
2119890minus08
1119890minus08
5119890minus09
288007
1214
Delaunay
◻Vo
lumes
30
346
1133
8560
332
1070
7119890minus09
3119890minus09
2119890minus09
3910895
1314
Delaunay
◻Vo
lumes
20
308
1131
15576
332
1947
1119890minus08
7119890minus09
3119890minus09
541564
514
14Delaunay
◻Vo
lumes
10177
1122
60774
332
7596
8119890minus09
4119890minus09
3119890minus09
167
50115
1514
Delaunay
◻Vo
lumes
05
199
1120
244936
332
30617
4119890minus09
2119890minus09
2119890minus09
698
215678
1614
Delaunay
◻Elem
ents
40
196
1100
5222
332
652
3119890minus08
1119890minus08
8119890minus09
4713098
1714
Delaunay
◻Elem
ents
30
123
1107
8810
332
1101
2119890minus08
9119890minus09
7119890minus09
6918598
1814
Delaunay
◻Elem
ents
20
901112
15922
332
1990
1119890minus08
5119890minus09
5119890minus09
107
30591
1914
Delaunay
◻Elem
ents
1063
1116
61456
332
7682
5119890minus09
2119890minus09
4119890minus09
389
113237
2014
Delaunay
◻Elem
ents
05
58
1117
246332
332
30791
8119890minus10
4119890minus10
1119890minus09
1676
498192
2114
Delq
uad
998779Vo
lumes
40
minus45
1144
6228
332
778
4119890minus08
2119890minus08
7119890minus09
308495
2214
Delq
uad
998779Vo
lumes
30
570
1153
11820
332
1477
3119890minus08
1119890minus08
4119890minus09
4010879
2314
Delq
uad
998779Vo
lumes
20
459
1103
24100
332
3012
2119890minus08
1119890minus08
5119890minus09
6217715
2414
Delq
uad
998779Vo
lumes
10512
1104
9640
03
3212050
2119890minus08
1119890minus08
9119890minus09
207
61714
2514
Delq
uad
998779Vo
lumes
05
528
1106
385600
332
48200
2119890minus09
1119890minus09
2119890minus09
838
255735
2614
Delqu
ad998779
Elem
ents
40
220
1093
3288
332
411
3119890minus08
2119890minus08
6119890minus09
308495
2714
Delq
uad
998779Elem
ents
30
140
1104
6148
332
768
4119890minus08
2119890minus08
8119890minus09
39110
0028
14Delq
uad
998779Elem
ents
20
82
1110
12392
332
1549
2119890minus08
9119890minus09
6119890minus09
6117067
2914
Delqu
ad998779
Elem
ents
1063
1115
48882
332
6110
2119890minus09
1119890minus09
1119890minus09
197
5804
630
14Delq
uad
998779Elem
ents
05
58
1116
194162
332
24270
2119890minus09
9119890minus10
2119890minus09
790
238769
3114
Delq
uad
◻Vo
lumes
40
minus416
1174
3132
332
391
5119890minus08
3119890minus08
8119890minus09
267322
3214
Delq
uad
◻Vo
lumes
30
minus248
1151
5922
332
740
9119890minus09
4119890minus09
2119890minus09
318779
3314
Delq
uad
◻Vo
lumes
20
minus132
1139
12050
332
1506
1119890minus08
7119890minus09
4119890minus09
4412239
3414
Delq
uad
◻Vo
lumes
10minus49
1126
48200
332
6025
6119890minus09
3119890minus09
2119890minus09
122
36094
3514
Delq
uad
◻Vo
lumes
05
minus23
1123
192800
332
24100
5119890minus09
2119890minus09
3119890minus09
464
140228
3614
Delq
uad
◻Elem
ents
40
224
1093
3294
332
411
3119890minus08
1119890minus08
7119890minus09
359852
3714
Delq
uad
◻Elem
ents
30
141
1104
6144
332
768
3119890minus08
2119890minus08
9119890minus09
5114018
3814
Delq
uad
◻Elem
ents
20
921111
12392
332
1549
1119890minus08
6119890minus09
5119890minus09
8122526
3914
Delq
uad
◻Elem
ents
1065
1115
48882
332
6110
5119890minus09
3119890minus09
4119890minus09
276
78431
22 Science and Technology of Nuclear Installations
Table2Con
tinued
Case
no
Symm-
etry
Mesh
algorithm
Basic
shape
Solutio
nmetho
dℓ119888
(cm)
Δ120588
(pcm
)max1206012
(mdash)
Total
unkn
owns
Outer
iter
Linear
iter
Inner
iter
Resid
ual
norm
Relativ
eerror
Error
estim
ate
Mem
ory
(Mb)
Page
faults
4014
Delq
uad
◻Elem
ents
05
59
1117
194162
332
24270
1119890minus09
5119890minus10
1119890minus09
1104
318967
4118
Delaunay
998779Vo
lumes
40
minus99
1139
4228
332
528
4119890minus12
2119890minus12
1119890minus12
359630
4218
Delaunay
998779Vo
lumes
30
50
1132
7712
332
964
1119890minus11
6119890minus12
2119890minus12
4211578
4318
Delaunay
998779Vo
lumes
20
177
1120
15776
332
1972
2119890minus12
1119890minus12
7119890minus13
5515436
4418
Delaunay
998779Vo
lumes
10142
1118
63902
332
7987
3119890minus12
1119890minus12
1119890minus12
160
46691
4518
Delaunay
998779Vo
lumes
05
176
1115
254612
332
31826
2119890minus12
8119890minus13
1119890minus12
555
168526
4618
Delaunay
998779Elem
ents
40
183
1096
2252
332
281
6119890minus12
3119890minus12
2119890minus12
359589
4718
Delaunay
998779Elem
ents
30
119
1104
4040
224
505
2119890minus08
1119890minus08
7119890minus09
41114
3748
18Delaunay
998779Elem
ents
20
85
1108
8156
332
1019
2119890minus12
8119890minus13
6119890minus13
5515053
4918
Delaunay
998779Elem
ents
1063
1112
32480
332
4060
1119890minus12
6119890minus13
8119890minus13
151
43671
5018
Delaunay
998779Elem
ents
05
58
1113
128358
332
1604
47119890minus13
3119890minus13
7119890minus13
527
157581
5118
Delaunay
◻Vo
lumes
40
37
1144
2380
224
297
5119890minus08
2119890minus08
1119890minus08
5314173
5218
Delaunay
◻Vo
lumes
30
218
1120
4194
332
524
7119890minus12
4119890minus12
2119890minus12
3610034
5318
Delaunay
◻Vo
lumes
20
345
1110
7960
332
995
5119890minus12
3119890minus12
1119890minus12
4712879
5418
Delaunay
◻Vo
lumes
10165
1117
30652
332
3831
2119890minus12
1119890minus12
7119890minus13
100
28851
5518
Delaunay
◻Vo
lumes
05
191
1117
122066
224
15258
2119890minus08
8119890minus09
9119890minus09
343
103825
5618
Delaunay
◻Elem
ents
40
175
1097
2526
332
315
4119890minus12
2119890minus12
1119890minus12
6016159
5718
Delaunay
◻Elem
ents
30
114
1104
4386
332
548
4119890minus12
2119890minus12
2119890minus12
5013768
5818
Delaunay
◻Elem
ents
20
901108
8234
332
1029
2119890minus12
1119890minus12
1119890minus12
7320094
5918
Delaunay
◻Elem
ents
1062
1112
31186
224
3898
2119890minus08
7119890minus09
9119890minus09
208
59589
6018
Delaunay
◻Elem
ents
05
58
1113
123122
224
15390
9119890minus09
5119890minus09
1119890minus08
807
2366
7061
18Delq
uad
998779Vo
lumes
40
minus210
1168
3224
332
403
1119890minus11
5119890minus12
2119890minus12
43117
8462
18Delq
uad
998779Vo
lumes
30
369
1150
6000
332
750
1119890minus11
6119890minus12
3119890minus12
5013502
6318
Delq
uad
998779Vo
lumes
20
312
1104
12316
332
1539
2119890minus11
8119890minus12
5119890minus12
6016395
6418
Delq
uad
998779Vo
lumes
1044
01094
48714
332
6089
9119890minus12
4119890minus12
4119890minus12
1183444
165
18Delq
uad
998779Vo
lumes
05
534
1081
193980
332
24247
5119890minus12
2119890minus12
4119890minus12
423
126768
6618
Delq
uad
998779Elem
ents
40
240
1090
1750
332
218
9119890minus12
4119890minus12
2119890minus12
43117
8667
18Delqu
ad998779
Elem
ents
30
150
1100
3184
332
398
1119890minus11
5119890minus12
2119890minus12
5013509
6818
Delq
uad
998779Elem
ents
20
89
1106
6426
332
803
5119890minus12
3119890minus12
2119890minus12
5916047
6918
Delq
uad
998779Elem
ents
1064
1111
24886
332
3110
2119890minus12
1119890minus12
1119890minus12
113
32788
7018
Delq
uad
998779Elem
ents
05
59
1113
98052
224
12256
1119890minus08
5119890minus09
8119890minus09
398
118153
7118
Delq
uad
◻Vo
lumes
40
minus398
1172
1650
332
206
1119890minus11
5119890minus12
2119890minus12
369907
7218
Delq
uad
◻Vo
lumes
30
minus249
1146
3058
332
382
4119890minus12
2119890minus12
1119890minus12
349349
7318
Delq
uad
◻Vo
lumes
20
minus131
1136
6228
224
778
3119890minus08
1119890minus08
5119890minus09
4010930
7418
Delq
uad
◻Vo
lumes
10minus51
1123
24536
224
3067
3119890minus08
2119890minus08
1119890minus08
7822716
7618
Delq
uad
◻Elem
ents
40
237
1090
1752
332
219
6119890minus12
3119890minus12
2119890minus12
4111213
7718
Delq
uad
◻Elem
ents
30
145
1101
3184
332
398
6119890minus12
3119890minus12
2119890minus12
43118
1178
18Delq
uad
◻Elem
ents
20
941107
6426
332
803
3119890minus12
1119890minus12
1119890minus12
6016220
7918
Delqu
ad◻
Elem
ents
1065
1111
24874
332
3109
1119890minus12
6119890minus13
8119890minus13
156
4400
080
18Delq
uad
◻Elem
ents
05
59
1113
9804
22
2412255
8119890minus09
4119890minus09
8119890minus09
567
162168
Science and Technology of Nuclear Installations 23
Tim
e con
sum
ed to
bui
ld th
e mat
rices
(sec
onds
)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay trianglesvolumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
10
30
100
300
1
3
Figure 23 Time needed to build the matrices versus number of unknowns
was considerably larger than the solution time itself Theeigenvalue problem was solved using a multilayer iterativeKrylov-Schur method [26] with a tolerance relative to thematrices norm
1003817100381710038171003817119877 sdot 120601 minus (1119896eff) sdot 119865 sdot 1206011003817100381710038171003817
119877 + (1119896eff) sdot 119865lt 10minus8 (11)
The reported residual norm and relative error are10038171003817100381710038171003817100381710038171003817119877 sdot 120601 minus
1
119896effsdot 119865 sdot 120601
10038171003817100381710038171003817100381710038171003817
1003817100381710038171003817119877 sdot 120601 minus (1119896eff) sdot 119865 sdot 1206011003817100381710038171003817
1003817100381710038171003817(1119896eff) sdot 1206011003817100381710038171003817
(12)
respectively The fields marked as outer linear and inneriterations refer to the number of steps needed to attainthe requested tolerance in each layer of the Krylov-Schuralgorithm The computer used to solve the problem has anIntel i7 920 267GHz processor with 4Gb of RAM runningDebian GNULinux Wheezy
When using a finite volumes-based scheme over anunstructured mesh the solver has to gather information
about which cells are neighbors and which are not Currentlygmsh does not write this kind of lists in its output filesso milonga has to explicitly solve the neighbors problemPerforming a linear search is an 119874(119873
2) task which is
unacceptable for problem sizes 119873 of interest The code usesa search based on a 119896-dimensional tree [27] which can inprinciple be performed in119874(119873) steps Still for large values of119873 the time needed to read and parse the mesh (number twopreviously mentioned) is the bottleneck of the solution Thisstep is not needed in finite elements although the construc-tion of the elementary matrices involves the computationof the multidimensional Jacobians and integrals which thenhave to be assembled Again for large 119873 this step (numberthree) takes up most of the time needed to solve the problem
The Delaunay algorithm is a standard method for gen-erating two-dimensional grids [16] by tessellating a planewith triangles If the mesh needs to be based on quadranglesinstead a recombination algorithm can be used to transformtwo adjacent triangles in one quadrangle whenever is possi-ble However for geometries which are based on rectangularshapes there exist other algorithms [25] both for meshingand for recombining the triangles that give rise to elementaryentities with right angles almost everywhere which may be a
24 Science and Technology of Nuclear Installations
Tim
e con
sum
ed to
solv
e the
eige
nval
ue p
robl
em (s
econ
ds)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
10
30
100
300
1
3
Figure 24 Time needed to solve the eigenvalue problem versus number of unknowns
desired property of the resulting grid For finite elements thesteps needed to build the matrices depend on the selectionof triangles or quadrangles as the basic elementary geometrybecause the shape functions change However the number ofunknowns is the same as the number of nodes that does notchange after a recombination procedure On the other handthe number of unknowns in the finite volumes schemes withtriangles is roughly twice as the number of unknowns withquadrangles for the same grid
Figure 19 shows how the computed static reactivity (ie1 minus 1119896eff) depends on the number of unknowns for eachof the sixteen combinations of geometry algorithm shapescheme The four accepted results published in the originalreference [10] are included for reference although it shouldbe taken into account that said reactivities were computedalmost forty years ago Figure 20 shows the total wall timeneeded to solve the problem as a function of 119873119866 whileFigures 21 22 23 and 24 show the times needed for eachof the first four steps involved in the solution maintainingthe same logarithmic scale for both the abscissas and theordinates Green data represent finite volumes whilst bluebullets correspond to finite elements Solid lines indicatequarter-core and dashed lines eighth-symmetry Fillet bullets
are results obtained by theDelaunay triangulation and emptybullets were computed with the delquad algorithm Finallytriangle-shaped data correspond to triangles and squares anddiamonds denote quadrangles as the basic geometry of thegrid
It can be seen that finite elements produce amuch smallerdispersion of eigenvalues 119896eff than finite volumes with therefinement of the mesh This can be explained because finitevolumes methods rely on a geometric condition of the meshwhich may change abruptly if the meshing algorithm decidesto allocate the cells in a rather different form for small changesin ℓ119888 Finite elements methods are less influenced by thesediscontinuous lay-out changes of the elements As expectedeighth-core symmetries give almost the same results as thequarter-core geometries with roughly half the unknownsFor small problems finite volumes run faster that finiteelements because the time needed to solve the neighborproblem is negligible When the problem size grows thistime increases and exceeds the overhead implied in the con-struction and assembly of the finite elements matrices Alsoat least for this configuration it is seen that the eigenvalueproblem is solved faster for finite volumes than for finiteelements
Science and Technology of Nuclear Installations 25
4 Conclusions
Unstructured grids provide the cognizant engineer witha wide variety of possibilities to deal with the design oranalysis of nuclear reactor cores These kinds of grids cansuccessfully reproduce continuous geometries commonlyfound in reactor cores such as cylinders and thereforenot only can the diffusion equation be better approximatedinside the domain of definition but also the fulfillment ofboundary conditions is improved A free computer codewas written from scratch that is able to completely solvethe 2D IAEA PWR Benchmark using unstructured gridsfor sixteen combinations of geometry meshing algorithmbasic shape and discretization scheme plus any value ofthe gridrsquos characteristic length by using a single input fileThe complete set of input files and codemdashexecutable andsourcemdashis available either online or upon request withcomments experiences suggestions and corrections beingmore than welcome Further development should includetackling full three-dimensional geometries with completethermal hydraulic feedback in order to analyze how thesolutions of the coupled neutronic-thermal problem dependon the spatial discretization scheme of the neutron leakageterm Parallelization of the computation and assembly of thematrices and of the solution of the eigenvalue problem andits implementation using GPUs are also desired features toimplement A problemwith direct applications that the futureversions of milonga ought to solve is the analysis of how thegeometry of the absorbing materials should be taken intoaccount in structured coarse grids in order to mitigate effectssuch as the rod-cusp problem
Suitable schemes for approximating the continuous dif-ferential operators by discrete matrix expressions includefinite volumes and finite elements families Finite volumesmethods compute cell mean values whilst finite elementsgive functional values at the gridrsquos nodes In general finiteelements are less sensitive to changes in the mesh so theresults they provide do not change significantly for differentmeshing algorithms or elementary shapes Small problemsare best solved by finite volumes as the neighbor-findingproblem is faster than the process of building and assemblingthe eigenvalue-problem matrices For a large number ofunknowns the process of finding which cell is neighbor ofwhichmdasheven based on a 119896-dimensional treemdashoverwhelmsthe computation and assembly of elementary matrices andfinite-element methods perform better
References
[1] G Glasstone and S Bell Nuclear Reactor Theory KriegerPublishing Company 1970
[2] J J Duderstadt and L J Hamilton Nuclear Reactor AnalysisJohn Wiley amp Sons New York NY USA 1976
[3] C Gho Reactor Physics Undergraduate Course Lecture NotesInstituto Balseiro 2006
[4] G G Theler and F J Bonetto ldquoOn the stability of the pointreactor kinetics equationsrdquoNuclear Engineering andDesign vol240 no 6 pp 1443ndash1449 2010
[5] E E Lewis andW F Miller Computational Methods of NeutronTransport John Wiley amp Sons 1984
[6] E Masiello ldquoAnalytical stability analysis of Coarse-Mesh finitedifference methodrdquo in Proceedings of the International Confer-ence on the Physics of Reactors (PHYSOR rsquo08) Nuclear Power ASustainable Resource pp 456ndash464 September 2008
[7] M L Zerkle Development of a polynomial nodal methodwith flux and current discontinuity factors [PhD thesis] Mas-sachusets Institute of Technology 1992
[8] J I Yoon andHG Joo ldquoTwo-level coarsemesh finite differenceformulation with multigroup source expansion nodal kernelsrdquoJournal of Nuclear Science andTechnology vol 45 no 7 pp 668ndash682 2008
[9] O Mazzantini M Schivo J Di Cesare R Garbero M Riveroand G Theler ldquoA coupled calculation suite for Atucha II oper-ational transients analysisrdquo Science and Technology of NuclearInstallations vol 2011 Article ID 785304 12 pages 2011
[10] Computational Benchmark Problem Comitee for the Mathe-matics and Computation Division of the American NuclearSociety ldquoArgonne Code Center Benchmark problem bookrdquoTech Rep ANL-7416 Supplement 2 Argonne National Labo-ratory 1977
[11] A F Henry Nuclear Reactor Analysis MIT Cambridge UK1975
[12] O C Zienkiewicz R L Taylor and J Z ZhuTheFinite ElementMethod Its Basis and Fundamentals vol 1 Elsevier 6th edition2005
[13] G Theler Difusion de neutrones en mallas no estructuradascomparacion entre volumenes y elementos finitos AcademicMonograph Universidad de Buenos Aires 2013
[14] G Theler ldquoMilonga a free nuclear reactor core analysis coderdquoAvailable online 2011
[15] G Theler F J Bonetto and A Clausse ldquoOptimizacion deparametros en reactores de potencia base de diseno delcodigo neutronico milongardquo in Reunion Anual de la AsociacionArgentina de Tecnologia Nuclear XXXVII 2010
[16] C Geuzaine and J-F Remacle ldquoGmsh a 3-D finite elementmesh generator with built-in pre- and post-processing facili-tiesrdquo International Journal for Numerical Methods in Engineer-ing vol 79 no 11 pp 1309ndash1331 2009
[17] S Balay J Brown K Buschelman et al ldquoPETSc users manualrdquoTech Rep ANL-9511-Revision 34 Argonne National Labora-tory 2013
[18] S Balay W D Gropp L Curfman McInnes and B F SmithldquoEfficient management of parallelism in object oriented numer-ical software librariesrdquo in Modern Software Tools in ScientificComputing E Arge A M Bruaset and H P Langtangen Edspp 163ndash202 Birkhauser 1997
[19] V Hernandez J E Roman and V Vidal ldquoSLEPC a scalable andflexible toolkit for the solution of eigenvalue problemsrdquo ACMTransactions on Mathematical Software vol 31 no 3 pp 351ndash362 2005
[20] M Galassi J Davies J Theiler et al GNU Scientific LibraryReference Manual 3rd edition 2009
[21] O C Zienkiewicz and R L Taylor The Finite Element MethodFor Solid and Structural Mechanics vol 3 Elsevier 6th edition2005
[22] A Imelda and K Doddy Evaluation of the IAEA 3-D PWRbenchmark problem using NESTLE code 2004
[23] R Mosteller ldquoStatic benchmarking of the NESTLE advancednodal coderdquo in Proceedings of the Joint International Conference
26 Science and Technology of Nuclear Installations
on Mathematical Methods and Supercomputing for NuclearApplications vol 2 pp 1596ndash1605 1997
[24] GTheler F J Bonetto andAClausse ldquoSolution of the 2D IAEAPWR Benchmark with the neutronic code milongardquo in Actasde la Reunion Anual de la Asociacion Argentina de TecnologıaNuclear XXXVIII 2011
[25] J F Remacle F Henrotte T Carrier-Baudouin et al ldquoAfrontal delaunay quad mesh generator using the 119871
infin normrdquoInternational Journal For Numerical Methods in Engineeringvol 94 no 5 pp 494ndash512 2013
[26] G W Stewart ldquoA Krylov-Schur algorithm for large eigenprob-lemsrdquo SIAM Journal on Matrix Analysis and Applications vol23 no 3 pp 601ndash614 2002
[27] J L Bentley ldquoMultidimensional binary search trees used forassociative searchingrdquo Communications of the ACM vol 18 no9 pp 509ndash517 1975
TribologyAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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FuelsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofPetroleum Engineering
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Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
CombustionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Renewable Energy
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
StructuresJournal of
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear InstallationsScience and Technology of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solar EnergyJournal of
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Wind EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear EnergyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
22 Science and Technology of Nuclear Installations
Table2Con
tinued
Case
no
Symm-
etry
Mesh
algorithm
Basic
shape
Solutio
nmetho
dℓ119888
(cm)
Δ120588
(pcm
)max1206012
(mdash)
Total
unkn
owns
Outer
iter
Linear
iter
Inner
iter
Resid
ual
norm
Relativ
eerror
Error
estim
ate
Mem
ory
(Mb)
Page
faults
4014
Delq
uad
◻Elem
ents
05
59
1117
194162
332
24270
1119890minus09
5119890minus10
1119890minus09
1104
318967
4118
Delaunay
998779Vo
lumes
40
minus99
1139
4228
332
528
4119890minus12
2119890minus12
1119890minus12
359630
4218
Delaunay
998779Vo
lumes
30
50
1132
7712
332
964
1119890minus11
6119890minus12
2119890minus12
4211578
4318
Delaunay
998779Vo
lumes
20
177
1120
15776
332
1972
2119890minus12
1119890minus12
7119890minus13
5515436
4418
Delaunay
998779Vo
lumes
10142
1118
63902
332
7987
3119890minus12
1119890minus12
1119890minus12
160
46691
4518
Delaunay
998779Vo
lumes
05
176
1115
254612
332
31826
2119890minus12
8119890minus13
1119890minus12
555
168526
4618
Delaunay
998779Elem
ents
40
183
1096
2252
332
281
6119890minus12
3119890minus12
2119890minus12
359589
4718
Delaunay
998779Elem
ents
30
119
1104
4040
224
505
2119890minus08
1119890minus08
7119890minus09
41114
3748
18Delaunay
998779Elem
ents
20
85
1108
8156
332
1019
2119890minus12
8119890minus13
6119890minus13
5515053
4918
Delaunay
998779Elem
ents
1063
1112
32480
332
4060
1119890minus12
6119890minus13
8119890minus13
151
43671
5018
Delaunay
998779Elem
ents
05
58
1113
128358
332
1604
47119890minus13
3119890minus13
7119890minus13
527
157581
5118
Delaunay
◻Vo
lumes
40
37
1144
2380
224
297
5119890minus08
2119890minus08
1119890minus08
5314173
5218
Delaunay
◻Vo
lumes
30
218
1120
4194
332
524
7119890minus12
4119890minus12
2119890minus12
3610034
5318
Delaunay
◻Vo
lumes
20
345
1110
7960
332
995
5119890minus12
3119890minus12
1119890minus12
4712879
5418
Delaunay
◻Vo
lumes
10165
1117
30652
332
3831
2119890minus12
1119890minus12
7119890minus13
100
28851
5518
Delaunay
◻Vo
lumes
05
191
1117
122066
224
15258
2119890minus08
8119890minus09
9119890minus09
343
103825
5618
Delaunay
◻Elem
ents
40
175
1097
2526
332
315
4119890minus12
2119890minus12
1119890minus12
6016159
5718
Delaunay
◻Elem
ents
30
114
1104
4386
332
548
4119890minus12
2119890minus12
2119890minus12
5013768
5818
Delaunay
◻Elem
ents
20
901108
8234
332
1029
2119890minus12
1119890minus12
1119890minus12
7320094
5918
Delaunay
◻Elem
ents
1062
1112
31186
224
3898
2119890minus08
7119890minus09
9119890minus09
208
59589
6018
Delaunay
◻Elem
ents
05
58
1113
123122
224
15390
9119890minus09
5119890minus09
1119890minus08
807
2366
7061
18Delq
uad
998779Vo
lumes
40
minus210
1168
3224
332
403
1119890minus11
5119890minus12
2119890minus12
43117
8462
18Delq
uad
998779Vo
lumes
30
369
1150
6000
332
750
1119890minus11
6119890minus12
3119890minus12
5013502
6318
Delq
uad
998779Vo
lumes
20
312
1104
12316
332
1539
2119890minus11
8119890minus12
5119890minus12
6016395
6418
Delq
uad
998779Vo
lumes
1044
01094
48714
332
6089
9119890minus12
4119890minus12
4119890minus12
1183444
165
18Delq
uad
998779Vo
lumes
05
534
1081
193980
332
24247
5119890minus12
2119890minus12
4119890minus12
423
126768
6618
Delq
uad
998779Elem
ents
40
240
1090
1750
332
218
9119890minus12
4119890minus12
2119890minus12
43117
8667
18Delqu
ad998779
Elem
ents
30
150
1100
3184
332
398
1119890minus11
5119890minus12
2119890minus12
5013509
6818
Delq
uad
998779Elem
ents
20
89
1106
6426
332
803
5119890minus12
3119890minus12
2119890minus12
5916047
6918
Delq
uad
998779Elem
ents
1064
1111
24886
332
3110
2119890minus12
1119890minus12
1119890minus12
113
32788
7018
Delq
uad
998779Elem
ents
05
59
1113
98052
224
12256
1119890minus08
5119890minus09
8119890minus09
398
118153
7118
Delq
uad
◻Vo
lumes
40
minus398
1172
1650
332
206
1119890minus11
5119890minus12
2119890minus12
369907
7218
Delq
uad
◻Vo
lumes
30
minus249
1146
3058
332
382
4119890minus12
2119890minus12
1119890minus12
349349
7318
Delq
uad
◻Vo
lumes
20
minus131
1136
6228
224
778
3119890minus08
1119890minus08
5119890minus09
4010930
7418
Delq
uad
◻Vo
lumes
10minus51
1123
24536
224
3067
3119890minus08
2119890minus08
1119890minus08
7822716
7618
Delq
uad
◻Elem
ents
40
237
1090
1752
332
219
6119890minus12
3119890minus12
2119890minus12
4111213
7718
Delq
uad
◻Elem
ents
30
145
1101
3184
332
398
6119890minus12
3119890minus12
2119890minus12
43118
1178
18Delq
uad
◻Elem
ents
20
941107
6426
332
803
3119890minus12
1119890minus12
1119890minus12
6016220
7918
Delqu
ad◻
Elem
ents
1065
1111
24874
332
3109
1119890minus12
6119890minus13
8119890minus13
156
4400
080
18Delq
uad
◻Elem
ents
05
59
1113
9804
22
2412255
8119890minus09
4119890minus09
8119890minus09
567
162168
Science and Technology of Nuclear Installations 23
Tim
e con
sum
ed to
bui
ld th
e mat
rices
(sec
onds
)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay trianglesvolumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
10
30
100
300
1
3
Figure 23 Time needed to build the matrices versus number of unknowns
was considerably larger than the solution time itself Theeigenvalue problem was solved using a multilayer iterativeKrylov-Schur method [26] with a tolerance relative to thematrices norm
1003817100381710038171003817119877 sdot 120601 minus (1119896eff) sdot 119865 sdot 1206011003817100381710038171003817
119877 + (1119896eff) sdot 119865lt 10minus8 (11)
The reported residual norm and relative error are10038171003817100381710038171003817100381710038171003817119877 sdot 120601 minus
1
119896effsdot 119865 sdot 120601
10038171003817100381710038171003817100381710038171003817
1003817100381710038171003817119877 sdot 120601 minus (1119896eff) sdot 119865 sdot 1206011003817100381710038171003817
1003817100381710038171003817(1119896eff) sdot 1206011003817100381710038171003817
(12)
respectively The fields marked as outer linear and inneriterations refer to the number of steps needed to attainthe requested tolerance in each layer of the Krylov-Schuralgorithm The computer used to solve the problem has anIntel i7 920 267GHz processor with 4Gb of RAM runningDebian GNULinux Wheezy
When using a finite volumes-based scheme over anunstructured mesh the solver has to gather information
about which cells are neighbors and which are not Currentlygmsh does not write this kind of lists in its output filesso milonga has to explicitly solve the neighbors problemPerforming a linear search is an 119874(119873
2) task which is
unacceptable for problem sizes 119873 of interest The code usesa search based on a 119896-dimensional tree [27] which can inprinciple be performed in119874(119873) steps Still for large values of119873 the time needed to read and parse the mesh (number twopreviously mentioned) is the bottleneck of the solution Thisstep is not needed in finite elements although the construc-tion of the elementary matrices involves the computationof the multidimensional Jacobians and integrals which thenhave to be assembled Again for large 119873 this step (numberthree) takes up most of the time needed to solve the problem
The Delaunay algorithm is a standard method for gen-erating two-dimensional grids [16] by tessellating a planewith triangles If the mesh needs to be based on quadranglesinstead a recombination algorithm can be used to transformtwo adjacent triangles in one quadrangle whenever is possi-ble However for geometries which are based on rectangularshapes there exist other algorithms [25] both for meshingand for recombining the triangles that give rise to elementaryentities with right angles almost everywhere which may be a
24 Science and Technology of Nuclear Installations
Tim
e con
sum
ed to
solv
e the
eige
nval
ue p
robl
em (s
econ
ds)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
10
30
100
300
1
3
Figure 24 Time needed to solve the eigenvalue problem versus number of unknowns
desired property of the resulting grid For finite elements thesteps needed to build the matrices depend on the selectionof triangles or quadrangles as the basic elementary geometrybecause the shape functions change However the number ofunknowns is the same as the number of nodes that does notchange after a recombination procedure On the other handthe number of unknowns in the finite volumes schemes withtriangles is roughly twice as the number of unknowns withquadrangles for the same grid
Figure 19 shows how the computed static reactivity (ie1 minus 1119896eff) depends on the number of unknowns for eachof the sixteen combinations of geometry algorithm shapescheme The four accepted results published in the originalreference [10] are included for reference although it shouldbe taken into account that said reactivities were computedalmost forty years ago Figure 20 shows the total wall timeneeded to solve the problem as a function of 119873119866 whileFigures 21 22 23 and 24 show the times needed for eachof the first four steps involved in the solution maintainingthe same logarithmic scale for both the abscissas and theordinates Green data represent finite volumes whilst bluebullets correspond to finite elements Solid lines indicatequarter-core and dashed lines eighth-symmetry Fillet bullets
are results obtained by theDelaunay triangulation and emptybullets were computed with the delquad algorithm Finallytriangle-shaped data correspond to triangles and squares anddiamonds denote quadrangles as the basic geometry of thegrid
It can be seen that finite elements produce amuch smallerdispersion of eigenvalues 119896eff than finite volumes with therefinement of the mesh This can be explained because finitevolumes methods rely on a geometric condition of the meshwhich may change abruptly if the meshing algorithm decidesto allocate the cells in a rather different form for small changesin ℓ119888 Finite elements methods are less influenced by thesediscontinuous lay-out changes of the elements As expectedeighth-core symmetries give almost the same results as thequarter-core geometries with roughly half the unknownsFor small problems finite volumes run faster that finiteelements because the time needed to solve the neighborproblem is negligible When the problem size grows thistime increases and exceeds the overhead implied in the con-struction and assembly of the finite elements matrices Alsoat least for this configuration it is seen that the eigenvalueproblem is solved faster for finite volumes than for finiteelements
Science and Technology of Nuclear Installations 25
4 Conclusions
Unstructured grids provide the cognizant engineer witha wide variety of possibilities to deal with the design oranalysis of nuclear reactor cores These kinds of grids cansuccessfully reproduce continuous geometries commonlyfound in reactor cores such as cylinders and thereforenot only can the diffusion equation be better approximatedinside the domain of definition but also the fulfillment ofboundary conditions is improved A free computer codewas written from scratch that is able to completely solvethe 2D IAEA PWR Benchmark using unstructured gridsfor sixteen combinations of geometry meshing algorithmbasic shape and discretization scheme plus any value ofthe gridrsquos characteristic length by using a single input fileThe complete set of input files and codemdashexecutable andsourcemdashis available either online or upon request withcomments experiences suggestions and corrections beingmore than welcome Further development should includetackling full three-dimensional geometries with completethermal hydraulic feedback in order to analyze how thesolutions of the coupled neutronic-thermal problem dependon the spatial discretization scheme of the neutron leakageterm Parallelization of the computation and assembly of thematrices and of the solution of the eigenvalue problem andits implementation using GPUs are also desired features toimplement A problemwith direct applications that the futureversions of milonga ought to solve is the analysis of how thegeometry of the absorbing materials should be taken intoaccount in structured coarse grids in order to mitigate effectssuch as the rod-cusp problem
Suitable schemes for approximating the continuous dif-ferential operators by discrete matrix expressions includefinite volumes and finite elements families Finite volumesmethods compute cell mean values whilst finite elementsgive functional values at the gridrsquos nodes In general finiteelements are less sensitive to changes in the mesh so theresults they provide do not change significantly for differentmeshing algorithms or elementary shapes Small problemsare best solved by finite volumes as the neighbor-findingproblem is faster than the process of building and assemblingthe eigenvalue-problem matrices For a large number ofunknowns the process of finding which cell is neighbor ofwhichmdasheven based on a 119896-dimensional treemdashoverwhelmsthe computation and assembly of elementary matrices andfinite-element methods perform better
References
[1] G Glasstone and S Bell Nuclear Reactor Theory KriegerPublishing Company 1970
[2] J J Duderstadt and L J Hamilton Nuclear Reactor AnalysisJohn Wiley amp Sons New York NY USA 1976
[3] C Gho Reactor Physics Undergraduate Course Lecture NotesInstituto Balseiro 2006
[4] G G Theler and F J Bonetto ldquoOn the stability of the pointreactor kinetics equationsrdquoNuclear Engineering andDesign vol240 no 6 pp 1443ndash1449 2010
[5] E E Lewis andW F Miller Computational Methods of NeutronTransport John Wiley amp Sons 1984
[6] E Masiello ldquoAnalytical stability analysis of Coarse-Mesh finitedifference methodrdquo in Proceedings of the International Confer-ence on the Physics of Reactors (PHYSOR rsquo08) Nuclear Power ASustainable Resource pp 456ndash464 September 2008
[7] M L Zerkle Development of a polynomial nodal methodwith flux and current discontinuity factors [PhD thesis] Mas-sachusets Institute of Technology 1992
[8] J I Yoon andHG Joo ldquoTwo-level coarsemesh finite differenceformulation with multigroup source expansion nodal kernelsrdquoJournal of Nuclear Science andTechnology vol 45 no 7 pp 668ndash682 2008
[9] O Mazzantini M Schivo J Di Cesare R Garbero M Riveroand G Theler ldquoA coupled calculation suite for Atucha II oper-ational transients analysisrdquo Science and Technology of NuclearInstallations vol 2011 Article ID 785304 12 pages 2011
[10] Computational Benchmark Problem Comitee for the Mathe-matics and Computation Division of the American NuclearSociety ldquoArgonne Code Center Benchmark problem bookrdquoTech Rep ANL-7416 Supplement 2 Argonne National Labo-ratory 1977
[11] A F Henry Nuclear Reactor Analysis MIT Cambridge UK1975
[12] O C Zienkiewicz R L Taylor and J Z ZhuTheFinite ElementMethod Its Basis and Fundamentals vol 1 Elsevier 6th edition2005
[13] G Theler Difusion de neutrones en mallas no estructuradascomparacion entre volumenes y elementos finitos AcademicMonograph Universidad de Buenos Aires 2013
[14] G Theler ldquoMilonga a free nuclear reactor core analysis coderdquoAvailable online 2011
[15] G Theler F J Bonetto and A Clausse ldquoOptimizacion deparametros en reactores de potencia base de diseno delcodigo neutronico milongardquo in Reunion Anual de la AsociacionArgentina de Tecnologia Nuclear XXXVII 2010
[16] C Geuzaine and J-F Remacle ldquoGmsh a 3-D finite elementmesh generator with built-in pre- and post-processing facili-tiesrdquo International Journal for Numerical Methods in Engineer-ing vol 79 no 11 pp 1309ndash1331 2009
[17] S Balay J Brown K Buschelman et al ldquoPETSc users manualrdquoTech Rep ANL-9511-Revision 34 Argonne National Labora-tory 2013
[18] S Balay W D Gropp L Curfman McInnes and B F SmithldquoEfficient management of parallelism in object oriented numer-ical software librariesrdquo in Modern Software Tools in ScientificComputing E Arge A M Bruaset and H P Langtangen Edspp 163ndash202 Birkhauser 1997
[19] V Hernandez J E Roman and V Vidal ldquoSLEPC a scalable andflexible toolkit for the solution of eigenvalue problemsrdquo ACMTransactions on Mathematical Software vol 31 no 3 pp 351ndash362 2005
[20] M Galassi J Davies J Theiler et al GNU Scientific LibraryReference Manual 3rd edition 2009
[21] O C Zienkiewicz and R L Taylor The Finite Element MethodFor Solid and Structural Mechanics vol 3 Elsevier 6th edition2005
[22] A Imelda and K Doddy Evaluation of the IAEA 3-D PWRbenchmark problem using NESTLE code 2004
[23] R Mosteller ldquoStatic benchmarking of the NESTLE advancednodal coderdquo in Proceedings of the Joint International Conference
26 Science and Technology of Nuclear Installations
on Mathematical Methods and Supercomputing for NuclearApplications vol 2 pp 1596ndash1605 1997
[24] GTheler F J Bonetto andAClausse ldquoSolution of the 2D IAEAPWR Benchmark with the neutronic code milongardquo in Actasde la Reunion Anual de la Asociacion Argentina de TecnologıaNuclear XXXVIII 2011
[25] J F Remacle F Henrotte T Carrier-Baudouin et al ldquoAfrontal delaunay quad mesh generator using the 119871
infin normrdquoInternational Journal For Numerical Methods in Engineeringvol 94 no 5 pp 494ndash512 2013
[26] G W Stewart ldquoA Krylov-Schur algorithm for large eigenprob-lemsrdquo SIAM Journal on Matrix Analysis and Applications vol23 no 3 pp 601ndash614 2002
[27] J L Bentley ldquoMultidimensional binary search trees used forassociative searchingrdquo Communications of the ACM vol 18 no9 pp 509ndash517 1975
TribologyAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FuelsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofPetroleum Engineering
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
CombustionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Renewable Energy
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
StructuresJournal of
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear InstallationsScience and Technology of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solar EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Wind EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear EnergyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Science and Technology of Nuclear Installations 23
Tim
e con
sum
ed to
bui
ld th
e mat
rices
(sec
onds
)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay trianglesvolumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
10
30
100
300
1
3
Figure 23 Time needed to build the matrices versus number of unknowns
was considerably larger than the solution time itself Theeigenvalue problem was solved using a multilayer iterativeKrylov-Schur method [26] with a tolerance relative to thematrices norm
1003817100381710038171003817119877 sdot 120601 minus (1119896eff) sdot 119865 sdot 1206011003817100381710038171003817
119877 + (1119896eff) sdot 119865lt 10minus8 (11)
The reported residual norm and relative error are10038171003817100381710038171003817100381710038171003817119877 sdot 120601 minus
1
119896effsdot 119865 sdot 120601
10038171003817100381710038171003817100381710038171003817
1003817100381710038171003817119877 sdot 120601 minus (1119896eff) sdot 119865 sdot 1206011003817100381710038171003817
1003817100381710038171003817(1119896eff) sdot 1206011003817100381710038171003817
(12)
respectively The fields marked as outer linear and inneriterations refer to the number of steps needed to attainthe requested tolerance in each layer of the Krylov-Schuralgorithm The computer used to solve the problem has anIntel i7 920 267GHz processor with 4Gb of RAM runningDebian GNULinux Wheezy
When using a finite volumes-based scheme over anunstructured mesh the solver has to gather information
about which cells are neighbors and which are not Currentlygmsh does not write this kind of lists in its output filesso milonga has to explicitly solve the neighbors problemPerforming a linear search is an 119874(119873
2) task which is
unacceptable for problem sizes 119873 of interest The code usesa search based on a 119896-dimensional tree [27] which can inprinciple be performed in119874(119873) steps Still for large values of119873 the time needed to read and parse the mesh (number twopreviously mentioned) is the bottleneck of the solution Thisstep is not needed in finite elements although the construc-tion of the elementary matrices involves the computationof the multidimensional Jacobians and integrals which thenhave to be assembled Again for large 119873 this step (numberthree) takes up most of the time needed to solve the problem
The Delaunay algorithm is a standard method for gen-erating two-dimensional grids [16] by tessellating a planewith triangles If the mesh needs to be based on quadranglesinstead a recombination algorithm can be used to transformtwo adjacent triangles in one quadrangle whenever is possi-ble However for geometries which are based on rectangularshapes there exist other algorithms [25] both for meshingand for recombining the triangles that give rise to elementaryentities with right angles almost everywhere which may be a
24 Science and Technology of Nuclear Installations
Tim
e con
sum
ed to
solv
e the
eige
nval
ue p
robl
em (s
econ
ds)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
10
30
100
300
1
3
Figure 24 Time needed to solve the eigenvalue problem versus number of unknowns
desired property of the resulting grid For finite elements thesteps needed to build the matrices depend on the selectionof triangles or quadrangles as the basic elementary geometrybecause the shape functions change However the number ofunknowns is the same as the number of nodes that does notchange after a recombination procedure On the other handthe number of unknowns in the finite volumes schemes withtriangles is roughly twice as the number of unknowns withquadrangles for the same grid
Figure 19 shows how the computed static reactivity (ie1 minus 1119896eff) depends on the number of unknowns for eachof the sixteen combinations of geometry algorithm shapescheme The four accepted results published in the originalreference [10] are included for reference although it shouldbe taken into account that said reactivities were computedalmost forty years ago Figure 20 shows the total wall timeneeded to solve the problem as a function of 119873119866 whileFigures 21 22 23 and 24 show the times needed for eachof the first four steps involved in the solution maintainingthe same logarithmic scale for both the abscissas and theordinates Green data represent finite volumes whilst bluebullets correspond to finite elements Solid lines indicatequarter-core and dashed lines eighth-symmetry Fillet bullets
are results obtained by theDelaunay triangulation and emptybullets were computed with the delquad algorithm Finallytriangle-shaped data correspond to triangles and squares anddiamonds denote quadrangles as the basic geometry of thegrid
It can be seen that finite elements produce amuch smallerdispersion of eigenvalues 119896eff than finite volumes with therefinement of the mesh This can be explained because finitevolumes methods rely on a geometric condition of the meshwhich may change abruptly if the meshing algorithm decidesto allocate the cells in a rather different form for small changesin ℓ119888 Finite elements methods are less influenced by thesediscontinuous lay-out changes of the elements As expectedeighth-core symmetries give almost the same results as thequarter-core geometries with roughly half the unknownsFor small problems finite volumes run faster that finiteelements because the time needed to solve the neighborproblem is negligible When the problem size grows thistime increases and exceeds the overhead implied in the con-struction and assembly of the finite elements matrices Alsoat least for this configuration it is seen that the eigenvalueproblem is solved faster for finite volumes than for finiteelements
Science and Technology of Nuclear Installations 25
4 Conclusions
Unstructured grids provide the cognizant engineer witha wide variety of possibilities to deal with the design oranalysis of nuclear reactor cores These kinds of grids cansuccessfully reproduce continuous geometries commonlyfound in reactor cores such as cylinders and thereforenot only can the diffusion equation be better approximatedinside the domain of definition but also the fulfillment ofboundary conditions is improved A free computer codewas written from scratch that is able to completely solvethe 2D IAEA PWR Benchmark using unstructured gridsfor sixteen combinations of geometry meshing algorithmbasic shape and discretization scheme plus any value ofthe gridrsquos characteristic length by using a single input fileThe complete set of input files and codemdashexecutable andsourcemdashis available either online or upon request withcomments experiences suggestions and corrections beingmore than welcome Further development should includetackling full three-dimensional geometries with completethermal hydraulic feedback in order to analyze how thesolutions of the coupled neutronic-thermal problem dependon the spatial discretization scheme of the neutron leakageterm Parallelization of the computation and assembly of thematrices and of the solution of the eigenvalue problem andits implementation using GPUs are also desired features toimplement A problemwith direct applications that the futureversions of milonga ought to solve is the analysis of how thegeometry of the absorbing materials should be taken intoaccount in structured coarse grids in order to mitigate effectssuch as the rod-cusp problem
Suitable schemes for approximating the continuous dif-ferential operators by discrete matrix expressions includefinite volumes and finite elements families Finite volumesmethods compute cell mean values whilst finite elementsgive functional values at the gridrsquos nodes In general finiteelements are less sensitive to changes in the mesh so theresults they provide do not change significantly for differentmeshing algorithms or elementary shapes Small problemsare best solved by finite volumes as the neighbor-findingproblem is faster than the process of building and assemblingthe eigenvalue-problem matrices For a large number ofunknowns the process of finding which cell is neighbor ofwhichmdasheven based on a 119896-dimensional treemdashoverwhelmsthe computation and assembly of elementary matrices andfinite-element methods perform better
References
[1] G Glasstone and S Bell Nuclear Reactor Theory KriegerPublishing Company 1970
[2] J J Duderstadt and L J Hamilton Nuclear Reactor AnalysisJohn Wiley amp Sons New York NY USA 1976
[3] C Gho Reactor Physics Undergraduate Course Lecture NotesInstituto Balseiro 2006
[4] G G Theler and F J Bonetto ldquoOn the stability of the pointreactor kinetics equationsrdquoNuclear Engineering andDesign vol240 no 6 pp 1443ndash1449 2010
[5] E E Lewis andW F Miller Computational Methods of NeutronTransport John Wiley amp Sons 1984
[6] E Masiello ldquoAnalytical stability analysis of Coarse-Mesh finitedifference methodrdquo in Proceedings of the International Confer-ence on the Physics of Reactors (PHYSOR rsquo08) Nuclear Power ASustainable Resource pp 456ndash464 September 2008
[7] M L Zerkle Development of a polynomial nodal methodwith flux and current discontinuity factors [PhD thesis] Mas-sachusets Institute of Technology 1992
[8] J I Yoon andHG Joo ldquoTwo-level coarsemesh finite differenceformulation with multigroup source expansion nodal kernelsrdquoJournal of Nuclear Science andTechnology vol 45 no 7 pp 668ndash682 2008
[9] O Mazzantini M Schivo J Di Cesare R Garbero M Riveroand G Theler ldquoA coupled calculation suite for Atucha II oper-ational transients analysisrdquo Science and Technology of NuclearInstallations vol 2011 Article ID 785304 12 pages 2011
[10] Computational Benchmark Problem Comitee for the Mathe-matics and Computation Division of the American NuclearSociety ldquoArgonne Code Center Benchmark problem bookrdquoTech Rep ANL-7416 Supplement 2 Argonne National Labo-ratory 1977
[11] A F Henry Nuclear Reactor Analysis MIT Cambridge UK1975
[12] O C Zienkiewicz R L Taylor and J Z ZhuTheFinite ElementMethod Its Basis and Fundamentals vol 1 Elsevier 6th edition2005
[13] G Theler Difusion de neutrones en mallas no estructuradascomparacion entre volumenes y elementos finitos AcademicMonograph Universidad de Buenos Aires 2013
[14] G Theler ldquoMilonga a free nuclear reactor core analysis coderdquoAvailable online 2011
[15] G Theler F J Bonetto and A Clausse ldquoOptimizacion deparametros en reactores de potencia base de diseno delcodigo neutronico milongardquo in Reunion Anual de la AsociacionArgentina de Tecnologia Nuclear XXXVII 2010
[16] C Geuzaine and J-F Remacle ldquoGmsh a 3-D finite elementmesh generator with built-in pre- and post-processing facili-tiesrdquo International Journal for Numerical Methods in Engineer-ing vol 79 no 11 pp 1309ndash1331 2009
[17] S Balay J Brown K Buschelman et al ldquoPETSc users manualrdquoTech Rep ANL-9511-Revision 34 Argonne National Labora-tory 2013
[18] S Balay W D Gropp L Curfman McInnes and B F SmithldquoEfficient management of parallelism in object oriented numer-ical software librariesrdquo in Modern Software Tools in ScientificComputing E Arge A M Bruaset and H P Langtangen Edspp 163ndash202 Birkhauser 1997
[19] V Hernandez J E Roman and V Vidal ldquoSLEPC a scalable andflexible toolkit for the solution of eigenvalue problemsrdquo ACMTransactions on Mathematical Software vol 31 no 3 pp 351ndash362 2005
[20] M Galassi J Davies J Theiler et al GNU Scientific LibraryReference Manual 3rd edition 2009
[21] O C Zienkiewicz and R L Taylor The Finite Element MethodFor Solid and Structural Mechanics vol 3 Elsevier 6th edition2005
[22] A Imelda and K Doddy Evaluation of the IAEA 3-D PWRbenchmark problem using NESTLE code 2004
[23] R Mosteller ldquoStatic benchmarking of the NESTLE advancednodal coderdquo in Proceedings of the Joint International Conference
26 Science and Technology of Nuclear Installations
on Mathematical Methods and Supercomputing for NuclearApplications vol 2 pp 1596ndash1605 1997
[24] GTheler F J Bonetto andAClausse ldquoSolution of the 2D IAEAPWR Benchmark with the neutronic code milongardquo in Actasde la Reunion Anual de la Asociacion Argentina de TecnologıaNuclear XXXVIII 2011
[25] J F Remacle F Henrotte T Carrier-Baudouin et al ldquoAfrontal delaunay quad mesh generator using the 119871
infin normrdquoInternational Journal For Numerical Methods in Engineeringvol 94 no 5 pp 494ndash512 2013
[26] G W Stewart ldquoA Krylov-Schur algorithm for large eigenprob-lemsrdquo SIAM Journal on Matrix Analysis and Applications vol23 no 3 pp 601ndash614 2002
[27] J L Bentley ldquoMultidimensional binary search trees used forassociative searchingrdquo Communications of the ACM vol 18 no9 pp 509ndash517 1975
TribologyAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FuelsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofPetroleum Engineering
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
CombustionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Renewable Energy
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
StructuresJournal of
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear InstallationsScience and Technology of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solar EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Wind EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear EnergyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
24 Science and Technology of Nuclear Installations
Tim
e con
sum
ed to
solv
e the
eige
nval
ue p
robl
em (s
econ
ds)
2000 3000 6000 10000 20000 30000 60000Number of unknowns
105 2 times 105 3 times 105
Quarter Delaunay triangles volumesQuarter Delaunay quads volumesQuarter delquad triangles volumesQuarter delquad quads volumesQuarter Delaunay triangles elementsQuarter Delaunay quads elementsQuarter delquad triangles elementsQuarter delquad quads elements
Eighth Delaunay triangles volumesEighth Delaunay quads volumesEighth delquad triangles volumesEighth delquad quads volumesEighth Delaunay triangles elementsEighth Delaunay quads elementsEighth delquad triangles elementsEighth delquad quads elements
01
003
001
03
10
30
100
300
1
3
Figure 24 Time needed to solve the eigenvalue problem versus number of unknowns
desired property of the resulting grid For finite elements thesteps needed to build the matrices depend on the selectionof triangles or quadrangles as the basic elementary geometrybecause the shape functions change However the number ofunknowns is the same as the number of nodes that does notchange after a recombination procedure On the other handthe number of unknowns in the finite volumes schemes withtriangles is roughly twice as the number of unknowns withquadrangles for the same grid
Figure 19 shows how the computed static reactivity (ie1 minus 1119896eff) depends on the number of unknowns for eachof the sixteen combinations of geometry algorithm shapescheme The four accepted results published in the originalreference [10] are included for reference although it shouldbe taken into account that said reactivities were computedalmost forty years ago Figure 20 shows the total wall timeneeded to solve the problem as a function of 119873119866 whileFigures 21 22 23 and 24 show the times needed for eachof the first four steps involved in the solution maintainingthe same logarithmic scale for both the abscissas and theordinates Green data represent finite volumes whilst bluebullets correspond to finite elements Solid lines indicatequarter-core and dashed lines eighth-symmetry Fillet bullets
are results obtained by theDelaunay triangulation and emptybullets were computed with the delquad algorithm Finallytriangle-shaped data correspond to triangles and squares anddiamonds denote quadrangles as the basic geometry of thegrid
It can be seen that finite elements produce amuch smallerdispersion of eigenvalues 119896eff than finite volumes with therefinement of the mesh This can be explained because finitevolumes methods rely on a geometric condition of the meshwhich may change abruptly if the meshing algorithm decidesto allocate the cells in a rather different form for small changesin ℓ119888 Finite elements methods are less influenced by thesediscontinuous lay-out changes of the elements As expectedeighth-core symmetries give almost the same results as thequarter-core geometries with roughly half the unknownsFor small problems finite volumes run faster that finiteelements because the time needed to solve the neighborproblem is negligible When the problem size grows thistime increases and exceeds the overhead implied in the con-struction and assembly of the finite elements matrices Alsoat least for this configuration it is seen that the eigenvalueproblem is solved faster for finite volumes than for finiteelements
Science and Technology of Nuclear Installations 25
4 Conclusions
Unstructured grids provide the cognizant engineer witha wide variety of possibilities to deal with the design oranalysis of nuclear reactor cores These kinds of grids cansuccessfully reproduce continuous geometries commonlyfound in reactor cores such as cylinders and thereforenot only can the diffusion equation be better approximatedinside the domain of definition but also the fulfillment ofboundary conditions is improved A free computer codewas written from scratch that is able to completely solvethe 2D IAEA PWR Benchmark using unstructured gridsfor sixteen combinations of geometry meshing algorithmbasic shape and discretization scheme plus any value ofthe gridrsquos characteristic length by using a single input fileThe complete set of input files and codemdashexecutable andsourcemdashis available either online or upon request withcomments experiences suggestions and corrections beingmore than welcome Further development should includetackling full three-dimensional geometries with completethermal hydraulic feedback in order to analyze how thesolutions of the coupled neutronic-thermal problem dependon the spatial discretization scheme of the neutron leakageterm Parallelization of the computation and assembly of thematrices and of the solution of the eigenvalue problem andits implementation using GPUs are also desired features toimplement A problemwith direct applications that the futureversions of milonga ought to solve is the analysis of how thegeometry of the absorbing materials should be taken intoaccount in structured coarse grids in order to mitigate effectssuch as the rod-cusp problem
Suitable schemes for approximating the continuous dif-ferential operators by discrete matrix expressions includefinite volumes and finite elements families Finite volumesmethods compute cell mean values whilst finite elementsgive functional values at the gridrsquos nodes In general finiteelements are less sensitive to changes in the mesh so theresults they provide do not change significantly for differentmeshing algorithms or elementary shapes Small problemsare best solved by finite volumes as the neighbor-findingproblem is faster than the process of building and assemblingthe eigenvalue-problem matrices For a large number ofunknowns the process of finding which cell is neighbor ofwhichmdasheven based on a 119896-dimensional treemdashoverwhelmsthe computation and assembly of elementary matrices andfinite-element methods perform better
References
[1] G Glasstone and S Bell Nuclear Reactor Theory KriegerPublishing Company 1970
[2] J J Duderstadt and L J Hamilton Nuclear Reactor AnalysisJohn Wiley amp Sons New York NY USA 1976
[3] C Gho Reactor Physics Undergraduate Course Lecture NotesInstituto Balseiro 2006
[4] G G Theler and F J Bonetto ldquoOn the stability of the pointreactor kinetics equationsrdquoNuclear Engineering andDesign vol240 no 6 pp 1443ndash1449 2010
[5] E E Lewis andW F Miller Computational Methods of NeutronTransport John Wiley amp Sons 1984
[6] E Masiello ldquoAnalytical stability analysis of Coarse-Mesh finitedifference methodrdquo in Proceedings of the International Confer-ence on the Physics of Reactors (PHYSOR rsquo08) Nuclear Power ASustainable Resource pp 456ndash464 September 2008
[7] M L Zerkle Development of a polynomial nodal methodwith flux and current discontinuity factors [PhD thesis] Mas-sachusets Institute of Technology 1992
[8] J I Yoon andHG Joo ldquoTwo-level coarsemesh finite differenceformulation with multigroup source expansion nodal kernelsrdquoJournal of Nuclear Science andTechnology vol 45 no 7 pp 668ndash682 2008
[9] O Mazzantini M Schivo J Di Cesare R Garbero M Riveroand G Theler ldquoA coupled calculation suite for Atucha II oper-ational transients analysisrdquo Science and Technology of NuclearInstallations vol 2011 Article ID 785304 12 pages 2011
[10] Computational Benchmark Problem Comitee for the Mathe-matics and Computation Division of the American NuclearSociety ldquoArgonne Code Center Benchmark problem bookrdquoTech Rep ANL-7416 Supplement 2 Argonne National Labo-ratory 1977
[11] A F Henry Nuclear Reactor Analysis MIT Cambridge UK1975
[12] O C Zienkiewicz R L Taylor and J Z ZhuTheFinite ElementMethod Its Basis and Fundamentals vol 1 Elsevier 6th edition2005
[13] G Theler Difusion de neutrones en mallas no estructuradascomparacion entre volumenes y elementos finitos AcademicMonograph Universidad de Buenos Aires 2013
[14] G Theler ldquoMilonga a free nuclear reactor core analysis coderdquoAvailable online 2011
[15] G Theler F J Bonetto and A Clausse ldquoOptimizacion deparametros en reactores de potencia base de diseno delcodigo neutronico milongardquo in Reunion Anual de la AsociacionArgentina de Tecnologia Nuclear XXXVII 2010
[16] C Geuzaine and J-F Remacle ldquoGmsh a 3-D finite elementmesh generator with built-in pre- and post-processing facili-tiesrdquo International Journal for Numerical Methods in Engineer-ing vol 79 no 11 pp 1309ndash1331 2009
[17] S Balay J Brown K Buschelman et al ldquoPETSc users manualrdquoTech Rep ANL-9511-Revision 34 Argonne National Labora-tory 2013
[18] S Balay W D Gropp L Curfman McInnes and B F SmithldquoEfficient management of parallelism in object oriented numer-ical software librariesrdquo in Modern Software Tools in ScientificComputing E Arge A M Bruaset and H P Langtangen Edspp 163ndash202 Birkhauser 1997
[19] V Hernandez J E Roman and V Vidal ldquoSLEPC a scalable andflexible toolkit for the solution of eigenvalue problemsrdquo ACMTransactions on Mathematical Software vol 31 no 3 pp 351ndash362 2005
[20] M Galassi J Davies J Theiler et al GNU Scientific LibraryReference Manual 3rd edition 2009
[21] O C Zienkiewicz and R L Taylor The Finite Element MethodFor Solid and Structural Mechanics vol 3 Elsevier 6th edition2005
[22] A Imelda and K Doddy Evaluation of the IAEA 3-D PWRbenchmark problem using NESTLE code 2004
[23] R Mosteller ldquoStatic benchmarking of the NESTLE advancednodal coderdquo in Proceedings of the Joint International Conference
26 Science and Technology of Nuclear Installations
on Mathematical Methods and Supercomputing for NuclearApplications vol 2 pp 1596ndash1605 1997
[24] GTheler F J Bonetto andAClausse ldquoSolution of the 2D IAEAPWR Benchmark with the neutronic code milongardquo in Actasde la Reunion Anual de la Asociacion Argentina de TecnologıaNuclear XXXVIII 2011
[25] J F Remacle F Henrotte T Carrier-Baudouin et al ldquoAfrontal delaunay quad mesh generator using the 119871
infin normrdquoInternational Journal For Numerical Methods in Engineeringvol 94 no 5 pp 494ndash512 2013
[26] G W Stewart ldquoA Krylov-Schur algorithm for large eigenprob-lemsrdquo SIAM Journal on Matrix Analysis and Applications vol23 no 3 pp 601ndash614 2002
[27] J L Bentley ldquoMultidimensional binary search trees used forassociative searchingrdquo Communications of the ACM vol 18 no9 pp 509ndash517 1975
TribologyAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FuelsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofPetroleum Engineering
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
CombustionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Renewable Energy
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
StructuresJournal of
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear InstallationsScience and Technology of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solar EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Wind EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear EnergyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Science and Technology of Nuclear Installations 25
4 Conclusions
Unstructured grids provide the cognizant engineer witha wide variety of possibilities to deal with the design oranalysis of nuclear reactor cores These kinds of grids cansuccessfully reproduce continuous geometries commonlyfound in reactor cores such as cylinders and thereforenot only can the diffusion equation be better approximatedinside the domain of definition but also the fulfillment ofboundary conditions is improved A free computer codewas written from scratch that is able to completely solvethe 2D IAEA PWR Benchmark using unstructured gridsfor sixteen combinations of geometry meshing algorithmbasic shape and discretization scheme plus any value ofthe gridrsquos characteristic length by using a single input fileThe complete set of input files and codemdashexecutable andsourcemdashis available either online or upon request withcomments experiences suggestions and corrections beingmore than welcome Further development should includetackling full three-dimensional geometries with completethermal hydraulic feedback in order to analyze how thesolutions of the coupled neutronic-thermal problem dependon the spatial discretization scheme of the neutron leakageterm Parallelization of the computation and assembly of thematrices and of the solution of the eigenvalue problem andits implementation using GPUs are also desired features toimplement A problemwith direct applications that the futureversions of milonga ought to solve is the analysis of how thegeometry of the absorbing materials should be taken intoaccount in structured coarse grids in order to mitigate effectssuch as the rod-cusp problem
Suitable schemes for approximating the continuous dif-ferential operators by discrete matrix expressions includefinite volumes and finite elements families Finite volumesmethods compute cell mean values whilst finite elementsgive functional values at the gridrsquos nodes In general finiteelements are less sensitive to changes in the mesh so theresults they provide do not change significantly for differentmeshing algorithms or elementary shapes Small problemsare best solved by finite volumes as the neighbor-findingproblem is faster than the process of building and assemblingthe eigenvalue-problem matrices For a large number ofunknowns the process of finding which cell is neighbor ofwhichmdasheven based on a 119896-dimensional treemdashoverwhelmsthe computation and assembly of elementary matrices andfinite-element methods perform better
References
[1] G Glasstone and S Bell Nuclear Reactor Theory KriegerPublishing Company 1970
[2] J J Duderstadt and L J Hamilton Nuclear Reactor AnalysisJohn Wiley amp Sons New York NY USA 1976
[3] C Gho Reactor Physics Undergraduate Course Lecture NotesInstituto Balseiro 2006
[4] G G Theler and F J Bonetto ldquoOn the stability of the pointreactor kinetics equationsrdquoNuclear Engineering andDesign vol240 no 6 pp 1443ndash1449 2010
[5] E E Lewis andW F Miller Computational Methods of NeutronTransport John Wiley amp Sons 1984
[6] E Masiello ldquoAnalytical stability analysis of Coarse-Mesh finitedifference methodrdquo in Proceedings of the International Confer-ence on the Physics of Reactors (PHYSOR rsquo08) Nuclear Power ASustainable Resource pp 456ndash464 September 2008
[7] M L Zerkle Development of a polynomial nodal methodwith flux and current discontinuity factors [PhD thesis] Mas-sachusets Institute of Technology 1992
[8] J I Yoon andHG Joo ldquoTwo-level coarsemesh finite differenceformulation with multigroup source expansion nodal kernelsrdquoJournal of Nuclear Science andTechnology vol 45 no 7 pp 668ndash682 2008
[9] O Mazzantini M Schivo J Di Cesare R Garbero M Riveroand G Theler ldquoA coupled calculation suite for Atucha II oper-ational transients analysisrdquo Science and Technology of NuclearInstallations vol 2011 Article ID 785304 12 pages 2011
[10] Computational Benchmark Problem Comitee for the Mathe-matics and Computation Division of the American NuclearSociety ldquoArgonne Code Center Benchmark problem bookrdquoTech Rep ANL-7416 Supplement 2 Argonne National Labo-ratory 1977
[11] A F Henry Nuclear Reactor Analysis MIT Cambridge UK1975
[12] O C Zienkiewicz R L Taylor and J Z ZhuTheFinite ElementMethod Its Basis and Fundamentals vol 1 Elsevier 6th edition2005
[13] G Theler Difusion de neutrones en mallas no estructuradascomparacion entre volumenes y elementos finitos AcademicMonograph Universidad de Buenos Aires 2013
[14] G Theler ldquoMilonga a free nuclear reactor core analysis coderdquoAvailable online 2011
[15] G Theler F J Bonetto and A Clausse ldquoOptimizacion deparametros en reactores de potencia base de diseno delcodigo neutronico milongardquo in Reunion Anual de la AsociacionArgentina de Tecnologia Nuclear XXXVII 2010
[16] C Geuzaine and J-F Remacle ldquoGmsh a 3-D finite elementmesh generator with built-in pre- and post-processing facili-tiesrdquo International Journal for Numerical Methods in Engineer-ing vol 79 no 11 pp 1309ndash1331 2009
[17] S Balay J Brown K Buschelman et al ldquoPETSc users manualrdquoTech Rep ANL-9511-Revision 34 Argonne National Labora-tory 2013
[18] S Balay W D Gropp L Curfman McInnes and B F SmithldquoEfficient management of parallelism in object oriented numer-ical software librariesrdquo in Modern Software Tools in ScientificComputing E Arge A M Bruaset and H P Langtangen Edspp 163ndash202 Birkhauser 1997
[19] V Hernandez J E Roman and V Vidal ldquoSLEPC a scalable andflexible toolkit for the solution of eigenvalue problemsrdquo ACMTransactions on Mathematical Software vol 31 no 3 pp 351ndash362 2005
[20] M Galassi J Davies J Theiler et al GNU Scientific LibraryReference Manual 3rd edition 2009
[21] O C Zienkiewicz and R L Taylor The Finite Element MethodFor Solid and Structural Mechanics vol 3 Elsevier 6th edition2005
[22] A Imelda and K Doddy Evaluation of the IAEA 3-D PWRbenchmark problem using NESTLE code 2004
[23] R Mosteller ldquoStatic benchmarking of the NESTLE advancednodal coderdquo in Proceedings of the Joint International Conference
26 Science and Technology of Nuclear Installations
on Mathematical Methods and Supercomputing for NuclearApplications vol 2 pp 1596ndash1605 1997
[24] GTheler F J Bonetto andAClausse ldquoSolution of the 2D IAEAPWR Benchmark with the neutronic code milongardquo in Actasde la Reunion Anual de la Asociacion Argentina de TecnologıaNuclear XXXVIII 2011
[25] J F Remacle F Henrotte T Carrier-Baudouin et al ldquoAfrontal delaunay quad mesh generator using the 119871
infin normrdquoInternational Journal For Numerical Methods in Engineeringvol 94 no 5 pp 494ndash512 2013
[26] G W Stewart ldquoA Krylov-Schur algorithm for large eigenprob-lemsrdquo SIAM Journal on Matrix Analysis and Applications vol23 no 3 pp 601ndash614 2002
[27] J L Bentley ldquoMultidimensional binary search trees used forassociative searchingrdquo Communications of the ACM vol 18 no9 pp 509ndash517 1975
TribologyAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FuelsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofPetroleum Engineering
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
CombustionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Renewable Energy
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
StructuresJournal of
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear InstallationsScience and Technology of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solar EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Wind EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear EnergyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
26 Science and Technology of Nuclear Installations
on Mathematical Methods and Supercomputing for NuclearApplications vol 2 pp 1596ndash1605 1997
[24] GTheler F J Bonetto andAClausse ldquoSolution of the 2D IAEAPWR Benchmark with the neutronic code milongardquo in Actasde la Reunion Anual de la Asociacion Argentina de TecnologıaNuclear XXXVIII 2011
[25] J F Remacle F Henrotte T Carrier-Baudouin et al ldquoAfrontal delaunay quad mesh generator using the 119871
infin normrdquoInternational Journal For Numerical Methods in Engineeringvol 94 no 5 pp 494ndash512 2013
[26] G W Stewart ldquoA Krylov-Schur algorithm for large eigenprob-lemsrdquo SIAM Journal on Matrix Analysis and Applications vol23 no 3 pp 601ndash614 2002
[27] J L Bentley ldquoMultidimensional binary search trees used forassociative searchingrdquo Communications of the ACM vol 18 no9 pp 509ndash517 1975
TribologyAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FuelsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofPetroleum Engineering
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
CombustionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Renewable Energy
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
StructuresJournal of
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear InstallationsScience and Technology of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solar EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Wind EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear EnergyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
TribologyAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FuelsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofPetroleum Engineering
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
CombustionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Renewable Energy
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
StructuresJournal of
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear InstallationsScience and Technology of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solar EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Wind EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear EnergyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014