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The 22nd International Conference on Transport Theory (ICTT-22)Portland, Oregon, September 11-15, 2011

ICTT-22 Abstracts

This document contains abstracts of the sixty papers being presented at the 22nd International Conference

on Transport Theory in Portland, Oregon, September 12-16, 2011. The conference was organized into 15

sessions, including a special panel presentation celebrating the 50th anniversary of Ken Case’s important

paper entitled ”Elementary Solutions of the Transport Equation and Their Applications”.

Table I. Session Schedule

Monday

Morning 1 Radiative Transfer IMorning 2 Combustion/Plasma Physics I

Afternoon 1 Reactor Physics IAfternoon 2 Deterministic Transport I

Tuesday

Morning 1 Uncertainty Quantification/Perturbation Theory IMorning 2 Radiative Transfer II

Afternoon 1 Deterministic Transport IIAfternoon 2 Monte Carlo I

Wednesday

Morning 1 Kinetics IMorning 2 Quantum Transport IAfternoon Columbia River Gorge Tour

Thursday

Morning 1 Reactor Physics IIMorning 2 Analytic Transport Solutions I

Afternoon 1 Radiative Transfer III/Fluid Dynamics IAfternoon 2 Special Session

Evening Dinner Cruise/Conference Banquet

Friday

Morning 1 Deterministic Transport IIIMorning 2 Transport Applications I/Reactor Physics III

8:00 AM

8:30 AM

9:00 AM RADIATIVE TRANSFER RECONSIDERED N.R. Corngold

UNCERTAINTY ANALYSIS OF RADIATION TRANSPORT PROBLEMS USING NON-‐INTRUSIVE POLYNOMIAL CHAOS TECHNIQUES

L. Gilli, D. Lathouwers, J. Kloosterman

NEUTRON INVERSE KINETICS VIA GAUSSIAN PROCESSES

P. Picca, R. FurfaroA STOCHASTIC THEORY OF THE NUMBER OF FISSIONS

A.K. Prinja

AN ANGULAR MULTIGRID ACCELERATION METHOD FOR SN EQUATIONS WITH HIGHLY FORWARD-‐PEAKED SCATTERING

B. Turcksin, J.C. Ragusa, J.E. Morel

9:25 AMMULTIPLE SCATTER IN SPATIALLY VARIABLE MEDIA IMPLEMENTED IN OPEN CL

S. Casacio, J. Tessendorf, R. Geist

INITIAL CONDITIONS STATE-‐BASED PERTURBATION THEORY

Y. Bang, H. S. Abdel-‐Khalik

AN ACCURATE SOLUTION TO MASTER/MOMENTS EQUATIONS FOR THE KINETICS OF BREAKABLE FILAMENT SELF-‐ASSEMBLY

B. Ganapol

SP3 SOLUTION VERSUS DIFFUSION SOLUTION IN NODAL CODES —WHICH IMPROVEMENT CAN BE EXPECTED

B. Merk, S. Duerigen

METHODOLOGY FOR DECOMPOSITION INTO TRANSPORT AND DIFFUSIVE SUBDOMAINS FOR THE LINEAR DISCONTINUOUS METHOD

N.D. Stehle, D.Y. Anistratov

9:50 AMPOLARIZED RADIATIVE TRANSFER IN A MULTI-‐LAYER MEDIUM SUBJECT TO FRESNEL BOUNDARY AND INTERFACE CONDITIONS

R.D.M. GarciaEXACT-‐TO-‐PRECISION GENERALIZED PERTURBATION THEORY

C. Wang, H.S. Abdel-‐Khalik

APPLICATION OF THE HYBRID TRANSPORT/POINT KINETICS TO TIME-‐DEPENDENT SOURCE-‐DRIVEN PROBLEMS

P. Picca, R. Furfaro, B.D. Ganapol

CONSISTENT RECONDENSATION THEORY

S. Douglas and F. Rahnema

A PIECEWISE LINEAR DISCONTINUOUS FINITE ELEMENT SPATIAL DISCRETIZATION FOR POLYHEDRAL GRIDS IN 3D CARTESIAN GEOMETRY

T. S. Bailey, W. D. Hawkins, M. L. Adams

10:15 AM

10:45 AMMODELING RADIATIVE THERMAL TRANSPORT IN A SCRAMJET COMBUSTOR

A.G. Irvine, I.D. Boyd, N.A. Genale, A.J. Crow

ANALITICAL DISCRETE ORDINATE METHOD FOR RADIATIVE TRANSFER IN VEGETATION CANOPIES

P. Picca, R. Furfaro, B.D. Ganapol

QUANTUM CORRECTIONS ON THE RADIATIVE TRANSFER EQUATION

J. Rosato

A NEW ANALYTIC SOLUTION OF THE ONE-‐SPEED NEUTRON TRANSPORT EQUATION FOR ADJACENT HALF-‐SPACES WITH ISOTROPIC SCATTERING

R. P. Smedley-‐Stevenson

DOSIMETRY CALCULATION OF TWO COMMERCIALLY AVAILABLE IODINE BRACHYTHERAPY SEEDS USING SPENCER-‐LEWIS 3D MULTI-‐GROUP SN TRANSPORT CODE

N. Ayoobian

11:10 AMTURBULENCE RADIATION INTERACTIONS IN PARTICULATE LADEN FLOW

M. Cleveland, T. Palmer, S. Apte

MOMENT-‐BASED, MULTISCALE SOLUTION APPROACH FOR THERMAL RADIATION TRANSPORT

H. Park and DA. Knoll

ATOM-‐ATOM RELAXATION WITH QUANTUM DIFFERENTIAL ELASTIC SCATTERING CROSS SECTIONS; DISTRIBUTION FUNCTION RELAXATION AND THE KULLBACK-‐LEIBLER ENTROPY

R. Sospedra-‐Alfonso, B.D. Shizgal

ENERGY-‐DEPENDENT ANALYTICAL SOLUTIONS FOR THE CHARGED PARTICLE TRANSPORT EQUATION

T. Geback, M. Asadzadeh

RADIATION FIELD CHARACTERIZATION, SHIELDING ASSESSMENT AND ACTIVATION CALCULATIONS FOR THE MYRRHA DESIGN

A. Ferrari, B. Merk, and J. Konheiser

11:35 AM KINETIC EQUATIONS FOR STARK LINE SHAPES

J. Rosato, H. Capes, R. Stamm, A Mekkaoui, Y. Marandet

REDUCING THE SPATIAL DISCRETIZATION ERROR OF THERMAL EMISSION IN IMPLICIT MONTE CARLO SIMULATIONS

A.G. Irvine, I.D. Boyd, N.A. Genale

SHAPE RELAXATION IN ELECTRON ATOM RELAXATION; THE KULLBACK-‐LEIBLER RELATIVE ENTROPY AND RELAXATION TIMES

R. Sospedra-‐Alfonso, B.D. Shizgal

SPATIAL MOMENTS OF CONTINUOUS TRANSPORT PROBLEMS COMPUTED ON GRIDS

J.D. DensmoreTWO-‐REGION DIFFUSION MODEL FOR IMPROVED ANALYSIS OF ADS EXPERIMENTS

V. Glivici-‐Cotruta, B. Merk

12:00 PMPLASMA DENSITY FLUCTUATION EFFECTS ON THE SCREENING OF NEUTRAL SPECIES IN TOKMAKS

A. Mekkaoui, Y. Marandet, J. Rosato, R. Stamm, H. Capes, M. Koubia, L. Gobert-‐Mouret, D. Reiter

DERIVING THE ASYMPTOTIC P1 APPROXIMATION FOR THERMAL RADIATIVE TRANSFER

S. I. HeizlerDIFFUSIVE LIMITS FOR A QUANTUM TRANSPORT MODEL WITH WEAK AND STRONG FIELDS

L. Barled, G. FrosaliEIGENVALUES OF THE ANISOTROPIC TRANSPORT EQUATION IN A SLAB

E. Sauter, F. de Azevedo, M. Thompson, M.T. Vilhena

A MESH-‐FREE APPROXIMATION OF SPATIAL NEUTRON FLUX DISTRIBUTION

D. Alaparmakov

12:25 PM

2:00 PMA METHOD FOR IMPROVING THE EIGENVALUE IN TRANSPORT PROBLEMS

S.R. Merton, R. Smedley-‐Stevenson, C.C. Pain

MIXED VARIATIONAL FORMULATION OF THE TRANSPORT EQUATION

J. Caraer, M. Peybernes

IN-‐WATER OCEAN OPTICS INVERSION ALGORITHM

N.J. McCormick, E. Rehm

2:25 PMNEUTRON TRANSPORT IN MOLTEN SALT REACTORS

I. Pazsit

APPLICATION OF SPECTRAL ELEMENTS FOR 1D NEUTRON TRANSPORT AND COMPARISON TO MANUFACTURED SOLUTIONS

A. Barbarino, S. Dulla, P. Raveeo, E.H. Mund, B. Ganapol

ON BOUNDEDNESS OF HIGHER VELOCITY MOMENTS FOR THE LINEAR BOLTZMANN EQUATION WITH DIFFUSE BOUNDARY CONDITIONS

R. Peeersson

2:50 PM

NEUTRON THERMAL SCATTERING LAWS FOR LIGHT AND HEAVY WATER FOR MODELING CRITICAL ASSEMBLIES AND TOF EXPERIMENTAL SET-‐UPS WITH NEUTRON TRANSPORT CODES

D. Roubtsov, K. Kozier, B. Becker, Y. Danon

ANALOG COMPUTING TO THE TIME-‐DEPENDENT SECOND-‐ORDER FORM OF NEUTRON TRANSPORT EQUATION IN X-‐Y GEOMETRY

A. Pirouzmanda, K. Hadadb, P. Raveeo

ON THE SPEED OF HEAT WAVES M. Mikai

3:15 PM

RECENT DEVELOPMENTS ON EXPLICIT FORMULATIONS FOR NODAL SCHEMES OF TWO-‐DIMENSIONAL NEUTRON TRANSPORT PROBLEMS

J. F. P. Filho, L. C. Cabrera and L.B. Barichello

P2-‐EQUIVALENT FORM OF THE SP2 EQUATIONS -‐ Including boundary and interface condiaons

R. McClerranDENSITY DISTRIBUTION OF THE MOLECULES OF A LIQUID IN A SEMINFINITE SPACE

V. Molinari, B.D. Ganapol, D. Mostacci

3:40 PM

4:10 PM

A NUMERICAL METHOD FOR ONE-‐SPEED SLAB-‐GEOMETRY ADJOINT DISCRETE ORDINATES PROBLEMS WITH NO SPATIAL TRUNCATION ERROR

D.S. Militao, H.A. Filho, R.C. Barros

VARIANCE REDUCTIONS FOR FORWARD AND INVERSE TRANSPORT PROBLEMS

G. Bal

4:35 PM ON SN-‐PN EQUIVALENCE R. SanchezMATERIAL MOTION CORRECTIONS FOR IMPLICIT MONTE CARLO RADIATION TRANSPORT

N. Genale, J. Morel

5:00 PMBOUNDARY CONDITION ANALYSIS FOR THE SP1 APPROXIMATION OF THE RADIATIVE-‐CONDUCTIVE EQUATION

F. de Azevedo, M. Thompson, E. Sauter, M.T. Vilhena

NECESSARY AND SUFFICIENT CONDITIONS FOR AN IMPLICIT MONTE CARLO DISCRETE MAXIMUM PRINCIPLE

A. Wollaber

5:25 PM

FULLY DISCRETE FINITE ELEMENT APPROXIMATION OF A BI-‐PARTITION MODEL FOR THE ENERGY DEPENDENT TRANSPORT EQUATION

M. Asadzadeh, T. Geback

A MODIFIED TREATMENT OF SOURCES IN IMPLICIT MONTE CARLO RADIATION TRANSPORT

N.A.Genale, T.J. Trahan

5:50 PMIMPROVED MIXED AND HYBRID DISCRETIZATION OF THE TRANSPORT EQUATION IN SLAB GEOMETRY

J. Caraer, M. Peybernes

A COARSE GRAINED PARTICLE TRANSPORT SOLVER DESIGNED SPECIFICALLY FOR GRAPHICS PROCESSING UNITS

F. A. van Heerden

6:15 PM6:30 PM7:15 PM

9:30 PM

End of Conference

Tour of Columbia River Gorge Special Session

Dinner Cruise/Conference Banquet

Commemoraang the 50th Anniversary of Ken Case's Paper: "“Elementary Soluaons of the Transport Equaaon

and Their Applicaaons” -‐ Paul Zweifel

Coffee Break

Dinner on your own Dinner on your own

No Host Wine/Brew Tasang No Host Wine/Brew TasangDinner on your own

Thursday 9/15

Breakfast

Coffee BreakAnaly:c Transport Solu:ons I

Wednesday 9/14

Breakfast

Coffee BreakQuantum Transport I

Kine:cs I

KINETIC MODELING OF NITRATE REMOVAL USING A NITRATE SELECTIVE RESIN: THE ROLE OF MASS TRANSER

Transport Applica:ons I/Reactor Physics III

Lunch

Uncertainty Quan:fica:on/Perturba:on Theory I

LunchLunchReactor Physics I Radia:ve Transfer III/Fluid Dynamics I

Lunch on your own

N. Talebbeydokha, A. A. Hekmatzadeh, A. Karimi-‐Jashani, K. Hadad

Coffee Break

Breakfast

Coffee Break Coffee Break

Breakfast/Welcome

Radia:ve Transfer I

Determinis:c Transport III

Determinis:c Transport I Monte Carlo ICoffee Break

Radia:ve Transfer II

Coffee Break

Determinis:c Transport II

Combus:on/Plasma Physics I

Friday 9/16

A. Seubert, A. Sureda, J. Lapins, J. Bader

EXPLORATION OF AN ADAPTIVE ANGULAR SOLVER IN NEUTRON TRANSPORT

D. Lathouwers, D.J. Koeze

Monday 9/12

Sign In Breakfast

Reactor Physics II

THE TRANSIENT 3-‐D TRANSPORT COUPLED CODE TORT-‐TD/ATTICA3D FOR HIGH-‐FIDELITY PEBBLE-‐BED HTGR ANALYSES

Tuesday 9/13


Monday, September 12, 2011

Radiative Transfer I

9:00 am Radiative Transfer Reconsidered - N.R. Corngold

9:25 am Multiple Scatter in Spatially Variable Media Implemented in Open CL - S. Casacio, J. Tessendorf, R.

Geist

9:50 am Polarized Radiative Transfer in a Multi-layer Medium Subject to Fresnel Boundary and Interface

Conditions - R.D.M. Garcia

Combustion/Plasma Physics I

10:45 am Modeling Radiative Thermal Transport in a Scramjet Combustor - A.G. Irvine, I.D. Boyd, N.A. Gen-

tile, A.J. Crow

11:10 am Turbulence Radiation Interactions in Particulate Laden Flow - M. Cleveland, T. Palmer and S. Apte

11:35 am Kinetic Equations for Stark Line Shapes - J. Rosato, H. Capes, R. Stamm, A Mekkaoui, Y. Marandet

12:00 pm Plasma Density Fluctuation Effects on the Screening of Neutral Species in Tokamaks - A Mekkaoui,

Y. Marandet, J. Rosato, R. Stamm, H. Capes, M. Koubiti, L. Gobert-Mouret, D. Reiter

Reactor Physics I

2:00 pm A Method for Improving the Eigenvalue in Transport Problems - S. R. Merton, R. Smedly-Stevenson,

C.C. Pain

2:25 pm Neutron Transport in Molten Salt Reactors - I. Pazsit

2:50 pm Neutron Thermal Scattering Laws for Light and Heavy Water for Modeling Critical Assemblies and

TOF Experimental Set-ups with Neutron Transport Codes - D. Roubtsov, K. Kozier, B. Becker, Y.

Danon

3:15 pm Recent Developments on Explicit Formulations for Nodal Schemes of Two-Dimensional Neutron Trans-

port Problems - J.F.P. Filho, L.C. Cabrera, L.B. Barichello


Deterministic Transport I

4:10 pm A Numerical Method for One-Speed Slab-Geometry Adjoint Discrete Ordinates Problems with No

Spatial Truncation Error - D.S. Militao, H.A. Filho, R.C. Barros

4:35 pm On SN -PN Equivalence - R. Sanchez

5:00 pm Boundary Condition Analysis for the SP1 Approximation of the Radiative-Conduction Equation - F.

de Azevedo, M. Thompson, E. Sauter, M.T. Vilhena

5:25 pm Fully Discrete Finite Element Approximation of a Bi-Partition Model for the Energy-Dependent Trans-

port Equation - M. Asadzadeh, T. Geback

5:50 pm Improved Mixed and Hybrid Discretization of the Transport Equation in Slab Geometry - J. Cartier,

M. Peybernes

The 22nd International Conference on Transport Theory (ICTT-22) Portland, Oregon, September 11-15, 2011

Radiative Transfer Reconsidered

Noel R. Corngold Watson Laboratories of Applied Physics

California Institute of Technology Pasadena, California 91125

[email protected]

Transport theory as we know it was parented by radiative transfer, flourished as neutronics and has, recently, with the advent of the laser, exploded into applications to climate science, marine science, medical science and every variety of remote sensing. Crucial to these laser-driven scenarios is the notion of coherence, which presents itself naturally when the wave nature of transported quantities is considered. The response of the transport community? One classic text [1] ignores the issue entirely, a more recent one [2] acknowledges it, but agrees to ignore or defer it. Overwhelmingly, theorists in our community have thought it adequate to treat laser-driven radiative transfer by patching the (neutron) transport equation. That equation, which is based on a ballistic view of transport, is maintained, but boundary conditions are altered; the wave-nature is acknowledged by including reflection and refraction. But recent elegant experiments [3] and developments in nearby fields [4] convince one that these simple “fixes” can no longer be maintained. The proper derivation of the radiative transport equation, based on a wave theory, reaches back-in the Western world-to a classic essay by Leslie Foldy [5] published many decades ago. Later, a Nobel-Prize winning notion [6] triggered an avalanche of papers of some relevance. In this paper we summarize the key ideas, the strategy, that lead to a proper transport equation and to natural, indisputable boundary conditions. Our approach is concise, and avoids the elaborate counting of “diagrams” one meets so often in the literature. It also indicates the limitations imposed on a typical derivation. This more fundamental approach should challenge the transport community to refine, alter, improve the analysis to bring it ever closer to the results of experiment, to Nature.

REFERENCES

[1] S. Chandrasekhar, Radiative Transfer, Oxford University Press, Oxford, UK (1950) Dover reprint (1960) [2] G. C. Pomraning, Equations of Radiative Hydrodynamics, Pergamon, Oxford, UK (1973). Dover reprint (2005).

mailto:[email protected]

REFERENCES (cont.)

[3] Z. Yaqoob et al, "Optical Phase Conjugation for Turbidity Suppression…," Nature Photonics 2, pp.110-115 (2008) [4] M.C.W. van Rossum and Th. M. Nieuwenhuizen, “Multiple Scattering of Classical Waves,” Revs. Mod. Phys. 71, pp. 313-372 (1999). [5] L.L. Foldy, "The Multiple Scattering of Waves," Phys. Rev. 67, pp.107-119 (1945). [6] A computer search for "Anderson Localization" (P.W. Anderson) will evoke a blizzard of papers. Our situation is close to what is called the "weak localization" of waves.


Multiple Scatter in Spactially Variable Media Implemented in OpenCL

Samuel CasacioSchool of ComputingClemson UniversityClemson, SC [email protected]

Jerry TessendorfDigital Production ArtsSchool of ComputingClemson UniversityClemson, SC 29634

[email protected]

Robert GeistSchool of ComputingClemson UniversityClemson, SC [email protected]

The steepest-descents approximation of the Feynman Path Integral for the radiative transfer equation for a

spatially uniform medium was obtained in [2]. Re-examining the approximation process, the general case

of a spatially variable medium follows by replacing the number of scattering lengths by a more rigorous

definition. In the original steepest-descents approach, the scattering coefficient b is multiplied by the

arclength s of a given path, so that the important factor is the number of scattering lengths bs of a path. In a

spatially variable medium, the scattering coefficient is a field b(x) and the number of scattering lengths is

`(s) =

∫ s

0ds′ b

(x′ +

∫ s′

0ds′′ β

(`(s′′)

))(1)

where β(`′) describes a steepest-descents path as a function of scattering length, i.e. a path that minimizes

the curvature κ, where

κ(`′) =

∣∣∣∣∣dβ(`′)d`′

∣∣∣∣∣ (2)

In single-scatter ray marching, each path is weighted by the transmissivity between the camera and the ray

march point. The multiple scatter paths are weighted by a factor

W (s, β0) = exp(−`(1 +N

(2η2q − 1/q

))(3)

with

η2 =(cos−1(n · n′))2

2µ`2N2, (4)

N is the normalization of the phase function, µ is related to the phase function asymmetry, n and n′ are the

viewing and source angles, and q is defined implicitly as

q = exp(−q2η2

)(5)

Samuel Casacio, Jerry Tessendorf, and Robert Geist

Multiple scattering in the steepest-descents approximation also induces spatial blurring of the volumetric

material as a function of the number of scattering lengths, i.e. the density distribution. A useful numerical

implementation approach is to use a three-dimensional version of texture mipmapping.

Multiple scatter rendering was previously implemented in OpenCL using Lattice Boltzmann methods [1],

and achieved interactive frame rates on reasonable scenes. This steepest-descents approach is

implementated using a ray marching scheme that extends the traditional straight line march to curved

march paths. This is parallelized in OpenCL by one-dimensional distribution over the number of scatter

lengths. The curved paths are derived from the spatial variation of the density field, the optical properties,

and the location of the camera and light source. Instead of accumulating light along discrete points of a

straight line, the curved paths satisfy the steepest-decent approximation, with a corresponding path weight

for the light accumulation. As a ray marching algorithm, the numerical implementation on a GPU using

OpenCL exploits the parallelization of multiple marches across GPU cores. We demonstrate the

implementation with cloud-like volumes.

REFERENCES

[1] Robert Geist, Karl Rasche, James Westall, and Robert Schalkoff. Lattice-boltzmann lighting.

Eurographics Symposium on Rendering (2004), 2004.

[2] Tessendorf J. Angular smoothing and spatial diffusion from the feynman path integral representation

of radiative transfer. Journal of Quantitative Spectroscopy & Radiative Transfer, pages 751–760, 2011.

2/2


POLARIZED RADIATIVE TRANSFER IN A MULTI-LAYER MEDIUMSUBJECT TO FRESNEL BOUNDARY AND INTERFACE CONDITIONS

R. D. M. GarciaInstituto de Estudos Avancados

Trevo Cel. Av. Jose Alberto Albano do Amarante n o 112228-001 Sao Jose dos Campos, SP, Brazil

[email protected]

In a recent work [1], an alternative approach for solving the scalar albedo problem for a multi-layer

medium subject to Fresnel boundary and interface conditions and uniform illumination by obliquely

incident parallel rays was proposed. The approach is based on the use of a nascent delta function (rather

than the Dirac delta distribution) to model the polar-angle dependence of the incident beam. Differently

than the classical approach [2], where both the polar-angle and the azimuthal-angle dependencies of the

incident beam are modeled by Dirac delta distributions, the alternative approach facilitates the

implementation of the analytical discrete-ordinates (ADO) method [3], since a particular solution is not

required when a nascent delta function is used. Numerical results obtained with the new approach [1] were

found to be as accurate as those from the classical (exact) approach, when a sufficiently small “narrowness”

parameter was used in the definition of the nascent delta function. That work [1] was successfully extended

for polarized radiative transfer in a single layer subject to a uniform beam of parallel rays striking one of its

surfaces obliquely and Lambert ground reflection on the other [4].

In this work, we extend the approach of Ref. [1] to polarized radiative transfer in a multi-layer medium

subject to Fresnel boundary and interface conditions and uniform external illumination in the form of

obliquely incident parallel rays. This topic is important to many areas of study, as for example remote

sensing [5], radiative transfer in the atmosphere-ocean system [6], and biomedical optics [7]. The problem

is formulated by the equation of transfer, for layers k = 1, 2, . . . ,K,

µ∂

∂τIk(τ, µ, φ) + Ik(τ, µ, φ) =

$k

4π

∫ 2π

0

∫ 1

−1Pk(µ, µ′, φ− φ′)Ik(τ, µ′, φ′) dµ′dφ′, (1)

where Ik(τ, µ, φ) is the Stokes vector that has the four Stokes parameters Ik(τ, µ, φ), Qk(τ, µ, φ),

Uk(τ, µ, φ), and Vk(τ, µ, φ) as components [8], τ ∈ (ak−1, ak) is the optical variable that measures the

position in layer k, and µ ∈ [−1, 1] and φ ∈ [0, 2π] are, respectively, the cosine of the polar angle and the

azimuthal angle that give the direction of propagation of the radiation. In our notation, a0 gives the location

of the surface of the first layer, a1, a2, . . . , aK−1 give the locations of the interfaces between the layers, and

aK the location of the surface of the last layer. To each layer, we associate an index of refraction nk, an

albedo for single scattering $k, and a phase matrix Pk(µ, µ′, φ− φ′).

The boundary and interface conditions subject to which we must solve Eq. (1) for k = 1, 2, . . . ,K were

derived from first principles [9]. Assuming that an external medium (τ < a0), characterized by an index of

R. D. M. Garcia

refraction n0, has a beam of radiation with Stokes parameters

F =

FIFQFUFV

(2)

traveling towards the surface located at τ = a0 along a direction defined by (µ0, φ0), we can write the

boundary condition for the first layer as [9]

I1(a0, µ, φ) = X(n1,0, µ)I1(a0,−µ, φ) + Y(n1,0, µ)F δ[f(n1,0, µ)− µ0]δ(φ− φ0), (3)

for µ ∈ (0, 1] and φ ∈ [0, 2π]. In this expression, we make use of some general definitions: the ratio

between indices of refraction nk,k′ = nk/nk′ , the refraction function f(n, µ) = [1− n2(1− µ2)]1/2, the

4× 4 reflection matrix X(n, µ), and the 4× 4 transmission matrix Y(n, µ) (see Ref. [9] for explicit

expressions for the elements of these matrices). At the interfaces between the layers, we have [9]

Ik(ak,−µ, φ) = X(nk,k+1, µ)Ik(ak, µ, φ) + Y(nk,k+1, µ)Ik+1[ak,−f(nk,k+1, µ), φ] (4a)

and

Ik+1(ak, µ, φ) = X(nk+1,k, µ)Ik+1(ak,−µ, φ) + Y(nk+1,k, µ)Ik[ak, f(nk+1,k, µ), φ], (4b)

for µ ∈ (0, 1], φ ∈ [0, 2π], and k = 1, 2, ...,K − 1. With regard to the boundary condition for the surface

of the last layer, located at τ = aK , we assume that there is no radiation coming from an external medium

(τ > aK) with index of refraction nK+1, and so we can write [9]

IK(aK ,−µ, φ) = X(nK,K+1, µ)IK(aK , µ, φ), (5)

for µ ∈ (0, 1] and φ ∈ [0, 2π].

The fact that the various polar-angle arguments ±µ, −f(nk,k+1, µ), and f(nk+1,k, µ) are present in

Eqs. (4) is a complication that must be overcome before an ADO solution to the problem is implemented.

Clearly, if ξi denotes the collection of N “half-range” polar quadrature points to be used in the ADO

method, then, in addition to the Stokes vectors evaluated at ±ξi, Stokes vectors are also needed at the

points −f(nk,k+1, ξi) and f(nk+1,k, ξi) which, in general, do not belong to the quadrature set. A

procedure based on cubic spline interpolation has been proposed to avoid this difficulty in the scalar

case [10], but at the expense of introducing an approximation. Our way of dealing with this complication is

based on what we call “pre-processing of the interface conditions” [11]. On generalizing such a procedure

for the case with polarization, we get from Eqs. (3)–(5):

Ik(ak−1, µ, φ)− Z−k (µ)Ik(ak−1,−µ, φ) = Fk δ(µ− µk)δ(φ− φ0) + W−k (µ, φ) (6a)

and

Ik(ak,−µ, φ)− Z+k (µ)Ik(ak, µ, φ) = W+

k (µ, φ), (6b)

2/4

Polarized radiative transfer subject to Fresnel conditions

for µ ∈ (0, 1], φ ∈ [0, 2π], and k = 1, 2, . . . ,K. In these expressions, the 4× 4 matrices Z±k (µ) and the

4-vectors Fk and W±k (µ, φ) are defined and computed by recurrence along the layers, and we define

µk = f(n0,k, µ0). It should be noted that Eqs. (6) for a given layer are coupled by way of the W±k (µ, φ)

vectors, which depend on the Stokes vectors for the other layers. Although a direct ADO solution is

possible, as discussed in Ref. [11] for the scalar case, the use of an iterative approach for solving the

problem is usually simpler and more convenient, especially when the number of layers is large.

We thus consider that Eqs. (1) and (6) constitute the mathematical formulation of the problem. Our

solution was based on the ADO method and can be summarized by the following steps:

• A decomposition of the original problem into unscattered and scattered problems was performed.

• An analytical solution for the unscattered problem was derived.

• A Fourier decomposition was used to reduce the scattered problem in each layer to a set of

2(Lmax + 1) azimuthally-independent problems, where Lmax denotes the maximum Legendre

expansion order used to represent the phase matrices in the layers.

• The Dirac delta distribution δ(µ− µk) that appears in Eq. (6a) was approximated by the rectangular

nascent delta function [1]

δε(µ− µk) =

(µmax − µmin)−1, µmin ≤ µ ≤ µmax,0, otherwise,

(7)

where µmin = max0, µk − ε/2, µmax = minµk + ε/2, 1, and ε is the “narrowness” parameter.

• The ADO method of Ref. [12] was used to solve the 2(Lmax + 1) azimuthally-independent problems

obtained for each layer. A layer-dependent, composite quadrature scheme [13] was employed, to

avoid the discontinuities that may occur in the derivative of the Stokes vector with respect to the

cosine of the polar angle, due to the phenomenon of total reflection.

• For each of the 2(Lmax + 1) azimuthally-independent problems, an iterative scheme based on

sweeps through the layers was used to find the solutions of the resulting K sets of coupled linear

systems for the superposition coefficients of the ADO method. Only one LU decomposition per layer

is required and the iterations are fast since they consist of matrix-vector multiplications.

• Post-processing [12] was used to obtain an improved solution for the Stokes vector that is continuous

in all variables (τ , µ, and φ).

The ADO solution just outlined was implemented on a personal computer and used to generate tabular

results of benchmark quality for several test cases.

3/4

R. D. M. Garcia

ACKNOWLEDGMENTS

The work of the author was supported by CNPq.

References

[1] R. D. M. Garcia and C. E. Siewert, “On the use of a nascent delta function in radiative-transfercalculations for multi-layer media subject to Fresnel boundary and interface conditions,” J. Quant.Spectrosc. Radiat. Transfer, 111, 128–133 (2010).

[2] R. D. M. Garcia, C. E. Siewert, and A. M. Yacout, “Radiative transfer in a multi-layer mediumsubject to Fresnel boundary and interface conditions and uniform illumination by obliquely incidentparallel rays,” J. Quant. Spectrosc. Radiat. Transfer, 109, 2151–2170 (2008).

[3] L. B. Barichello and C. E. Siewert, “A discrete-ordinates solution for a non-grey model with completefrequency redistribution,” J. Quant. Spectrosc. Radiat. Transfer, 62, 665–675 (1999).

[4] R. D. M. Garcia and C. E. Siewert, “A simplified implementation of the discrete-ordinates method forradiative transfer with polarization,” in preparation (2011).

[5] L. Tsang and J. A. Kong, “Radiative transfer theory for active remote sensing of half-space randommedia,” Radio Sci., 13, 763–773 (1978).

[6] G. W. Kattawar and C. N. Adams, “Stokes vector calculations of the submarine light field in anatmosphere-ocean with scattering according to a Rayleigh phase matrix: Effect of interface refractiveindex on radiance and polarization,” Limnol. Oceanogr., 34, 1453–1472 (1989).

[7] C. Bordier, C. Andraud, E. Charron, and J. Lafait, “Radiative transfer model with polarization effectsapplied to organic matter,” Physica B, 394, 301–305 (2007).

[8] S. Chandrasekhar, Radiative Transfer, Oxford University Press, London (1950).

[9] R. D. M. Garcia, “Boundary and interface conditions for polarized radiation transport in a multilayermedium,” Proc. Int. Conf. on Mathematics and Computational Methods Applied to Nuclear Scienceand Engineering (M&C 2011), May 8–12, 2011, Rio de Janeiro, Brazil (2011).

[10] B.-T. Liou and C.-Y. Wu, “Radiative transfer in a multi-layer medium with Fresnel interfaces,” HeatMass Transfer, 32, 103–107 (1996).

[11] R. D. M. Garcia, C. E. Siewert, and A. M. Yacout, “On the use of Fresnel boundary and interfaceconditions in radiative transfer calculations for multilayered media,” J. Quant. Spectrosc. Radiat.Transfer, 109, 752–769 (2008).

[12] C. E. Siewert, “A discrete-ordinates solution for radiative-transfer models that include polarizationeffects,” J. Quant. Spectrosc. Radiat. Transfer, 64, 227–254 (2000).

[13] R. D. M. Garcia, “On the discontinuities in the angular derivative of the radiation intensity forradiative-transfer problems subject to Fresnel boundary and interface conditions,” Proc. ofEurotherm83—Computational Thermal Radiation in Participating Media III, April 15–17, 2009,Lisbon, Portugal (2009).

4/4


Modeling Radiative Thermal Transport in a Scramjet Combustor

Adam G. IrvineDepartment of Aerospace Engineering

University of Michigan1320 Beal Avenue

Ann Arbor, MI [email protected]

Iain D. [email protected]

Andrew J. [email protected]

Supersonic combustion ramjet (scramjet) combustors produce combustion products that are strong

radiators at temperatures where radiative thermal transport is a significant mode of energy transport. For a

hydrogen fueled scramjet operating in the temperature range of 2000-3000K, H2O and OH are the primary

participants in radiative thermal transport. There can also be an influence on the flow field due to radiation,

in particular shock locations, which can lead to changes in convective heat transfer. In the design of

scramjet engines it is critical to have an accurate heat flux distribution to the combustor walls and nozzle.

Numerical simulation can provide heat transfer data which would be otherwise difficult to obtain in the

hypersonic operating regime of scramjet engines.

Simulation of the radiative thermal transfer of a scramjet combustor requires consideration of a

non-isothermal, non-grey, heterogeneous gaseous system that could have a complex geometry.

Additionally an accurate spectral model must be employed to provide absorption, scattering, and emission

characteristics of the gaseous medium which is comprised of numerous species. These requirements

narrow the number of applicable modeling techniques considerably. In support of the Predictive Science

Academic Alliance Program (PSAAP) the primary goal of this study is to quantify the thermal energy

transfer by radiation in a scramjet engine, specifically the Hyshot II, while quantifying the error of the

radiative thermal transport simulation. Flow field solutions of the Hyshot II are used by the radiative

transport model, and treated as constant with the exception of thermal energy.

This work uses an Implicit Monte Carlo (IMC) to model radiative thermal transport as described by Fleck

and Cummings [1][2][3]. The code is multi-dimensional, non-isothermal, time dependent, and non-grey. A

validation case for an initially cold one dimensional slab heated by a blackbody is shown in Figure (1)

Adam G. Irvine, Iain D. Boyd, and Andrew J. Crow

(a) Fleck and Cummings results [1] (b) Validation results

Figure 1: 1D non-grey problem heated by blackbody at x = 0 at various time steps

showing the temperature distribution through the slab at different times, and shows agreement with the

results shown in by Fleck and Cummings[1]. The radiative transport model requires inputs of the flow field

properties as well as spectral properties that are based on the composition, temperature, and pressure of the

flow field.

Spectral models used in literature are typically band models that integrate regions of the spectrum with

assumptions about the distribution of the spectral lines in these regions [4][5][6]. A Line by Line (LBL)

model is capable of computing the spectrum accurately, but is seldom used due to high computational

costs. However, this cost can be mitigated by using a sufficiently detailed table of the absorption coefficient

without compromising accuracy[7]. The tables are discretized across wavenumber, temperature, pressure,

and species mole fraction. This work uses a LBL model that has been validated as shown in figure 2, using

the HITEMP 2010 database [7] which contains data for spectral lines necessary to compute the absorption

coefficient.

The coupling of the radiative transport and spectral modeling methods will be shown taking into

consideration spectral correlations [6]. The completed code will be compared against other radiative

transport models in order to determine what physics are important in minimizing epistemic uncertainty. A

model with acceptable epistemic uncertainty balanced against computational efficiency will be used in the

PSAAP program as part of an effort to quantify uncertainty in the simulation of a scramjet.

2/4

Modeling Radiative Thermal Transport in a Scramjet Combustor

(a) Model comparison from [8] (b) Validation results

Figure 2: Comparison of spectral model to literature results

ACKNOWLEDGMENTS

This material is based upon work supported by the Department of Energy [National Nuclear Security

Administration] under Award Number NA28614.

Disclaimer: This report was prepared as an account of work sponsored by an agency of the United States

Government. Neither the United States Government nor any agency thereof, nor any of their employees,

makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy,

completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that

its use would not infringe privately owned rights. Reference herein to any specific commercial product,

process, or service by trade name,trademark, manufacturer, or otherwise does not necessarily constitute or

imply its endorsement, recommendation, or favoring by the United States Government or any agency

thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the

United States Government or any agency thereof.

References

[1] Fleck, J.A., and Cummings, J.D., “An Implicit Monte Carlo Scheme for Calculating Timeand Frequency Dependent Nonlinear Radiation Transport,” J. Comp. Phys., 8, pp. 313-342(1971).

[2] Fleck, J.A., and Canfield, E.H., “A Random Walk Procedure for Improving theComputational Efficiency of the Implicit Monte Carlo Method for Nonlinear RadiationTransport,” 54, pp. 508-523 (1984).

[3] Larsen, E.W., and Mercier, B., “Analysis of a Monte Carlo Method Nonlinear RadiativeTransfer ,” J. Comp. Phys., 71, pp. 50-64 (1987).

3/4

Adam G. Irvine, Iain D. Boyd, and Andrew J. Crow

[4] Ludwig, C. B., Malkmus, W., Reardon, J. E., and Thompson, J. A. L., “Handbook of InfraredRadiation from Combustion Gases,” NASA SP-3080 (1973).

[5] Nelson, H.F., “Radiative Heating in Scramjet Combustors,” J. Thermophysics and HeatTransfer, 11, pp. 59-64 (1997).

[6] Liu, J., Tiwari, S. N., “Radiative Heat Transfer Effects in Chemically Reacting NozzleFlows,” J. Thermophysics and Heat Transfer, 10, pp. 436-444 (1996).

[7] Wang, A.,Modest, M.F., “Spectral Monte Carlo models for nongray radiation analyses ininhomogeneous participating media,” Intern. J. Heat and Mass Transfer, 50, pp. 3877-3889(2007)

[8] Rothman, L. S., Gordon. I. E., Barber, R.J., Dothe, H. Gamache, R.R., Goldman, A.,Perevalov, V.I., Tashkun, S.A., Tennyson, J., “HITEMP, the High-Temperature MolecularSpectroscopic Database,” J. Quant. Spectrosc. Radiat. Transfer, Vol 111, pp. 2139-2150(2010).

[9] Modest, M. F., Radiative Heat Transfer, McGraw-Hill, Inc., (1993).

4/4


TURBULENCE RADIATION INTERACTIONS IN PARTICULATE LADENFLOW

Mathew ClevelandDepartment of Nuclear Engineering

Oregon State University116 Radiation CenterCorvallis, OR 97331

[email protected]

Sourabh ApteOregon State University

Rodgers HallCorvallis, OR [email protected]

Todd PalmerDepartment of Nuclear Engineering

Oregon State University116 Radiation CenterCorvallis, OR 97331

[email protected]

The effects from Turbulence Radiation Interactions (TRI) in particulate laden flows can have a significant

effect on thermal radiation fields and corresponding material heating[1]. Radiative heat transfer has been

extensively studied in a variety of stochastic media including, but not limited to, combustion

problems[1–4]. Most combustion problems contain strong heterogeneities which can be treated

stochastically. In pulverized coal combustion these heterogeneities include particulate such as coal, fly-ash,

and char[1, 5]. These materials are typically accounted for stochastically using an atomic mix model. TRI

effects have been shown to be very sensitive to the presence of soot in turbulent flames, significantly

decreasing mean flame temperatures[6].

This work will expand upon a simplified test case, developed by Deshmukh et al.[7], to highlight the

effects of fuel particulate on TRI phenomena. This will include the addition of a simplified solid fuel

combustion problem. All problems will be grey and the material temperatures will be uncoupled with the

radiation field. This work will provide insight into the sensitivity of TRI uncertainties such as thermal

radiation fields and thermal emission to the presence of scattering and absorbing fuel particulate. This will

also provide insight into the effects of particulate on species dispersion and turbulence dissipation. Finally,

this research will look at the use of the atomic mix model as compared to individual particulate tracking

and its effect on radiative emission and absorption predictions.

This work will look at the combustion of particulate fuel in a statistically homogeneous non-premixed

system[7]. The thermal heating will be simulated using the following simplified combustion model

Todd S. Palmer and Some O. Guy

consisting of solid fuel and oxidizer;

O + F → P + Q (1)

In this reaction O, F , and P denote the concentration of oxidizer, fuel, and the resulting product P . The

variable Q represents the amount of energy created during the reaction. If a constant surface area reaction

rate is used to describe this combustion reaction and the oxidizer is in great excess to the fuel it is possible

to make a variety of useful observations about the means of the problem. This allows for the analytic

representation of particle evaporation on a per-particle basis as a function of time. The material properties

of the particles will be treated as constant as a function of time. The material properties of the gas will be

accounted for using similar simplified models based on the product concentration in the system. This will

allow for the comparison of different probability distribution functions describing particle distributions in

the turbulent flow. This can be demonstrated by comparing the numerically evaluated means as compared

to the means estimated by the particle distributions used.

< f(xp) >=∫ 1

0f(xp)P (xp) (2)

The function f(xp) defines some material property as a function of product concentration (xp) and the

function P (xp) is the probability of concentration xp existing in any given cell of the problem.

ACKNOWLEDGMENTS

This work was supported by Department of Energy-National Energy Technology Laboratory Contract No.

41817M4077. We wish to thank Dr. Cathy Summers (NETL, Albany) for monitoring this work.

REFERENCES

[1] D. Haworth, Progress in probability density function methods for turbulent reacting flows, Progress inEnergy and Combustion Science, 36, 2, pp. 168–259 (Apr. 2010).

[2] G. L. Olson, D. S. Miller, E. W. Larsen, and J. E. Morel, Chord length distributions in binary stochasticmedia in two and three dimensions, Journal of Quantitative Spectroscopy and Radiative Transfer, 101,2, pp. 269–283 (Sep. 2006).

[3] D. S. Miller, F. Graziani, and G. Rodrigue, Benchmarks and models for time-dependent grey radiationtransport with material temperature in binary stochastic media, Journal of Quantitative Spectroscopyand Radiative Transfer, 70, 1, pp. 115–128 (Jul. 2001).

[4] A. Wang, M. F. Modest, D. C. Haworth, and L. Wang, Monte Carlo simulation of radiative heattransfer and turbulence interactions in methane/air jet flames, Journal of Quantitative Spectroscopyand Radiative Transfer, 109, 2, pp. 269–279 (Jan. 2008).

2/3

Sample ICTT-22 Abstract

[5] J. G. Marakis, C. Papapavlou, and E. Kakaras, A parametric study of radiative heat transfer inpulverised coal furnaces, International Journal of Heat and Mass Transfer, 43, 16, pp. 2961–2971(Aug. 2000).

[6] L. Tess, F. Dupoirieux, and J. Taine, Monte Carlo modeling of radiative transfer in a turbulent sootyflame, International Journal of Heat and Mass Transfer, 47, 3, pp. 555–572 (Jan. 2004).

[7] K. Deshmukh, D. Haworth, and M. Modest, Direct numerical simulation of turbulence-radiationinteractions in homogeneous nonpremixed combustion systems, Proceedings of the CombustionInstitute, 31, 1, pp. 1641–1648 (Jan. 2007).

3/3


KINETIC EQUATIONS FOR STARK LINE SHAPES

J. Rosato, H. Capes, Y. Marandet, A. Mekkaoui, and R. Stamm Laboratoire PIIM

UMR 6633 Université de Provence / CNRS Centre de St-Jérôme, Case 232 F-13397 Marseille Cedex 20 [email protected]

We address the issue of line shape modeling using kinetic equations for plasma spectroscopy purposes. Line shapes – the probability density functions for an atom emitting / absorbing a photon at a given frequency ω – are proportional to the Fourier transform of the dipole autocorrelation function. This quantity involves the statistical average of the atomic evolution operator U(t) performed over the perturbers’ states. Assuming classical kinetic plasma, these perturbers are characterized by the classical N-particle phase space distribution fN(1…N,t). One of the most widely used models for Stark broadening at low density is the so-called impact approximation [1]: it is assumed that the atom interacts briefly with one perturber at once (like collisions) and the resulting line shape is a Lorentzian function, whose width is determined from matrix elements of an operator K characterizing the collision frequency. In this work, we examine the role of nonbinary collisions by developing a generalization of the model for the collision operator. This is done in the framework of the so-called unified theory for Stark broadening [2, 3]. We introduce a hierarchy of operators φs(1…s,t) = fs(1…s,t)U(t), s = 0…N, where fs is the s-particle reduced phase space distribution, and we obtain a BBGKY-like hierarchy of evolution equations for the operators φs. In the most simplified treatment of these equations, triple correlations between the atom and the perturbers are neglected and the usual impact collision operator is obtained. Here we relax this assumption and retain triple correlations using a generalization of the Kirkwood truncature hypothesis [4] to quantum operator. Applications to hydrogen lines are done in the framework of tokamak plasma spectroscopy. We show that the neglect of triple correlations can lead to a significant overestimate of the width of lines with a high principal quantum number (see Fig. 1). This is confirmed by comparisons to numerical simulations [5].


2

6 7 8 9 10

5.0x10-5

1.0x10-4

1.5x10-4

Nonbinary model

Binary approximation

HW

HM

(a.

u.)

Principal quantum number

Figure 1. Half width at half maximum (HWHM) of hydrogen lines when triple

correlations are retained (circles), compared to the binary approximation

(squares). The binary approximation overestimates the broadening as the

principal quantum number increases.

ACKNOWLEDGMENTS

This work is supported by the French National Research Agency (contract ANR-07-

BLAN-0187-01) and by the collaboration (LRC DSM99-14) between the PIIM

laboratory and the CEA Cadarache (Euratom Association) within the framework of

the French Research Federation on Magnetic Fusion (FRFCM).

REFERENCES

[1] H. R. Griem, Spectral Line Broadening by Plasmas, Academic Press, London, UK

(1974).

[2] D. Voslamber, “Unified Model for Stark Broadening,” Z. Naturforsch., 24a, pp.

1458-1472 (1969).

[3] E. W. Smith, J. Cooper, and C. R. Vidal, “Unified Classical-Path Treatment of Stark

Broadening in Plasmas,” Phys. Rev., 185, pp. 140-151 (1969).

[4] J. G. Kirkwood, “Statistical Mechanics of Fluid Mixtures,” J. Chem. Phys., 3, pp.

300-313 (1935).

[5] R. Stamm, E. W. Smith, and B. Talin, “Study of Hydrogen Stark Profiles by means

of Computer Simulation,” Phys. Rev. A, 30, pp. 2039-2046 (1984).


Plasma Density Fluctuation Effects on the Screening of Neutral Species inTokamaks

A. Mekkaoui, Y. Marandet, J. Rosato, R. Stamm, H. Capes, M. Koubiti and L. Godbert-MouretPIIM, CNRS/Universite de Provence

Centre de Saint JeromeMarseille F-13397 Cedex 20, France

[email protected]

D. ReiterIEF-4 Plasmaphysik

Forschungszentrum Julich GmbHTEC Euratom association, D-52425 Julich, Germany

[email protected]

The dynamic of neutral species in plasmas is very important in the design and operation of future fusion

devices like ITER. Neutral particles (atoms and molecules) originate mainly from the recycling of ions

impinging on the wall, and their density is high only in the outer regions of the plasma, close to the wall. In

this part of the plasma, called the scrape-off layer (SOL), strong turbulent fluctuations are observed [1].

The effects of plasma density fluctuations on neutral particles transport have initially been studied by Prinja

[2], assuming Gaussian statistics. The latter choice leads to an exact effective transport model, but negative

density realizations implies nuisances [3][4], which we tried to alleviate using multivariate Gamma

statistics for the fluctuations. Indeed, recent experimental measurements showed that the plasma density in

the SOL is Gamma distributed [5]. The charge exchange (scattering) and ionization (absorption) processes

are taken into account, and the analytical solution of the transport equation obtained by Smirnov [6] is

averaged over fluctuations, leading to an exact result for the average neutral particles density (cf. Fig. 1 and

Fig. 2) and its higher moments. Expressions for the average ionization source and the cross correlation of

neutral particles density are obtained. The probability density functions of the optical depth and of the

neutral density are derived. For a given fluctuations rate (R = ne/ne), the Kubo number [7] is identified as

a statistical control parameter. The role of stochastic boundary conditions, which are correlated to the

plasma density statistics through ions recycling process, is investigated and an effective Boltzmann-like

equation is obtained.

Mekkaoui A. et al.

10-4

10-3

10-2

10-1

100

<N>/

N 0

8642x/λmfp

K=0 K=0.8 K=4 K=∞

Λ=0.8R=80%

Figure 1. Deterministic source. Neutral penetration depth in SOL plasma normalized to the neutral meanfree path (λmpf) vs Kubo number (K = Rλ/λmfp), where λ is the correlation length of fluctuations andR = ne/ne. Λ = νcx/(νcx + νie), is the ratio of the charge exchange (scattering) rate (νcx) to the total rate,where νie is the ionization (absorption) rate. The zero Kubo number corresponds to the turbulence-free case(R = 0).

10-3

10-2

10-1

100

<N>/

N 0

8642x/λmfp

K=0 K=0.8 K= ∞

Λ=0R=80%

Figure 2. Stochastic source. Neutral penetration depth in the SOL within the scattering-free approximation(Λ = 0).

ACKNOWLEDGMENTS

This work is the result of collaboration (LRC DSM99-14) between the PIIM laboratory and the CEACadarache (Euratom Association) within the framework of the French Research Federation on MagneticFusion (FRFCM), whith the support of the French National Research Agency (contract ANR-07-

2/3

Screening of Neutral Species

BLAN-0187-01).

REFERENCES

[1] J. A. Boedo, “Edge turbulence and sol transport in tokamaks”, J. Nucl. Mater., 390, pp. 29-37,(2009).

[2] Anil. K. Prinja, “The effect of turbulent density fluctuations on neutral particle transport,”. In Proc.U.S. Edge Plasma Physics : Theory and Application Workshop, Albuquerque, New Mexico, April26-28 (1993).

[3] Anil. K. Prinja and A. Gonzalez-Aller, “ Particle Transport in the presence of Parametric Noise,”Nucl. Sci. Eng., 124, pp. 89-96 (1996).

[4] V. Bezak, “ Stationary Particle Transport in a Randomly Inhomogeneous Rod,” Acta Phys. Slovaca,52, pp. 11-22 (2002).

[5] J. P. Graves, et al., “Self-similar density turbulence in the TCV tokamak scrape-off layer”, PlasmaPhys. Control. Fus., 47, pp. 1-9 (2005).

[6] S. Rehker and H. Wobig, “A Kinetic Model for the neutral gas between plasma and wall,” PlasmaPhys., 15, pp. 1083-1097 (1973).

[7] R. Kubo, “Stochastic Liouville Equation”, J. Math. Phys., 4, pp. 174183 (1963).

3/3


A METHOD FOR IMPROVING THE EIGENVALUE IN TRANSPORTPROBLEMS

Simon R. Merton, Richard Smedley-StevensonComputational Physics Group

AWE AldermastonReading, United [email protected]

Chris C. PainDepartment of Earth Science and Engineering

Imperial College LondonLondon, United Kingdom

[email protected]

A method is described for improving the accuracy of the eigenvalue in neutron transport problems. Using

the adjoint system of equations to the forward model, an approximation to the error in the eigenvalue is

derived. This is achieved by convolving the residual of the governing equation with the adjoint eigenvector.

A defect iteration on the eigenvalue is then performed in which the approximation to the error is removed

from the eigenvalue. This is shown to improve the estimate of the eigenvalue on coarse computational

meshes. Alternatively, the scheme can generate a mesh-wise sensitivity map of the eigenvector. This may

be used as a metric to automate a mesh adaption algorithm reliably enhancing fidelity of the solution.

One may expand the ith eigenvalue λ(ψ) and the residual R(ψ) in a first-order Taylor series, in which ψ

and ψ are the ith eigenvector and exact ith eigenvector respectively, in the continuum:

λ(ψ)− λ(ψ) =∂λ(ψ)

∂ψ

(ψ − ψ

)(1)

R(ψ)− R(ψ) =∂R(ψ)

∂ψ

(ψ − ψ

)(2)

Eqs. (1)–(2) are exact provided all higher derivatives are zero. This can be achieved by applying certain

normalisations to the eigenvector and has the advantage that the functional Hessian is not needed.

Combining Eqs. (1)–(2) and using the fact that R(ψ) = 0 one readily obtains an expression for the error in

the eigenvalue:

λ(ψ)− λ(ψ) = −∂λ(ψ)∂ψ

(∂R(ψ)

∂ψ

)−1

R(ψ)

Simon R. Merton, Richard Smedley-Stevenson, Chris C. Pain

= −R(ψ)T(∂R(ψ)

∂ψ

)−T (∂λ(ψ)

∂ψ

)T

= −R(ψ)Tψ∗ (3)

in which the definition of the adjoint ith eigenvector ψ∗ has been inserted. One obtains a first estimate of

the eigenvalue, for example via inverse power iteration on the fission source. The adjoint system of

equations to the forward model is then computed. A defect iteration on the eigenvalue is then performed in

which Eq. (3) is used to obtain a computational estimate of λ(ψ).

A simple test case is used to demonstrate this correction procedure applied to the Boltzmann Transport

Equation, discretised using linear discontinuous finite elements in Cartesian (X-Y) geometry. Spherical

Harmonics (PN ) bases are used to perform the angular expansion. However, the framework is completely

general and discrete ordinates (SN ) or wavelets (LWN ) may alternatively be used. The materials used in

the test case are illustrated in Table I.

Table I. Material parameters for test problem 1Region σt σf σs

1 10.0 10.0 0.02 10.0 0.0 0.0

Absolute error in the eigenvalue is plotted against the number of elements in Fig. 1.

Figure 1. Defect correction in test problem 1

Three element types (p=1, p=2 and p=3) are plotted without correction. The p=1 result with the correction

2/3

A Method for Improving the Eigenvalue in Transport Problems

is included. Correction is seen to improve the convergence of the eigenvalue, producing a slope that is

better than the uncorrected eigenvalue on a p=1 element mesh but not quite as good as the uncorrected

eigenvalue on a p=2 element mesh. However, the computational cost of obtaining the eigenvalue on

higher-order meshes is significantly greater than the cost of the correction procedure on a linear element

mesh.

ACKNOWLEDGMENTS

The work of the first author has been performed as part of a PhD in computational physics at Imperial

College London. This PhD has received funding from AWE plc.

REFERENCES

[D. A. Venditti and D. L. Darmofal, 2002] D. A. Venditti and D. L. Darmofal. “Grid Adaptation forFunctional Outputs: Application to Two-Dimensional Inviscid Flows” Journal of ComputationalPhysics., 176, pp. 40-69 (2002).

[D. A. Venditti and D. L. Darmofal, 2003] D. A. Venditti and D. L. Darmofal. Anisotropic GridAdaptation for Functional Outputs: Application to Two-Dimensional Viscous Flows. Journal ofComputational Physics., 187, pp. 22-46 (2003).

[D. A. Venditti and D. L. Darmofal, 2000] D. A. Venditti and D. L. Darmofal. Adjoint Error Estimationand Grid Adaptation for Functional Outputs: Application to Quasi-One-Dimensional Flow. Journalof Computational Physics., 164, pp. 204-227 (2000).

[N. A. Pierce and M. B. Giles, 2004] N. A. Pierce and M. B. Giles. Adjoint and Defect Error Boundingand Correction for Functional Estimates. Journal of Computational Physics., 200, pp. 769-794(2004).

[M. B. Giles and N. A. Pierce and E. Suli, 2004] M. B. Giles and N. A. Pierce and E. Suli. Progress inAdjoint Error Correction for Integral Functionals. Computing and Visualization in Science., 6, pp.113-121 (2004).

[M. Ainsworth and J. T. Oden, 1997] M. Ainsworth and J. T. Oden. A Posteriori Error Estimation in FiniteElement Analysis. Computer Methods in Applied Mechanics and Engineering., 142, pp. 1-88(1997).

[P. W. Power, 2007] P. W. Power PhD Thesis:Error Measures for Finite Element Ocean Modelling.Imperial College, London United Kingdom (2007).

3/3


NEUTRON TRANSPORT IN MOLTEN SALT REACTORS

Imre PazsitDepartment of Nuclear EngineeringChalmers University of Technology

SE-412 96 Goteborg, [email protected]

Molten Salt Reactors (MSR) are one of the six reactor types which were selected as prospective

Generation-IV systems. An MSR consists of a fuel circulating in a primary circuit in form of a molten

fluoride salt, such that it passes a moderating core where the the critical neutron multiplication takes place.

Such reactors open up a new type of physics and neutron transport problems due to the fact that the delayed

neutron precursors move together with the fuel.

We have investigated some of the properties of the static and dynamic equations and their solutions in a

simple model system. Regarding the static equations, the presence of directed convection of the delayed

neutron precursors means the appearance of yet another term in the neutron transport equation which

makes the equation non self-adjoint. Construction of the adjoint equation and the adjoint flux requires, in

addition to the usual manipulations, reversing the flow direction of the fuel and a reversion of the boundary

conditions for the delayed neutron precursors. The presence of the fuel convection term results in the fact

that for an MSR, not even the one-group diffusion equation is self-adjoint.

Elimination of the equation for the delayed neutron precursors by quadrature will lead to an

integro-differential equation in one-group diffusion theory with no closed form solution. Analytical

solution can be obtained in the special case of assuming infinite fuel circulation velocity, in which case also

the equation becomes self-adjoint.

We have also considered the dynamic response of the system in the frequency domain to a fluctuation of the

absorption cross sections. For infinite fuel velocity, the Green’s function in the frequency domain reads as

∇2G(x, x0, ω) +B2(ω)G(x, x0, ω) +η(ω)

T

∫ a

−aG(x, x0, ω)dx′ = δ(x− x0) (1)

where

η(ω) =λνΣfβ

D(λ+ iω)(2)

An analytical solution of this equation can be obtained by a method similar to the elimination of the

uncollided flux, i.e. to search the solution in the sum of two terms, one satifying the inomogeneous r.h.s.

but without the integral term, and the other containing the integral term but with a zero r.h.s. This solution


method can be used to construct solutions for the considerably more complicated case of finite fuel

circulation velocities. These solutions will be given and discussed in the talk.


NEUTRON THERMAL SCATTERING LAWS FOR LIGHT AND HEAVY WATER FOR MODELING CRITICAL ASSEMBLIES AND TOF EXPERIMENTAL SET-UPS WITH NEUTRON TRANSPORT

CODES

Dan Roubtsov, Ken Kozier Atomic Energy of Canada Limited

Chalk River Laboratories Chalk River, Ontario K0J 1J0, Canada

[email protected], [email protected]

Björn Becker, Yaron Danon Department of Mechanical, Aerospace, and Nuclear Engineering

Rensselaer Polytechnic Institute Troy, New York 12180, USA

[email protected],[email protected]

Seven years have passed since the seminal work, “How Accurately Can We Calculate Thermal Systems?” by D.E. Cullen et al. was issued [1]. Meanwhile, the new Thermal Scattering Laws (TSL) for H2O and D2O have been available in the evaluated nuclear data libraries, such as JEFF 3.1 (2005, [2]) and ENDF/B-VII.0 (2006, [3]); see also multi-group libraries developed by Kyoto University group [4]. Consequently one can ask whether we can now do a better (more accurate) modeling of the thermal systems with the new TSL. To address this question, we will discuss the theoretical models and methods used to produce the neutron TSL for H2O and D2O that are accepted in the modern evaluated nuclear data libraries, such as, JEFF 3.1, ENDF/B-VII.0, and JENDL 4.0. These TSL are usually used to produce the thermal scattering data files for Monte Carlo transport codes, such as MCNP(X), and multi-group deterministic transport codes, such as, WIMS, HELIOS, etc. The standard way to produce a TSL in the ENDF format is to use LEAPR module of the nuclear data processing code NJOY developed at LANL, USA [5]. As with any computer code designed to model physical phenomena, LEAPR has some limitations from both the theory and computational standpoints [6, 7]. For example, it is generally assumed that the self-diffusion, librations (hindered translations and hindered rotations), and intra-molecular vibrations in liquids are independent [8]. It is also assumed that one can disregard coherence effects in neutron scattering on H2O and use Vineyard’s or Sköld’s approximation for the coherent scattering component of the thermal scattering kernel of D2O [2]. Nevertheless, within the current LEAPR capabilities and NJOY structure, there is a way to improve the accuracy of TSL for H2O and D2O by using, for example, the temperature dependent libration spectra and static structure factors based on recent advances in simulations and experiments on liquids. Then, can we distinguish different models of neutron scattering on light/heavy water in reactor physics applications and in modeling neutron scattering set-ups using standard neutron transport codes and the nuclear data libraries? In addressing this question, we


2

will discuss how the results of simulations of the ZED-2 reactor at the Chalk River Laboratories (CRL) are sensitive to the different TSL’s of H2O and D2O in terms of keff bias. For this study, we use MCNP5 [9] and a multi-temperature ENDF/B-VII.0-based MCNP library generated at CRL [10]. In the ZED-2 reactor, one can assemble different critical cores by using different types of coolant (light water/heavy water/air), different lattice pitch, etc.; the cores are always moderated by heavy water. As there is a large amount of heavy water moderator and reflector in the ZED-2 cores, the neutron spectrum in such a moderator is always dominated by its Maxwellian component; see Figure 1. We found that the TSL reactivity worth of D-in-D2O is not very large, keff(D-in-D2O s(E,T)) keff(Free-Gas-H s(E,T)) ~ 1 mk (= 100 pcm), and so the sensitivity of keff to the details of D-in-D2O scattering data is small. Here, the TSL reactivity worth for a nuclide A is eff(TSL A) eff(Free-Gas-Model A) keff(TSL A) keff(Free-Gas-Model A) for modeling the critical cores (keff 1.0). For the coolant, on the other hand, the situation is different and, for example, for H2O-cooled cases, the TSL reactivity worth of H-in-H2O is significant, keff(H-in-H2O s(E,T)) keff(Free-Gas-H s(E,T)) ~ 10 mk (= 1000 pcm), and keff is sensitive enough to H-in-H2O scattering data (see Figure 2), such as the position of the hindered rotation peak in the libration spectrum of H2O. Then we will discuss whether one can obtain significant deviations from the Maxwellian asymptotic behavior of the neutron fluxes (r, E) at low energies (E < 0.1 eV). For testing of TSL’s and transport calculations, we discuss the MCNP simulation of time-of-flight (TOF) measurement set-up, in which thermal neutrons are incident on a thick slab of water and the thermal neutron leakage is measured; see Figure 3. Although it is more an integral-type of experiment, experiments with water slabs could be performed to address the question of how accurately can we predict the responses of simple thermal systems using the available TSL’s and Monte Carlo neutron transport codes.


3

Figure 1. Thermal scattering cross sections for D-in-D2O at T= 293.6 K and a typical neutron spectrum in ZED-2 (D2O-cooled and D2O-moderated reactor

lattice)

Figure 2. Thermal scattering cross sections for H-in-H2O at T= 300.0 K and a typical neutron spectrum in ZED-2 (H2O-cooled and D2O-moderated reactor

lattice)


4

Vacuum

ElectronBeam

Neutron Source(Ta Target)

NeutronDetector

TOF(15 m)

Moderator Slab(H2O, D2O or

CH2)

Figure 3. Neutron leakage experiment (neutron beam is slowed down in a slab of water)

REFERENCES

[1]D.E.Cullen,R.N.Blomquist,Ch.Dean,D.Heinrichs,M.A.Kalugin,M.Lee,

Y.‐K.Lee,R.MacFarlane,Y.Nagaya,andA.Trkov,“HowAccuratelyCanWeCalculateThermalSystems?”,ReportINDC(USA)‐107,UCRL‐TR‐203892,(2004).

[2]M.Mattes,J.Keinert,“ThermalNeutronScatteringDatafortheModeratorMaterialsH2O,D2OandZrHxinENDF‐6FormatandasACELibraryforMCNP(X)Codes”,ReportINDC(NDS)-0470, IAEA, (2005).

[3]M.B.Chadwick,P.Obložinskýetal.(CSEWGcollaboration),“ENDF/B‐VII.0:NextGenerationEvaluatedNuclearDataLibraryforNuclearScienceandTechnology”,Nuclear Data Sheets,107,pp.2931‐3060(2006).

[4]N.Morishima,S.Ito,“NeutronMulti‐GroupCross‐sectionsofModeratorMaterialsforColdandUltra‐ColdNeutronProduction”,Nuclear Instruments and Methods in PhysicsResearch Section A,572,pp.1071‐1082(2007).

[5]R.E.MacFarlane,D.W.Muir,“TheNJOYNuclearDataProcessingSystem,Version91”,LANLreportLA‐12740‐M,(1994).

[6]R.E.MacFarlane, A.C.Kahler,“MethodsofProcessingENDF/B‐VIIwithNJOY”,Nuclear Data Sheets,111,pp.2739‐2890(2010).

[7]A.Yamamoto,N.Sugimura,“ImprovementonMulti‐GroupScatteringMatrixinThermalEnergyRangeGeneratedbyNJOY”,Annals of Nuclear Energy,33,pp.555‐559(2006).

[8]R.E.MacFarlane,“NewThermalNeutronScatteringFilesforENDF/B‐VIRelease2”,LANLreportLA‐12639‐MS,(1994).

[9]X‐5MonteCarloTeam,“MCNP—AGeneralMonteCarloN‐ParticleTransportCode”,Version5,LANLreportLA‐UR‐03‐1987(2003,Revised10/3/2005).

[10]D.Altiparmakov,“ENDF/B‐VII.0versusENDF/B‐VI.8inCANDUCalculations”,Proceedings of PHYSOR 2010 – Advances in Reactor Physics to Power the Nuclear Renaissance, Pittsburgh,Pennsylvania,USA,May9‐14,2010,onCD‐ROM,AmericanNuclearSociety,LaGrangePark,IL(2010).


RECENT DEVELOPMENTS ON EXPLICIT FORMULATIONS FORNODAL SCHEMES OF TWO-DIMENSIONAL NEUTRON

TRANSPORT PROBLEMS

Joao Francisco Prolo FilhoInstituto de Matematica, Estatıstica e Fısica

Universidade Federal do Rio GrandeAv. Italia, Km 8

96202-900 Rio Grande, RS, [email protected]

Luciana Chimendes CabreraInstituto de Fısica e MatematicaUniversidade Federal de Pelotas

96001-970 Capao do Leao, RS, [email protected]

Liliane Basso BarichelloInstituto de Matematica

Universidade Federal do Rio Grande do SulAv. Bento Goncalves 9500

91509-900 Porto Alegre, RS, [email protected]

In this work, we develop closed form solutions for the discrete ordinates integrated equations, derived from

the application of a nodal scheme in a two-dimensional neutron transport problem. The formulation [1] is

based on the idea of the ADO method [2] and due to the analytical feature of the approach it is not

necessary the spatial domain division into cells. In addition, the properties of the level symmetric

quadrature scheme [3] are used to derive eigenvalue systems of reduced order, when compared with other

discrete ordinates approaches. The above mentioned features seem to contribute for a more efficient

formulation in the computational point of view. Here, different schemes are analyzed for defining auxiliary

equations which relate the integrated fluxes with the unknown fluxes at the boundary, introduced into the

model from the integration process. Numerical results are presented and compared with test cases available

in the literature [4, 5] for rectangular regions with an unitary source located in the center of the

homogeneous medium.

ACKNOWLEDGMENTS

The authors would like to thank to CNPq of Brazil for partial financial support to this work.

J. F. Prolo Filho, L. C. Cabrera and L. B. Barichello

REFERENCES

[1] L. B. Barichello, L.C. Cabrera and J. F. Prolo Filho, “An analytical approach for a nodal scheme oftwo-dimensional neutron transport problems,” Ann. Nucl. Energy, 38, pp. 1310-1317 (2011).

[2] L. B. Barichello and C. E. Siewert, “A discrete-ordinates solution for a non-Grey model withcomplete frequency redistribution,” J. Quant. Spectrosc. Radiat. Transfer, 62, pp. 665-675 (1999).

[3] E. E. Lewis and W. F. Miller, Computational Methods of Neutron Transport, John Wiley & Sons,New York USA (1984).

[4] R. W. Tsai and S. K. Loyalka, “A numerical method for solving the integral equation of neutrontransport: III,” Nucl.Sci. Eng., 59, pp. 536-540 (1976).

[5] Y. Watanabe and C. W. Maynard, “The discrete cones method for two-dimensional neutron transportcalculations,” Transp. Theory Stat. Physics, 15, pp. 135-156 (1986).

2/2

The 22nd International Conference on Transport Theory (ICTT-22)

Portland, Oregon, September 11-15, 2011

A NUMERICAL METHOD FOR ONE-SPEED SLAB-GEOMETRY

ADJOINT DISCRETE ORDINATES PROBLEMS WITH NO

SPATIAL TRUNCATION ERROR

Damiano S. Militão, Hermes Alves Filho, Ricardo C. Barros

Programa de Pós-graduação em Modelagem Computacional

Universidade do Estado do Rio de Janeiro

Rua Alberto Rangel s/n

28630-050 Nova Friburgo, RJ, Brasil

[email protected]

It is well known that the adjoint transport operator plays a very important role in

deterministic and stochastic particle transport calculations both in non-multiplying and in

multiplying media [1-3].

Presented here is an application of the one-speed spectral Green’s function (SGF) method

[4] for adjoint discrete ordinates (SN) problems in slab geometry. The adjoint SGF

method is free from spatial truncation errors; it generates numerical values for the node-

edge and node-average adjoint angular fluxes that agree with the analytic solution of the

adjoint SN equations, apart from computational finite arithmetic considerations.

Therefore, let us consider a spatial grid on a slab of thickness L where each spatial node

Гj has width hj = xj+1/2 – xj-1/2, total and scattering macroscopic cross sections σTj and σSj

respectively, with j = 1 : J, cf. Fig. 1. The adjoint SN equations in Гj appear as

,:1,2

)()( †

1

††† NmQxxdx

dj

N

n

nn

Sj

mTjmm

(1)

where we have defined )(† xm as the adjoint angular flux and †

jQ as the adjoint

source [5].

Figure 1: Spatial Grid By performing a spectral analysis of Eq. (1) in a way which is very similar to the spectral analysis described in Ref. [4], we write

0 L

x1/2 x3/2

Г1

xj-1/2 xj+1/2 XJ-1/2 xJ+1/2

Гj ГJ

hj



2

NmxQ

eax j

SjTj

jx

lm

N

l

lmlTj :1,,

)()()(

†

/†

1

†

(2)

as the general solution of Eq. (1) within Гj, where l are arbitrary constants and l ,

l = 1 : N, are N real eigenvalues that appear in pairs. Now we integrate Eq. (1) within node Гj and divide the result by hj to obtain the conventional discretized spatial balance adjoint SN equations, which together with the offered SGF adjoint auxiliary equations

)( ††

2/1,0

,

†

2/1,0

,

†

, jjnnmjnnmjm QHnn

(3)

form the adjoint SGF nodal equations. Furthermore, we substitute Eq. (2) into Eq. (3) to obtain the N2 values of Λm,n by solving

.)()()()2

sinh(2

0

†

,

2/

0

†

,

2/†

n

ljTj

n

ljTj

lnnm

h

lnnm

h

lm

l

Tjj

Tjj

l aeaeah

h

(4)

Here, for fixed m, we vary l = 1 : N to obtain a system of N linear equations in N unknowns Λm,n, n = 1 : N. Therefore, for m = 1 : N, we solve N systems for the N2 values of Λm,n. Moreover, we obtain the expression

SjTj

j

N

n

nm

j

Q

QH

†

1

,†

)1(

)( . (5)

We use the values of Λm,n so obtained and Eq. (5) in the SGF adjoint auxiliary equations (3); then we substitute the result into the collision and scattering terms of the discretized spatial balance adjoint SN equations to write the adjoint one-node block inversion (NBI†) matrix equation

0,††††††† mj

OUT

1/2j

OUT

1/2j

IN

1/2j QSψGψGψ . (6)

We use Eq. (6) to sweep from left to right (μm > 0) across the nodes Гj of the spatial grid set up on the slab, j = 1 : J. We remark that we use the most recent estimates for the node exiting adjoint angular fluxes to calculate the incident adjoint angular fluxes in the directions of the SN transport sweep; in this case from left to right



3

(μm > 0). Therefore, the NBI† iterative scheme is based on sweeping from left to right (μm > 0) and from right to left (μm < 0) until a prescribed convergence criterion is satisfied. A negative feature of coarse-mesh methods is that they do not generate detailed profile of the solution; therefore, to remedy this drawback, we substitute the converged adjoint SGF numerical solution into Eq. (2) and solve the resulting system for the expansion coefficients βl , l = 1 : N. With these, we are able to reconstruct the adjoint solution at any point within each spatial node Гj, j = 1 : J. Now we present some numerical results to a test problem, which consists of a 30 cm slab with three distinct regions: region 1 (10 cm, σT = 1.0 cm-1, σS = 0.9 cm-1); region 2 (5 cm, σT = 1.0 cm-1, σS = 0.9 cm-1) and region 3 (15 cm, σT = 0.9 cm-1, σS = 0.8 cm-1).

gamma rays decrease in intensity with the 267.361-day half-life characteristic

of . Therefore, by storing 1022 atoms of

(approximately 1g) in region 1 of the slab, we estimate a measurement of a detector response (5 cm, σa = 0.5 cm-1) that we place in region 3 (25 < x < 30) to evaluate the gamma ray leakage at the moment of the storage as well as one year and five years later, when the source has leaked to region 2. Table I displays the numerical results for the S128 Gauss-Legendre model.

Table I. Numerical results to the model problem

Time

Absorption rate in the detector region (measurement of the detector response)

Forward S128 DDa method

Forward S128 SGFb method

Adjoint S128 DD method

Adjoint S128 SGFc method

t = 0 3.98 x 1019

(15 sec)d

3.98 x 1019

(0.094 sec) 3.98 x 1019 3.98 x 1019

t = 1 year

2.13 x 1020

(15 sec) 2.13 x 1020

(0.093 sec) 2.12 x 1020 2.13 x 1020

t = 5 years

4.78 x 1018

(15 sec) 4.78 x 1018

(0.078 sec) 4.78 x 1018 4.78 x 1018

Total execution time

45 sec Total execution time

0.265 sec Total execution time

3.300 sec Total execution time

0.130 sec a. Diamond Difference [2] on a spatial grid composed of 3000 cells. b. Spectral Green’s function (SGF) [1] on a spatial grid composed of 1 node per region. c. Present SGF method for adjoint SN on a spatial grid composed of 1 node per region. d. The execution time of this run was 15 seconds.

For the forward S128 model we applied the classical reflective boundary conditions at x = 0 and vacuum boundary conditions at x = 30 cm. For the adjoint S128 we applied no-leakage boundary conditions at x = 30 cm and set Q† = 0.5 at 25 < x < 30, which is the detector absorption cross section. We remark that to estimate the detector response when gamma source changes and leaks to the adjacent region by using the forward SN model, we need to run the SN code over again, since by changing the gamma source location and/or intensity, different forward SN



4

problems arise. On the other hand, by using the adjoint SN model, we simply need to integrate the source weighted by accurate adjoint SN solution over all the independent variables used in the model; i.e., x-coordinate direction and N angular directions in the present model. Therefore, the adjoint SGF method, as described here, is very efficient at estimating measurements of detector responses due to uncharged radiation sources. As future work, we propose a generalization of the present adjoint SGF method for energy multigroup adjoint SN problems.

ACKNOWLEDGEMENTS This research work is part of the project of the National Institute of Science and

Technology on Innovative Nuclear Reactors (INCT/CNPq – FAPERJ, Brazil). The first

author (DSM) acknowledges the financial support given by Fundação Coordenação de

Aperfoiçoamento de Pessoal de Nível Superior (CAPES, Brazil) for the development of

this work.

REFERENCES

[1] Q. Zhang and H. S. Abdel-Khalik, “Adjoint-Based Global Variance Reduction

Approach for Reactor Analysis Problems”, International Conference on Mathematics

and Computational Methods Applied to Nuclear Science and Engineering (M&C

2011) Rio de Janeiro, RJ, Brazil, May 8-12, 2011, on CD-ROM, Latin American

Section (LAS) / American Nuclear Society (ANS) ISBN 978-85-63688-00-2 (2011).

[2] T. D. McLaughlin, G. E. Sjoden and K. L. Manalo, “Detector Placement Optimization

for Cargo Containers using Deterministic Adjoint Transport Examination for SNM

Detection”, International Conference on Mathematics and Computational Methods

Applied to Nuclear Science and Engineering (M&C 2011) Rio de Janeiro, RJ, Brazil,

May 8-12, 2011, on CD-ROM, Latin American Section (LAS) / American Nuclear

Society (ANS) ISBN 978-85-63688-00-2 (2011).

[3] J. M. Hykes and Y. Y. Azmy, “Radiation Source Reconstruction with Known

Geometry and Materials using the Adjoint”, International Conference on Mathematics

and Computational Methods Applied to Nuclear Science and Engineering (M&C

2011) Rio de Janeiro, RJ, Brazil, May 8-12, 2011, on CD-ROM, Latin American

Section (LAS) / American Nuclear Society (ANS) ISBN 978-85-63688-00-2 (2011). [4] R. C. Barros and E. W. Larsen, “A Numerical Method for One-Group Slab-

Geometry Discrete Ordinates Problems with no Spatial Truncation Error,” Nuclear Science and Engineering, 104, pp. 199-208 (1990).

[5] E.E. Lewis and W.F. Miller Jr., Computational Methods of Neutron Transport, John

Wiley & Sons, New York (1993).


ON SN-PN EQUIVALENCE

Richard Sanchez

Commissariat a l’Energie Nucleaire et aux Energies AlternativesCEA de Saclay, DM2S SERMA

91191 Gif-sur-Yvette cedexFrance

[email protected]

Ray effects are a consequence of the fact that the discrete set of directions of the SN quadrature formula isnot invariant under arbitrary rotations. A significant number of techniques have been proposed to diminish orcompletely eliminate ray effects,[1],[2],[3],[4] and the modern consensus is to reformulate the discrete ordinates(DO) equation as a projection over a set of angular interpolation functions in a space F and on completingthe PN basis so as to obtain an orthonormal basis in F . It is then possible to add a fictitious source to the SNequations and force these equations to yield a PN -like solution. We note that the reformulation of the SNequations leads to a discrete-ordinates-like collocation formulation where the scattering term is calculatedwith a Galerkin quadrature.[5] That is, to eliminate ray effects one has to modify the scattering term and addan artificial source.

In this paper we discuss SN−PN equivalence and analyze the potential of the collocation equations to reduceray effects. The main result in this paper is a general expression for the artificial source which generalizesthe analysis in [3].

We note by E [NK ] = span−→AK(Ω) ⊆ F [Nd] the highest rotationally invariant subspace in F and

introduce two projectors, PK : L2(S2) −→ E and P : L2(S2) −→ F , such that, for ψ ∈ L2(S2),(PKψ)(Ω) =

−→AK(Ω) · (

−→AK , ψ) and (Pψ)(Ω) =

−→f (Ω) ·

−→ψ , where

−→ψ [Nd] = vecψ(Ωd). We note that

P acts as the identity on F , and therefore on E , but PK acts as the identity only on E .

We consider the one-group transport equation Bψ = S with ψ, S ∈ L2(S2) and B = Ω · ∇ + σ − H anddefine the approximate equations

PKBψK = PKS, ψK ∈ E (1)

andPBψ = PS, ψ ∈ F . (2)

The first equation is the result of an orthogonal (Galerkin) projection and gives the PN equations in E forthe angular moments in the expansion ψK =

−→AK(Ω) ·

−→φ K , while the second is the result of an oblique

projection, based on a set of interpolation functions fd(Ω), over F and results in the DO-like collocationequation discussed in [5], whose solution can be written as ψ =

−→f (Ω) ·

−→ψ =

−→A (Ω) ·

−→φ . However, even

in the hypothetical case when E = F , these two equations would produce radically different numericalapproximations because the projections of the term Ωψ /∈ F are different.

We now turn to the quest for a DO-like approximation that accepts a PN solution and consider the modifiedcollocation equation

PBψ = PS + S∗, ψ, S∗ ∈ F , (3)

Richard Sanchez

where the artificial source S∗ has been added so that ψK = PKψ is also a solution of PN equations (1) or,in other words, that the first NK moments of the solution of (3) are equal to the

−→φ K solution of (1).

We note that F = E+E⊥, where E⊥ = P⊥F is the orthogonal complement of E in F and P⊥ : L2(S2) −→E⊥ is the orthogonal projector associated to E⊥ so that PK+P⊥ is the unit operator in F . Hence, any h ∈ Fcan be written as h = PKh+P⊥h = hK+h⊥. Then, it follows from Eq. (3) that S∗ = P(BψK−S)+PBψ⊥and, because ψK is the unique solution of (1) with appropriate boundary conditions, we see that the finalform of S∗ depends only on PBψ⊥. Moreover, a simple algebraic calculation shows that PBE⊥ 6= 0 sowe may conclude that the source S∗ is not unique and, indeed, a number of different artificial sources havebeen proposed in the literature. Our next task is to derive a general form for S∗.

Expression (3) shows that S∗ depends linearly on ψK and ψ⊥ plus a source term. Hence, we shall considerthe general form

S∗ = P(Bψ − S)− αQK − βQ⊥, (4)

where α and β are non zero reals and

QK = PKT ψ − hK , Q⊥ = P⊥Rψ − h⊥. (5)

In this last expression T ,R : F → F are two unspecified linear operators and h ∈ F . Replacing (4) into (3)gives two equations, QK = 0 in E and Q⊥ = 0 in E⊥. Next, we request that the first equation reduces to thePN equations for ψK and use the second equation to provide an equation for ψ⊥. The constraint in QK = 0yields T = B and hk = PK(T ψ⊥+S). By using this result we find, as expected, thatQK = Pk(BψK−S)and, by inserting this expression in (4), we can write

S∗ = (P − αPK)(BψK − S) + PBψ⊥ − βQ⊥ (6)

We turn next to the equation Q⊥ = 0 which reads

P⊥RψK + P⊥Rψ⊥ = h⊥. (7)

With ψK defined by the PN equations, this last equation provides a relation between ψ⊥ and h⊥. Wepass now to explore several ways to define Q⊥ and compare our results with results in the literature.[3] Forreasons to be discussed later, the sources proposed in the literature assume that S ∈ E and take α = 1 so thecontribution from S in (6) vanishes. Except if specified, all the sources mentioned hereafter can be found inRef. [3].

First, we analyze the case when ψ⊥ remains undefined. In view of Eq. (7), this requires h⊥ = P⊥RψK andP⊥Rψ⊥ = 0. Hence, the resulting source is that in (6) with Q⊥ = 0. This is the Reed-Jung source whichin our notation reads (S∗)RJ = (PB − PKBPK)ψ.

Next, we consider the case when Eq. (7) fully or partially definesψ⊥. We note that forψ⊥ to be fully defined,operator P⊥RP⊥ : E⊥ → E⊥ must be invertible. One possibility is to define ψ⊥ to be independent of ψK .This entails ψ⊥ = ψ⊥0 and h⊥ = P⊥R(ψK + ψ⊥0). The last expression gives Q⊥ = P⊥R(ψ⊥ − ψ⊥0)and using this result in (6) yields

S∗ = (P − αPK)(BψK − S) + PBψ⊥ − βP⊥R(ψ⊥ − ψ⊥0). (8)

2/7

On SN-PN Equivalence

For β = σ and P⊥RP⊥ = P⊥ (the identity in E⊥) this formula gives the Jung source and the modifiedReed-Jung source, (S∗)J = P(1 − PK)(B − σ)ψ and (S∗)MRJ = [PB − PKBPK − σ(1 − PK)]ψ,respectively. The only difference with (8) is that the Jung source contains the term (P − PK)Bψ⊥ insteadof the term PBψ⊥. Notice that both sources impose ψ⊥ = 0 and, therefore, these spurious terms vanish inthe final solution. The interest of spurious terms in the source will be discussed hereafter.

A second possibility is to have ψ⊥ partially or completely specified in terms of ψK and a given h⊥. Inthe former case the constraint is given by Eq. (7), while in the latter case P⊥RP⊥ is invertible and ψ⊥ =(P⊥RP⊥)−1(h⊥ − P⊥RψK). The corresponding S∗ is that in (6) with Q⊥ from (5). For β = 1, h⊥ = 0andR⊥ = PBP⊥, this gives the Miller-Reed source, (S∗)MR = PK(PB−BPK)ψ, and, as we show next,the Lathrop source, (S∗)L = PKPB(1−PK)ψ. We note that P⊥PBP⊥ is invertible and that therefore ψ⊥is fully constrained.

The source proposed by Lathrop is of particular interest because Lathrop uses a basis−→U (Ω)[Nd] = vecUm(Ω)

of spherical functions orthonormal with respect to the scalar product defined by the quadrature formula

(f, g) =∑d

wdf(Ωd)g(Ωd). (9)

The Um functions are determined via the Gram Schmidt method and the construction process ensures thatthe space F [Nd] = spanUm(Ω) contains also a basis of spherical harmonics, but the latter are notorthonormal with respect to scalar product (9). Moreover, the functions Um with the same k(m) generatethe same rotationally invariant subspace than the homologous Am spherical harmonics. The orthonormalityof the basis functions guarantees that DM = Id, where D and M are the DO matrices with elementsDmd = wdUm(Ωd) and Mdm = Um(Ωd). Hence, we also have MD = Id or, in an explicit form,

wd∑m

Um(Ωd)Um(Ωd′) = δdd′ .

This relation shows that one can construct a collocation basis−→f (Ω) = Dt−→U (Ω) such that fd(Ωd′) = δdd′ .

Similarly as before we introduce the projectors P and PK but it is worth noting that the latter is not anymorean orthogonal projector inL2(S2) because (9) is a degenerate scalar product over functions ψ /∈ E . However,by observing that (Um, ψ) =

∑dDmdψd we have (Pψ)(Ω) =

−→f (Ω) ·

−→ψ =

−→U (Ω) ·D

−→ψ =

−→U (Ω) ·(

−→U ,ψ)

and so we have P = PK +P⊥, where PK and P⊥ are the projection over the respective Um functions. Withthis result we can write Lathrop’s source as (S∗)L = PKBψ⊥ and prove that this source is of the form in (6).It is worth noting that the PN equations of Lathrop are not the usual PN equations because the projectionPK is with respect to the DO ‘scalar’ product in (9).

We end this discussion on artificial sources by considering a recent work by Brown et al.[4] who adopteda technique closely related to that of Lathrop. We shall discuss this work in our operator notation. Brownet al. used a scalar product defined by a DO quadrature of order Nd that exactly integrates (Am, Am′) forall spherical harmonics in a rotationally invariant space E [NK ]. Then, using the DO scalar product (9) theyapplied a singular value decomposition technique to complete space E with an orthogonal space E⊥[N⊥],where N⊥ = Nd − NK , so that the total space F is the orthogonal sum of E and E⊥. And, indeed, theartificial source proposed by Brown et al., (S∗)BCH = PKΩ · ∇P⊥ψ, is that of Lathrop for the casewhen the scattering is confined to space E : PKBψ⊥ = PK(Ω · ∇ + σ − H)ψ⊥ = (S∗)BCH becauseHψ⊥ = PKψ⊥ = 0. The essential difference with Lathrop’s work is that Brown et al. constructed theirorthonormal basis by completing a space of spherical harmonics that were already orthonormal with respect

3/7

Richard Sanchez

to both the scalar product (A1) and the DO scalar product in (9). Moreover, besides defining a set of truePN equations, the SVD approach of Brown et al. automatically constructs a set of unspecified orthonormalspherical functions, while that of Lathrop is prone to the degeneracy discussed in [5] and, therefore, requiresthe use of selection rules in the choice of the spherical functions.

Which so many possibilities to construct artificial sources the question raises as which one is best. By itsvery nature the source S∗ increases the number of iterations and demands more storage than those for thetypical DOA. We should therefore look for a source that minimizes these drawbacks and limit our analysisto typical applications where scattering and sources occur in space E . In order to reduce coupling one seeksto minimize the number of moments in the artificial source which brings to set α = 1 in Eq. (6) letting to

S∗ = (P − PK)(Ω · ∇ψK) + P(Ω · ∇+ σ)ψ⊥ − βQ⊥.

The first term on the right hand side introduces the moments φm with k(m) = K, while the second dependson all the higher moments with k(m) > K. If we let Q⊥ = 0 (the Reed-Jung source), then the iterationsmay increase because the higher moments are not controlled by the iteration process. Setting ψ⊥ = ψ⊥0constrains the iterations to converge to a set of unrelated solutions ψK and ψ⊥ but, according to numericaltests,[3],[4] it seems that the most efficient iterations are obtained by letting ψ⊥ depend on ψK .

In this work we are interested on the potential of the uncorrected DO-like collocation equations, which offersa better angular quadrature than the typical DOA, to mitigate ray effects. The discussion of the structure ofthe artificial source provides a view in this matter. Indeed, because the correction introduced by source S∗depends only on the moments of the solution of (3) with k(m) ≥ K, one would expect that the uncorrectedDO-like Eq. (2) will reduce ray effects for problems for which these moments and their gradients are small.But this will be possible only if scattering dominated absorption. Thus, our finding joints others’ in showingthat the ability to reduce ray effects is problem dependent.

Regarding this point, it is instructive to compare the correction S∗ with the corresponding correction SDO∗for the ordinary DO equation. The DO equation can be written as in (2) with the difference that the Galerkinscattering operator H must be replaced by H + (HDO − H), where HDO is based on the DO angularquadrature formula. By making this replacement we may write the DO equation as in (2) with the changePS → PS + (HDO − H)ψ and compute, as before, the corresponding corrective source SDO∗ . As onewould expect, the result is SDO∗ = S∗+ (H−HDO)ψ. Hence, in the frequent case when the DO quadratureformula is not exact for all the spherical harmonics in the scattering term, the extra term (HDO −H)ψ willdepend on lower angular moments with k(m) < Ktrunc, where Ktrunc ≤ K is the order of truncature ofthe scattering operator. The result is the well-known fact that the uncorrected DO equation exhibit strongray effects.

REFERENCES

[1] K. D. Lathrop, ‘Remedies for Ray Effects,’ Nucl. Sci. Eng., 45, 255, 1971.

[2] Wm.H. Reed, ‘Spherical harmonic solutions of the neutron transport equation from discrete ordinatecodes,’ Nucl. Sci. Eng. 49, 10-19, 1972.

[3] W.F. Miller, Jr. and Wm H. Reed, ‘Ray-Effect Mitigation Methods for Two-Dimensional Neutron Trans-port Theory,’ Nucl. Sci. Eng. 62, 391-411, 1977.

4/7


[4] P.N. Brown, B. Chang and U.R. Hanebutte, ‘Spherical harmonic solutions of the Boltzmann transportequation via discrete ordinates,’ Progress in Nuclear Energy, 39, 263-284, 2001.

[5] R. Sanchez and J. Ragusa, ‘On the construction of Galerkin angular quadratures,’ Nucl. Sci. Eng.,accepted for publication, April 2011.

[6] B.G. Carlson and C.E. Lee, ‘Mechanical quadrature and the transport equation,’ Los Alamos ScientificLaboratory, LAMS-2573, 1961.

[7] S. Lang, Linear Algebra, Undergraduate Texts in Mathematics, Springer, 2000.

Appendix A: Precision of the LS angular quadrature

The weights of the level symmetric (LS) angular quadrature are computed so as to exactly evaluate integralsof the form

Ipqr =1

4π

∫(4π)

dΩµpxµqyµ

rz ∼

∑d

wd(µpx)d(µ

qy)d(µ

rz)d.

Because of the invariance of the direction set with respect to planar symmetries, this formula shows that allmoments with at least an odd power are exactly integrated by the LS quadrature. Hence, one only considersthe quadratures for p, q and r even. Moreover, since the angular directions are invariant under rotations ofangle π/2 with axes x, y and z, the direction set can be decomposed into subsets of directions which areconnected via rotations and symmetries and, therefore, have the same weight. There at most three classes ofsubsets. For directions in the first octant, Class I comprises a single set for a single direction with identicaldirector cosines (α, α, α), class II comprises one or more sets of 3 directions that have two identical cosinedirectors, (β, α, α), (α, β, α) and (α, α, β), and class III contains one or more sets of 6 vectors, each setbeing defined by the directions whose components are the permutations of three different cosine directorsα, β and γ. Therefore, one can write Ipqr =

∑Nwn=1wnanRpqr(Ωn), where Nw = 1 +E[4Nµ(Nµ + 4)−

1]/48 is the number of different weights, an = 4/3, 4 and 8 for class I, II and III, respectively, and

Rpqr(Ωn) =∑σ

µσ(p)x µσ(q)y µσ(r)z .

In this expression the sum in σ is over the six permutations of three elements and the director cosines arethose corresponding to one of the directions Ωn of the direction set.

Hence, one may only consider the integrals Rpqr(Ωn) for p ≥ q ≥ r even. However, the integrals Rpqr forgiven s = p + q + r are not linearly independent. By using the relation µ2x + µ2y + µ2z = 1 one can showthat Rpqr can be written as a linear combinations of integrals of the form Rppr with p ≥ r and the same svalue and of integrals Rp′q′r′ with lower values of s. Therefore, for a given value of s one have to consideronly the integrals Rppr for p ≥ r. We note λ(s) the number of independent integrals with s = 2p + r andΛ(s) =

∑s′≤s λ(s′). Table I gives the values of the Rppr to be used for the determination of the weights for

s = 0 to 30.

Table I: Integrals Rppr to be calculated for s = 2p+ r.

5/7

Richard Sanchez

s λ(s) Λ(s) (p, r) s λ(s) Λ(s) (p, r)

0 1 1 (0,0) 16 2 10 (6,4),(8,0)2 0 1 18 2 12 (6,6),(8,2)4 1 2 (2,0) 20 2 14 (8,4),(10,0)6 1 3 (2,2) 22 2 16 (8,6),(10,2)8 1 4 (4,0) 24 3 19 (8,8),(10,4),(12,0)

10 1 5 (4,2) 26 2 21 (10,6),(12,2)12 2 7 (4,4),(6,0) 28 3 24 (10,8),(12,4),(14,0)14 1 8 (6,2) 30 3 27 (10,10),(12,6),(14,2)

The spherical harmonics Akl(Ω) of order k are linear combinations of products of the form µpxµqyµrz with

p+ q + r = k. Hence, in order to exactly integrate all the products of two spherical harmonics with ordersk, k′ ≤ K, one needs to exactly evaluate all the integrals Rppr for s = 2p+ r = k+ k′ for k+ k′ even. Forconservation we have k = 0 and want to preserve as many integrals for 0 ≤ k′ ≤ Kcon as possible. Hence,for an LS formula with Nw independent weights, the highest conservation order Kcon equals the greatests such that Λ(s) ≤ Nw. Also, the order K of the greatest rotationally invariant subspace EK is given byK = s/2, for the greatest s such that Λ(s) ≤ Nw. The corresponding K values are compared to those fromthe Galerkin quadrature, KGal = 2Nµ − 1, in Table II.

Table II: Orders Kcon and K for given Nµ. Nw = number of weights.KGal = K for Galerkin quadrature.

Nµ 1 2 3 4 5 6 7 8 9 10

Nw 1 1 2 3 4 5 7 8 10 12

Kcon 3 3 5 7 9 11 13 15 17 19

K 1 1 2 3 4 5 6 7 8 9

KGal 1 3 5 7 9 11 13 15 17 19

The predictedKcon andK values have also been observed in our numerical calculations with one exception.For Nµ = 4 we obtain Kcon = 9 and K = 4. Clearly, this result must be related to the recursion relationsbetween spherical harmonics but it is nevertheless a coincidence. Indeed, in our calculations we use Lee’sprescription µ21 = 1/[3(2Nµ − 1)] to define the first axial cosine. As soon as we changed the initial µ1 theresults dropped down to the theoretically predicted values Kcon = 7 and K = 3.

Appendix B: Construction of F

Let E [NK ] = spanUm(Ω), k(m) ≤ K be a subspace of F [Nd] ⊂ L2(S2) such that the functions Um(Ω)are orthonormal with respect to the scalar product in (9) and let N⊥ = Nd − NK . In order to define acollocation basis

−→f (Ω) = setfd(Ω), 1 ≤ d ≤ Nd in F we need to find the orthogonal complement

E⊥[N⊥] of E in F . We note that there exist two linear mappings

E ιK→ F πK→ E , (B1)

6/7


where ιK is the canonical injection of E in F and πK is the orthogonal projection of F on E , which satisfyπKιK = 1K and (πK)t = ιK , where 1K is the identity in E . Moreover, the null space NK of πK isorthogonal to the image E of ιK so that E⊥ = NK .

We shall use the basis−→U K(Ω) = setUm(Ω), 1 ≤ m ≤ NK in E and wish to construct a collocation

basis−→f (Ω) in F . In these bases ιK(

−→U K)(Ω) = M t

K

−→f (Ω) and (πK

−→f )(Ω) =

−→U K(Ω) · (

−→U K ,

−→f ) =

DtK

−→f (Ω), where MK [Nd, NK ] = matMK)dm = Um(Ωd) and DK [NK , Nd] = mat(DK)md =

(Um, fd) = wdUm(Ωd). Hence, matrices MK and DK realize injection ιK and projection πK andDKMK = IK . However, DK = M t

KW , where W [Nd, Nd] = diagwd, so DK is not the transposeof MK and therefore the null space of DK is not orthogonal to the image of MK . The reason of course isthat−→f (Ω) is not an orthonormal basis,[7] indeed (fd, fd′) = wdδdd′ .

In order to construct E⊥ it is more convenient to go back to the natural situation in (B1) and adopt anorthonormal basis

−→f ∗(Ω) = W−1/2

−→f (Ω). In this basis, operators ιK and πK are represented by ma-

trices M∗K = W 1/2MK and (M∗K)t, respectively, and E⊥ can be directly found using a singular valuedecomposition (SVD) of matrix (M∗K)t. The SVD algorithm provides matrix X[Nd, N⊥], whose columnscontain the components over the basis

−→f (Ω) of the orthonormal basis vectors of E⊥. Thus, the vectors

−→U (Ω)[N⊥] = Um(Ω),m > NK = Xt−→f (Ω) form an orthonormal basis in E⊥. We can write nowF = E⊥E⊥, where the ⊥ indicates a direct sum of orthogonal spaces, and extend (B1) to the entire spaceF :

E⊥E⊥ = F i→ F π→ F = E⊥E⊥,

where i and π are the canonical injection and the orthogonal projection (note that theses maps equal theunit operator in F). Clearly, in the orthonormal bases

−→U (Ω)[Nd] = Um(Ω) and

−→f ∗(Ω) the maps i

and π are represented by matrices M∗[Nd, Nd] = (M∗K , X) and its transpose, respectively, and we have(M∗)tM∗ = Id. Finally, changing basis back to the

−→f (Ω) we get the matrices

M = (MK ,W−1/2X), D = M tW =

(M tKW

XtW 1/2

)such that MD = Id.

7/7


Boundary Condition Analysis For the SP1 Approximation of theRadiative-Conductive Equation

Fabio de S. Azevedo1, Mark Thompson2, Esequia Sauter3 and Marco Tulio B. M. Vilhena4

PPGMApUniversidade Federal do Rio Grande do Sul

Bento Goncalves, P.O. Box 15080Porto Alegre - RS, [email protected]@ufrgs.br

[email protected]@ufrgs.br

The nonlinear coupling between the temperature and radiative transport has been a major topic in the

literature. Applications are manifold and range between several fields of physics and engineering including

gas turbines chambers, ceramic devices, furnaces, annealing and cooling of glass, solidification of metal

oxide melt droplets and high-temperature nuclear reactors. Virtually any model dealing with high

temperature will need to consider radiative effects. However these conductive-radiative problems are often

too complex to be solved numerically and analytical solutions are available only for very particular

problems ([11]). Thus several approximations, which are less expensive, have been proposed and studied in

the literature. One important family of such approximate models is called the Simplified PN (or SPN )

approximations. The SPN was first proposed by Gelbart ([4] and [5]) and theoretically examined in [8]. A

good review of this methodology was given in [9]. Rigorous work concerning the mathematical analysis of

these problems is available for some special cases of equations ([1] and [2]), but the analysis of more

complete models is still missing as was noted in [6].

In [8] a full asymptotic expansion is given for a fluid without thermal conduction, while in [9] and [6],

which consider the phenomenon, the boundary conditions given have been derived not using an asymptotic

approach but a variational approximation. According to the authors of [9], the result they obtained works

better than they expected suggesting the existence of “theoretical principles at work here that we do not yet

fully understand”. Extensive numerical investigations have been carried out in the literature (eg. [7] and

[10]) validating the usage of the methodology. The aim of this work is to advance the comprehension of

this problem by providing the boundary condition through the asymptotic analysis.

The evolution of the temperature of a radiative compressible fluid inside a convex region D is modeled by

the following simplified dimensionless equation: (see [3] for further details):

∂T

∂t+ u · ∇T −∇ · k0∇T =

∫ ∞ν0

∫S2

κ′

ε2(I −B(ν, T )) dΩdν, (1a)

εΩ · ∇I +(σ′ + κ′

)I =

14πσ′∫S2IdΩ + k′B(ν, T ). (1b)

Fabio S. de Azevedo, Mark Thompson, Esequia Sauter and Marco Tulio M. B. Vilhena

Here the function B(ν, T ) is the adimensionalised Planck function given by

B(ν, T ) = η21

ν3ref

Iref

2hpν3

c2

[exp

(hpννrefkBTTref

)− 1

]−1

.

Here kB , hp and c are constant and stand respectively for the Boltzmann constant, Planck constant and the

speed of light in the vacuum. η1 is the refractive index of internal medium. k0 is the heat diffusibility and u

is the velocity field, which is assumed to be known and sufficiently regular. The boundary of the opaque

part of the radiative spectrum is denoted by ν0 and ε represents a dimensionless parameter describing the

distance from an optically thick medium, ε = 1Xrefkref

, where Xref and kref are reference length scale and

reference absorption rate respectively. Tref , Iref and νref are arbitrary reference values for the

temperature, intensity of radiation and frequency. The reference values for time and absorption are given

by:

tref = cvκrefx2ref

TrefIrefνref

and kref =IrefνrefκrefTref

.

Here cv is the volumetric specific heat capacity; the coefficients κ′, σ′ and λ′ are allowed to depend only on

the frequency ν. κ′ is strict positive, σ′ is nonnegative and λ′ = σ′ + κ′.

The boundary conditions are given by

εk0∂T

∂η= h(Tb − T ) + αsπ

(η2

η1

)2 ∫ ν0

0[B(ν, Tb)−B(ν, T )] dν, (2)

here Tb is the temperature on the boundary, α and h are positive constants and η2 is the reflective index of

outer medium.

I(Ω) = ρ(µ)I(Ω′) + (1− ρ(µ))B(ν, Tb(x)), η · Ω < 0. (3)

The angle of reflection is given by Ω′ = Ω− 2(η · Ω)η and µ is defined as µ = Ω · η, where η is the

outward vector.

Finally, the initial condition is given by:

T (x, 0) = T0(x), t = 0 (4)

Since in this work we want to deal with the comparison between this problem and its simplified version

given by the SP1 approximation, we shall call the system (1)-(4) as the “original problem” in contrast to the

“reduced problem” given by SP1.

In the reduced problem, the temperature T is approximated by θ and the average intensity14π

∫S2 I(x, t,Ω)dΩ is approximated by Φ(x, t). The great advantage of this formulation is that Φ does not

depend on Ω and satisfies a simple elliptic equation given by:

2/4

Boundary Condition Analysis For the SP1 Approximation of the Radiative-Conductive Equation

(− ε2

3κ′λ′4+ 1

)Φ = B(ν, θ), x ∈ D

εb∂

∂ηΦ + Φ = B(ν, θb), x ∈ ∂D

(5)

where θb = Tb and b is positive (not necessarily constant). The equation for the temperature is obtained

from (1a) and (2) replacing T by θ and 14π

∫S2 I(x, t,Ω)dΩ by Φ:

∂θ

∂t+ u · ∇θ −∇ · k0∇θ = 4π

∫ ∞ν0

κ′

ε2(Φ−B(ν, θ)) dν (6a)

εk0∂θ

∂η= h(θb − θ) + αsπ

(η2

η1

)2 ∫ ν0

0[B(ν, θb)−B(ν, θ)] dν, (6b)

The initial condition is given by the equation (4) replacing T by θ:

θ(x, 0) = θ0(x), t = 0 (7)

where θ0(x) = T0(x). The value of b in (5) proposed in [9] is given by:

b =1 + 3r21− 2r1

· 23λ′

, rk =∫ 1

0ρ(−µ)µkdµ, k = 1, 2. (8)

We note that if 0 ≤ ρ ≤ 1 and ρ 6≡ 1, r1 <∫ 10 µdµ = 1/2, which implies b is well-defined and positive.

When ρ ≡ 1, the boundary condition in (5) must be replaced by the homogeneous Neumann boundary

condition, nevertheless, in this work we will not cover this singular case.

In order to obtain a value for b via asymptotic analysis, we will use the theory of singular perturbations,

considering both problems as ε-perturbations of the Rosseland equation:

∂T (0)

∂t + u · ∇T (0) −∇ · k0∇T (0) = 4π3λ′∫∞ν0

∆B(ν, T (0))dν, x ∈ D t > 0,

T (0)(x, 0) = T(0)0 (x), x ∈ D, t = 0,

T (0)(x, t) = Tb, x ∈ ∂D, t > 0.

(9)

here we have assumed that the initial condition is given in the form T0(x) = T(0)0 + εT

(1)0 +O(ε2).

Carrying out the asymptotic analysis, we derive an algebraic-operational equation on b in order to make the

expansions of the reduced problem and the original problem coincide up to the first order term. We show

that a low order approximation of this algebraic-operational equation produces the same results previously

obtained in the literature via a variational approximation. Then we calculate numerically the value of b for

several different values of ρ, σ′ and β, our numerical experiments suggest looking at the limit for high

values of β. We derive a methodology to calculate its value numerically. When we look at our numerical

experiments, we find interestingly that the asymptotic value of b deviates only a few percentage points from

the variational approximation.

3/4

Fabio S. de Azevedo, Mark Thompson, Esequia Sauter and Marco Tulio M. B. Vilhena

ACKNOWLEDGMENTS

This work was conducted as part of a project of the National Institute of Science and Technology on

Innovative Nuclear Reactors (INCT-Brazil). F. Azevedo was supported by a postdoctoral fellowship of the

CAPES(Brazil). E. Sauter was supported by a doctoral fellowship of CNPq(Brazil). Professor Vilhena

thanks the CNPq for the partial financial support of this work.

REFERENCES

[1] C. Bardos, F. Golse, and B. Perthame, “The Rosseland approximation for the radiative transferequations,” Comm. Pure Appl. Math., 40(6), pp. 691-721 (1987).

[2] C. Bardos, F. Golse, B. Perthame, and R. Sentis, “The nonaccretive radiative transfer equations:Existence of solutions and rosseland approximation,” J. funct. Anal., 77(2), pp. 434-460 (1988).

[3] M. Frank, M. Seaıd, A. Klar, R. Rinnam, and G. Thommes, “A comparison of approximate modelsfor radiation in gas turbines,”Prog. Comput. Fluid Dyn., 4, pp. 191-197 (2004).

[4] E. M. Gelbart, “Applications of spherical harmonics methods to reactor problems,” WAPD-BT-20(1960).

[5] E. M. Gelbart, J. Davis, and J. Pearson, “Iterative solutions to the P1 and double Pl equations,” Nucl.Sci. Eng., 5, pp. 36-44 (1958).

[6] A. Klar and N. Siedow. “Boundary layers and domain decomposition for radiative transfer anddiffusion equation: applications to glass manufacturing process,” Eur. J. App. Math., 9, pp. 351-372(1998).

[7] A. Klar, J. Lang, and M. Seaıd, “Adaptive solutions of spn-approximations to radiative heat transfer inglass” Int. J. Therm. Sci., 44(11), pp.1013-1023 (2005).

[8] E. W. Larsen, G. C. Pomraning, and V. C. Badham, “Asymptotic analysis of radiative transferproblems,” J. Quantit. Spectrosc. Radiat. Transf., 29(4), pp.285-310, 1983.

[9] E. W. Larsen, G. Thommes, A. Klar, M. Seaıd, and T. Gotz, “Simplified pn approximations to theequations of radiative heat transfer and applications,” J. Comput. Phys., 183(2), pp. 652-675 (2002).

[10] E. Schneider, M. Sead, J. Janicka, and A. Klar, “Validation of simplified pn models for radiativetransfer in combustion systems,” Commun. Numer. Methods Eng., 24(2), pp 85-96 (2008).

[11] B. Su and G. L. Olson, “An analytical benchmark for non-equilibrium radiative transfer in anisotropically scattering medium,” Ann. Nucl. Energy, 24(13), pp. 1035-1055 (1997).

4/4


Fully discrete finite element approximation of a bi-partition modelfor the energy dependent transport equation

M. Asadzadeh and T. GebackDepartment of Mathematics

Chalmers University of TechnologySE-412 96 Goteborg, SWEDEN


In this paper the objective is to construct, analyze and implement some realistic particle transport models

relevant in applications in radiation therapy. Our goal is to develop a flexible model incorporated with the

analytic theory of the particle transport based on bipartition and Fokker-Planck developments (cf [1]-[5]).

Using the spherical harmonics we derive an energy dependentproblem. We show stability and convergence

for semidiscrete (discrete energy) and fully discrete (spatial/radial discretizations) finite element schemes

for the fluence differential:f(x, r,Ω, E) of charged particles symmetrically distributed around thex-axis,

at the distancer from it, and travelling in directionΩ ∈ S2 with energyE. The equation is derived under

thecontinuous slowing down assumption and retainedenergy-loss straggling term(diffusion in energy):

Ω · ∇f −1

2

∂2ω(E)f

∂E2−

∂S(E)f

∂E= Cf (x, r,E) + Q(x, r,Ω, E), (1)

whereQ is a source term andCf is the collision factor, depending on the elastic scattering cross-sectionσs:

Cf (x, r,Ω, E) =

∫

4π

σs(E,Ω · Ω′)(

f(x, r,Ω′, E) − f(x, r,Ω, E))

dΩ′. (2)

W use spherical coordinatesΩ = (cos θ, sin θ cos ϕ, sin θ sin ϕ) and expandf in spherical harmonics as

f(x, r,Ω, E) =

∞∑

n=0

n∑

m=0

(n − m)!

(n + m)!

2n + 1

4παman,m(x, r,E) cos(mϕ)Pm

n (cos θ). (3)

Hereθ is the angle from thex-axis,α0 = 1, αm = 2, for m ≥ 1. Furtherf is assumed to be symmetric inϕ

so that thesin(mϕ) terms vanish (the analysis are valid with retainedsin(mϕ)). The coefficientsan,m are:

an,m(x, r,E) =

∫ 1

−1

∫ 2π

0f(x, r,Ω, E)Pm

n (cos θ) cos(mϕ) dϕ d(cos θ).

We may expand the collision and source terms (fp below) in spherical harmonics, viz:

Cf (x, r,Ω, E) =

∞∑

n=0

n∑

m=0

Cn,mf (x, r,E)Y m

n (Ω),

Q(x, r,Ω, E) =

∫

4π

∫ E0

2E

σc(E′, E)

1

2πδ(Ω · Ω′ − φ(E′, E))fp(x, r,Ω′, E′) dE′ dΩ′.

(4)

Then, insertingf , Cf andQ from (3) and (4) in (1), we end up with

A∂u

∂x+ B

∂u

∂r−

1

2

∂2(ω(E)u)

∂E2−

∂(S(E)u)

∂E= C(E)u + q(x, r,E), (5)

Asadzadeh, M. and Geback, T.

where the vectoru(x, r,E) contain the coefficientsan,m(x, r,E), A andB are matrices containing the

coefficientsAkn,j andB

±,kn,j , (cos θ, sin θ moments forpk

j andpkn) respectively, andC is a diagonal matrix.

Let nowΓ := (A,B), ∇xr = (∂x, ∂r), D(E) = γE(E)ω(E) + C(E), and write (5) as a 1st order system,

Γ · ∇xru −1

2vE − γ(E)v = D(E)u + q,

v = (ω(E)u)E .

(6)

We define a semi-discrete (discretization inE) approximationuh, vh : Ix × Ir −→ Sh ×Wh for u,v,

in the finite element space,Sh × Wh. To derive error estimates for the semi-discrete case, we use the split:

u− uh = (u − uh) − (uh − uh) := η − ε

v − vh = (v − vh) − (vh − vh) := ξ − ν.

Lemma 1 Let (ζ,w) = (η,u) or (ζ,w) = (ξ,v). Then, for j = 0, 1, we have the interpolation estimates

‖ζ‖j + ‖∇xvζ‖j ≤ Chℓ+1−j(

‖w‖ℓ+1 + ‖∇xvw‖ℓ+1

)

, ℓ = 0, 1, . . . , (Note: 2 estimates) (7)

‖ζ‖W j,p(IE) ≤ Chℓ+1−j‖w‖W ℓ+1,p(IE), 1 ≤ p ≤ ∞, ℓ = 0, 1, . . . , (Note: 2 estimates). (8)

Theorem 1 Using the above interpolation estimates, the errors u − uh and v − vh can be estimated as

‖(u − uh)(x, r)‖ + ‖(v − vh)(x, r)‖ + h‖(v − vh)(x, r)‖1 ≤ C(E0)hmin(ℓ+1,s+1)×

×(

‖u‖L∞xr(Hℓ+1) + ‖v‖L∞xr(Hs+1) + ‖∇xr · v‖L2xr(Hs+1)

)

.(9)

Hereh is the mesh size andHt is the Sobolev space ofL2(IE) functions with all (partial) derivatives of

order≤ t in L2(IE). Further‖ · ‖t is the sobolevHt-norm,‖ · ‖0, denoted by‖ · ‖ is the usualL2-norm,

Lpxr(Ht) denotes theLp-norm inx andr andHt-norm inE. LikewiseW t,p areLp-based Sobolev spaces.

The fully discrete problem is the study of the equation (5) which is also discretized in(x, r) variables.

REFERENCES

[1] Asadzadeh, M., Brahme, A., Xin, J.,Galerkin methods for primary ion transport in inhomogeneousmedia, Kinet. Relat. Models 3 (2010), 373-394.

[2] Asadzadeh, M., Brahme, A., Kempe, J.,Ion transport in inhomogeneous media based on thebipartition model for primary ions. Computer and Mathematics with Applications. 60 (2010)2445-2459.

[3] Asadzadeh, M.,The Fokker-Planck Operator as an Asymptotic Limit in Anisotropic Media, Math.Comput. Modelling, 35 (2002) pp 1119-1133.

[4] Larsen, E. W. and Liang L.,The Atomic Mix Approximation for Charged Particle Transport, SIAM,J. Appl. Math., 68 (2007) pp.43-58.

[5] Pomraning, G. C.,The Fokker-Planck Operator as an Asymptotic Limit., Math. Models MethodsAppl. Sci. 2(1992), pp 21-36.

2/2


IMPROVED MIXED AND HYBRID DISCRETIZATION OF THETRANSPORT EQUATION IN SLAB GEOMETRY

J. CartierCEA, DAM, DIF

F-91297 Arpajon, [email protected]

M. PeybernesCEA, DAM, DIF


Introduction

In this paper we deal with a mixed and hybrid finite element method (MHFEM) slab geometry

discretization of the transport equation arising from the new variational formulation introduced in

[1]. The aim of this study is to construct such a discretization by preserving the diffusion limit

both in the entire diffusive region, close to the boundaries and for internal interface problems.

Statement of the problem

Let us introduce a quadrature set QM = µm, wm; m = 1, . . . ,M for discretization in angle µ

in such a way that∑M

m=1 wm = 1,∑M

m=1 µmwm = 0,∑M

m=1 µ2mwm = 1

3 . We cut the

slab-geometry domain interval D = [0, L] in I intervals [xi− 12, xi+ 1

2] where

0 = x 12< · · · < xi+ 1

2< · · · < xI+ 1

2= L. We now consider the discrete ordinates, lumped mixed

and hybrid finite elements form of mono-energetic transport problem:

i) gli+ 1

2,m

+ gri− 1

2,m

+ σt,ihiui,m = hiσs,iφi + hiqi,

ii)σt,ihi2

gli+ 1

2,m

+ µ2m

(ui+ 1

2,m − ui,m

)=µmσs,ihi

2φli+ 1

2

+µmhi2

qi+ 12,

iii)σt,ihi2

gri− 1

2,m

+ µ2m

(ui− 1

2,m − ui,m

)= −µmσs,ihi

2φri− 1

2

− µmhi2

qi− 12,

iv) gli− 1

2,m

+ gri− 1

2,m

= 0, v) gli+ 1

2,m

+ gri+ 1

2,m

= 0.

(1)

J. Cartier and M. Peybernes

with boundary conditions u 12,m = αm, for µm > 0, uI+ 1

2,m = βm, for µm < 0, gr1

2,m

=

−µmu 12,m, for µm < 0 and gl

I+ 12,m

= µmuI+ 12,m, for µm > 0. We have chosen a spatial mesh

such that hi = xi+ 12− xi− 1

2is the width of cell i, ui,m represents the value of the angular flux at

cell i in the direction µm, ui− 12,m corresponds to the value of the angular flux at edge i− 1

2 in the

direction µm, ui+ 12,m is the value of the angular flux at edge i+ 1

2 in the direction µm, ggi+ 1

2,m

is

the value of the angular current density in the cell i at edge i+ 12 in the direction µm, gd

i− 12,m

is

the value of the angular current density in the cell i at edge i− 12 in the direction µm. We have

defined qi+ 12= 1

2(qi + qi+1) for 1 ≤ i ≤ I − 1, q 12= q1, qI+ 1

2= qI , φi =

∑Mm=1wmui,m,

φli+ 1

2

=∑M

m=1wmµmgli+ 1

2,m

and φri+ 1

2

=∑M

m=1wmµmgri+ 1

2,m

.

We proved in [2] that transport leading-order MHFEM numerical solution in the interior of a

diffusive region satisfies a MHFEM discretization of the same diffusion problem that the transport

leading-order exact solution satisfies. Close to boundaries, we get a correct Chandrasekhar

weighted integral of the incident flux. This is no more the case for an internal interface problem

between a non-diffusive region and a diffusive region. Thus we propose in this paper a way to

improve MHFEM discretization in order to circumvent this major drawback. The other

advantages (diffusion limit in the whole domain and at the boundaries, symmetric positive definite

linear system, etc...) of MHFEM are not affected by these new improvements.

Internal interface analysis

We consider here a problem consisting of a non-diffusive region on the left side of the edge xi+ 12

and a diffusive region on the right side of xi+ 12. Assuming that the angular flux entering in the

diffusive region from the transport region is known, we examine the expression of the scalar flux

at the interface, we look for an expression of the Dirichlet condition at the interface for the

diffusion asymptotic problem on the right side by computing Ui+ 12=∑

m 3µ2mwmu(0)

i+ 12,m.

Standard MHFEM

We confess that we can not obtain this expression with the standard MHFEM scheme. Actually,

equation governing the behavior at the interface is (for each µm) gli+ 1

2,m

+ gri+ 1

2,m

= 0. Thus, the

upstream and downstream current are always coupled at the interface. Moreover, the correct

outgoing equation at interface for solving transport problem in the left side for positive µm should

2/4

IMPROVED MIXED AND HYBRID DISCRETIZATION OF THE TRANSPORT EQUATION IN SLAB GEOMETRY

be gli+ 1

2,m

= µmui+ 12,m but it is no longer the case with the MHFEM scheme. Numerical results

shows that MHFEM scheme exhibits a 3µ2 weighting at the interface (see figure 1 and numerical

results from [2, 3]). We propose hereafter a technique to obtain correct results at the interface.

Improved MHFEM

A mean to avoid this problem is to replace equations gli+ 1

2,m

+ gri+ 1

2,m

= 0 by the following

conditions gli+ 1

2,m

+σt,iσt,i+1

gri+ 1

2,m

=σt,i+1−σt,iσt,i+1

µmui+ 12,m for µm > 0 and

gli+ 1

2,m

+σt,iσt,i+1

gri+ 1

2,m

= −σt,i+1−σt,iσt,i+1

µmui+ 12,m for µm < 0. This method consists in breaking

the angular current continuity equation and using the diffusive properties of materials in order to

solve the correct transport problem in each domain. Then, denoting by uei+ 1

2,m

the angular flux

incident on the diffusive region from the transport region (i.e. for µm > 0) and assuming that the

transport region is finely zoned, using some algebraic operations and an asymptotic analysis of the

scheme, we get Ui+ 12= 2

∑µm>0

(32µ

2m + µm

)wmu

ei+ 1

2,m. Hence, we obtain a correct

approximation of the Dirichlet condition for the diffusive domain by using this improved method

as we see in the figure 1.

Numerical results : internal interface btest problem

We now present the result on a problem intoduced by C.J.Gesh in [3] and inspired from the

Larsen-Morel’s benchmark [4]. It consists in a slab domain divided in two regions. The first

region D1 = [0, 1] contains a purely absorbant medium with a free source, the second region

D2 = [1, 2] consists in a purely diffusive region. The problem is defined by:µ∂u(x, µ)

∂x+ σtu(x, µ) =

σs2

∫ 1

−1u(x, µ′) dµ′, for x ∈ [0, 2] and µ ∈ [−1, 1],

u(0, µ) = 1 for µ > 0, u(2, µ) = 0 for µ < 0,

σt(x) =

2, for 0 < x < 1,100, for 1 < x < 2,

σs(x) =

0, for 0 < x < 1,100, for 1 < x < 2.

(2)

This problem corresponds to a two mean free path purely absorbing problem adjoining an

hundred mean free path highly diffusive problem. An isotropic incoming flux u(x, µ) = 1 at the

left side decreases in the non-diffusive free source transport region to an anisotropic angular flux

u(x, µ) = e− 2µ . The exact solution of this problem exhibits a boundary layer at the interface due

to the anisotropy of the interface angular flux entering into a diffusive region.

3/4


0.06

0.07

0.08

0.09

0.1

0.11

0.12

0.13

0.14

0.15

0.16

0.6 0.8 1 1.2 1.4

referenceimproved MHFEM

MHFEM

Figure 1. Scalar flux for internal interface test problem

In the figure 1, we plot a reference solution obtained by diamond-difference discrete ordinates

scheme, standard MHFEM solution and improved MHFEM solution with 100 cells in D1 and

only 10 in D2. We can see that the standard MHFEM scheme exhibits a 3µ2 weighting whereas

we obtain correct 32µ

2 + µ weighting as we predicted using improved MHFEM.

Conclusions and prospects

We have derived an improved discretization of MHFEM for the transport equation in slabgeometry. This new discretization satisfies the diffusion limit away to boundary layers, close toboundaries and at the interface between diffusive and non-diffusive regions. These improvementsdo not affect the main advantages of MHFEM (second order scheme, symmetric positive definitelinear system, etc ...). We expect to extend our improvement to two-dimensional problems.

REFERENCES

[1] J.Cartier, M.Peybernes, Mixed Variational Formulation and Mixed-Hybrid Discretization ofthe Transport Equation, Transport Theory and Statistical Physics, vol. 39, issue 1, pp. 1-46,(2010).

[2] J.Cartier, G.Samba, Mixed-Hybrid Finite Element Method for the Transport Equation,Nucl. Sci. Engineering, vol. 154, number 1, pp. 28-47, (2006).

[3] C.J.Gesh, Finite Element Method for Second Order Forms of the Transport Equation,Ph.Thesis, Texas A&M University, (1999).

[4] E.W.Larsen, J.E.Morel, W.F.Miller Jr., Asymptotic Solutions of Numerical TransportProblems in Optically Thick, Diffusive Regimes, Journal of Computational Physics, 69,283-324, (1987).

4/4


Tuesday, September 13, 2011

Uncertainty Quantification/Perturbation Theory I

9:00 am Uncertainty Analysis of Radiation Transport Problems Using Non-Instrusive Polynomial Chaos Tech-

niques - L. Gilli, D. Lauthouwers, J. Kloosterman

9:25 am Initial Conditions State-Based Perturbation Theory - Y. Bang, H.S. Abdel-Khalik

9:50 am Exact-to-Precision Generalized Perturbation Theory - C. Wang, H.S. Abdel-Khalik

Radiative Transfer II

10:45 am Analytical Discrete Ordinate Method for Radiative Transfer in Vegetation Canopies - P. Picca, R.

Furfaro, B.D. Ganapol

11:10 am Moment-Based, Multiscale Solution Approach for Thermal Radiation Transport - H. Park, D.A. Knoll

11:35 am Reducing the Spatial Discretization Error of Thermal Emission in Implicit Monte Carlo Simulations

- A.G. Irvine, I.D. Boyd, N.A. Gentile

12:00 pm Deriving the Asymptotic P1 Approximation for Thermal Radiative Transfer - S.I. Heizler

Deterministic Transport II

2:00 pm Mixed Variational Formulation of the Transport Equation - J. Cartier, M. Peybernes

2:25 pm Application of Spectral Elements for 1D Neutron Transport and Comparison to Manufactured Solu-

tions - A. Barbarino, S. Dulla, P. Ravetto, E.H. Mund, B. Ganapol

2:50 pm Analog Computing to the Time-Dependent Second-Order Form of the Neutron Transport Equation in

X-Y Geometry - A. Pirouzmanda, K. Hadadb, P. Ravetto

3:15 pm P2-Equivalent Form of the SP2 Equations - Including Boundary and Interface Conditions - R. McClar-

ren


Monte Carlo I

4:10 pm Variance Reductions for Forward and Inverse Transport Problems - G. Bal

4:35 pm Material Motion Corrections for Implicit Monte Carlo Radiation Transport - N.A Gentile, J.E. Morel

5:00 pm Necessary and Sufficient Conditions for an Implicit Monte Carlo Discrete Maximum Principle - A.

Wollaber

5:25 pm A Modified reatment of Sources in Implicit Monte Carlo Radiation Transport - N.A. Gentile, T.J.

Trahan

5:50 pm A Coarse-Grained Particle Transport Solver Designed Specifically for Graphics Processing Units -

F.A. van Heerden


UNCERTAINTY ANALYSIS OF RADIATION TRANSPORT PROBLEMS USING NON-INTRUSIVE POLYNOMIAL CHAOS

TECHNIQUES

Luca Gilli, Danny Lathouwers and Jan-Leen Kloosterman Physics of Nuclear Reactors

Delft University of Technology Mekelweg 15, 2629JB Delft

The Netherlands [email protected], [email protected], [email protected]

Uncertainty analysis methodologies represent an important tool in the radiation transport field due to the uncertainty which is usually present in the input parameters of the problem such as cross sections and material compositions. Taking this uncertainty into account means treating the transport equation as a stochastic differential equation whose solution can be used to find the statistical moments for the quantities of interest. Different techniques are available that can be used to solve this stochastic equation. Generally, they can be divided in two main categories based on whether they follow a statistical or a deterministic approach. Statistical methods, as the Monte Carlo technique, are exact and require a large computational effort while deterministic methods usually rely on model approximations which make the technique faster compared to the first approach. In this paper we present the application of a spectral deterministic method, based on the Polynomial Chaos Expansion first introduced by Wiener [1], to a problem modelled by a multi-group transport equation. The application of PCE based techniques for Uncertainty Quantification was first proposed by Ghanem [2] as a way to achieve the accuracy of statistical methods in a deterministic fashion. In the present work a PCE based technique is applied to a criticality problem in order to determine the uncertainty on the multiplication factor caused by a given uncertainty on some of the input cross sections. By using a Polynomial Chaos Expansion it is possible to express the stochastic multiplication factor [3] in the following spectral form

0

( ) ( ( ))eff p pp

k k

ξ

where is an orthogonal polynomial basis from the Wiener-Askey scheme [4] depending on the set of random variables used to define the stochastic problem. The set of coefficients kp is determined in a non-intrusive way by projecting the spectral expansion on each term of the orthogonal basis. The projection results in an integral over the stochastic space that is solved using quadrature formulas which require the value of the stochastic quantity on the corresponding stochastic abscissa. In this way it is possible to solve the Boltzmann equation independently for each different realization of the stochastic input parameter set to determine the stochastic multiplication factor.


2

The work focuses on the derivation of a polynomial basis suitable to represent the energy correlation in the cross-section uncertainties. The results are compared with the ones obtained with a standard Monte Carlo approach and with a first order propagation. The dimensionality of the stochastic problem is discussed and two ways to reduce the issue are proposed: the first is the use of a sparse grid and the second is the implementation of a perturbation technique used to calculate part of the stochastic realizations of the output. We show that for transport problems non-intrusive PCE methods can be applied in a relatively easy and versatile way and can achieve the precision of Monte Carlo approaches.

REFERENCES

[1] N.Wiener,“TheHomogeneousChaos”,American Journal of Mathematics,60,pp.897‐936(1938).

[2] R.G.GhanemandP.D. Spanos,Stochastic Finite Elements: a Spectral Approach,Doverpublications(1991).

[3] M.M.R. Williams, “Polynomial Chaos functions and stochastic differentialequations”,Annals of Nuclear Energy,33,pp.774‐785(2006).

[4] D.XiuandG.E.Karniadakis,“TheWiener–Askeypolynomialchaosforstochasticdifferentialequations”,SIAM J. Sci. Comput.,24,pp.619‐644(2002).


Initial Conditions State-Based Perturbation Theory

Youngsuk Bang, and Hany S. Abdel-Khalik Department of Nuclear Engineering

North Carolina State University P.O.Box 7909

Raleigh, NC 27695-7909 [email protected], [email protected]

We have derived and tested a new state-based approach to estimate the change of state variable due to initial value variation [1] and compared it original model and one-shot Hessian-based algorithm which is response-based approach [2]. The new approach employs a state-based rather than response-based approach to perturbation theory. By that, we means that instead of setting an adjoint problem that is dependent on the response of interest, a set of adjoint problems are employed with source terms that are independent of the responses of interest. The source terms for the adjoint problems are based on estimating all the possible variations in the state (flux) resulting from all possible variations in the initial conditions. It is found that the state variations has much few degrees of freedom that the size of the state space. By solving only a few adjoint problems, one can estimate the variations in any response. This addresses one of the challenges of existing perturbation theory, where the number of adjoint executions is dependent on the number of responses of interest. As a representative problem, the following simplified 1-dimensional time-dependent decoupled one-group diffusion equation is considered (refer to Ref. [3] for terminology) :

1 2

2

2

, t td x t dDe e

dt dx (1)

where 2.0D , 1 , 1 0.01 , 2 0.02 , 0,10x , 0,1t . Using the Finite

Differencing Method (FDM, central differencing for spatial discretization and forward differencing for temporal discretization), the spatial location, x , is divided into 20 points, which means the number of initial values are 20. In Figure 1, the comparison between actual solution, state-based approach and response-based approach at 20th time step ( 1t ) is presented. One-shot algorithm [2] requires n r forward runs to extract the basis for state variable and reduce the system of governing equations. In this simple problem, the basis size of state variables could be reduced from 20 to 10 with criteria of capturing 99.9999% eigenvalues of the Hessian matrix. Note that to get a new response output value with perturbed initial values, the reduced governing equation should be solved again, and if one needs to examine another response output, the whole algorithm should be repeated. On the other hand, our new method [1] requires only solving equation 2n times (forward and adjoint) to construct the state variable basis and the adjoint matrix, and then, the new state variable due to differentiated initial values can be computed by only matrix-matrix and matrix-vector operation without performing the algorithm repeatedly. In addition, there is no approximation in our method so that the computed state variables are same with actual solution within machine precision.


2

ACKNOWLEDGMENTS

This work was supported by the Consortium for Advanced Simulation of Light Water Reactors (www.casl.gov), an Energy Innovation Hub (http://www.energy.gov/hubs) for Modeling and Simulation of Nuclear Reactors under U.S. Department of Energy Contract No. DE-AC05-00OR22725.

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

StateVariable

IndependentVariable(spatiallocation)

InitialConditionActualSolutionState‐BasedApproach(ref.[1])Response‐BasedApproach(ref.[2])

Figure 1. Comparison of State-Variable Calculation after 20th time step

REFERENCES

[1] Youngsuk Bang and Hany S. Abdel-Khalik, “State-Based Adjoint Model Reduction

for Large Scale Control Problems”, 4th International Symposium on Resilient Control Systems, August 9-11, 2011 (submitted)

[2] Bashir, O., Willcox, K., Ghattas, O., Van Bloemen Waanders, B., and Hill, J., “Hessian Based-Model Reduction for Large-Scale Systems with Initial Condition Inputs”, International Journal for Numerical Methods in Engineering, Vol. 73, Issue 6, pp. 844-868, 2008

[3] M. L. Williams, “Development of Depletion Perturbation Theory for Coupled Neutron/Nuclide Fields”, Nuclear Science and Engineering, 70, pp. 20-36, 1979


Exact-to-Precision Generalized Perturbation Theory Applied to Source-Driven Systems

Congjian Wang and Hany S. Abdel-Khalik

Department of Nuclear Engineering NC State University Raleigh, NC 27695

[email protected] and [email protected]

This work introduces a new development in generalized perturbation theory (GPT) intended to address some of the challenges currently facing its state-of-the-art techniques. Some of these challenges include the difficulty in assessing the accuracy of GPT estimates for general nonlinear models; the overwhelming computational burden required to estimate the high order variations of all responses [1, 2]; and finally the explosion in the state variables phase space resulting from mesh refinement [3]. We have derived an exact-to-precision generalized perturbation theory (EpGPT). Unlike existing GPT techniques, where variational estimates are determined to a prescribed order of accuracy, e.g., 1st, 2nd, etc., EpGPT retains all high orders necessary to meet the accuracy requirements prescribed by the user. The estimates of EpGPT are compared with the exact estimates obtained by direction forward perturbations as well as first-order GPT. A simple one dimensional diffusion model is employed to compare these estimates. Consider a source driven transport model described by:

( ) ( )p Q p H (1) where Q represents the source term; n is a vector that describes the state, often

referred to as the flux in reactor calculations; n nH is a matrix operator that describes the discretized version of the continuous transport operator; and finally both Q and H

are dependent on the model’s k input parameters that are described by a vector kp .

Next, consider an integral response R that is a function of the state variables and the input parameters p, taking here for simplicity as inner product with the state vector,

TR S (2)

where S is a vector whose elements are dependent on the parameters p. Perturbing the input parameters by some amounts yields the following equation:

2

0 1 2 0( )( )ii Q Q H H (3)

In this expression, the perturbed flux is given by ( 1 ):

0 1 2,


2

Combining terms with common powers of results in a series of equations given by:

0 0Q H (4)

1 0Q H H (5) The exact response variation is given by:

0 1 0 0 1 1 2( ) ( ) ( ) ( )T T T TR S S S S S (6)

The first-order variation of response is given by:

*

1 0 1 0 0( ),T T T TR S S S Q H (7) where * is given by:

*T S H (8)

The EpGPT variation of response is given by:

0

* * 1 *0 0[ ( ) ]( ),

T T T

T T T Tn r

R S S S

S w Q

Γ I U I Γ U Γ H (9)

where nw is a source coefficient vector, n rV which columns can span the subspace of state variations, and * ,n r n r Γ U , are given by:

*, T U HV H Γ V (10)

The setup is shown in Fig. 1, and the cross sections and input parameters are defined in Table I. We also show the direct calculation, first-order GPT approximation, and EpGPT approximation in Fig. 2. The variation of detector response could be estimated with a four-dimensional subspace of variation of state variables compared to the 40-dimensional full space. This means we only need 4 forward perturbation calculations to obtain the subspace of variation of state variables, and 4 adjoint calculations to get higher-order approximation of our detector response.

Figure 1. Sample Model Description

Symmetric Boundary

Symmetric Boundary

1cm 1cm 1cm 1cm

Fuel‐1 Scatter Fuel‐2 Absorb


3

Table I. Parameters and Cross Section Data

Region Domain(cm)

1( )t cm

1( )s cm

1( )f cm

1( )fv cm

10 ( )cm

3 sec

nS

cm

ih

1 0,1 3 1 0.5 1.5 0 1 0.12 1,2 10 8 0 0 0 0 0.13 2,3 3 1 1.025 2 0 0 0.14 3,4 10 1 0 0 0 0 0.1

Figure 2. Comparison of EpGPT, 1st order GPT, and

Direct Perturbations Estimates

REFERENCES

[1]A.Gandini,“ImplicitandExplicitHigherOrderPerturbationMethodsforNuclearReactorAnalysis,”Nucl. Sci. Eng.,67,pp.347‐355(1978).

[2]E.Greenspan,Y.Karni,andD.Gilai,“Higherordereffectsincross‐sectionsensitivityanalysis,”Seminar-Workshop on the Theory and Applications of Sensitivity and Uncertainty,ORNL/RSIC‐42,OakRidgeNationalLaboratory,pp.231‐248(1979).

[3]M.S.Mckinley,F.Rahnema,“Higher‐OrderBoundaryConditionPerturbationTheoryforNeutronTransportEquation,”Nucl. Sci. Eng.,140,pp.285‐294(2002)

0 0.5 1 1.5 2 2.5 3 3.5 4-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

Position (cm)

R

Forward Calculation

1st-order Adjoint ApproximationEpGPT


ANALITICAL DISCRETE ORDINATE METHOD FOR RADIATIVE TRANSFER IN VEGETATION CANOPIES

Paolo Picca

Dipartimento di Energetica, Politecnico di Torino Corso Duca degli Abruzzi, 24 - 10029 Torino, Italy

[email protected]

Roberto Furfaro, Department of Systems and Industrial Engineering

University of Arizona, Tucson AZ 85721 [email protected]

Barry D. Ganapol

Aerospace and Mechanical Engineering, University of Arizona, Tucson AZ 85721

[email protected]

The study of radiative transfer (RT) in vegetation canopies plays an important role in quantitative remote sensing of Earth’s ecosystems [1]. When compared to other classical linear transport problems, the description of photon propagation through foliage exhibits special features [2]. In fact, because of the directionality of the leaves, the total cross section becomes a function of the angle and the scattering kernel turns out to be not rotationally invariant [1]. More precisely, the one-angle, azimuthally averaged RT equation for photon transport in a homogeneous canopy writes:

∫+

−

→Γ=+∂∂ 1

1

')',()'(),()(),( µµτµµµτµµττ

µ dIIGI . (1)

For its solution, classical methods developed for linear Boltzmann problems needs to be adapted to account for the specific features of leaf canopies (e.g. [3]-[5]). In this paper, we show how to extend a well-known transport algorithm for the solution of 1D linear Boltzmann equation, namely the Analytical Discrete Ordinate (ADO) method [6], to account for the peculiarities of vegetation canopies. The most interesting feature of ADO is that its analyticity in space that makes it suitable also for simulations in optically thick media. Furthermore, the ADO method is fast as it does not require iterative cycles and is known to be highly accurate for conventional RT problems [6]. However, the ADO method cannot be applied to a generic anisotropic medium, i.e. a material whose absorption and scattering properties are arbitrary functions of the angle. Taking advantage of the symmetries which characterize vegetation canopies [1], i.e.: )()( µµ −= GG (2a) and:


2

)'()'()'( µµµµµµ −→−Γ=→Γ=→Γ , (2b) it is possible to show that the transport equation can still be reduced to an eigenvalue problem as in classical ADO approach [6]. Moreover, the boundary conditions that are typically implemented to simulate the soil reflectance at the bottom of the canopy (i.e. Lambertian reflectance) are shown to be easily implemented in the ADO framework.

The ADO method for canopy transport has been developed and numerically implemented to test its accuracy. Results are compared with benchmark available in literature [3]. Table I reports the comparison of the ADO results with the FN and SN solutions. The simulations in Table I are performed in the red region of the spectrum and results confirm the first 3 digits reported in literature [3]. Table II reports the hemispherical reflectance for a variety of configurations, including various leaf area distributions (LAD). Again the ADO results are in accordance with the literature values [3].

Table I. Reflectance and transmittance for a single-angle canopy (0.5-0.6 µm, τL=0.03, ρL=0.07, rs=0.1, LAI=1 and θ*=25.31°).

Reflected Radiance Transmitted Radiance

µ FN (N=13) SN (N=12)

ADO (N=12) FN (N= 13) SN (N=12)

ADO (N=12)

9.21968E-03 9.3246E-02 9.3243E-02 9.3250E-02 4.2747E-02 4.2746E-02 4.2747E-02 4.79414E-02 8.7581E-02 8.7579E-02 8.7584E-02 4.5798E-02 4.5798E-02 4.5799E-02 1.15049E-01 8.4287E-02 8.4286E-02 8.4290E-02 4.3816E-02 4.3817E-02 4.3813E-02 2.06341E-01 8.5781E-02 8.5780E-02 8.5784E-02 3.4649E-02 3.4650E+02 3.4644E-02 3.16084E-01 8.7904E-02 8.7903E-02 8.7906E-02 2.6841E-02 2.6841E-02 2.6837E-02 4.37383E-01 8.8902E-02 8.8901E-02 8.8902E-02 2.3502E-02 2.3502E-02 2.3503E-02 5.62617E-01 8.8902E-02 8.8901E-02 8.8902E-02 2.3502E-02 2.3502E+02 2.3503E-02 6.83916E-01 8.8902E-02 8.8901E-02 8.8902E-02 2.3502E-02 2.3502E-02 2.3503E-02 7.93659E-01 8.8902E-02 8.8901E-02 8.8902E-02 2.3502E-02 2.3502E-02 2.3503E-02 8.84951E-01 8.8902E-02 8.8901E-02 8.8902E-02 2.3502E-02 2.3502E-02 2.3503E-02 9.52059E-01 8.8902E-02 8.8901E-02 8.8902E-02 2.3502E-02 2.3502E-02 2.3503E-02 9.90780E-01 8.8902E-02 8.8901E-02 8.8902E-02 2.3502E+02 2.3502E+02 2.3503E-02


3

Table II. Comparison of albedo computed with SN, FN and ADO (µ0=1.0, LAI=1.0; r s=0.0).

albedo LAD ρL τL

SN FN ADO 0.07 0.03 1.3401E-02 1.3400E-02 1.34007E-02

Erectophile 0.45 0.45 1.4777E-01 1.4774E-01 1.47760E-01 0.07 0.03 2.6427E-02 2.6427E-02 2.62513E-02

Planophile 0.45 0.45 2.5411E-01 2.5411E-01 2.51837E-01 0.07 0.03 2.0947E-02 2.0947E-02 2.09476E-02

Plagiophile 0.45 0.45 2.1537E-01 2.1536E-01 2.15370E-01 0.07 0.03 1.6022E-02 1.6021E-02 1.60213E-02

Unophile 0.45 0.45 1.6952E-01 1.6950E-01 1.69510E-01

REFERENCES

[1] K. J. Shultis, R. B. Myneni, Radiative transfer in vegetation canopy with anisotropic scattering, Journal of Quantitative Spectroscopy and Radiative Transfer, 39, 115-129 (1988).

[2] P. Picca, R. Furfaro, On the special features of photon transport in canopies, International Conference in Transport Theory - ICTT21, Torino, 12-17 July 2009.

[3] B. D. Ganapol, R. B. Myneni, The FN method for the one-angle radiative transfer applied to plant canopies, Remote Sens. Environ., 39, 213-231 (1992).

[4] W. Verhoef, Light scattering by leaf layers with application to canopy reflectance modelling: The SAIL model, Remote Sens. Environ., 16,125-141 (1984).

[5] B. D. Ganapol, R. B. Myneni, The application of the principles invariance to the radiative transfer equation in plant canopies, Journal of Quantitative Spectroscopy and Radiative Transfer, 48, 321-339 (1992).

[6] C. E. Siewert. A concise and accurate solution to Chandrasekhar’s basic problem in radiative transfer. Journal of Quantitative Spectroscopy and Radiative Transfer. 64 (3). 109-130 (2000).


MOMENT-BASED, MULTISCALE SOLUTION APPROACH FORTHERMAL RADIATION TRANSPORT

H. Park and D.A. KnollT-3 Fluid Dynamics and Solid Mechanics

Los Alamos National LaboaratoryP.O. Box 1663 T-X MS-B216

Los Alamos, NM [email protected], [email protected]

We discuss a moment-based, multiscale solution algorithm for thermal radiation transport (TRT) equation.

We solve gray TRT equation coupled to temperature equation. This work focuses on the formulation of

low-order equation, importance of consistency, and time accuracy. 1D gray TRT equation is expressed as,

1

c

∂I

∂t+ µ

∂I

∂x+ σI =

ac

2σT 4 +

Q

2, (1)

cv∂T

∂t=

∫ +1

−1σ

(I − ac

2T 4)dµ, (2)

where, I and T are the angular flux and temperature, a, c, σ and cv are the radiation constant, speed of

light, opacity and specific heat of mateiral, respectively. Due to the radiation re-emission term (i.e.,acσ2 T 4), Eq (1) and Eq. (2) are strongly coupled.

Many deterministic approaches for solving the nonlinar system employ lineariztion [1, 2]. Since the

re-emission term can be considered as the effective scattering source, the conventional linearization often

laggs the scattering terms. This may result in slow convergence when the scattering ratio is close to unity.

Recent work by [3] solves the system with an inexact-Newton method, and show large advantage over

traditional semi-implicit methods.

Our approach differs from conventional approaches in the following way. In order to solve the coupled

system, we first rewrite the Eq. (1) in moment-based equation either with quasi-diffusion (QD)[4] or

nonlinear diffusion acceleration (NDA) [5, 6] form. The closure relations are obtained from the solution of

HO TRT equation. The LO moment-based equation is solved toghether with Eq. (2) in tightly coupled

manner via nonlinear elimination. Thus, our overall algorithm becomes nested iterative scheme which

employ outer Picard iteration and inner Newton iteration. To obtain LO equation, we take 0th moment of

Eq. (1) (i.e., integrate over µ.)1

c

∂E

∂t+∂F

∂x= acσT 4 +Q, (3)

where, E =∫Idµ and F =

∫µIdµ are the 0th and 1st moments of angular flux. For QD formulation. We

also utilize the 1st moment of Eq. (1),

1

c

∂F

∂t+∂EE∂x

+ σF = 0, (4)

H. Park and D. A. Knoll

where E is the Eddington tensor defined by

E ≡∫µ2Idµ∫Idµ

. (5)

Instead of Eq. (4), NDA formulation assumes the current F has the following form,

F = −D∂E∂x

+ DE, (6)

where D is the drift velocity term that provides the consistency in cell-averaged scalar flux between LO

and HO equations.

The closure relationships E or D are computed from HO solution. Fig. 1 shows the comparision among

NDA, consistent QD and inconsistent QD methods for the gray Marshak wave problem. The solution

labeled “Source Iteration” represents the HO solution that is obtained from direct coupling of Eq. (1) and

(2). As can be seen from Fig. 1, the inconsistent QD formulation has a wave front slightly behind.

However, once the consistency term is added in Eq. (4), the wave front of the QD soluion matches with the

HO solution.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6 7 8 9 10

Tem

pera

ture

distance (cm)

Source IterationNDA Consistent

QD ConsistentQD Inconsistent

(a) Radition Flux

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6 7 8 9 10

Tem

pera

ture

distance (cm)

Source IterationNDA Consistent

QD ConsistentQD Inconsistent

(b) Temperature

Figure 1. Comparision of scalar flux and temperature with different method (t=1000)

ACKNOWLEDGMENTS

ADD ACKNOWLEDGEMENTS

REFERENCES

[1] J. Morel, T. Wareing, K. Smith, A linear-discontinuous spatial difference scheme for Sn radiative

transfer calculations, Journal of Computational Physics 128 (1996) 445–462.

2/3

Moment-based TRT

[2] E. Larsen, A grey transport acceleration method for time-dependent radiative transfer problems,

Journal of Computational Physics 2 (1988) 459–480.

[3] B. Chang, A deterministic photon free method to solve radiation transfer equations, Journal of

Computational Physics 222 (2007) 71–85.

[4] V.Ya.Gol’din, A quasi-diffusion method of solving the kinetic equation, USSR Computational

Mathematics and Physics 4 (1967) 136–149.

[5] K. Smith, J. Rhodes III, Full-core, 2-d LWR core calculations with CASMO-4E, 2002, PHYSOR 2002,

Seoul, Korea.

[6] D. Knoll, H. Park, K. Smith, Application of the Jacobian-free Newton-Krylov method to nonlinear

acceleration of transport source iteration in slub geometry, Nuclear Science and Engineering 167

(2011) 122–132.

3/3


Reducing the Spatial Discretization Error of Thermal Emission in ImplicitMonte Carlo Simulations

Adam G. IrvineDepartment of Aerospace Engineering

University of Michigan1320 Beal Avenue

Ann Arbor, MI [email protected]

Iain D. [email protected]

Nicholas A. GentileLawrence Livermore National Laboratory

Livermore, CA [email protected]

Implicit Monte Carlo (IMC), as described by Fleck and Cummings [1], is often used to simulate radiative

transport problems. In problems with strong coupling between radiation and matter, spatial discretization

in IMC can cause what is known as teleportation error [2]. Teleportation error is seen where energy

deposited on one side of a zone is redistributed throughout the zone in the next time step which will cause

thermal emissions to occur too far into the zone. This has the net effect of transporting energy too quickly.

Furthermore, decreasing the time step will increase the number of these redistributions over a set time and

increase the teleportation error. In IMC, this error is reduced by effective scattering where energy that

would be deposited by physical absorption is carried by the photon. However, effective scattering decreases

as the time step decreases. Since teleportation error is dependent upon both spatial and temporal

discretization, grids may need to be refined as time steps are changed in order to ensure the teleportation

error is small.

It is common to use an approximation of the temperature profile through the zone in order to more

accurately distribute emissions [3] and reduce, but not eliminate, the teleportation error when using large

zones as seen in figure 1. This approximation is commonly referred to as source tilting, and when zone

sizes are small, the IMC results using source tilting line up well with those found using diffusion, but show

faster energy transport as zone size is decreased. There is also work that handles photons in a

“semi-analog” fashion, tracking photons from emission to absorption and following the bundle of energy

which is reemitted after a delay [4].

For this work, we use IMC but sample locations along the photon paths for use as emission locations. Since

IMC does not deposit energy at points, but along paths, it is necessary to first select a path based on it’s

probability of absorption, which is proportional to the fraction of it’s energy lost along a path which is

shown in Eg.(1). Once a path is selected, we then sample a point along that path using equation 2.

Adam G. Irvine, Iain D. Boyd, and Nicholas A. Gentile

0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Position

Tem

pera

ture

Diffusion with20 Zones w/Tilt and dt =1Diffusion with200 Zones w/Tilt and dt =1IMC with20 Zones w/Tilt and dt =1IMC with200 Zones w/Tilt and dt =1

Figure 1: 1D grey problem with source at x = 0, σ = 10T 3 , tfinal = 500, cv = 7.14, a = c = 1

4E = Eo

(1− e−fσd

)(1)

X = Xo + Ω−ln

(1− ξ

(1− e−fσd

))fσ

(2)

where E is the photon energy, f is the Fleck factor, σ is the absorption cross section, d is the path length,

X is the photon location Ω is the direction cosine, and ξ is a random number uniformly distributed between

zero and one. The points found in equation 2 are stored for use in the next time step as emission locations.

This ensures that, with the exception of the first time step, thermal reemission only occurs in locations

where photons have deposited energy. Applying this Teleportation Correction (TC) method to the same test

problem as above for various zone sizes, it is shown in figure 2 that the speed at which energy is

transported through the system does not change with zone size.

We quantify the error obtained with source tilting and with the TC method compared to a finely resolved

solution for this unit problem across time and space discretizations in figure 3. The error peaks found at

high temporal discretization and at a large number of zones are due to instabilities in IMC. Since the TC

method requires only the additional calculation of equation 2, the storage of points and their retrieval the

additional computational cost is minimal with the exception of the additional memory required. The use of

TC amounts to an increase of approximately 25% in memory required, but this could be reduced if TC is

only used in regions with large opacity gradients.

We also apply the TC method to a similar one dimensional problem with a real material and a multigroup

opacity. Next we use a two dimensional Graziani crooked pipe test [5] in which teleportation error causes a

2/4

Reducing the Spatial Discretization Error of Thermal Emission in Implicit Monte Carlo Simulations

0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Position

Tem

pera

ture

10 Zones w/Tilt and dt =120 Zones50 Zones100 Zones200 Zones

(a) IMC using source tilting

0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Position

Tem

pera

ture

10 Zones w/TC and dt =120 Zones50 Zones100 Zones200 Zones

(b) IMC using new Teleportation Correction (TC) method

Figure 2: 1D grey problem with source at x = 0, σ = 10T 3 , tfinal = 500, cv = 7.14, a = c = 1

101

102

103

10−3

10−2

10−1

100

0

0.2

0.4

0.6

0.8

1

1.2

1.4

dt

Approximate Errors withTilt Correction

Number of Zones

(a) IMC using source tilting

101

102

103

10−3

10−2

10−1

100

0

0.2

0.4

0.6

0.8

1

1.2

1.4

dt

Approximate Errors withTC Correction

Number of Zones

(b) IMC using new Teleportation Correction (TC) method

Figure 3: Sum of releative errors for a 1D grey problem with source at x = 0, σ = 10T 3 , tfinal = 500, cv =

7.14, a = c = 1

3/4

Adam G. Irvine, Iain D. Boyd, and Nicholas A. Gentile

slower energy transport through the thin regions due to faster transport through the thick regions. Finally

we test the TC method in a three dimensional domain replicated simulation.

ACKNOWLEDGMENTS

This material is based upon work supported by the Department of Energy [National Nuclear Security

Administration] under Award Number NA28614.

The work of Nicholas A. Gentile performed under the auspices of the U.S. Department of Energy by

Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.

Disclaimer: This report was prepared as an account of work sponsored by an agency of the United States

Government. Neither the United States Government nor any agency thereof, nor any of their employees,

makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy,

completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that

its use would not infringe privately owned rights. Reference herein to any specific commercial product,

process, or service by trade name,trademark, manufacturer, or otherwise does not necessarily constitute or

imply its endorsement, recommendation, or favoring by the United States Government or any agency

thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the

United States Government or any agency thereof.

References

[1] Fleck, J.A., and Cummings, J.D., “An Implicit Monte Carlo Scheme for Calculating Timeand Frequency Dependent Nonlinear Radiation Transport,” J. Comp. Phys., 8, pp. 313-342(1971).

[2] McKinley M.S., and Brooks, E.D III, Szoke A., “Comparison of Implicit and SymbolicMonte Carlo Line Transport with Frequency Weight Vector Extension,” J. Comp. Phys., 189,pp. 330-349 (2003).

[3] Fleck, J.A., and Canfield, E.H., “A Random Walk Procedure for Improving theComputational Efficiency of the Implicit Monte Carlo Method for Nonlinear RadiationTransport,” 54, pp. 508-523 (1984).

[4] C. Ahrens, and E.W. Larsen, “A ‘Semi-analog’ Monte Carlo method for grey radiativetransfer problems,” Proceedings of the American Nuclear Society Mathematics andComputations Topical Meeting, (2001)

[5] Graziani, F., and LeBlanc, J. “The crooked pipe test problem,” UCRL-MI-143393 (2000).

4/4


DERIVING THE ASYMPTOTIC P1 APPROXIMATION FOR THERMALRADIATIVE TRANSFER

Shay I. HeizlerDepartment of Physics

Bar-Ilan UniversityRamat-Gan, [email protected]

We study the capabilities of the asymptoticP1 approximation for the problem of radiative transfer. The

classicP1 approximation (which gives rise to the Telegrapher’s equation) replacesthe problem of infinite

particle velocity of the diffusion approximation with the wrong finite value of particle velocity,c/√3.

Careful derivation of the asymptoticP1 equations, directly from the time-dependent Boltzmann equation,

yields the correct steady-state solution and the (almost) exact time evolution, at the cost of solving theP1

equations. We test the asymptoticP1 approximation with the well-known Su-Olson benchmark for

radiative transfer [1, 2], for which we obtain a semi-analytical solution. We found that the asymptoticP1

approximation yields a better solution than both the diffusion approximation and the classicP1

approximation, yields the correct steady-state behavior for the energy density and the (almost) correct

particle velocity.

Recently, a new approach was offered, based on an asymptotic development (both on space and time) of

the time-dependent Boltzmann equation, called the asymptoticP1 approximation (or the asymptotic

Telegrapher’s equation approximation) [3, 4]. This approximation yields the correct steady-state eigenvalue

of theasymptotic diffusion approximation [5, 6], and the almost correct time evolution, such as the particle

velocity (the wave-front behavior). This approximation was developed mainly to deal with neutron

transport, and it is worthwhile to develop and test it in radiative transfer problems, where the use of

diffusion-models is vast.

In this study we follow the methodology in [7], but with an opposite rationale. In[7] analytic solutions of

the classicP1 were found for benchmarking a general spherical harmonics (PN ) code (an exact transport

solution in the limitN → ∞). Here, we intend to show via a semi-analytical solution for the Su-olson

benchmark [1, 2], that the asymptoticP1 approximation reproduces the main features of a full transport

solution in the cost of solvingP1 equations (PN with N = 1).

The radiation transport equation in on one dimensional slab geometry for grey radiation is:

1

c

∂I

∂t+ µ

∂I

∂z= σ (B − I) + S, (1)

whereI is the specific intensity,σ is the opacity,S is an external source,µ is the cosine of the angle

between the particle’s trajectory and thez-axis andc is the speed of light.B is the local black-body

Shay I. Heizler

radiation emitted from the material. If we integrate Eq. (1) over all solid angle, we get the zero moment of

the transport equation,the conservation law:

1

c

∂E

∂t+

1

c

∂F

∂z= σ

(

aT 4m − E

)

+ S, (2)

whereE is the energy density and is defined as the zero’s moment of the specific intensity I andF is the

radiation flux, the first moment of the specific intensity. Assuming that the specific intensity is decomposed

from its two first moments (by the Legendre series) and operating∫

1

−1µdµ over Eq. (1) yields [5]:

1

c

∂F

∂t+

c

3

∂E

∂z+ σF = 0 (3)

Eq. (3) is the first moment of the transport equation and it is an approximate equation. Together with the

conservation law (Eq. (2)) which is exact, these two equation are decomposed the classicP1 approximation

yields the wrong particle velocity,c/√3 [5, 7]. The second equation is approximate and contains the factor

of 3. Thus, the rationale of the asymptoticP1 approximation [3, 4] urge us to find a modified equation (like

the asymptotic diffusion approximation developed a modified Fick’s law [6]) ofthis form:

Ac

∂F

∂t+ c

∂E

∂z+ σBF = 0, (4)

whereA andB are two media-dependent parameters that should be determined from an asymptotic

derivation of the exact time-dependent Boltzmann equation.B determines the steady-state behavior and

equals toB = 1/D0 of the asymptotic diffusion approximation. In our caseB = 3 but it can be modified to

include general media features [5].A determines the time-evolution, including the particle velocity. Of

course,A = B = 3 reproduces the classicP1 approximation.

Applying the Laplace transform to Eq. (4) yields a Fick’s law shape:

F = −cD(s) · ∂E∂z

(5)

with thiss-dependent diffusion coefficient:

D(s) ≡ c

As+ Bcσ (6)

Eq. (6) is helpful in finding the asymptotic constantsA andB.

Applying the Laplace transform to the exacttime-dependent Boltzmann equation yields atime-independent

Boltzmann equation with as-dependent cross-sections. We develop the usual asymptotic treatment of this

equation, yields a Fick’s-law relation between the first two moments with a modifieds-dependent diffusion

coefficient:

D′ (c′(s))

= cs

(cσ + s)2κ′20(s)

≡ cD0(c

′(s))cσ + s

(7)

2/4

AsymptoticP1 Approximation for Radiaive Transfer

with κ′20(s) as the exact Case’s-eigenvalues for the asymptotic regime [6] andc′(s) is thes-dependent

albedo (weff or γ(z) in [5]).

Now we compare the resulting modifieds-dependent diffusion coefficient in Eq. (7) with Eq. (6) and solve

for A andB, by using the tabulated (or approximated) value ofκ′20(s) orD0(c

′(s)) [3, 6]. Since this value

is complex, we expand the modified diffusion coefficient in a Taylor series:c

D′ (c′(s))=

cσ + s

D0(c′(s)),≈ Bcσ +As+O(s2) + . . . (8)

remembering thats → 0 corresponds tot → ∞ according to the final-value theorem. That means an

asymptotic treatment for the time, in addition to the derivation on space. Substitutingthe value ofκ′20(s) or

D0(c′(s)) (in our local-thermodynamic-equilibrium case,c = 1 since all the photons absorbed in the

material are emitted immediately through the black-body radiation) yieldsB = 3 andA = 3/5. For a

general media problems that includes scattering events or optically thin media weuse the definition of the

albedoc and find by Eq. (8), a generalA(c) andB(c) (For a wide discussion, see[3]).

Next, we find the Green function for the asymptoticP1 equations, which will be helpful for a general

source (in specific, the Su-Olson benchmark). Using similar technique as inRef. [7] applying the Laplace

transform on time and Fourier transform on space, usingQ = δ(x)δ(t). For the case of similar material

heat capacity and radiation heat capacity (ε = 1 in [1, 2, 7] terminology), and local thermodynamic

equilibrium (LTE) we get (in dimensionless unitsx ≡ σz andτ ≡ εcσt):

G(x, x0, τ) =1

2√2A

e−B2A τ

[

Aδ(

τ −√2A|x− x0|

)

+ (9)

B2H

(

τ −√2A|x− x0|

)

I0

( B2A

√

τ2 − 2A(x− x0)2)

+

τI1

(

B2A

√

τ2 − 2A(x− x0)2)

√

τ2 − 2A(x− x0)2

whereIn(x) is the modified-Bessel function of first kind of ordern andH(x, τ) is the Heaviside-step

function. We can immediately see that in contrast to the infinite particle velocity of the diffusion

approximation and the wrong finite velocity of the grey classicP1 approximation,c/√3 ≈ 0.58c, the LTE

asymptoticP1 approximation yields almost the correct particle velocity,√

5/6c ≈ 0.91c. The LTE classic

P1 approximation yields a poor agreement in the particle velocity,c/√6 ≈ 0.41c.

The (dimensionless) energy density in that case is forτ 6 10 for the Su-Olson benchmark is thus:

E (x, τ) =1

2√2A

∫

0.5

−0.5

dy[

Ae− B√

2A |x−y|H

(

τ −√2A|x− y|

)

+ (10)

B2

∫ τ

0

dτ ′e−B2A τ ′H

(

τ ′ −√2A|x− y|

)

I0

( B2A

√

τ ′2 − 2A(x− y)2)

+

τ ′I1(

B2A

√

τ ′2 − 2A(x− y)2)

√

τ ′2 − 2A(x− y)2

3/4

Shay I. Heizler

Eq. (10) is similar to the solutions in [7], with the appropriate Green function. The integrals in Eq. (10) can

be solved numerically. The radiation energy of the asymptoticP1 approximation (Eq. (10)) along with the

grey-transport, the diffusion and the classic LTEP1 approximation is shown in Fig. 1(a).

0.1 1 10x

0.001

0.01

0.1

1

Rad

iatio

n E

nerg

y D

ensi

ty

TransportAsymptotic P

1 (LTE

Diffusion (LTE)Classic P

1 (LTE)

τ=103.161

0.1

0.1 1 10x

0.001

0.01

0.1

1

Ene

rgy

Den

sity

Radiation - transportMateriel - transportP

1 (Grey)

Asymptotic P1 (LTE)

τ=10

3.1610.1

Figure 1. (a) The radiation energy of the asymptotic P1 approximation (Eq. (10)) along with thetransport, the diffusion and the classic LTE P1 approximation (b) The same along with the greyclassic P1 approximation

We can see that besides the wave front, the asymptoticP1 approximation yields the best approximation for

the exact transport solution. In addition, it is even better from the grey classicP1 approximation that was

derived in [7] (see Fig. 1(b)), except in very early times inside the source (|x| 6 0.5), when there is still a

large difference between the material and the radiation energy.

REFERENCES

[1] B. Su and G.L. Olson, “An Analytical Benchmark for Non-Equilibrium Radiative Transfer in anIsotropically Scattering Medium”,Ann. Nucl. Energy, 13, pp. 1035-1055 (1997).

[2] G.L. Olson, L.H. Auer and M.L. Hall,“Diffusion,P1, and Other Approximate Forms of RadiationTransport”,J. Quant. Spectrosc. Radiat. Transfer, 64, pp. 619-634 (2000).

[3] S.I. Heizler, “Asymptotic Telegrapher’s Equation (P1) Approximation for the Transport Equation”,Nucl. Sci. Eng., 166, pp. 17-35 (2010).

[4] S.I. Heizler, “Deriving a Modified Asymptotic Telegrapher’s Equation(P1) Approximation”,Proceedings of PHYSOR 2010, Pittsburgh, Pennsylvania, USA, May 9-14, 2010 (on CD-ROM).

[5] G.C. Pomraning,The Equations of radiation hydrodynamics, Pergamon Press (1973).

[6] K.M. Case, F.De. Hoffmann, G. Placzek, B.Carlson and M. Goldstein,Introduction to the theory ofneutron diffusion - volume I, Los Alamos Scientific Laboratory (1953).

[7] R.G. McClarren, J.P. Holloway and T.A. Brunner, “AnalyticP1 Solutions for the Time-Dependent,Thermal Radiative Transfer in Several Geometries,J. Quant. Spectrosc. Radiat. Transfer, 109, pp.389-403 (2008).

4/4


MIXED VARIATIONAL FORMULATION OF THE TRANSPORTEQUATION

J. CartierCEA, DAM, DIF


M. PeybernesCEA, DAM, DIF


The purpose of this abstract is to present the new variational formulation of the transport equation

introduced by the authors in the paper [1]. This formulation is based on a mixed form of the second order

transport equation (see C.J. Gesh [2] and J. Cartier et al. [1]).

In order to derive rigorous transport numerical methods based on finite elements, we must introduce and

study some variational formulations related to the transport equation. This has been done first by

C.G.Pomraning and M.Clark [3] by considering the first order transport equation and more recently by

L.Bourhrara [4], who have presented most of variational formulations related to the first order transport

equation and the second-order self adjoint angular flux transport equation (SAAF) studied by J.Morel and

J.M.McGhee [5]. L.Bourhrara [4] presented some variational formulations of the first order transport

equation which are obtained by multiplying the transport equation by trial functions, leading to a

symmetric variational problem. The well-posedness of these variational formulations has also been proven.

Our main idea is to derive mixed forms of the transport equation with both angular flux and angular current

density from the transport equation in order to apply mixed and hybrid finite element method (MHFEM).

We introduce proper functional settings and approximation spaces adapted to transport problems. Notice

that mixed form of the transport equation can be derived directly from the SAAF second-order transport

equation in the same way than the mixed form arising from elliptic partial differential equations theory.

First we consider the standard first-order form of the mono-energetic stationary transport equation:~Ω · ~∇u+ σtu = Ku+ q in X,u = ub on Γ−.

(1)

where u represents the angular flux, σt is the macroscopic total cross section, ub is an inhomogeneous

incoming angular flux, q is an inhomogeneous source. D is a bounded open subspace of R3 and its

boundary ∂D, S2 the unit sphere of R3, x ∈ D is the position variable, dx represents the Lebesgue

measure on R3 and ds is the measure induced by dx on ∂D, ~Ω ∈ S2 is the directional variable,

X = D × S2 and Γ = ∂D × S2,

Γ− =

(x, ~Ω) ∈ Γ such that ~Ω · ~n(x) < 0, and Γ+ =

(x, ~Ω) ∈ Γ such that ~Ω · ~n(x) > 0

,


~n = ~n(x) indicates the outward unit normal vector at x to the boundary ∂D. The scattering operator is

given by

Ku =

∫S2

σs(~Ω · ~Ω′)u(~Ω′)dν(~Ω′), (2)

where σs(x, ~Ω′ · ~Ω) is the differential scattering cross section.

Then, in order to derive the mixed variational formulation of the transport equation, we introduce the

classical transport functional Hilbert space W (see Dautray-Lions [7] for more details) as

W =u ∈ L2, ~Ω · ~∇u ∈ L2, u ∈ L2

+

,

with the norm

||u||2W = ||u||2L2 + ||~Ω · ~∇u||2L2 + ||u||2L2+.

In this study, we assume that q ∈ L2 , ub ∈ L2− and σt ∈ L∞(X), where L2 = L2(X, dx dν),

L2− = L2(Γ−, |~Ω · ~n|ds dν), and L2

+ = L2(Γ+, |~Ω · ~n|ds dν) and the following notations:

PΩ = ~Ω⊗ ~Ω = (ΩiΩj)1≤i,j≤3, (3)

where PΩ represents the projection tensor over any given direction ~Ω whose components are Ωi and IR3 is

the identity tensor of R3 (then we have PΩ~g = ~Ω(~Ω · ~g)). For the study of transport mixed formulation, we

need to introduce the following space:

Y =~g = u~Ω;u ∈W

. (4)

and we verify that Y is an Hilbert space with the following norm:

||~g||Y = ||~Ω · ~g||W . (5)

To establish the mixed formulation of the transport equation, we introduce the angular current density ~g as

~g = ~Ωu and noticing that ~Ω (~Ω · ~∇u) = PΩ~∇u, σt~Ωu = σt~g, and Ku = K(~Ω · ~g), we obtain by

introducing ~g into equation (1) a coupled set of equations with the angular flux and the angular current

density as their unknowns: σt~g + PΩ

~∇u = ~ΩK(~g · ~Ω

)+ ~Ωq in X,

σtu+ ~∇ · (PΩ~g) = Ku+ q in X.

(6)

where the specific related boundary conditions are:u = ub for (x, ~Ω) ∈ Γ−,

~g = ~Ωu for (x, ~Ω) ∈ Γ+.

(7)

Then we prove the equivalence between mixed problem (6)-(7) and the initial transport problem (1) with

the following proposition:

2/4

Mixed variational formulation of the transport equation

Proposition 1 A solution (u,~g) ∈W × Y of the mixed formulation of the transport equation (6)-(7) is

solution of the standard first order form of the transport equation (1), and reciprocally.

The existence and uniqueness of the mixed transport problem is proved in the following proposition

Proposition 2 The problem

~∇ · ~g + σtu = Ku+ q in X,

PΩ~∇u+ σt~g = ~Ω (Ku+ q) in X,

u = ub for (x, ~Ω) ∈ Γ−,

~g = ~Ωu for (x, ~Ω) ∈ Γ+.

(8)

has a unique solution (~g, u) ∈ Y ×W and we get

||~g||2Y = ||u||2W ≤ C(||q||2L2(X) + ||ub||2L2(Γ−)

)(9)

where C is a nonnegative constant.

To establish the mixed variational formulation we introduce some bilinear form (we only consider the case

where σs ≡ 0 for the sake of simplicity):

For all (~g,~h) ∈ Y × Y :

a(~g,~h) = (A~g,~h) =

∫Xσt (~g · ~h) dx dν(~Ω) +

∫Γ+

(~Ω · ~n) (~g · ~h) ds dν(~Ω). (10)

For all (u,~h) ∈ L2 × Y :

b(u,~h) = (u,B~h) = −∫Xu ~∇ ·

[PΩ~h]dx dν(~Ω). (11)

For all (u, v) ∈ L2 × L2:

d(u, v) =

∫Xσt u v dx dν(~Ω), (12)

We also introduce some linear forms, for all ~h ∈ Y :

L1(~h) =

∫Xq(x, ~Ω) (~Ω · ~h) dx dν(~Ω)−

∫Γ−

(~Ω · ~n) ub (~Ω · ~h) ds dν(~Ω), (13)

and for all v ∈ L2:

L2(v) = −∫Xq(x, ~Ω) v dx dν(~Ω). (14)

And we obtain the following mixed variational problem: find (~g, u) ∈ Y × L2 such that:a(~g,~h) + b(u,~h) = L1(~h) for all ~h ∈ Y

b(v,~g)− d(u, v) = L2(v) for all v ∈ L2.(15)

3/4


Proposition 3 Problem (15) is equivalent to initial transport problem (1), i.e., if (u,~g) ∈W × Y is a

solution of (15) then u is a solution of (1) and ~g = ~Ωu; reciprocally, if u is a solution of (1) then u and ~Ωu

are solutions of (15).

Finally, we prove existence and uniqueness of solution for this abstract problem by using some results

arising from Brezzi and Fortin [6].

Theorem 1 Let q ∈ L2 and ub ∈ L2−. Under appropriate assumptions from Brezzi and Fortin [6], problem

(15) has a unique solution (u,~g) in L2 × Y/M , where

M = KerBt ∩KerA. (16)

Moreover, we have the bound:

||u||L2 + ||~g||Y/KerBt ≤ C(||q||L2 + ||ub||L2−

), (17)

where C is a constant.

We have introduced a mixed variational formulation of the transport equation. We prove existence and

uniqueness of a solution for the related mixed abstract transport problem.

REFERENCES

[1] J.Cartier, M.Peybernes, Mixed Variational Formulation and Mixed-Hybrid Discretization of theTransport Equation, Transport Theory and Statistical Physics, vol. 39, issue 1, pp. 1-46, (2010).

[2] C.J.Gesh, Finite Element Method for Second Order Forms of the Transport Equation, Ph.Thesis,Texas A&M University, (1999).

[3] G.C.Pomraning, M.Clark, The Variational Method Applied to the Monoenergetic BoltzmannEquation. Part I,II, Nucl. Sci. Eng. 16, 147-164, (1963).

[4] L.Bourhrara, New Variational Formulations for the Neutron Transport Equation ,Transport Theoryand Statistical Physics, vol. 33, issue 2, pp. 93-124, (2004).

[5] J.E.Morel, J.M.McGhee, A Self-Adjoint Angular Flux Equation, Nucl. Sci. Eng. 132, 312-325,(1999).

[6] F.Brezzi and M.Fortin, Mixed and hybrid finite element methods , Springer-Verlag, New York (1991).

[7] R.Dautray, J.L.Lions, Analyse mathematique et calcul numerique pour les sciences et les techniques,Masson, (1985).

4/4


APPLICATION OF SPECTRAL ELEMENTS FOR 1D NEUTRONTRANSPORT AND COMPARISON TO MANUFACTURED SOLUTIONS

A. Barbarino, S. Dulla, P. RavettoPolitecnico di Torino

Torino, [email protected], [email protected], [email protected]

E.H. MundUniversite Libre de Bruxelles

Brussels, [email protected]

B.G. GanapolUniversity of Arizona

Tucson, AZ, [email protected]

1 Introduction

Spectral element methods (SEM) provide highly accurate solutions to elliptic ODEs and PDEs in a

variational framework using the Galerkin or Petrov-Galerkin methods [1].Unfortunately, neutron transport

problems, being first order, do not fitstricto sensu into this framework. There are notable exceptions

however. If scattering is isotropic, the Boltzmann equation may be cast into a second order

integro-differential equation for the even component of the angular flux, the so-called even parity

formulation of neutron transport due to Vladimirov (see [2]). Even with anisotropic scattering, thePN

formulation of neutron transport (or one of its variants) leads to a set of coupled elliptic equations, and as a

recent study shows, highly accurate transport solutions can indeed beobtained with spectral elements [3].

In deterministic transport computations theSN discrete ordinates method is also widespreadly adopted.

The purpose of this study is to apply the spectral element approach to neutron transport in both thePN and

SN formulations, in order to probe the accuracy and convergence trends of SEM. To monitor the evolution

of the error, a reference benchmark solution is required. Therefore, themanufactured transport solution

approach is adopted for a one-dimensional configuration with isotropic scattering. This name has been

coined to refer to problems having both an external neutron source and boundary conditions tailored to a

predetermined solution [4]. The reference solutions are compared to the results obtained with the SEM

approach and with a standardSN solver (diamond-difference in space). AnSN solution converged in space

and angle [5] will serve as a reference for more complicated benchmark evaluations that will be performed

in the future.

The problem considered is the one-velocity Boltzmann transport equation inplanar geometry with

Barbarino et al.

isotropic scattering:

µ∂ψ(x, µ)

∂x+ Σt(x)ψ(x, µ) =

Σs(x)

2

∫

+1

−1

dµ′ ψ(x, µ′) + S(x, µ), (1)

where(x, µ) ∈ D := (a, b)× [−1,+1]. The unknownψ(x, µ) is the angular neutron flux andS(x, µ) an

external neutron source. As usual, the space dependence of the cross sectionsΣs(x) andΣt(x) is assumed

to be piecewise constant. Despite the space dependent notation - left only as a reminder - it is therefore

quite legitimate to extract these parameters from differentiation as we shall consistently do below.

Equation (1) must be supplemented with boundary conditions. For the sake of simplicity, we assume

vacuum conditions at both ends of the spatial domain:

ψ(a, µ) = 0, ∀µ ∈ [0, 1]; ψ(b, µ) = 0, ∀µ ∈ [−1, 0], (2)

and we introduce the scalar flux:

Φ (x) :=

∫

+1

−1

dµψ(x, µ). (3)

It turns out that Eq. (1) is quite amenable to solution using Vladimirov’s even-parity formulation of

transport (see [2]). Letψ+(x, µ) andψ−(x, µ) denote the even- and odd parity components ofψ(x, µ)

with respect to the angular variableµ:

ψ±(x, µ) =1

2(ψ(x, µ) ± ψ(x,−µ)). (4)

Replacingµ in (1) by−µ, adding and subtracting, one obtains two equations involvingψ+ andψ−:

µ∂ψ−(x, µ)

∂x+ Σt ψ

+(x, µ) =Σs φ(x)

2+ S+(x, µ), (5)

µ∂ψ+(x, µ)

∂x+ Σt ψ

−(x, µ) = S−(x, µ), (6)

where, by analogy with the neutron flux,S+ andS− denote the even- and odd-parity components of the

external source. Elimination ofψ−(x, µ) through (6) gives a second-order equation forψ+(x, µ):

−µ2

Σt

∂2ψ+

∂x2(x, µ) + Σt ψ

+(x, µ) =Σs

2φ (x) + Q(x, µ), (7)

with

Q(x, µ) = S+(x, µ) −µ

Σt

∂S−

∂x(x, µ). (8)

We observe that the r.h.s. of (8) is an even function ofµ, henceQ+(x, µ) = Q(x, µ). Assuming there is no

external neutron source on the boundary,i.e. S(a, µ) = S(b, µ) = 0, and using (4), one can easily show

that the vacuum boundary conditions forψ+(x, µ) are transformed into:

µ∂ψ+

∂x(a, µ) − Σt ψ

+(a, µ) = 0, ∀µ ∈ [0, 1].

µ∂ψ+

∂x(b, µ) − Σt ψ

+(b, µ) = 0, ∀µ ∈ [−1, 0], (9)

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SEM for neutron transport

that is, a Robin type condition. Equations (7-9) constitute the basic formulationof the transport problem.

The angular treatment can be dealt with adopting thePN Legendre polynomial method, thus leading to a

set of coupled equations for the even moments of the flux (φ2n(x) with n = 0, . . . , (N − 1)/2).

Alternatively, theSN discrete ordinates method solves for the angular fluxes according to the

Gauss-Legendre quadrature set, although only half of the directions are considered, due to the even-parity

formulation of the problem (ϕn(x) for n = 1, . . . , N/2). Once either thePN or theSN formulation has

been established, SEM is adopted to treat the spatial dependence.

2 Spectral element formulation of the transport problem

The spectral element method is based on the application of the Galerkin method: having introduced a set of

test functionsθ(x) := (θ0(x), . . . , θ2N (x))T with components belonging to the Sobolev spaceH1(a, b),

the balance model under consideration is projected on such test functions. The solution is sought in a

subspace ofH1(a, b), such that the weighted balance equation is satisfied for anyθ(x) belonging to this

subspace.

The full details on the numerical formulation of the problem with SEM can be found in [3, 6]. However,

the result of the adoption of the PN or SN techniques to the neutron balance problem in conjunction with

the implementation of SEM leads to a set of linear equations for the problem unknownsϕ:

Lϕ := (D ⊗ K + Ic ⊗M)ϕ = M q, (10)

which are characterized, for PN and SN , by a different structure and sparsity of the coefficient matrixL.

The structure of such matrices is presented in Figure 1, comparing a P5 and S6 formulation. In slab

geometry and with consistent boundary conditions they are proven to be equivalent in the reproduction of

the scalar flux.

3 Manufactured solutions to study SEM

The principle of manufactured solutions lies in the possibility, for simple geometrical and material

configurations, to have a reference benchmark solution for a source problem by imposing a certain flux

distribution within the domain and identifying the corresponding source from the balance equation. Such a

solution is then to be compared with the results of numerical codes and allows one to analyze the

appropriateness and the convergence rates of the numerical schemes adopted.

Some results based on the manufactured solution approach are presentedhere, in order to show the

improved performance that can be obtained with SEM with respect to standard diamond-difference like SNalgorithms.

At first, the angular flux is defined in slab geometry, considering a reference domain(−1,+1). For

simplicity, one single spectral element is adopted and the medium is assumed homogeneous (Σt = 1).

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Barbarino et al.

P5 formulation

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ϕ1(x)

ϕ2(x)

ϕ3(x)

Figure 1. Structure of matrixL fro theP5 (left) andS6 (right) formulation of the transport model usingSEM with two subdomains (number of block sub-matrices appearing in correspondence to each unknown)and order 5 for the Gauss-Lobatto-Legendre quadrature (dimension of the block sub-matrices).

Computations will be performed for different valuesc of the number of secondaries per collision. The

angular flux is taken in the form:

ϕ(x, µ) = f(x)g(µ) (11)

and the following expression for the spatial and angular dependence is considered for the results presented

here:

f(x) = (1− x2)2 exp(−0.2x2), g(µ) = 3µ2 + 5µ4, (12)

leading to a corresponding source:

S(x, µ) =(

1− x2)

exp(−0.2x2)

Σt

(

1− x2) (

3µ2 + 5µ4)

− 2Σs

(

1− x2)

(13)

−(

3µ3 + 5µ5) [

4x+ 0.4x(

1− x2)]

. (14)

The error in the scalar flux with respect to the reference solution is of the following form:

ε(N,K) =

(

∑Kk=1

(ΦN,K(xk)− Φref (xk))2)1/2

(

∑Kk=1

Φ2ref (xk)

)1/2, (15)

depending on the orderN of thePN / SN methods and on the number of spatial nodesK.

The results obtained with the various approaches adopted are summarized inTable 3. The improved

convergence characteristics of SEM with respect to standard discretization schemes for SN is clearly

visible. The consistency of results obtained with SN and PN−1 is also verified. It must be noticed, however,

that in the P3-S4 case relevant differences are experienced, even if the same SEM approach for the spatial

dependence is used. This is due to the fact that in the S4 calculation the real angular dependence of the

source is considered, taken in the directions of the discrete ordinates, while PN in general adopts a

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SEM for neutron transport

Table I. DiscreteL2 error in the scalar flux(c = 0.5). The reference, analytical result is compared tosolution in PN and SN , adopting SEM and the SN with standard diamond differencing (DD). The parameterK represents the degree of the polynomials for all SEM approaches, leading toK + 1 equations, and iscompared with theSN /DD method adoptingK + 1 nodes (same number of equations solved).

S2 P1 S4 P3

K DD SEM SEM DD SEM SEM

4 3.3963e− 01 3.4878e− 01 5.1453e− 01 8.6324e− 02 3.5534e− 02 8.0136e− 026 3.3667e− 01 3.4222e− 01 5.4432E − 01 4.4790e− 02 7.4531e− 04 5.3218e− 028 3.3616e− 01 3.4208e− 01 5.4250E − 01 2.7262e− 02 1.1277e− 05 5.2859e− 0210 3.3608e− 01 3.4200e− 01 5.4132E − 01 1.8304e− 02 1.3300e− 07 5.2657e− 0212 3.3608e− 01 3.4191e− 01 5.4052E − 01 1.3127e− 02 1.2796e− 09 5.2515e− 0214 3.3610e− 01 3.4183e− 01 5.3995E − 01 9.8696e− 03 1.0375e− 11 5.2410e− 0220 3.3617e− 01 3.4185e− 01 5.3890E − 01 5.0430e− 03 1.4691e− 14 5.2215e− 02

S6 P5 S8 P7

K DD SEM SEM DD SEM SEM

4 8.8018e− 02 3.6567e− 02 3.6567e− 02 8.8908e− 02 3.6973e− 02 3.6973e− 026 4.5713e− 02 7.6314e− 04 7.6314e− 04 4.6162e− 02 7.7386e− 04 7.7386e− 048 2.7832e− 02 1.1464e− 05 1.1464e− 05 2.8101e− 02 1.1601e− 05 1.1601e− 0510 1.8689e− 02 1.3454e− 07 1.3454e− 07 1.8868e− 02 1.3580e− 07 1.3580e− 0712 1.3404e− 02 1.2904e− 09 1.2904e− 09 1.3531e− 02 1.2997e− 09 1.2997e− 0914 1.0078e− 02 1.0440e− 11 1.0440e− 11 1.0174e− 02 1.0499e− 11 1.0499e− 1120 5.1498e− 03 8.1638e− 15 8.2447e− 15 5.1984e− 03 7.5617e− 15 9.7083e− 15

truncated expansion in the angle variable; that is, the P3 case is not sufficiently accurate to describe the

physical behavior of the actual source. This problem is overcome in the P5 solution, since the angular

dependence of the source (4-th order polynomial) is represented exactly.

In the final presentation, results for various manufactured solutions will be presented. A set of more

complicated benchmark cases will also be studied, where an SN calculation converged in space and angle

[5] will be adopted as reference.

REFERENCES

[1] M.O. Deville, P.F. Fischer, E.H. and Mund,High-Order Methods for Incompressible Fluid Flow,Cambridge University Press, Cambridge (2002).

[2] E.E. Lewis, W.F. Miller,Computational Methods of Neutron Transport, Amer. Nucl. Soc., LaGrange Park (1993).

[3] E.H. Mund, Spectral element solutions for thePN neutron transport equation,Computers andFluids, 43(1), 102-106 (2011).

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Barbarino et al.

[4] J.S. Warsa, J.D. Densmore, A.K. Prinja, J.E.Morel, Manufactured Solutions in the Thick DiffusionLimit, Nucl. Sci. Eng., 166, 36 (2010).

[5] B.D. Ganapol, D.E. Kornreich, Mining the multigroup-discrete ordinatesalgorithm for high qualitysolutions,M&C 2005, Avignon (2005).

[6] A. Barbarino, S. Dulla, P. Ravetto, E.H. Mund, The spectral elementapproach for the solution ofneutron transport problems,M&C 2011, Rio de Janeiro, Brazil (2011).

6/6


ANALOG COMPUTING TO THE TIME-DEPENDENT SECOND-ORDER FORM OF NEUTRON TRANSPORT EQUATION IN X-Y

GEOMETRY

Ahmad Pirouzmanda, Kamal Hadadb

Department of Nuclear Engineering Shiraz University, Shiraz, Iran

[email protected] [email protected]

Piero Ravetto

Politecnico di Torino, Dipartimento di Energetica, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

[email protected]

This paper is a continuation of our previous study which utilized a novel method using cellular neural networks (CNN) coupled with spherical harmonics method to solve the neutron transport equation in x-y geometry [1]. In this work we extend the CNN application to time-dependent solution of neutron transport equation. To achieve this goal, an equivalent electrical circuit based on time-dependent second-order form of neutron transport equation and relevant boundary conditions is obtained using CNN method. The CNN architecture and the modeling of neutron transport equation using CNN method thoroughly have been explained in previous studies [1]-[4]. The advantages of the CNN simulator over the numerical methods have to do with its analog and parallel processing algorithm. They include, but are not necessarily limited to, reduction in the computational time and circumvention of the exponential increase in complexity as the scale and dimension of problem increase. It is also possible to implement the complete algorithm using VLSI. To show the ability of CNN in time-dependent condition, we simulate 2D time-dependent TWIGL problem using P3 approximation. Our CNN model includes two-energy group time-dependent second-order form of neutron transport equation and one-group of delayed neutron precursor density equation. We use the CNN model to simulate step and ramp perturbation transients in the core. The 2D TWIGL problem models a 160.0 cm square reactor consisting of three material regions: unperturbed seed regions that contain the primary fissile material, an identically composed perturbed seed region to which time-dependent properties will be introduced, and a blanket region that contains fissile material and surrounds the core on all sides. The 2D model is laid out in quarter-core symmetry as shown in Figure 1. In initial state, this hypothetical TWIGL reactor is slightly subcritical and the perturbed and unperturbed seed regions have identical material properties. The initial two-group properties for materials, the one delayed group constants, and additional parameters that will be required to perform a time-dependent calculation have been given in reference [5].


2

Unperturbed Seed

Perturbed Seed

Blanket

Reflected Boundary

Vacuum Boundary

x [cm]

y [cm]

0.0 24.0 56.0 80.0

24.0

56.0

80.0

Figure 1. Problem schematic for 2D TWIGL seed/blanket problem

Table I. Cell and lattice sizes used in CNN model for 2D TWIGL problem Model Cell Size (cm) keff

1 8.0 0.9143 2 4.0 0.9084 3 2.0 0.9075

A delayed supercritical transient is initiated in the seed/blanket reactor by decreasing the thermal macroscopic transport cross-section, σt2, in the perturbed seed region from the initial value of 0.83333 cm-1 to a final value of 0.82983 cm-1. For each model presented in Table I two separate transient calculations are performed. In the first case, the perturbation is introduced as a step change when the core is in steady state condition. In the second case, a ramp change is introduced over the time period 0.2 sec. For all models presented in Table I, both step and ramp changes are introduced at t=1000 seconds. The normalized power traces for Model 1, Model 2 and Model 3 of 2D TWIGT problem are shown in Figure 2 for the step transient and Figure 3 for the ramp transient. Figure 4 compares the transient result calculated by CNN (Model 1) with reference transient resulting from step change in σt2 [5]. There is some error between CNN results and reference results for step perturbation insertion (Figure 4). The main reason for this error is due to different approaches that have been utilized by two models. While CNN model uses spherical harmonic expansion of angular flux, the reference’s solution is base on method of characteristic (MOC). However with decreasing the time step size in MOC method this error reduces [5]. The CNN results show that for all presented models in Table I, the normalized power traces converge to some limiting values. In addition, our CNN model can follow up step and ramp transients in the core for different cell sizes.


3

Figure 2. CNN normalized power for simulation of 2D TWIGL step transient.

Figure 3. CNN normalized power for simulation of 2D TWIGL ramp transient.


4

Figure 4. Comparison of normalized power calculated by CNN and MOC methods

for the 2D TWIGL problem.

REFERENCES

[1] A. Pirouzmand, and K. Hadad, “Cellular neural network to the spherical harmonics

approximation of neutron transport equation in x–y geometry. Part I: Modeling and verification for time-independent solution,” Annals of Nuclear Energy, 38, pp. 1288–1299 (2011).

[2] L.O Chua and L. Yang, “Cellular Neural Networks: Theory,” IEEE Trans. On Circuits and Systems, 35, pp. 1257-1272 (1988).

[3] K. Hadad, and A. Piroozmand, “Application of cellular neural network (CNN) method to the nuclear reactor dynamics equations,” Annals of Nuclear Energy, 34, pp. 406-416 (2007).

[4] K. Hadad, A. Piroozmand, and N. Ayobian "Cellular Neural Networks (CNN) Simulation for the TN Approximation of Time Dependent Neutron Transport Equation in Slab Geometry,” Annals of Nuclear Energy, 35, pp. 2313–2320 (2008).

[5] J. B. Taylor, “The Development of a Three-Dimensional Nuclear Reactor Kinetics Methodology Based on the Method of Characteristics,” Ph.D. Thesis in Nuclear Engineering, Pennsylvania State University (2007).


P2-EQUIVALENT FORM OF THE SP2 EQUATIONSIncluding boundary and interface conditions

Ryan G. McClarrenDepartment of Nuclear Engineering

Texas A&M University3133 TAMU

College Station, TX [email protected]

The spherical harmonics (PN ) equations are a moment-based method to solve the Boltzmann transport

equation by expanding the angular variable Ω in terms of spherical harmonics functions and then truncating

the expansion at some order with a closure. These methods have been shown to work well in problems

with moderate amounts of scattering or with appropriate closures [1–4]. The SPN equations, or simplified

PN equations, were originally derived by Gelbard through taking a spherical harmonics expansion to the

1-D slab geometry transport equation and making some ad hoc substitutions to make the equations “look”

3-D. Gelbard was able to show that under many situations, the most general being an infinite medium with

a constant cross-section, the SPN solution for the scalar flux would be the same as the scalar flux solution

from the full, and much more complicated, PN equations. Later variational and asymptotic derivations of

the SPN equations were presented [5–9]. These derivations made it clear that the SPN equations, in the form

they are most commonly solved, do not give the same solution as the PN equations. Also, there has never

been an interpretation of the SPN unknowns in terms of spherical harmonics moments, except for the scalar

flux and current where the intrepretation is trivial. Much of the current knowledge of can be found by the

interested reader in a recent special issue of TTSP commenorating the 50th anniversary of the SPN equations

[10].

In the 1970’s Selengut presented in an inscrutably terse ANS transactions paper a derivation of a P3-

equivalent form of the SP3 equations with appropriate interface conditions. The trail of this work apparently

went cold thereafter, and no numerical solutions or in depth derivations of these equations have surfaced in

the literature. The lack of derivation details has made it difficult to extend Selengut’s work and verify its

correctness, in a similar vein to Fermat’s last theorem in that we have the result, but not how it was arrived

at∗. Of course there would be a large impact of a P3 -equivalent SP3 method in computational transport in

that SP3 is a workhorse method for reactor calculations and it has only 3 angular unknowns compared to

the 16 unknowns of the P3 equations in first-order form.

In an ongoing research program we are endeavoring to find PN -equivalent SPN methods, and this abstract

presents some initial, though theoretically important, results to that end. Specifically, we present a P2-

equivalent form of the SP2 equations in 2-D Cartesian geometry. We were able to find an interpretation of

∗It might be beyond the pale to call this Selengut’s last theorem as I believe he did much work after this.

McClarren

all the SP2 unknowns and show how it goes to the standard equation for the scalar flux in the case of an

infinite medium with constant cross-section.

We will be considering the linear, steady, and one-speed transport equation with isotropic scattering:

(Ω · ∇+ σt)ψ =1

4π(σsφ+Q) , (1)

where ψ(~x,Ω, t) is the angular flux with scalar flux given by

φ(~x) =∫4π

ψ(~x,Ω) dΩ. (2)

Also, in Eq. (1) σt is the macroscropic total cross-section, σs is the macroscopic scattering cross-section,

and Q is the isotropic, prescribed source.

The P2 equations as an approximation to Eq. (1) in 2-D x− z geometry, as derived previously [11, 12], are

σaψ00 +

1√3∂

∂zψ0

1 −√

23∂

∂xψ1

1 =Q√4π, (3a)

σtψ01 +

∂

∂z

(1√3ψ0

0 +2√15ψ0

2

)−√

25∂

∂xψ1

2 = 0, (3b)

σtψ11 +

∂

∂x

(− 1√

6ψ0

0 +1√30ψ0

2 −1√5ψ2

2

)+

1√5∂

∂zψ1

2 = 0, (3c)

σtψ02 +

2√15

∂

∂zψ0

1 +

√215

∂

∂xψ1

1 = 0, (3d)

σtψ12 +

1√5∂

∂zψ1

1 −1√10

∂

∂xψ0

1 = 0, (3e)

σtψ22 −

1√5∂

∂xψ1

1 = 0, (3f)

where

ψml (~x) =∫4π

Y ml (Ω)ψ(~x, Ω) dΩ,

with

Y ml (µ, ϕ) = (−1)m

√2l + 1

4π(l −m)!(l +m)!

P|m|l (µ)eimϕ. (4)

Now we will define a re-normalized moment by undoing the normalization constant used in the above

definition of the moments and removing the Condon-Shortley phase term:

ψlm = (−1)mψml

√4π(l +m)!

(2l + 1)(l −m)!, (5)

2/6


making

ψ00 =√

4πψ00, ψ10 =

4π√3ψ0

1, ψ11 = −√

8π3ψ1

1, ψ20 =

√4π5ψ0

2, ψ21 = −√

24π5ψ1

2, ψ22 =

√96π5ψ2

2.

Under this normalization ψ00 = φ and ~J = (ψ11, 0, ψ10)t. These definitions make the P2 equations

σaψ00 +∂

∂zψ10 +

∂

∂xψ11 = Q, (6a)

σtψ10 +∂

∂z

(13ψ00 +

23ψ20

)+

13∂

∂xψ21 = 0, (6b)

σtψ11 +∂

∂x

(13ψ00 −

13ψ20 +

16ψ22

)+

13∂

∂zψ21 = 0, (6c)

σtψ20 +25∂

∂zψ10 −

15∂

∂xψ11 = 0, (6d)

σtψ21 +35∂

∂zψ11 +

35∂

∂xψ10 = 0, (6e)

σtψ22 +65∂

∂xψ11 = 0. (6f)

These equations are starting to look like the SP2 equations, but there are still some algebraic hoops to jump

through.

The next step is to define a linear combination of ψ20 and ψ22 as a new unknown. If we take Eq. (6d) and

add it with one-half times Eq. (6f) we get that

σtφ2 +25∂

∂zψ10 +

25∂

∂xψ11 = 0, (7)

where

φ2 = ψ20 +ψ22

2.

From Eqs. (6d) and (6f) we also get that

32ψ20 +

14ψ22 = − 3

5σt

∂

∂zψ10. (8)

Next, we will eliminate ψ20 and ψ22 in favor of φ2 in Eqs. (6b) and (6c). We note that

23

(φ2 +

35σt

∂

∂zψ10

)=

23

(ψ20 +

12ψ22 −

32ψ20 −

14ψ22

)= −1

3ψ20 +

16ψ22, (9)

which is exactly what we need to write the x-derivative term in Eq. (6c) in terms of φ2. Using this result and

solving Eq. (6e) for ψ21 makes Eq. (6c):

σtψ11 +∂

∂x

(13ψ00 +

23φ2

)=

∂

∂z

15σt

∂

∂zψ11 +

∂

∂z

15σt

∂

∂xψ10 −

∂

∂x

25σt

∂

∂zψ10. (10)

To we will deal with Eq. (6b) we need to write ψ20 in terms of φ2 and ψ11. We do this by writing

23ψ20 =

23φ2 −

13ψ22 =

23φ2 +

25σt

∂

∂xψ11,

3/6

McClarren

where we have used Eq. (6f) to write ψ22 in terms of ψ11. This makes Eq. (6b)

σtψ10 +∂

∂z

(13ψ00 +

23φ2

)=

∂

∂x

15σt

∂

∂zψ11 +

∂

∂x

15σt

∂

∂xψ10 −

∂

∂z

25σt

∂

∂xψ11, (11)

The P2 equations can now be written in terms of 4 variables that can be interpreted as the SP2 unknowns:

σaφ0 +∇ · ~φ1 = Q, (12a)

σt~φ1 +13∇φ0 +

23∇φ2 =

∂∂z

15σt

∂∂zψ11 + ∂

∂z1

5σt

∂∂xψ10 − ∂

∂x2

5σt

∂∂zψ10

0∂∂x

15σt

∂∂zψ11 + ∂

∂x1

5σt

∂∂xψ10 − ∂

∂z2

5σt

∂∂xψ11

, (12b)

σtφ2 +25∇ · ~φ1 = 0, (12c)

where φ0 = ψ00 and ~φ1 = (ψ11, 0, ψ10)t, and ∂∂yψlm = 0. These equations are the SP2 equations for x− z

geometry with extra terms on the right-hand side of the φ1 equations. We can simplify these terms using

vector calculus operators. Here we will see the curl operator, an operator not commonly seen in transport

theory. Parsing the righthand side of Eq. (12b) yields

15

∂∂z

1σt

∂∂zψ11 + ∂

∂z1σt

∂∂xψ10 − ∂

∂x2σt

∂∂zψ10

0∂∂x

1σt

∂∂zψ11 + ∂

∂x1σt

∂∂xψ10 − ∂

∂z2σt

∂∂xψ11

=1

5σt

∂2

∂z2ψ11 − ∂2

∂x∂zψ10

0∂2

∂x2ψ10 − ∂2

∂x∂zψ11

− 2

5

(∂∂zψ10

)∂∂xσ

−1t

0(∂∂xψ11

)∂∂zσ

−1t

+15

(∂∂zψ11 + ∂

∂xψ10

)∂∂zσ

−1t

0(∂∂xψ10 + ∂

∂zψ11

)∂∂xσ

−1t .

(13)

Using the definition of the curl operator we get ∂2

∂z2ψ11 − ∂2

∂x∂zψ10

0∂2

∂x2ψ10 − ∂2

∂x∂zψ11

= −∇×∇× ~φ1. (14)

We can also make the simplification:

−2

(∂∂zψ10

)∂∂xσ

−1t

0(∂∂xψ11

)∂∂zσ

−1t

+

(∂∂zψ11 + ∂

∂xψ10

)∂∂zσ

−1t

0(∂∂xψ10 + ∂

∂zψ11

)∂∂xσ

−1t .

= −(∇×~φ1)×∇σ−1t +2

(∂∂zψ11

)∂∂zσ

−1t −

(∂∂zψ10

)∂∂xσ

−1t

0(∂∂xψ10

)∂∂xσ

−1t −

(∂∂xψ11

)∂∂zσ

−1t

(15)

Putting this all together gives the P2 equivalent SP2 equations:

σaφ0 +∇ · ~φ1 = Q, (16a)

σt~φ1+13∇φ0+

23∇φ2 = − 1

5σt∇×∇×~φ1−

15

(∇× ~φ1

)×∇σ−1

t +25

(∂∂zψ11

)∂∂zσ

−1t −

(∂∂zψ10

)∂∂xσ

−1t

0(∂∂xψ10

)∂∂xσ

−1t −

(∂∂xψ11

)∂∂zσ

−1t

,

(16b)

σtφ2 +25∇ · ~φ1 = 0. (16c)

It is entirely possible that the last term in Eq. (16b) can be simplified using some other operators, but this

simplification has to date escaped this author.

4/6


Properties of the P2 Equivalent SP2 equations

When σt is constant the ~φ1 equation becomes

σt~φ1 +13∇φ0 +

23∇φ2 = − 1

5σt∇×∇× ~φ1, (17)

which when we apply the divergence operator (∇·) becomes

∇ · ~φ1 = − 13σt∇2φ0 −

23σt∇2φ2, (18)

because∇· (∇× ~F ) = 0 for any vector field ~F . Substituting Eq. (18) into the equations for φ0 and φ2 gives

− 13σt∇2φ0 −

23σt∇2φ2 + σaφ0 = Q, (19a)

− 115σt∇2φ0 −

215σt∇2φ2 + σtφ2 = 0. (19b)

These are precisely the SP2 equations when σt is uniform.

The procedure to derive boundary and interface conditions that we are currently pursuing will take standard

P2 conditions and repeat the derivation above to get the proper conditions in terms of φ, ~φ1, and φ2.

ACKNOWLEDGMENTS

Thanks to Marvin L. Adams for many useful discussions of the SPN equations.

REFERENCES

[1] Ryan G. McClarren and C. D. Hauck. Robust and accurate filtered spherical harmonics expansions for

radiative transfer. Journal of Computational Physics, 229:5597–5614, 2010.

[2] Ryan G McClarren, James Paul Holloway, and Thomas A Brunner. On solutions to the Pn equations

for thermal radiative transfer. J. Comput. Phys., 227(5):2864–2885, Jan 2008.

[3] Cory D. Hauck and Ryan G. McClarren. Positive PN closures. SIAM Journal on Scientific Computing,

submitted for publication, 2009.

[4] M Schaefer, M Frank, and C Levermore. Diffusive corrections to PN approximations. Arxiv preprint

arXiv:0907.2099, Jan 2009.

[5] GC Pomraning. Asymptotic and variational derivations of the simplified Pn equations. Annals of

Nuclear Energy, 20(9):623–637, 1993.

[6] R. P. Rulko and Edward W. Larsen. Variational derivation and numerical analysis of P2 theory in

planar geometry. Nucl. Sci. Eng., 114:271–285, 1993.

5/6

McClarren

[7] EW Larsen, JE Morel, and JM McGhee. Asymptotic derivation of the multigroup P1 and simplified

PN equations with anisotropic scattering. Nuclear Science and Engineering, 123(3):328–342, 1996.

[8] D I Tomasevic and E W Larsen. The simplified P2 approximation. Nuclear Science and Engineering,

122(3):309–325, 1996.

[9] PS Brantley and EW Larsen. The simplified P3 approximation. Nuclear Science and Engineering,

134(1):1–21, 2000.

[10] Ryan G McClarren. Theoretical aspects of the simplified Pn equations. Transport Theory and Statis-

tical Physics.

[11] Thomas A. Brunner. Riemann Solvers for Time-Dependent Transport Based on the Maximum Entropy

and Spherical Harmonics Closures. PhD thesis, University of Michigan, 2000.

[12] Ryan G. McClarren. Spherical harmonics methods for thermal radiation transport. PhD thesis, Uni-

versity of Michigan, Ann Arbor, 2007.

6/6


Variance Reductions for Forward and Inverse Transport Problems.

Guillaume BalDepartment of Applied Physics & Applied Mathematics

Columbia University500 W. 120th St

New York, NY [email protected]

We consider an inverse transport problem with applications in atmospheric remote sensing. The forward

transport equation is solved by a Monte Carlo method with variance reduction techniques that aim to re-

direct photons toward the location of the detectors in an unbiased manner. The inverse transport problem

requires multiple forward solves. We present a path-recycling methodology to efficiently vary parameters in

the transport equation and provide sizable variance reductions in the solution of inverse transport.

Because in the inverse transport problem is severely ill-posed in the remote sensing configuration we con-

sider, it is recast in a convenient Bayesian framework. The full application of Bayesian inverse problems

requires exploration of a posterior density that typically does not possess a standard form. For that reason,

Markov Chain Monte Carlo (MCMC) methods are often used. These require many evaluations of a compu-

tationally expensive forward model to produce the equivalent of one uncorrelated sample from the posterior.

We consider applications where approximate forward models at multiple resolution-levels are available, each

with probabilistic posterior error estimates. This occurs for example when the forward model is a Monte

Carlo integral as is the case in remote sensing. We present a novel MCMC method called (MC)3 that

uses the low-resolution forward models to approximate draws from a posterior built with the high-resolution

forward model. The high-resolution models are rarely run and a significant speed up is achieved.

We apply the (MC)3 strategy and the path recycling methodology to solve the inverse transport problem as

it appears in atmospheric remote sensing. Examples of reconstructions as well as estimates of speed up will

be provided. This work is joint collaboration with Ian Langmore (Columbia University), Anthony Davis

(JPL), and Youssef Marzouk (M.I.T.). More details are available in the references [1–3].

ACKNOWLEDGMENTS

This work was supported in part by the National Science Foundation under grant DMS-0804696 and the

Department of Energy and the program NNSA-22 under Grant DE-FG52-08NA28779.

REFERENCES

[1] Bal, G., Davis A. and I. Langmore. A hybrid (monte carlo/deterministic) approach for multi-dimensional radiation transport. submitted, 2011.

Guillaume Bal

[2] I Langmore, Davis A. and Bal, G. Toward physics-based atmosphere/surface remote sensing in 3dgeometry: Proof-of-concept for an absorbing gaseous plume in a deep valley using reflected sunlight.In preparation.

[3] Bal, G., I. Langmore, and Y. Marzouk. Bayesian Inverse Problems with Monte Carlo Forward Mod-els. In preparation.

2/2


A MODIFIED TREATMENT OF SOURCES IN IMPLICIT MONTECARLO RADIATION TRANSPORT

N. A. GentileLawrence Livermore National Laboratory L-38

7000 East AvenueLivermore, CA 94551

[email protected]

Jim E. MorelTexas A&M University

Department of Nuclear EngineeringCollege Station, Texas 77843-3133

[email protected]

We describe changes to the Implicit Monte Carlo (IMC) algorithm [1] to include the effects of material

motion. These changes assume that the problem can be embedded in a global Lorentz frame. We also

assume that the material in each zone can be characterized by a single velocity. With this approximation,

we show how to make IMC Lorentz invariant, so that the material motion corrections are correct to all

orders of v/c. We develop thermal emission and face sources in moving material and discuss the coupling

of IMC to the non-relativistic hydrodynamics equations via operator splitting. We discuss the effect of this

coupling on the value of the ”Fleck factor” in IMC.

A considerable amount of effort has gone into modifying the transport equation to account for the effects of

material motion [2], [3], [4]. Efforts have been made to model the effects both exactly and approximately

to first order in vc . Here, v is the fluid velocity, and c is the speed of light. In this work, we will describe our

efforts at modifying the Implicit Monte Carlo (IMC) method [1] of radiation transport to account for

material motion.

We have decided to eschew approximations good only to first order in vc for several reasons. One is that we

would like to have a simulation code without these approximations to compare to other codes which use

these approximations. This would allow us to assess the accuracy of various vc approximations now in use.

A second is that vc approximations often make use of the derivative of the opacity with respect to

temperature. This derivative arises from doing Taylor series expansions of Doppler-shifted emission and

absorption terms. For tabular opacities, this quantity is often difficult to compute accurately for numerical

reasons. Because the Monte Carlo method can do Doppler shifts without approximation on particles in the

simulation, we can avoid the effects of these inaccuracies.

The streaming operator in the transport equation takes on its familiar, zero velocity, form in the lab frame

[2], [4]. This means that tracking lab frame particle positions in the Monte Carlo algorithm is

N. A. Gentile and Jim E. Morel

straightforward. While the streaming operator is easy to simulate in the lab frame, the source terms in the

transport equation become complicated by the need to account for material motion [2], [3], [4]. The

equation of state relates the temperature to the fluid frame energy density. The opacity is angle-independent

only in the fluid frame. The material and radiation temperatures at equilibrium are equal only in the fluid

frame. This means that the source terms in the lab frame have to be calculated from the fluid frame source

terms by a careful examination of physical invariants. A convenient list of the relationship between the lab

and fluid frame expressions for various terms in the transport equation is provided in chapter 6 of [2].

Transforming material properties into the lab frame yields the following form for the transport equation:

1

c

∂IL∂t

+ ΩL · ∇IL = −σa,LIL + σa,LB[νF , T ]

[γDL(ΩL)]3. (1)

Here, a subscript L indicates a lab frame quantity. or example, σa,L is the angle-dependent Lab frame

absorption opacity.

In a typical radiation hydrodynamics code, this equation is coupled with the non-relativistic hydrodynamics

conservation laws [2]∂ρ

∂t= −∇ · ρ~v , (2)

∂ρ~v

∂t= −∇P −∇ · (ρ~v~v) +

∆ ~Mr

∆t, (3)

and∂ρv2 + ρε

∂t= −∇ · (1

2ρv2~v)−∇ · (ρε~v)−∇ · (P~v) +

∆Er

∆t. (4)

Here ρ is the mass density, P is the matter pressure, ∆ ~Mr is the net momentum transferred from the

radiation to the matter during the time step, and ∆Er is the net energy transferred. This net energy includes

contributions from the radiation to both the internal energy and the kinetic energy of the matter.

When v in Eqs.(2)-(4) is large, the effects of material motion in the transport equation are significant. A

radiation hydrodynamics code must simulate material motion corrections to transport accurately. The Mach

45 radiating shock semi-analytic test problem [5] is an example of a calculation that stresses the material

motion correction in a radiation hydrodynamics code.

In Fig. 1, we see the results of an IMC simulation of the Mach 45 radiating shock test problem with no

radiation pressure comparted to the semi-analytic solution. In this problem, the radiation pressure in the hot

material is approximatey twice the mateiral pressure. Ignoring the radiation pressure causes the simulaton

to diverge from the correct answer. This simulation shows that including the effects of radiation correctly is

necessary to get an accurate answer.

In Fig. 2, we see the results of a simulation in which Eqs.(1)-(4) are solved. The effects of material motion

on the radiation field are simulated by using the modified thermal source term in Eq.(1), while the effects of

2/4

Material motion corrections for IMC

Figure 1. Matter and radiation temperature, and density and velocity, for Mach 45 radiating shocksimulated without radiation presure.

Figure 2. Matter and radiation temperature, and density and velocity, for Mach 45 radiating shocksimulated with material motion effects.

radiation pressure on the material are simulated by calculating the radiation momentum and energy

deposition and including those as sources, as in Eqs.(2)-(4). In this simulation, the values obtained for the

3/4

N. A. Gentile and Jim E. Morel

matter and radiation temperature, and the matter density and velocity, are more accurate.

ACKNOWLEDGMENTS

The work of the first author performed under the auspices of the U.S. Department of Energy by LawrenceLivermore National Laboratory under Contract DE-AC52-07NA27344.

REFERENCES

[1] J. A. Fleck, Jr., and J. D. Cummings, “An Implicit Monte Carlo Scheme for Calculating Time andFrequency Dependent Nonlinear Radiation Transport,” J. Comput. Phys., 8, pp. 313-342 (1971).

[2] G. C. Pomraning. The Equations of Radiation Hydrodynamics. Pergamon, New York U.S.A. (1973).

[3] D. Mihalas and B. Weible-Mihalas. Foundations of Radiation Hydrodynamics. Dover Publications,New York U.S.A. (1984).

[4] J. Castor. Radiation Hydrodynamics. Cambridge University Press, New York U.S.A. (2004).

[5] R. B. Lowrie and J. D. Edwards, “Radiative Shock Solutions with Nonequilibrium RadiationDiffusion,” Shock Waves, 18, pp. 129-143 (2007).

4/4


NECESSARY AND SUFFICIENT CONDITIONS FOR AN IMPLICITMONTE CARLO DISCRETE MAXIMUM PRINCIPLE

Allan B. WollaberComputational Physics and Methods (CCS-2)

Los Alamos National LaboratoryPO Box 1663

Los Alamos, NM [email protected]

The Implicit Monte Carlo (IMC) equations have long been known to be susceptible to overheating when

the time step ∆t is too large. In 1987, Larsen and Mercier developed a maximum principle that provided a

maximum ∆t such that the IMC temperature solution does not non-physically exceed the boundary

condition temperatures [1]. However, the limit was found to be overly conservative in practice. More

recently, Wollaber and Larsen developed an approximate discrete maximum principle that allowed for the

spatial grid effect of the IMC equations, which much more closely predicted the maximum ∆t for which

the IMC solutions would provide solutions that do not violate the maximum principle [2, 3]. In this paper,

we complement the recent work by providing rigorous necessary and sufficient conditions on the IMC

temperature solutions such that they do not violate the maximum principle. These necessary and sufficient

conditions are derived by constructing “special” IMC-like equations that adjust the effective scattering term

(without adjusting the corresponding term in the material energy equation) to achieve the desired goal.

We begin from the Fleck and Cummings Implicit Monte Carlo (IMC) equations for a 1-D, nonlinear,

frequency-dependent problem given by:

1

c

∂I

∂t+ µ

∂I

∂x+ σ0I =

1

2σ0(1− f)

∫ ∞0

∫ 1

−1I dµ dν + σ0f2πB0 , (1a)

cv∆t

(T1 − T0) =

∫∆t

∫ ∞0

∫ 1

−1σ0f(I − 2πB0)dµ dν dt , (1b)

I(x, µ, ν, 0) = 2πB0 , 0 ≤ x ≤ X, −1 ≤ µ ≤ 1 , 0 < ν , (1c)

I(0, µ, ν, t) = 2πBu , 0 < t , 0 < µ ≤ 1 , 0 < ν , (1d)

I(X,µ, ν, t) = 2πB0 , 0 < t , −1 ≤ µ < 0 , 0 < ν , (1e)

where I = I(x, µ, ν, t) is the specific intensity, B = B(T, ν) is the Planck function, T = T (x, t) is the

material temperature, 0 ≤ f ≤ 1 is the Fleck factor, and c is the speed of light. For now, we assume that the

specific heat cv = constant and the opacity σ0 = σ(T0, ν) for simplicity. Essentially, the above problem

corresponds to a cold slab that is suddenly exposed to incident Planckian radiation at temperature Tu on the

left side (a Marshak-like wave).

Allan B. Wollaber

A NECESSARY CONDITION

We first look for a lower bound on the intensity solutions given by Eqs. (1). The lower bound will

correspond to the minimum possible energy that may be deposited in the first cell during the first time step,

and will therefore become a necessary condition for the maximum principle of the original problem to be

met. Essentially, we formulate a “false” IMC equation in which the effective scattering term is removed:

1

c

∂I

∂t+ µ

∂I

∂x+ σ0I = σ0f2πB0 , (2a)

cv∆t

(T1 − T0

)=

∫∆t

∫ ∞0

∫ 1


I(x, µ, ν, 0) = 2πB0 , 0 ≤ x ≤ X, −1 ≤ µ ≤ 1 , 0 < ν , (2c)

I(0, µ, ν, t) = 2πBu , 0 < t , 0 < µ ≤ 1 , 0 < ν , (2d)

I(X,µ, ν, t) = 2πB0 , 0 < t , −1 ≤ µ < 0 , 0 < ν . (2e)

Next, we define ψ = I − I and subtract Eqs. (2) from Eqs. (1) to find:

1

c

∂ψ

∂t+ µ

∂ψ

∂x+ σ0ψ =

1

2σ0(1− f)

∫ 1

−1I dµ , (3a)

cv∆t

(T1 − T1

)=

∫∆t

∫ ∞0

∫ 1

−1σ0f ψ dµ dν dt , (3b)

ψ(x, µ, ν, 0) = 0 , 0 ≤ x ≤ X, −1 ≤ µ ≤ 1 , 0 < ν , (3c)

ψ(0, µ, ν, t) = 0 , 0 < t , 0 < µ ≤ 1 , 0 < ν , (3d)

ψ(X,µ, ν, t) = 0 , 0 < t , −1 ≤ µ < 0 , 0 < ν . (3e)

Because Eq. (1a) is a linear transport equation with a positive source, I is non-negative. Therefore, the

source term in Eq. (3a) is non-negative. Because the source term, initial condition, and boundary conditions

in Eqs. (3) are non-negative, ψ must also be non-negative. Therefore, Eq. (3b) implies that:

T1 > T1 , (4)

since

T1 = T1 +∆t

cv

∫∆t

∫ ∞0

∫ 1

−1σ0f ψ dµ dν dt .

We therefore conclude that an exact maximum principle generated from Eqs. (2) is a necessary condition

for T1 < Tu (i.e., if the modified temperature is greater than the boundary temperature (T1 > Tu), then the

true IMC temperature T1 > Tu). We note that as f → 1, for small time steps or opacities, this condition

becomes sharp, since Eqs. (2) limit to the original IMC equations [Eqs. (1)] and the inequality in Eq. (4)

becomes an equality. As f → 0 (for large ∆t and/or opacities) the condition gets weaker.

To make this practical, the next step would be to exactly solve the purely absorptive problem in Eq. (2a)

and use Eq. (2b) to obtain T1. This is tedious, but straightforward.

2/4

Necessary and Sufficient Conditions for an IMC Discrete Maximum Principle

A SUFFICIENT CONDITION

Previously, sufficient conditions have been presented that did not depend on the spatial grid [1]. In this

section, we derive a new sufficient condition that includes the grid effect. To do this, we look for an upper

bound on the intensity solution of Eq. (1a). Note that if we choose the maximal solution I = 2πBu (without

any grid dependence), then we should reproduce the earlier results due to Larsen and Mercier. Instead we

consider the following, modified IMC equations that contain a purely (effective) scattering term:

1

c

∂I

∂t+ µ

∂I

∂x+ σ0I =

1

2

∫ ∞0

∫ 1

−1σ0Idµ dν + 2πσ0fB0 , (5a)

cv∆t

(T1 − T0

)=

∫∆t

∫ ∞0

∫ 1


I(x, µ, ν, 0) = 2πB0 , 0 ≤ x ≤ X, −1 ≤ µ ≤ 1 , (5c)

I(0, µ, ν, t) = 2πBu , 0 < t , 0 < µ ≤ 1 , (5d)

I(X,µ, ν, t) = 2πB0 , 0 < t , −1 ≤ µ < 0 . (5e)

Next, we define Ψ = I − I and subtract Eqs. (1) from Eqs. (5) to find:

1

c

∂Ψ

∂t+ µ

∂Ψ

∂x+ σ0Ψ =

1

2

∫ ∞0

∫ 1

−1σ0Ψdµ dν +

1

2f

∫ ∞0

∫ 1

−1σ0Idµ dν, (6a)

cv∆t

(T1 − T1

)=

∫∆t

∫ ∞0

∫ 1

−1σ0f Ψ dµ dν dt , (6b)

Ψ(x, µ, ν, 0) = 0 , 0 ≤ x ≤ X, −1 ≤ µ ≤ 1 , 0 < ν , (6c)

Ψ(0, µ, ν, t) = 0 , 0 < t , 0 < µ ≤ 1 , 0 < ν , (6d)

Ψ(X,µ, ν, t) = 0 , 0 < t , −1 ≤ µ < 0 , 0 < ν . (6e)

As before, the source on the right side of Eq. (6a) is positive, the initial and boundary conditions for Eq. (6)

are non-negative, so Ψ must also be non-negative. From Eq. (6b) we may therefore conclude that:

T1 > T1 , (7)

since

T1 = T1 +∆t

cv

∫∆t

∫ ∞0

∫ 1

−1σ0f Ψ dµ dν, dt .

We therefore conclude that an exact maximum principle generated from Eqs. (5) is a sufficient condition for

T1 < Tu. Note that as ∆t →∞, then f → 0, so in this limit the sufficient condition becomes sharp (in the

right side of Eq. (7), both f and Ψ go to zero, making the inequality an equality). To make this practical,

the next step would be to solve the purely scattering problem in Eq. (5a) to obtain I and then use it to find

T from Eq. (5b). This can likely be achieved via 1-D, time-dependent Caseology [4], although other

3/4

Allan B. Wollaber

approximations may be necessary. We note that 1-D, time-dependent, monoenergetic problems and 1-D,

steady-state, multigroup problems have readily available solutions [4–6].

Altogether, the solutions of the necessary and sufficient conditions place lower and upper bounds on the

temperature update T1 < T1 < T1 that are grounded in transport theory and that do not rely on the

approximations made in previous work [2, 3]. The solution of the purely scattering problem may also

provide insight into a more accurate solution of the general problem with arbitrary effective scattering.

Depending upon the level of computational effort needed to generate these solutions and the sharpness of

these inequalities, these conditions may be useful in computing practical limits on the time step size to

prevent material overheating in IMC solutions.

ACKNOWLEDGMENTS

This research was performed under the auspices of Los Alamos National Security, LLC, for the National

Nuclear Security Administration of the U.S. Department of Energy under contract DE-AC52-06NA25396.

REFERENCES

[1] E. W. Larsen and B. Mercier, “Analysis of a Monte Carlo method for nonlinear radiative transfer,”

Journal of Computational Physics, vol. 71, no. 1, pp. 50–64, 1987.

[2] A. B. Wollaber and E. W. Larsen, “Towards a discrete maximum principle for the Implicit Monte Carlo

equations,” in Proceedings of NECDC 2010, no. LA-UR-10-05710, Los Alamos National Laboratory,

October 18-22 2010.

[3] A. B. Wollaber, E. W. Larsen, and J. D. Densmore, “Towards a frequency-dependent discrete

maximum principle for the Implicit Monte Carlo equations,” in Proc. ANS Topical Meeting,

International Topical Meeting on Mathematics and Computation, (Rio De Janeiro, Brazil), American

Nuclear Society, May 8–12 2011.

[4] K. M. Case and P. F. Zweifel, Linear Transport Theory. Addison-Wesley, Reading, MA, 1967.

[5] N. J. McCormick and I. Kuscer, “Singular eigenfunction expansions in neutron transport theory,”

Advances in Nuclear Science and Technology, vol. 7, pp. 181–282, 1973.

[6] B. Ganapol, “Multigroup Caseology in 1D via the Fourier transform,” Progress in Nuclear Energy,

vol. 50, no. 8, pp. 886 – 907, 2008.

4/4


A MODIFIED TREATMENT OF SOURCES IN IMPLICIT MONTECARLO RADIATION TRANSPORT

N. A. GentileLawrence Livermore National Laboratory L-38

7000 East AvenueLivermore, CA 94551

[email protected]

Travis J. TrahanDepartment of Nuclear Engineering & Radiological Sciences

University of Michigan1928 Cooley Building

2355 Bonisteel BoulevardAnn Arbor, Michigan USA 48109-2104

[email protected]

We describe a modification of the treatment of photon sources in the IMC algorithm [1]. We describe this

modified algorithm in the context of thermal emission in an infinite medium test problem at equilibrium

and show that it completely eliminates statistical noise.

Let us examine the case of an infinite medium problem with gray, constant opacities with matter and

radiation at equilibrium at temperature T, simulated with the IMC method. The census photons

representing the initial radiation all initially have time t = 0 and have a total energy aT 4. During a time

step of size ∆t, the energy of each census photon will decrease by a factor exp[−σc∆t]. The total energy

in census photons at the end of the time step will therefore be

Ec(t = ∆t) = aT 4 exp[−σc∆t]. (1)

To simulate thermal emission, we will make Ns thermal source photons, each with a different initial time

ti,p in [0,∆t]. We will assume all Ns photons have the same initial energy acσPT 40 V∆t/Ns. Since ti,p is

different for each thermal source photon, they will all reach time ∆t with different energies

Ep(t = ∆t) = Ep(t = 0) exp[−σc(∆t− ti,p)]. The sum of these energies will be

Et ≡Ns∑p=1

Ep(t = ∆t) = aT 40 V cσ∆t

1

Ns

Ns∑p=1

exp[−σc(∆t− ti,p)] . (2)

Since

limNs→∞

1

Ns

Ns∑p=1

exp[−σc(∆t− ti,p)] =1

∆t

∫ ∆t

0exp[−σc(∆t− τ)]dτ =

1− exp[−cσ∆t]

cσ∆t, (3)

the sum in Eq.(2) is a Monte Carlo estimate for an integral over all possible thermal emission times. Using

Eq.(2) and Eq.(3), we find that, in the limit of a large Ns, the radiation energy due to thermally emitted

N. A. Gentile and Travis J. Trahan

photons at t = ∆t will be

Et(t = ∆t) = aT 40 V (1− exp[−cσ∆t]), (4)

so Ec +Et = aT 40 V , which is the value necessary to maintain thermal equilibrium. The matter energy will

also be the same as the initial value, by energy conservation. With a finite number of photons, we will not

maintain thermal equilibrium exactly, because the sum in Eq.(2) will only approximate the integral, with an

error that is proportional to N− 1

2s [2].

This is illustrated in Fig. 1. This plot shows an IMC simulation using one zone, a cube with unit length in

each direction. All faces have reflecting boundaries, making it effectively an infinite medium problem. The

material and radiation temperatures were initialized to 1. The material has a heat capacity cv = 1.0, and an

absorption opacity σ = 10. The simulation used 100 photons per time step, and units were chosen so that

a = c = 1. The simulation used ∆t = 0.001 from t = 0 to t = 1, ∆t = 0.01 from t = 1 to t = 2, and

∆t = 0.1 for t > 2.

0.99

0.995

1

1.005

1.01

0 0.5 1 1.5 2 2.5 3

T

t

Tm and Tr in infinite medium problem

TmTr

Figure 1. Matter and radiation temperature for an infinite medium test problem simulated with IMCwith three different values for the time step ∆t.

In the IMC algorithm, we regard physical quantities like opacity and heat capacity as constant throughout

the time step. This means that the distance to scatter, the amount of absorption on a given photon path, etc.,

are independent of when they occur during the time step. We can therefore calculate the contribution that is

made for every emission time in [tn, tn+1] for each path taken by a source photon. We do this be replacing

2/4

Modified IMC sources

the absorbed and census energy for the photon on each path with the values, averaged over all possible

emission times, for a photon emitted with the same energy.

As in IMC, we sample a position, direction, frequency, and energy Ep for each source photon, and the

energy will decrease exponentially: Ep(s) = Ep(s0)exp[−σ(s− s0)]. However, we do not sample an

emission time. Instead of the source photon having a time, it will have a distance traveled, sp, with an

initial value of 0. The source photon will travel a total distance c∆t in the time step, unless it exits the

problem through a boundary. Physical source photons born after tn+1 − s/c will reach census before they

travel a distance s. Therefore, the radiation energy represented by the modified IMC source photon after it

has moved a distance s is Ep(s) times the fraction of physical source photons that have not reached census

after traveleing s:

Er(s) = Ep(s)∆t− s/c

∆t. (5)

The energy of the source photon will be either absorbed, leave the problem through a boundary, or reach

census as the photon reaches sp = c∆t.

First, we will calculate the amount of energy that reaches census on a given path from s0 to s1, averaged

over the possible emission times τ . This is

Ec(s1) =1

∆t

∫ tn+1

tndτ

∫ s1

s0ds Ep(s0) exp[−σ(s−s0)]δ[tn+1−(τ+s/c)] =

Ep(s0)

cσ∆t1− exp[−σ(s1 − s0)] ,

(6)

where the δ function enforces the fact that a particle emitted at τ reaches census after traveling a distance s

satisfying τ + s/c = tn+1.

Next, we will calculate the amount of energy that is absorbed on the path from s0 to s1. This could be done

by evaluating an integral over τ similar to Eq.(6), but it can be done with less effort by conservation of

energy. The radiation energy at s1 is related to the photon energy at s0 by

Ea(s0, s1) + Ec(s0, s1) + Er(s1) = Er(s0) , (7)

with Er(s) in given by Eq.(5). This yields

Ea(s0, s1) = Ep(s0)∆t− s0/c

∆t

1− exp[−σ(s1 − s0)]

∆t− s1/c

∆t− s0/c

− Ec(s0, s1). (8)

In the modified IMC algorithm, we use Eq.(6) to calculate a contribution to census, and Eq.(8) to calculate

a contribution to absorption, on each path for each source photon. Census photons are treated the same as

in standard IMC.

The modified IMC results for the test problem described earlier are show in Fig. 2. Tm = Tr = 1.0 holds

for all times, and there is no statistical noise in the modified IMC simulation for any value of ∆t. This

happens because the source photons in the modified IMC algorithm each contribute exactly the amount of

3/4

N. A. Gentile and Travis J. Trahan

energy calculated in Eq.(2) to census. In effect, the integrand in Eq.(3) is evaluated exactly, not

approximately as a sum over a finite number of emission times. The value of Tr at the end of the time step

is 1.0 to roundoff. By conservation of energy, Tm = 1.0 to roundoff at the end of the time step also, and so

these values are maintained in subsequent time steps of the calculation.

0.9999

0.99992

0.99994

0.99996

0.99998

1

1.00002

1.00004

1.00006

1.00008

1.0001

0 0.5 1 1.5 2 2.5 3

TmTr

Figure 2. Matter and radiation temperature for infinite medium test problem using modified IMCwith three different time steps

ACKNOWLEDGMENTS

The work of the first author performed under the auspices of the U.S. Department of Energy by LawrenceLivermore National Laboratory under Contract DE-AC52-07NA27344. The work of the second authorperformed with the support of the DOE Computational Science Graduate Fellowship, grant numberDE-FG02-97ER25308.

REFERENCES

[1] J. A. Fleck, Jr., and J. D. Cummings, “An Implicit Monte Carlo Scheme for Calculating Time andFrequency Dependent Nonlinear Radiation Transport,” J. Comput. Phys., 8, pp. 313-342 (1971).

[2] M. A. Kalos and P. A. Whitlock, Monte Carlo Methods, Second Edition, Wiley-VCH, WeinheimGermany. (2008).

4/4


A COARSE GRAINED PARTICLE TRANSPORT SOLVER DESIGNEDSPECIFICALLY FOR GRAPHICS PROCESSING UNITS

Francois A. van HeerdenRadiation and Reactor Theory

South African Nuclear Energy CorporationPO Box 582, Pretoria 0001, South Africa

[email protected]

Due to their accuracy, geometric flexibility and scalability, Monte Carlo simulations are ideally suited tosolving the Boltzmann transport equation in large heterogeneous systems. However, their slowconvergence requires a large number of histories, and the simulation of large problems is typically onlypractical, in terms of computational time, on expensive multi-node computing clusters. The use ofGraphics Processing Units (GPU’s) for general purpose computing has rapidly increased during the pastdecade. These powerful co-processors allow high performance, massively parallel computing on a singledesktop computer. However, they remain fairly specialized, and mapping any solution to this architecture isnot always a trivial task. This paper introduces a novel coarse grained Monte Carlo transport solver thatretains a lot of the attractive features of traditional Monte Carlo simulations, but introduces a computationalmodel that efficiently utilizes the power of modern, many-core, streaming processors.

In the coarse grained particle model, the spatial dependence of the particle flux is approximated usingcompactly supported kernels Kh,

φ(x) ≈

p

ωpKh(x− xp), (1)

where ωp denotes the particle weight, and h is a measure of the particle size (support of the kernel). Thekernel satisfies Kh(x− xp) → δ(x− xp) as h → 0, so that the point particle kernel is recovered in thelimit. This is similar in spirit to other mesh-free particle methods [1], and Smoothed ParticleHydrodynamics [2] . Since the source (or initial condition) can be approximated deterministically,

Q(x) ≈

p

qpKh(x− xp), (2)

in such a way that the entire volumetric source is covered, space is never sampled, which reduces theoverall variance in the problem. This does however introduce an approximation error, which depends onthe particle size h. Energy and angular dependence is still approximated using point kernels, and, as intraditional Monte Carlo schemes, are sampled during the simulation.

Modern GPU’s are capable of running thousands of concurrent threads, and excel at tasks exhibiting a highdegree of data parallelism. Although a Monte Carlo simulation is embarrassingly parallel, it does not mapwell to this type of architecture, which groups processing threads into a large number of SIMD (SingleInstruction Multiple Data) like instruction units. Indeed, the difficulties in mapping Monte Carlo

Francois A. van Heerden

simulation to vector architectures are well documented [3–6]. The problem arises due to a lack ofcoherence between particle histories: Two particles starting at the same position with the same velocity canquickly diverge in their execution path, undergoing different interactions and accessing different data. Thisstalls the entire SIMD unit, and yields only moderate speedups. The coarse particle model avoids thisproblem by using and entire SIMD unit to simulate a particle. More specifically, the spatial dependence ofthe particle kernel is further divided into a number of sub-nodes,

ωpKh(x− xp) =

ν

ωνpKh(x− xνp), (3)

and each processing thread in a vector unit is assigned a node ν. Since the particle is transported as onecoherent unit, there is no significant branching between nodes, and the full power of the streamingprocessor can be utilized. This leads to large speedups (100x - 1000x) as compared with serial (pointkernel) Monte Carlo codes, and improves significantly on the performance gains obtained in [6].

The simulation method was implemented using CUDA [7] in a code called CUCGP (CUda Course GrainedParticles). The code was tested on a number of fixed source transport benchmarks [8, 9], as well as theC5G7 MOX criticality benchmark [10]. Results typically exhibit the same degree of accuracy expectedfrom a Monte Carlo simulation, but with significantly reduced calculational time, even outperforming mostof its deterministic counterparts. Table I compares CUCGP, in terms of accuracy and performance, to thedeterministic codes that took part in the C5G7 benchmark. Results were taken from [11].

keff Runtime

MCNP (benchmark) 1.183810 DaysCUCGP 1.183730 ≈ 15 min

CRONOS2-SN 1.177230 HoursTORT-GRS 1.180450 Days

THREEDANT 1.183919 HoursDeCART 1.183860 Hours

CRX 1.185360 Not GivenMCCG3D 1.183450 Not GivenUNKGRO 1.181040 Days

VARIANT-SE 1.178243 DaysPARTISN 1.183620 MinutesATTILA 1.183480 Days

TORT-ORNL 1.182340 Not Given

Table I. C5G7 3-D benchmark results

REFERENCES

[1] S. Li and W.K. Liu. Meshfree Particle Methods. Springer, 2004.

[2] J. J. Monaghan. Smoothed particle hydrodynamics. Annual review of astronomy and astrophysics,30(1):543–574, 1992.

2/3

A Coarse Grained Particle Transport Solver Designed Specifically for GPU’s

[3] F.B. Brown and W.R. Martin. Monte carlo methods for radiation transport analysis on vectorcomputers. Progress in Nuclear Energy, 14(3):269–299, 1984.

[4] A. Badal and A. Badano. Monte carlo simulation of x-ray imaging using a graphics processing unit.In Nuclear Science Symposium Conference Record (NSS/MIC), 2009 IEEE, pages 4081–4084. IEEE,2009.

[5] W.C.Y. Lo, T.D. Han, J. Rose, and L. Lilge. GPU-accelerated Monte Carlo simulation forphotodynamic therapy treatment planning. In Proc. of SPIE-OSA Biomedical Optics, 2009.

[6] A. G. Nelson and K. N. Ivanov. Monte carlo methods for neutron transport on graphics processingunits using CUDA. In PHYSOR 2010 – Advances in Reactor Physics to Power the NuclearRenaissance, Pittsburgh, Pennsylvania, USA, May 9-14 2010.

[7] NVIDIA. CUDA programming guide. NVIDIA Corp., Santa Clara, CA, 2007.

[8] Y. Nagaya K. Kobayashi, N. Sugimura. 3-D radiation transport benchmarks for simple geometrieswith void region. In Mathematics and Computation, Reactor Physics and Environmental Analysis inNuclear Applications, September 27-30 1999.

[9] Y.Y. Azmy. Benchmarking the accuracy of solution of 3-dimensional transport codes and methodsover a range in parameter space. Technical Report NEA/NSC/DOC (2007)1/REV1, 2007.

[10] E. E. Lewis, M. A. Smith, N. Tsoulfanidis, G. Palmiotti, T. A. Taiwo, and R. N. Blomquist.Benchmark specification for deterministic 2D/3D MOX fuel assembly transport calculations withoutspatial homogenisation (C5G7 MOX). Technical Report NEA/NSC/DOC(2001)4, NEA, March 2001.

[11] E. E. Lewis, M. A. Smith, N. Tsoulfanidis, G. Palmiotti, T. A. Taiwo, and R. N. Blomquist.Benchmark on deterministic transport calculations without spatial homogenisation: MOX fuelassembly 3-D extension case. Technical Report NEA/NSC/DOC(2005)16, NEA, 2005.

3/3


Wednesday, September 14, 2011

Kinetics I

8:25 am In Memory of Carlo Cercignani - Giovanni Frosali

8:35 am Kinetic Modeling of Nitrate Removal Using a Nitrate Selective Resin: The Role of Mass Transfer - N.

Talebbeydokhti, A.A. Hekmatzedeh, A. Karimi-Jashani, K. Hadad

9:00 am Neutron Inverse Kinetics via Gaussian Processes - P. Picca, R. Furfaro

9:25 am An Accurate Solution to the Master/Moments Equations for the Kinetics of Breakable Filament Self-

Assembly - B.D. Ganapol

9:50 am Application of the Hybrid Transport/Point Kinetics Method to Time-Dependent Source-Driven Prob-

lems - P. Picca, R. Furfaro, B.D. Ganapol

Quantum Transport I

10:45 am Quantum Corrections on the Radiative Transfer Equation - J. Rosato

11:10 am Atom-Atom Relaxation with Quantum Differential Elastic Scattering Cross Sections; Distribution

Function Relaxation and the Kullback-Leibler Entropy - R. Sospedra-Alfonso, B.D. Shizgal

11:35 am Shape Relaxation in Electron Atom Relaxation; The Kullback-Leibler Relative Entropy and Relax-

ation Times - R. Sospedra-Alfonso, B.D. Shizgal

12:00 pm Diffusive Limits for a Quantum Transport Model with Weak and Strong Fields - L. Barletti, G. Frosali



KINETIC MODELING OF NITRATE REMOVAL USING A

NITRATE SELECTIVE RESIN: The ROLE OF MASS TRANSFER

Nasser Talebbeydokhti

Department of Civil and Environmental Engineering

Center for Environmental Research and Sustainable Development

Shiraz University

P.o. Box 7134851156, Shiraz, Iran

[email protected]

Ali A. Hekmatzadeh1, A. Karimi-Jashani

2, and Kamal Hadad

3

1 Department of Civil and Environmental Engineering,

Islamic Azad University, Branch of Eghlid 2

Department of Civil and Environmental Engineering, Shiraz University, Shiraz Iran.

3 School of Mechanical Engineering, Shiraz University, Shiraz Iran.

Shiraz University

[email protected] [email protected]

[email protected]

Nitrate is a common pollutant of groundwater in many regions around the world. Ion

exchange is one of the best feasible methods for nitrate removal from water suppliers due

to its high efficiency, simple operation, and relatively low cost. Determining the kinetic

mechanism of nitrate adsorption to this resin can be used to design, optimize, and predict

the nitrate breakthrough curve of fixed bed columns in a real water treatment processes.

Although considerable researches have been performed on nitrate adsorption to several

ion exchange resins, little attention has been given to the rate of nitrate adsorption and

determining which kinetic mechanism is the rate controlling. In this work, the overall rate

of nitrate adsorption from aqueous solutions using a nitrate selective ion exchange resin

was studied. Several batch kinetic tests were arranged including different initial nitrate

concentration and different adsorbent dosage. A full rate kinetic model including external

mass transfer, particle pore diffusion, and particle surface diffusion (Equation 1 [1]) were

developed and solved numerically using the Crank-Nicholson scheme in MATLAB

environment. An optimization technique was employed to optimize the model

parameters.

[

]

[

] (1)

In order to solve Equation 1, one more relationship is needed as it contains two unknown

variables: C and q. Therefore, by assuming local equilibrium between liquid phase and

solid phase in the pore space, equilibrium isotherm was used to explain the relationship

between C and q. several researchers used adsorption isotherms such as Langmuir model

to describe the equilibrium data whereas these isotherms are related to surface adsorption





2

[3-4]. Here, the real equilibrium isotherm for ion exchange process called Mass action

isotherm was used in the kinetic modeling [5].

In addition, the contribution of particle pore diffusion and particle surface diffusion were

considered in the rate modeling equation separately to obtain their importance in

modeling. The estimated model parameters are nearly the same for all batch kinetic tests.

The experimental kinetic data were well described by all theoretical models. However,

different values of pore diffusion and surface diffusion were obtained for each theoretical

model. Consequently, it was concluded that a full kinetic model containing all processes

of mass transfer is the best way to model the decay profile of nitrate adsorption with

respect to time. As an example, Fig. 1 shows the experimental and theoretical decay time

profiles of nitrate adsorption onto resin particles for three sets of kinetic tests. In this

figure, theoretical profile was obtained according to external mass transfer and pore

diffusion. The initial nitrate concentration, the resin mass, and the solution volume are

given in the figure. Finally, the results indicated that the kinetic process is initially

controlled by external mass transport and then by intraparticle diffusion. Moreover,

particle pore diffusion was more dominant than particle surface diffusion.

Fig. 1: experimental and theoretical decay time profiles of nitrate adsorption onto resin

particles for three sets of kinetic experiment

[1] K. Miyabe, G. Guiochon, " Kinetic study of the mass transfer of bovine serum albumin in anion-exchange chromatography'" Journal of Chromatography A, 866, pp. 147–171 (2000).

[2] Metcalf and eddy, water treatment: principle and design, John Wily & Son, Inc.,

2005

time (m)

C/C

0

0 40 80 120

0

0.2

0.4

0.6

0.8

1

C= 60 mg/l - M=0.15 g - V=0.2 l

C= 120 mg/l - M=0.05 g - V=0.2 l

C= 120 mg/l - M=0.50 g - V=0.2 l



3

[3] B. Alyüz, Sevil Veli, " Kinetics and equilibrium studies for the removal of nickel and zinc from aqueous solutions by ion exchange resins," Journal of Hazardous Materials, 167, pp. 482–488 (2009).

[4] V.M.T.M. Silva, A.E. Rodrigues, " Kinetic studies in a batch reactor using ion exchange resin catalysts for oxygenates production: Role of mass transfer mechanisms," Chemical Engineering Science, 61, pp. 316 – 331 (2006).

[5] N. Z. Misak, "Some aspects of the application of adsorption isotherms to ion exchange reactions," Reactive & Functional Polymers, 43, pp. 153–164 (2000).


NEUTRON INVERSE KINETICS VIA GAUSSIAN PROCESSES

Paolo Picca Dipartimento di Energetica, Politecnico di Torino

Corso Duca degli Abruzzi, 24 - 10029 Torino, Italy [email protected]

Roberto Furfaro,

Department of Systems and Industrial Engineering University of Arizona, Tucson AZ 85721

[email protected]

The problem of determining the neutron reactivity through the interpretation of experiments represents nowadays a major challenge for the development of Accelerator –Driven-Systems (ADSs). Over the past fifty years, several techniques have been developed to infer the subcriticality from measures in source pulse transients (e.g. [1]-[3])., Under some rather restrictive assumptions, such as point kinetic behavior of the reactor power, the inverse problem can be solved analytically. However, most of the current ADS designs are based on loosely coupled cores [4]-[5]. For these systems, the point approximation of reactor dynamics is rather inappropriate because spectral and spatial transients become rather significant [6]. Hence, to invert the kinetic equations in cases where spatial/spectral effects are important, special techniques have been subsequently proposed. One of such options is to maintain the use of the point kinetics and introduce ad hoc correction factors to capture the spatial and energy aspect transient phenomena [7]. Another possibility is to use more advanced kinetic models (e.g. multipoint kinetics) and to perform the inversion by means of artificial neural networks [8]. In this paper, we propose a novel inverse kinetics technique based on Gaussian Processes (GP). Over the past few years, GP have become attractive supervised machine learning techniques for both regression and classification problems [9]. GP have a strong theoretical foundation (rooted in the Bayesian inference framework) and have been considered as a replacement of neural networks. Here, we start by study a simple case, i.e. the determination of the subcriticality level through the interpretation of the thermal

energy released in a pulsed transient, namely ∫==T

dttPTEx0

')'();0( . A GP is used to

approximate the function )(xf=ρ . Under a GP model for the function )(⋅f , the joint

distribution of )(),...,( 1 kxfxf is multivariate Gaussian for any finite set of input points,

i.e. kxx ,...,1 . The properties of the GP are specified through the mean function, i.e.

[ ])(xfΕ=µ , and the covariance function [ ])'(),( xfxfCov . Using an appropriate training set, a GP model can learn the underlying inverse relationship between energy produced in a pulse transient and the reactivity. The training set is a collection of pairs of input and output, i.e. ix,ρ for Ii ,...,1= . The energy released in a pulse transient


2

starting from zero power is computed using a point kinetic model (without delayed emission), assuming that the neutron mean lifetime of the system is known (here 10-4s). As it is difficult to guess a priori a behavior of the reactivity as a function of the energy, a non-informative constant mean function is chosen. The squared exponential covariance function is considered, i.e.

[ ] [ ]232

2

2

21 exp);(),( γδ

γγ pq

qpqp

xxxfxfCov +

−

−=θ (1)

where the hyperparameter is defined as ),,( 321 γγγ=θ . The optimal set of the GP

hyperparameters can be computed using the training set, maximizing the log marginal likelihood [9].

Figure 1. Behavior of the multiplication constant as a function of the energy released

in a pulsed transient.

The GP is first trained on 50 pairs of reactivity/released energies (5% noise on GP input

data). Figure 1 reports the [ ] 11 −−= ρeffk as a function of the energy released in the

transient. In Fig. 1, the area around the crosses (training points) includes the 95% confidence interval. A cross-validation of the GP is carried out on values of );0( TE outside the training set. Table I shows a comparison between the values of these multiplication constants and the GP estimations. The agreement between the target and the GP values is rather interesting, considering the limited number of training cases and the significant superposed noise. The advantage of GP compared with neural network is


3

that information about the variance of the estimation is naturally made available through the Bayesian formalism.

Table I. Cross-validation of the GP results on 10 test cases out of the training set.

keff GP keff

estimate variance

0.80000 0.79137 4.634E-04

0.82111 0.81809 3.655E-04

0.84222 0.84387 3.575E-04

0.86333 0.86802 3.598E-04

0.88444 0.89008 3.582E-04

0.90556 0.91014 3.608E-04

0.92667 0.92901 3.654E-04

0.94778 0.94804 3.667E-04

0.96889 0.96787 3.752E-04

0.99000 0.98564 4.117E-04

REFERENCES

[1] Sjöstrand, N. G., Measurements on a subcritical reactor with a pulsed neutron source, Proceedings of the United Nations International Conference on the Peaceful Uses of Atomic Energy, Geneva, 1956.

[2] Gozani, T., A modified procedure for the evaluation of pulsed source experiments in subcritical reactors. Nukleonik, 4, 348, 1962.

[3] Garelis, E., Russell, J. L., Theory of pulsed neutron source measurements, Nuclear Science and Engineering, 16, 263, 1963.

[4] Soule. R. et al., Neutronic studies in support of accelerator-driven systems: the MUSE experiments in the MASURCA facility, Nuclear Science and Engineering, 148, 124–152, 2004.

[5] Kiyavitskaya, H., Bournos, V., Fokov, Y., Martsynkevich, B., Routkovskaia, C., Gohar, Y., Persson, C.-M., Gudowski, W., YALINA-Booster Benchmark Specifications for the IAEA Coordinated Research Projects on Analytical and Experimental Benchmark Analysis on Accelerator Driven Systems and Low Enriched Uranium Fuel Utilization in Accelerator Driven Sub-Critical Assembly Systems, IAEA Report, 2007.

[6] Dulla S., Ravetto P., Rostagno M.M., Bianchini G., Carta M., D’Angelo A., Some features of spatial neutron kinetics for multiplying systems, Nuclear Science and Engineering, 149, 88-100, 2005.

[7] Gabrielli F., Carta M., D’Angelo A., Maschek W., Rineiski A., Inferring the reactivity in accelerator driven systems: Corrective spatial factors for Source-Jerk and area methods, Progress in Nuclear Energy, 50, 370-376, 2008.

[8] Picca P., Furfaro R., Ganapol B. D., Dulla S. and Ravetto P., Application Of Artificial Neural Networks To Infer Subcriticality Level Through Kinetic Models, PHYSOR 2010, Pittsburgh, Pensylvania, 9-14 May 2010.


4

[9] Rasmussen C. E. and Williams C. K. I., Gaussian Processes for Machine Learning, MIT Press, Boston, 2006.


An Accurate Solution to Master/Moments Equations for the Kinetics of Breakable Filament Self-Assembly

Barry D. Ganapol

Department of Aerospace and Mechanical Engineering University of Arizona Tucson Arizona 85721

[email protected]

Proteinaceous aggregation occurs through self-assembly-- a process not entirely understood. In a recent article [1], an analytical theory for amyloid fibril growth via secondary rather than primary nucleation was presented. Remarkably, with only a single kinetic parameter, the authors were able to unify growth characteristics for a variety of experimental data. In essence, they seem to have uncovered the underlying allometric law governing the evolution of filament elongation simply from two coupled non-linear ordinary differential equations obtained originally from a master equation. While this work adds significantly to our understanding of filament self-assembly, it required an approximate analytical solution representation. If this were always true, the discovery of such scaling laws would be infrequent. Here, we show that the same results are found by purely numerical means. In addition, the numerical method features a highly accurate solution strategy for the coupled ODEs based on a fundamental finite difference scheme and convergence acceleration. Once a reliable numerical solution has been established, a dimensional analysis then provides the scaling law. 1. INTRODUCTION The prediction of proteinaceous aggregation may be the key to understanding the pathology of a host of degenerative transmittable diseases in mammals [2]. Whether a result of external initiation or a symptom of infection, protein misfolding seems to be a central element in prion protein disease progression. Fortunately, how macromolecules polymerize into long one-dimensional chains from monomer nucleation lends itself to simulation through kinetic equations that include self-assembly and disassembly [3]. As with any physical description and especially true for biophysical systems, the specification of meaningful rate constants is essential for a successful prediction of chain length distributions. More importantly however, is the discovery of underlying allometric principles governing fibrillogenesis. This requires forming appropriate combinations of rate constants and investigating their dynamic invariance to self-assembly. In the SCIENCE article [1], based on a master equation characterizing nucleated polymerization through conventional kinetics, the authors remark

“The lack of analytical solutions to such master equations has, however, represented a challenge to the quest to establish general principles and laws governing filamentous growth.”


2

The purpose of the present investigation is to demonstrate clearly that this is not the case and to indicate how the same fundamental scaling law can be uncovered through purely numerical means. The necessary elements to do so are (1) an accurate numerical solution to a set of non-linear ODEs and (2) a straightforward dimensional analysis. While presented in the context of microscopic filamentous growth, on its own, the numerical approach represents a new way to consider a numerical solution to coupled non-linear ODEs. In particular, a highly accurate numerical solution algorithm is proposed rivaling the accuracy of any analytical solution, in particular that found in ref. 1. We begin with the master equation describing filamentous growth through monomer nucleation. As with all physical investigation, the formulation of a rate equation, balancing creation and destruction of the physical elements, is central. At this point, we refer to the presentation of Ref. 1, where the following master equation for filament length appears:

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ,1

,2 , 1

2 , 1 ,

2 , .c

c

nn j n

i j

f t jk m t f t j

tk m t f t j k j f t j

k f t i k m t δ

+

+ −

∞

−= +

∂= − −

∂− − − +

+ +∑

(1)

The first term after the equals describes the creation of a filament of length j from the nucleation of monomers at either end of a filament of length j-1. The second term represents the loss of a filament of length j as it grows to length j+1 through secondary nucleation. The third term specifies the possibility of a filament of length j breaking at any of its j-1 links. The fourth term refers to the contribution from all filaments of length greater than j breaking at either end to form a filament of length j. The final term approximates the source of spontaneous and homogeneous monomer nucleation to length nc seeding the growth of all subsequent filaments and gives classical conclusions with no

fragmentation. ( )m t is the monomer concentration found from the following moment

equations:

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

2 1

2 1 / 2 ,

c

c

nc n tot

nc c c n

dP tk m t n P t k m t k m

dtdm t

k m t n n k P t n k m tdt

− −

+ −

= − + − + +

= − − − −

(2a)

where


3

( ) ( ),cj n

P t f t j∞

=≡ ∑

is the filament number density. If

( ) ( ),cj n

M t jf t j∞

== ∑ ,

is mass concentration, then

( ) ( )totm t m M t= − , (2b)

where mtot is the total protein concentration kept fixed. Equations (2a,b) are to be solved with initial conditions

( )( )

0

0

0

0 .

P P

M M

=

= (2c)

2. A FINITE DIFFERENCE NUMERICAL ALGORITHM

Equations (2) can be expressed in the finite difference form on a uniform grid of interval h

( ) ( )1

1 1 1 12 2 2j j j j j j j jh h hI I

−

+ + + + = − + + +

y A y A y y S S , (3a)

where

( ) ( )( )

P tt

m t

≡

y (3b)

( )( ) ( ) ( )( ) ( ) ( )

1

1

2 1

1 2

c

c

nc n

nc c c n

k n k k m tt

n n k k P t n k m t

−− −

−− +

− − − +≡ − − −

A y (3c)

( ) .0

totk mt − ≡

S (3d)

The initial conditions are

( ) 0

0

0tot

Pm M

≡ − y . (3e)


4

To this point, the development has followed along traditional lines from which we know the solution is of second order and will contain an iteration error as well because of the dependence of A on y. Our immediate goal therefore is to use both of these errors to our advantage to attain the highest possible accuracy. This is done through convergence acceleration. Here, we use two forms of acceleration, Romberg and Wynn-epsilon [4],[5]. The fundamental concept of convergence acceleration is to view a high order numerical solution as a limiting process. In this case, the limit is for the discretization h to approach zero. In this way, a sequence of solutions is developed and it is their convergence that will provide a highly accurate solution. Convergence acceleration is then the processes that, based on the asymptotic form of the error to convergence, extrapolates the error to give a more rapidly convergence sequence.

The above figure shows a comparison of the converged accelerated solution of Eqs(2) with the approximate solution indicating a more accurate representation of the near sigmoid growth of the protein population (inflection). The difference in the solutions has a profound effect on the onset of fibrillogenesis. In addition to the numerical solution of Eqs(2), a dimension analysis will be included to indeed show that an analytical solution is not required to uncover the allometric scaling laws. The final section of the presentation will be devoted to the solution of the master equation based on an alternative Taylor series expansion with convergence acceleration.


5

REFERENCES [1] T.P.J. Knowles, et. al., An Analytical Solution to the Kinetics of Breakable Filament

Assembly, Sci. 326, 1533-1537(2009). [2] F. Chiti, E.M. Dobson, Protein Misfolding, Functional Amyloid, and Human Disease, Annu. Rev. Biochem. 75,333(2006). [3] T. Pöschel, N. Brilliantov and C. Frömmel, Kinetics of Prion Growth, Biophys, J., 85, 3460-3474(3003). [4] A. Sidi, Practical Extrapolation Methods, Cambridge University Press, Cambridge, UK(2003). [5] P. Wynn, On the Convergence and Stability of the Epsilon Algorithm, SIAM J. Num, Anal., 3,#1 91-122(1966).


APPLICATION OF THE HYBRID TRANSPORT/POINT KINETICS TO TIME-DEPENDENT SOURCE-DRIVEN PROBLEMS

Paolo Picca

Dipartimento di Energetica, Politecnico di Torino Corso Duca degli Abruzzi, 24 - 10029 Torino, Italy

[email protected]

Roberto Furfaro, Department of Systems and Industrial Engineering

University of Arizona, Tucson AZ 85721 [email protected]

Barry D. Ganapol

Aerospace and Mechanical Engineering, University of Arizona, Tucson AZ 85721

[email protected]

The paper describes the novel kinetic methodology, recently proposed in [1] and known as hybrid transport point kinetic (HTPK) method. The method, based on a physical intuition, decouples the solution of time-dependent linear Boltzmann equation into the solution of a set of collisionless transport problems and an approximate kinetic model. More precisely, the first few collisions are described through a multi-generation approach. The generic equation for the j-th generation writes:

),,(),,()(1 ]1[][ txqtxx

xtvjj −=

Σ+∂∂+

∂∂ µψµ (1a)

where:

'.),',(2

)()(),(

1

1

]1[]1[ µµψυ

∫+

−

−− Σ+Σ= dtx

xxtxq jfsj (1b)

After the first T generations, the multiply-collided contribution is approximated via a lumped parameter method, here point kinetics [2]. The solution can be hence written as the sum of the transport contributions and a point kinetic evolution, i.e.:

),()(),,(),,(1

][ µϕµψµψ xtAtxtxT

j

jHTPK +=∑

=

(2)

Table I presents the material properties of the multiplying system, typical of an accelerator-driven system [3]. Figure 1 reports the behavior of the thermal power during the transient and compares different kinetic approximations. The reference solution is computed numerically solving the transport model. Indeed the HTPK method sensibly


2

improves the accuracy of the results, even at its lowest order (i.e. T=1). Table II provides the computational time for PK and HTPK as compared with reference transport solution. Even though the PK is the faster model, the hybrid approach provides a remarkable gain in accuracy with a limited increase in the computational time. Furthermore, HTPK remains convenient with respects to reference solution. In the final presentation, other simulations concerning different types of source transients (e.g. source pulse, step change in the boundary conditions) will be analyzed as well. Table I. Geometry and material properties of the system, with a total length a = 100

cm (keff=0.972480; v = 2.2 105 cm/s and neutron lifetime is ΛPK = 65.12 µs).

S: source region

M: multiplying

region

R: reflector region

L/2 [cm] 5 20 25

Σ [cm-1] 0.09 0.13 0.08

Σs [cm-1] 0.07 0.05 0.06

υΣf [cm-1] 0.00 0.09 0.00

Figure 1: Thermal power evolution for different model approximations. Bold curve:

reference; dotted curve: PK; dashed curve: HTPK.


3

Table II. Comparison of the computation time of full space kinetics (reference), point kinetics (PK) and different approximations of HTPK. The computational

times are normalized on the reference at tf=0.1 ms.

tf =0.1 ms 0.5 ms 2.5·ms 7.5·ms ref. 1.00 5.41 35.43 130.07 PK 5.43 5.47 5.51 5.52 T=1 6.22 7.77 11.69 22.49 T=2 6.39 8.63 15.87 33.50 T=3 6.58 9.54 20.09 45.74

REFERENCES

[1] Picca, P., Ganapol, B.D., Furfaro, R., 2010. A Hybrid Transport-Point Kinetic Method For The Analisis Of Source Transients In Subcritical Systems, Physor 2010, Pittsburgh, Pennsylvania, May 9-14.

[2] Akcasu, Z., Lellouche, G.S., Shotkin, L.M., 1971. Mathematical Methods in Nuclear Reactor Dynamics, Academic Press. New York.

[3] Soule, R., 2004. Neutronic studies in support of accelerator-driven systems: the MUSE experiments in the MASURCA facility, Nuclear Science and Engineering, 148, 124–152.


QUANTUM CORRECTIONS ON THE RADIATIVE TRANSFER EQUATION

J. Rosato

Laboratoire PIIM UMR 6633 Université de Provence / CNRS

Centre de St-Jérôme, Case 232 F-13397 Marseille Cedex 20 [email protected]

The quantum phase space formalism proposed by Wigner [1] is applied to radiation transport problems. In the radiative transfer theory [2], the specific intensity is commonly interpreted as being proportional to the phase space density of photons f(r,p,t) and this quantity obeys a semi-classical transport equation of Boltzmann type. By semi-classical, it is implied that the source and loss terms corresponding to photon emission and absorption are treated quantum mechanically whereas the transport is described through the usual differential operator ∂/∂t + v.∇∇∇∇ occurring in the classical Boltzmann equation (here with v = cp/p). In this work, we examine the validity of the semi-classical model by addressing the derivation of the radiation transport equation quantum mechanically from first principles. A suitable framework is provided by the Wigner phase space formalism adapted to second quantization (e.g. [3]). We show, through suitable assumptions [4], that the one-photon distribution obeys a closed transport equation. This equation reduces to the usual radiative transfer equation in the limiting case where both the wavelength λ and the coherence length λc = λ²/∆λ are small compared to the other spatial scales of interest. In the general case, however, we show that this transport equation contains integral terms denoting delocalization in both the r- and p-spaces. This is a feature of the wave nature of light or, equivalently, of the Heisenberg uncertainty principle. This suggests that the usual radiative transfer equation may be inaccurate in cases where the wave effects are significant. We try to illustrate this point through application to hydrogen lines. Figure 1 shows as an example the specific intensity of the Ly α radiation outgoing from a uniform sphere of radius equal to the photon mean free path mfp, for various values of λc/mfp, assuming Doppler broadening.


2

-3 -2 -1 0 1 2 3

0.0

0.2

0.4

0.6 λ

c/mfp = 0

λc/mfp = 1

λc/mfp = 2

∆ω/∆ωD

Spe

cific

inte

nsity

(a.

u.)

Figure 1. Specific intensity of the hydrogen Lyman αααα line for various λλλλc/mfp.

Here ∆ω∆ω∆ω∆ω/∆ω∆ω∆ω∆ωD denotes the frequency detuning in units of the Doppler width

∆ω∆ω∆ω∆ωD. The intensity maximum decreases as the coherence length increases.

ACKNOWLEDGMENTS

This work is partially supported by the French National Research Agency (contract

ANR-07-BLAN-0187-01) and by the collaboration (LRC DSM99-14) between the

PIIM laboratory and the CEA Cadarache (Euratom Association) within the

framework of the French Research Federation on Magnetic Fusion (FRFCM).

REFERENCES

[1] E. Wigner, “On the Quantum Corrections for Thermodynamic Equilibrium,” Phys.

Rev., 40, pp. 749-759 (1932).

[2] G. C. Pomraning, The Equations of Radiation Hydrodynamics, Dover (1973).

[3] W. E. Brittin and W. R. Chappell, “The Wigner Distribution Function and Second

Quantization in Phase Space,” Rev. Mod. Phys., 34, pp. 620-627 (1962).

[4] J. Rosato, “The Photon Picture of Radiation Transport,” Am. J. Phys., 78, pp. 851-

857 (2010).


Atom-Atom Relaxation with Quantum Differential Elastic Scattering CrossSections; Distribution Function Relaxation and the Kullback-Leibler Entropy

Reinel Sospedra-Alfonso and Bernie D. ShizgalDepartment of Chemistry and Institute of Applied Mathematics

University of British Columbia2036 Main Mall, Vancouver, British Columbia V6T 1Z1

[email protected]@chem.ubc.ca

A fundamental problem in gas kinetic theory is the relaxation to equilibrium of the distribution function of

suprathermal test particles dilutely dispersed in a second component at equilibrium. This problem has a

long history [1–3] with also important applications to laboratory measurements of hot atom relaxation

[4, 5] as well as in atmospheric science [6–10]. The Boltzmann equation defined by the quantum scattering

collision cross section for binary collisions between the two species describes the time evolution of an

initial isotropic nonequilibrium distribution function of an ensemble of test particles of mass m1 dilutely

dispersed in a background gas of particles of mass m2 and at equilibrium at temperature Tb. The main

objective of the present paper is to study the effect of the quantum differential cross section on the

relaxation behaviour [3, 8]. The systems considered are energetic He in N [8], He in Xe and Xe in He for

which realistic interaction potentials are known. The quantum mechanical differential cross sections that

define the collision operator in the Boltzmann equation are employed. The collision cross sections are

determined quantum mechanically with the semiclassical JWKB phase shifts [11].

The methodology employed for the solution of the Boltzmann equation is based on the expansion of the

distribution function in Sonine polynomials [12]. These basis functions have been used in the solution of

the Boltzmann equation for transport problems [13, 14], reactive systems [15], sound dispersion [16],

spectral properties of the Boltzmann collision operator [1, 17, 18] and the interpretation of Doppler profiles

that probe details of velocity distribution functions [19, 20]. The mathematics for employing this basis set

for arbitrary differential cross section has been developed recently [21]. The expansion of the distribution

function in the Sonine polynomial basis reduces the Boltzmann equation for this spatially homogenous

problem to a set of linear ordinary differential equations which are solved in terms of the eigenvalues and

eigenfunctions of the linear Boltzmann collision operator. The reciprocal of the eigenvalues is a measure of

the relaxation times to equilibrium.

This work is motivated by the suggestion in [9] that for the strong forward scattering quantum differential

cross section the relaxation to equilibrium is characterized by two time scales; a short time scale τg during

which the initial distribution function relaxes to a local Maxwellian with the time dependent temperature of

the minor species followed by a slow equilibration of the temperature to Tb with a time scale roughly 10τg.

In this work, we study the shape relaxation of the distribution function in terms of the Kullback-Leibler

entropy [22–25] of f(v, t) relative to the steady state Maxwellian distribution fSS(v) evaluated at the bath

Reinel Sospedra-Alfonso and Bernie D. Shizgal

temperature, namely

ΣSS(t) = −∫f(v, t) ln

f(v, t)

fSS(v)dv. (1)

We also measure the departure of the distribution function from a local Maxwellian evaluated at the time

dependent temperature by means of

ΣLM (t) = −∫f(v, t) ln

f(v, t)

fLM (v, T (t))dv. (2)

There are a several unique systems for which an initial Maxwellian distribution function at T (0) 6= Tb will

relax to the Maxwellian at Tb through a sequence of Maxwellians at T (t) [26]. These systems are said to

exhibit canonical invariance and ΣLM (t) = 0 for all t provided that the initial distribution is a Maxwellian.

In this work, we report on the extent to which the systems studied here exhibit any evidence for canonical

invariance that would validate the conclusions reported in [9].

ACKNOWLEDGMENTS

This research is supported by grants to BDS from the Natural Sciences and Engineering Research Council

of Canada and the Canadian Space Agency.

REFERENCES

[1] M. R. Hoare and C. H. Kaplinsky, J. Chem. Phys. 52, 3336, (1970).

[2] J. Park, N. Shafer and R. Bersohn, J. Chem. Phys. 91, 7861 (1989).

[3] A. S. Clarke and B. Shizgal, Phys. Rev. E49, 347 (1994).

[4] M. Anderlini and D. Guery-Odelin, Phys. Rev. A71, 032706 (2006).

[5] Y. Matsumi and A. M. S. Chowdhury, J. Chem. Phys. 104, 7036 (1996).

[6] T. Nakayama, K. Takahashi and Y. Matsumi, Geophys. Res. Letters, 32, L24803 (2005).

[7] B. D. Shizgal, Planet. Space Sci. 52, 915 (2004).

[8] P. Zhang, V. Kharchenkov and A. Dalgarno, Mol. Phys. 105, 1487 (2007).

[9] P. Zhang, V. Kharchenko, A. Dalgarno, Y. Matsumi, T. Nakayama and K. Takahashi, Phys. Rev.Letters 100, 103001 (2008).

[10] K. Kabin and B. D. Shizgal, J. Geophys. Research 107, 1029 (2002).

[11] J. S. Cohen, Rapid accurate calculation of JWKB phase shifts, J. Chem. Phys. 68, 1841 (1978).

[12] E. L. Tipton, R. V. Tompson and S. K. Loyalka, Euro. J. Mech. B/Fluids 28, 353 (2009).

[13] M. T. Bouazza and M. Bouledroua, Mol. Phys. 105, 51 (2007)

2/3

[14] J. E. Ramos, Mol. Phys. 103, 2323 (2005).

[15] B. D. Shizgal and A. Chikhaoui, Physica A365, 317 (2006).

[16] D. G. Napier and B. D. Shizgal, Physica A387, 4099 (2008).

[17] S. Kryszewski and J. Gondek, Phys. Rev. A56, 3923 (1997).

[18] B. Shizgal, M. J. Lindenfeld and R. Reeves, Chem. Phys. 56, 249 (1981).

[19] M. J. Bastian, C. P. Lauenstein, V. M. Bierbaum and S. R. Leone, J. Chem. Phys. 98, 9496 (1993).

[20] J. W. Nicholson, W. Rudolph and G. Hager, J. Chem. Phys. 104, 3537 (1996).

[21] B. D. Shizgal and R. Dridi, Comp. Phys. Comm. 181, 1633 (2010).

[22] B. D. Shizgal, Astrophys. Space Sci. 312, 227 (2007).

[23] P. Hick and G. Stevens, Astron. Astrophys. 172, 350 (1987).

[24] A. R. Plastino, H. G. Miller and A. Plastino, Phys. Rev. E56, 3927 (1997).

[25] H. Risken, The Fokker-Planck Equation, pp. 134137 (Springer, New York, 1989)

[26] H. C. Andersen, I. Oppenheim, K. E. Shuler and G. H. Weiss, J. Math. Phys. 5, 522 (1964).

3/3


Shape relaxation in Electron Atom Relaxation; the Kullback-Leibler RelativeEntropy and Relaxation Times

Reinel Sospedra-Alfonso and Bernie D. ShizgalDepartment of Chemistry and

Institute of Applied MathematicsUniversity of British Columbia

2036 Main MallVancouver, British Columbia V6T 1Z1

[email protected]@chem.ubc.ca

The relaxation of energetic electrons in a background gas at equilibrium at a fixed temperature Tb is an

important fundamental problem in kinetic theory with equally important applications to numerous devices

in plasma processing of materials, plasma displays and other technologies [1]. There has been extensive

work done to date for both atomic [2–6] and molecular moderators [7–9]. In atomic moderators, there have

been two noticeable effects that occur. One is the transient negative mobility effect predicted by McMahon

and Shizgal [2] and observed experimentally by Warman et al [10] and discussed since then by other

researchers [11, 12]. The other is the unexpected negative differential conductivity effect in mixtures of

inert gases [13] that was previously thought to occur only for polyatomic gases with internal degrees of

freedom [14, 15].

Recently, Zhang et al [16] suggested that the evolution of the shape of the distribution function in

electron-atom relaxation is analogous to atom-atom relaxation insofar as the establishment of an

intermediate local Maxwellian distribution parametrized by the electron temperature T (t) 6= Tb. For the

electron-atom relaxation this is known to occur if electron-electron collisions are included and dominate

the electron-atom collision rate [5]. The main objective of this paper is to study the electron-atom

relaxation process in terms of the shape of the electron distribution function and the relaxation time for the

approach of the distribution function to equilibrium. We consider two rare gas atom moderators, Argon and

Neon, characterized by two different electron-atom momentum transfer cross sections.

The dependence of the distribution function is expressed in reduced speed, x = v√me/2kBTb, where me

is the electron mass. The time t′ is express in units of t−10 = (nme/2M)σ0√

2kBTb/me, that is t = t′/t0,

where n and M are respectivly the density and mass of the moderator particles, and σ0 is a convenient hard

sphere cross section specified later on. In terms of these dimensionless variables, the isotropic portion of

the distribution function is given by the Fokker-Planck equation of the form [2, 3]

∂f(x, t)

∂t=

1

x2∂

∂x

[2x4σ(x)f + x2B(x)

∂f(x, t)

∂x

](1)

where B(x) = xσ(x) + α2/xσ(x) with σ(x) = σ(x)/σ0, and σ is the e-atom momentum transfer cross

section. The quantity α = (M/6me) [eE/nσ0kBTb]2 measures the strength of the uniform external

Reinel Sospedra-Alfonso and Bernie D. Shizgal

electric field E. The Fokker-Planck equation, Eq. (1), is solved with the finite difference algorithm

developed by Chang and Cooper [17]. The relaxation of the distribution function is determined together

with the temperature. A measure of the departure of the distribution function from the steady state

distribution fSS(x) is the Kullback-Leibler entropy [18–21], defined by

ΣSS(t) = −4π

∫ ∞0

x2f(x, t) lnf(x, t)

fSS(x)dx. (2)

In the absence of an external field α = 0, fSS(x) is a Maxwellian evaluated at the bath temperature. In this

regime, we also measure the departure of the distribution function from a local Maxwellian evaluated at the

temperature of the electrons, that is, we compute the quantity

ΣLM (t) = −4π

∫ ∞0

x2f(x, t) lnf(x, t)

fLM (x, T (t))dx. (3)

0 1 2 3 4 50.00

0.75

1.50

2.25

0.00 0.25 0.50 0.75 1.000

1

2

3

4

5

6

7

8

9

10

11

0.0 0.1 0.2 0.3 0.4 0.5 0.6

-12

-10

-8

-6

-4

-2

0

0.0 0.1 0.2 0.3 0.4 0.5 0.6-1.8

-1.6

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

(A)

0.5 0.22

0.15

0.09

0.03

t=0

x

(B)

(b)

(a)

t

(C)

(b)

(a)

t

(D)

(b)

(a)

t

Figure 1. The zero field time variation of f(x, t), T (t)/Tb, ΣSS(t) and ΣLM (t) for electrons in Ar withan analytic fit to the momentum transfer cross section reported by Mozumder [22] and an initial normalizedGaussian distribution, f(x, 0) = A exp[−β(x − x0)2], x0 = 4 and β = 15. The time is expressed in unitsof t0 = 6730s with σ0 = 0.23A2, Tb = 290.1K and n = 1011cm−3. Dashed lines correspond to (a) thehard sphere σ = 1 and (b) the Maxwell molecule σ(x) = (8/3

√π)/x momentum transfer cross sections.

Typical results of these calculations are shown in Figure 1. The main result of this work is that the

distribution function does not relax quickly to a local Maxwellian followed by a slower equilibration via a

continuous sequence of Maxwellians. Systems that preserve a local Maxwellian have been discussed by

Andersen et al [23] and includes the Lorentz Maxwell molecule system [24]. For such systems,

ΣLM (t) = 0 for all time if the initial distribution function is a Maxwellian. Additional results for initial

Maxwellian distributions and α 6= 0 as well as results for the e-Neon system will be discussed.

2/3

ACKNOWLEDGMENTS

This research is supported by grants to BDS from the Natural Sciences and Engineering Research Council

of Canada and the Canadian Space Agency.

REFERENCES

[1] Z. Lj. Petrovic, S. Dujko, D. Maric, G. Malovic, Z. Nikitovic, O. Sasic, J. Jovanovic, V. Stojanovicand M. Radmilovi-Radenovi, J. Phys. D: Appl. Phys. 42, 194002 (2009).

[2] D. R. A. McMahon and B. Shizgal, Phys. Rev. A31, 1894 (1985).

[3] B. Shizgal and D. R. A. McMahon, Phys. Rev. A32, 3669 (1985).

[4] I. K. Bronic and M. Kimura, J. Chem. Phys. 104, 8973 (1996).

[5] D. Trunec, P. Spanel and D. Smith, Chem. Phys. Letters 372, 728 (2003).

[6] R. Plasil, I. Korolov, T. Kotrik, P. Dohnal, G. Bano, Z. Donko, and J. Glosik, Eur. Phys. J. D54, 391(2009).

[7] K. Kowari, L. Demeio and B. Shizgal, J. Chem. Phys. 97,2061 (1992).

[8] K. Kowari and B. Shizgal, Chem. Phys. 185, 1 (1994).

[9] T. Shimada, Y. Nakamura, Z. Lj. Petrovic and T. Makabe, J. Phys. D: Appl. Phys. 36, 1936 (2003).

[10] J. M. Warman, U. Sowada, and M. P. De Haas, Phys. Rev. A31, 1974 (1985).

[11] R. E. Robson, Z. Lj. Petrovic, Z. M. Raspopovic and D. Loffhagen, J. Chem. Phys. 119, 11249(2003).

[12] N. A. Dyatko, J. Phys.: Conf. Series 71, 012005 (2007).

[13] B. Shizgal, Chem. Phys. 147, 271 (1990).

[14] S. R. Hunter, J. G. Carter and L. G. Christophorou, J. Appl. Phys. 58, 58 (1985).

[15] N. L. Aleksandrov, N. A. Dyatko, I. V. Kochetov, A. P. Napartovich, and D. Lo, Phys. Rev. E53,2730 (1996).

[16] P. Zhang, V. Kharchenko, A. Dalgarno, Y. Matsumi, T. Nakayama and K. Takahashi, Phys. Rev.Lett. 103, 159302 (2009).

[17] J. S. Chang and G. Cooper, J. Comput. Phys. 6, 1 (1970).

[18] B. D. Shizgal, Astrophys Space Sci. 312, 227 (2007).

[19] P. Hick and G. Stevens, Astron. Astrophys. 172, 350 (1987).

[20] A. R. Plastino, H. G. Miller and A. Plastino, Phys. Rev. E56, 3927 (1997).

[21] H. Risken, The Fokker-Planck Equation, pp. 134-137 (Springer, New York, 1989)

[22] A. Mozumder, J. Chem. Phys. 72, 6289 (1980).

[23] H. C. Andersen, K. E. Shuler, I. Oppenheim and G. H. Weiss, J. Math. Phys. 5, 522 (1964).

[24] H. Oser, K. E. Shuler and G. H. Weiss, J. Chem. Phys. 41, 2661 (1964).

3/3


DIFFUSIVE LIMITS FOR A QUANTUM TRANSPORT MODELWITH WEAK AND STRONG FIELDS

Luigi BarlettiDipartimento di Matematica ”U.Dini”

University of FlorenceViale Morgagni 67/A

I-50134 Firenze (Italy)[email protected]

Giovanni FrosaliDipartimento di Sistemi e Informatica

University of FlorenceVia Santa Marta 3

I-50139 Firenze (Italy)[email protected]

Quantum Fluid Dynamics has recently proved an essential tool for semiconductor device modeling in order

to incorporate quantum effects in models which are relatively simple, flexible and cheap from the

computational point of view. In this work we start from the kinetic-like formulation of quantum mechanics

due to Wigner and we write the equations for the moments of the Wigner functions for a (pseudo)spinorial

case of interest in semiconductor physics, namely the two-band k·p Hamiltonian

H =

(− ~2

2m∆+ γ −α · ∇α · ∇ − ~2

2m∆− γ

)describing electrons in a periodic potential with two available energy bands. Here, α is the k·p coupling

vector and γ > 0 is half the band gap. The k·p Hamiltonian acts on wavefunctions in L2(R3,C2) and,

therefore, the system has a spin-like degree of freedom (pseudospin), whose meaning is clearly related to

the energy bands.

The ultimate goal is to write down diffusive semiclassical equations for the densities, n+ and n−, of the

electrons in the two bands. To this aim we first write the kinetic (Wigner) equations for the Hamiltonian H ,

where interaction effects are introduced in the form of a BGK-like term, thus making the system relax to a

local equilibrium state. Following the Degond and Ringhofer’s technique [3, 4], who extended to the

quantum case Levermore’s entropy minimization principle [5], the local equilibrium state is assumed to be

given by the quantum entropy minimization principle, under the constraint of given moments n+ and n−.

After writing the Wigner-BGK equation in terms of dimensionless variables, two dimensionless parameters

appear which will be assumed to be small: a diffusive parameter τ (the scaled typical collision time) and a

semiclassical parameter ϵ (the scaled Planck constant). In particular, we shall consider two different

scalings, the first one corresponding to a weak external field (assumed to be of order τ ) and the second one

corresponding to a strong field (assumed, in this case, to be of order 1).

First, we look for the quantum diffusive limit (τ → 0, ϵ ∼ 1) when the field is weak, by applying to the

Wigner-BGK equations the first order Chapman-Enskog method. The result of this procedure consists of

L. Barletti and G. Frosali

two coupled equations for n+ and n−. However, these equations are only of theoretical interest, since they

are rather involved, because of the complicated dependence of the local equilibrium. Then, we pass to the

semiclassical limit, ϵ → 0, in the quantum diffusive equations. In order to do that, we need the explicit

expression of the local equilibrium state, at least at leading order in ϵ, which has been computed in [1] for

different scalings of the band parameters α and γ with respect to ϵ.

Finally, the whole procedure can be repeated in the strong field scaling. In this case, the leading order of

the Chapman-Enskog expansion is no longer represented by the BGK local equilibrium state alone, but is

the kernel of the BGK operator coupled with the field operator. However, explicit computations, even

though more complicated, are still possible.

We remark that the semiclassical limit ϵ → 0 keeps trace of the (pseudo)spinorial structure of the problem

but misses Bohm terms (also known as “quantum potential” or “quantum pressure”), which are of order ϵ2.

A two-band drift-diffusion model for a spin-orbit Hamiltonian, based on the quantum minimum entropy

principle, has been recently obtained in [2], with the inclusion of 2 terms.

ACKNOWLEDGMENTS

This work is performed under the auspices of the Italian Ministry of University (MIUR National Project

“Mathematical modelling of semiconductor devices, mathematical methods in kinetic theories and

applications”).

REFERENCES

[1] L. Barletti, G. Frosali, “Diffusive limit of the two-band k·p model for semiconductors,” J. Stat.Phys., 139, pp. 280-306 (2010).

[2] L. Barletti, F. Mehats, “Quantum drift-diffusion modeling of spin transport in nanostructures,” J.Math. Phys., 51, 053304 (2010).

[3] P. Degond, C. Ringhofer, “Quantum moment hydrodynamics and the entropy principle,” J. Stat.Phys., 112, 587–628 (2003).

[4] P. Degond, F. Mehats, C. Ringhofer, “Quantum energy-transport and drift-diffusion models,” J. Stat.Phys., 118, 625–667 (2005)

[5] C. D. Levermore, “Moment closure hierarchies for kinetic theories.” J. Stat. Phys. 83, 1021–1065(1996)

2/2


Thursday, September 15, 2011

Reactor Physics II

8:25 am In Memory of Vinicio Boffi - Barry Ganapol

8:35 am The Transient 3-D Transport Coupled Code TORT-3D/ATTICA3D for High-Fidelity Pebble-Bed HTGR

Analyses - A. Seubert, A. Sureda, J. Laupins, J. Bader

9:00 am A Stochastic Theory of the Number of Fissions - A.K. Prinja

9:25 am SP3 Solution Versus Diffusion Solution in Nodal Codes - Which Improvement Can Be Expected? - B.

Merk, S. Duerigen

9:50 am Consistent Recondensation Theory - S. Douglas, F. Rahnema

Analytic Transport Solutions I

10:45 am A New Analytic Solution of the One-Speed Neutron Transport Equation for Adjacent Half-Spaces with

Isotropic Scattering - R.P. Smedley-Stevenson

11:10 am Energy-Dependent Analytical Solutions for the Charged Particle Transport Equation - T. Geback, M.

Asadzadeh

11:35 am Spatial Moments of Continuous Transport Problems Computed on Grids - J.D. Densmore

12:00 pm Eigenvalues of the Anisotropic Transport Equation in a Slab - E. Sauter, F. de Azevedo, M. Thompson,

M.T. Vihena

Analytic Transport Solutions I

2:00 pm In-Water Ocean Optics Inversion Algorithm - N.J. McCormick, E. Rehm

2:25 pm On Boundedness of Higher Velocity Moments for the Linear Boltzmann Equation with Diffuse Bound-

ary Conditions - R. Petterson

2:50 pm On the Speed of Heat Waves - M. Mikai

3:15 pm Density Distribution of the Molecules of a Liquid in a Semi-Infinite Space - V. Molinari, B.D. Ganapol,

D. Mostacci


Special Session

4:10 pm Commemorating the 50th Anniversary of Ken Case’s Paper: “Elementary Solutions of the Transport

Equation and Their Applications” - Paul Zweifel



The Transient 3-D Transport Coupled Code TORT-TD/ATTICA3D

for High-Fidelity Pebble-bed HTGR Analyses

A. Seubert, A. Sureda

Gesellschaft für Anlagen- und Reaktorsicherheit (GRS) mbH

Forschungszentrum, D-85748 Garching, Germany

[email protected]

J. Lapins1, J. Bader

1,2, E. Laurien

1

1Institut für Kernenergetik und Energiesysteme (IKE), Universität Stuttgart

Pfaffenwaldring 31, D-70569 Stuttgart, Germany 2EnBW Kernkraft GmbH, Kernkraftwerk Philippsburg,

Rheinschanzinsel, D-76661 Philippsburg, Germany

INTRODUCTION

As most of the acceptance criteria are local core parameters, application of transient 3-D

fine mesh neutron transport and thermal hydraulics coupled codes is necessary for best

estimate evaluations of safety margins. This also applies to high-temperature gas cooled

reactors (HTGR) since available analysis tools and methods have, in many cases, lagged

behind the state of the art compared to other reactor technologies, e.g. LWR. Increasing

computing power supports the application of 3-D fine-mesh transient transport codes

using few energy groups. This paper describes the discrete ordinates based coupled code

system TORT-TD/ATTICA3D and its application to PBMR-400 transient analyses.

THE TORT-TD 3-D TRANSIENT DISCRETE-ORDINATES CODE COUPLED

WITH THE POROUS MEDIUM CODE ATTICA3D

The time-dependent 3-D fine-mesh few-group discrete ordinates (SN) neutron transport

code TORT-TD [1] is being developed at GRS. It is based on the DOORS steady-state

neutron transport code TORT [2][3] and solves the steady-state and time-dependent few-

group transport equation with an arbitrary number of prompt and delayed neutron

precursor groups in both Cartesian and cylindrical (r--z) geometry. Unconditional

stability in transient calculations is achieved using a fully implicit time discretization

scheme. Scattering anisotropy is treated in terms of a Pl Legendre scattering cross section

expansion. Computing time can be saved by extrapolating the angular fluxes to the next

time step using the space-energy resolved inverse reactor period g. The features of

TORT-TD further include:

• 64 bit encoding to meet imposed tight convergence criteria and to enable TORT-TD to

be applied to large realistic problems exceeding 32 bit RAM limitations;

• Movements of single control rods or control rod banks;

• Processing of parameterized tabulated cross section libraries for up to 5 state

parameters; interpolation either linear or with cubic spline polynomials, thus allowing

to study the impact of different interpolation schemes on cross section evaluation;

• Generalized Equivalence Theory (GET) at pin cell level to reduce homogenization

errors in LWR applications;



2

• Time-dependent anisotropic distributed external source;

• Leakage and buckling calculation over larger spatial regions (e.g. spectral zones)

using the neutron current density in discrete ordinates representation;

• Calculation of Xenon/Iodine equilibrium and transient distribution as a prerequisite for

operational transients;

• Fully integrated 3-D fine-mesh few-group diffusion solver (steady state and time-

dependent) in both Cartesian and cylindrical geometry for fast running scoping

calculations that can be employed as preconditioner for subsequent discrete-ordinates

calculations or future embedded transport-diffusion analyses.

The ATTICA3D Advanced Thermal hydraulics Tool for In-vessel and Core Analysis in 3

Dimensions, developed at IKE of Stuttgart University [4,5], applies the porous medium

approach in both cylindrical and Cartesian geometry. Subdivision between solid and fluid

fraction in a considered control volume is done via the porosity parameter ε. Thermal

non-equilibrium in-between solid and gas phase is considered. The finite volume method

is used for spatial discretization of the partial differential equations, and time integration

is done by a fully implicit time adaptive multi step backward differentiation formula. To

correctly capture the feedback of thermal hydraulics on neutronics, a heterogeneous

temperature model for pebble type fuel elements is available which correctly accounts for

heterogeneous heat generation by the fuel kernels.

The coupling of TORT-TD with ATTICA3D is established by adopting the so-called

internal coupling methodology. Therein, the thermal-hydraulics code models the entire

fluid-dynamics and heat transport phenomena of the system including the core region,

whereas the neutron kinetics is treated by TORT-TD. The coupled code system

TORT-TD/ATTICA3D is represented by a single executable in which ATTICA3D acts

as the main program and calls TORT-TD in terms of a subroutine whenever an update

calculation of the power distribution is requested. For the data exchange between

TORT-TD and ATTICA3D, already existing TORT-TD interface routines have been

utilized in combination with the ATTICA3D mesh overlay feature that transfers 3-D

distributions from its thermal-hydraulic mesh to a superimposed neutron-kinetics mesh

and vice versa. This allows for efficient data transfer via direct memory access of array

elements.

APPLICATION OF TORT-TD/ATTICA3D TO STEADY-STATE AND

TRANSIENT ANALYSES OF THE PBMR-400 DESIGN

As first application of the coupled TORT-TD/ATTICA3D, the PBMR-400 benchmark

[6] has been selected. This benchmark was initiated by the OECD/NEA/NSC as an

international activity for code-to-code comparison. TORT-TD and ATTICA3D models in

3-D cylindrical geometry have been prepared with the possibility to azimuthaly resolve

control rods for later simulation of, e.g., non-symmetric control rod movements. From the

various benchmark exercises, we present in this paper results for the total control rod

withdrawal (TCRW, exercise 5a) and the cold helium ingress (exercise 6) transients.



3

The TCRW transient involes simultaneous removal of all control rods at maximum

operational speed (1 cm/sec). The total fission power evolution obtained by

TORT-TD/ATTICA3D during the TCRW transient is shown in Figure 1 (left).

Obviously, the power evolution is in good agreement with other benchmark participants’

solutions. The well-known control rod cusping effect is also present in the TORT-TD

solution, though to a smaller extent, i.e. with a shorter period due to the finer axial

discretization. In addition, the maximum fuel temperature evolution shown in Figure 1

(right) also compares well with the other benchmark solutions.

Figure 1: Evolution of total fission power (left) and maximum fuel temperature (right)

during the TCRW transient obtained with TORT-TD/ATTICA3D (black line) in

comparison to other benchmark participants’ solutions.

Exercise 6 of the PBMR-400 benchmark simulates a cold helium ingress by reducing the

inlet temperature from 500°C to 450°C over 10 seconds. The temperature is kept at this

level for the next 290 seconds until it is brought back to its initial value of 500°C within

10 seconds. Figure 2 (left) shows the total power evolution obtained with

TORT-TD/ATTICA3D in light blue. Again, good agreement can be derived from the

comparison with other benchmark participants’ solutions. This also applies to the

maximum fuel temperature evolution that is depicted in Figure 2 (right).

Figure 2: Evolution of total fission power (left) and maximum fuel temperature (right)

during the cold helium ingress transient obtained with TORT-TD/ATTICA3D (light blue)

in comparison to other benchmark participants’ solutions.



4

SUMMARY

This paper describes the time-dependent 3-D discrete-ordinates based coupled code

system TORT-TD/ATTICA3D and its application to HTGR of pebble bed type. For the

coupled code system TORT-TD/ATTICA3D, represented by a single executable, the so-

called internal coupling approach has been implemented. The 3-D distributions of

temperatures from ATTICA3D and power density from TORT-TD are efficiently

exchanged by direct memory access of array elements via interface routines. As first

coupled TORT-TD/ATTICA3D applications, results for two transients (total control rod

withdrawal and cold helium ingress) as specified in the PBMR-400 benchmark are

presented. The results are very promising and demonstrate that the coupled code system

TORT-TD/ATTICA3D can be a key component in a future comprehensive 3-D code

system for HTGR of pebble bed type.

ACKNOWLEDGMENTS

This work was supported by the German Federal Ministry of Economics and Technology

due to a resolution of the German Bundestag. Additional support was provided by EnBW

Kernkraft GmbH.

REFERENCES

[1] A. Seubert, K. Velkov and S. Langenbuch, “The Time-Dependent 3-D Discrete

Ordinates Code TORT-TD with Thermal-Hydraulic Feedback by ATHLET Models”, Physor 2008, Interlaken, Switzerland, September 14-19, 2008.

[2] W. A. Rhoades, R. L. Childs, “The TORT Three dimensional Discrete Ordinates Neutron/Photon Transport Code”, Nucl. Sci. Eng., 107, p. 397 (1991).

[3] W. A. Rhoades, D. B. Simpson, “The TORT Three dimensional Discrete Ordinates Neutron/Photon Transport Code (TORT Version 3)”, ORNL/TM-13221, 1991.

[4] K. Hossain, M. Buck, N. Ben Said, W. Bernnat, G. Lohnert, “Development of a Fast 3D Thermal-Hydraulic Tool for Design and Safety Studies for HTRS”, Nucl. Eng. Design, 238, p. 2976 (2008).

[5] K. Hossain, M. Buck, W. Bernnat, G. Lohnert, “TH3D, a three-dimensional thermal hydraulics tool, for design and safety analysis of HTRs”, Proceedings of HTR2008, Washington D.C, USA, Sept 28 – Oct 1, 2008.

[6] F. Reitsma et al., “PBMR Coupled Neutronics/Thermal hydraulics Transient Benchmark The PBMR-400 Core Design”, NEA/NSC/DOC(2007), Draft-V07, http://www.nea.fr/science/wprs/pbmr400.


A STOCHASTIC THEORY OF THE NUMBER OF FISSIONS

Anil K. PrinjaChemical and Nuclear Engineering Department

University of New MexicoMSC01 1120, 209 Farris Engineering Center

Albuquerque, NM [email protected]

The stochastic theory of neutron populations in multiplying media is well established [1–4] and has been

used to describe the extinction and divergence of neutron chains in fissile systems as well as to develop an

understanding of the importance of the source strength (e.g., spontaneous fission, cosmic ray background,

start-up source) in the development of the asymptotic state of the neutron populations. However, in many

applications (such as criticality safety, weapons safety, fast-burst reactors) it is not the neutron population

that directly drives the phenomena of interest but rather the energy deposited in the medium as a result of

fission reactions. Under these conditions one is more interested in the stochastic distribution of the total

fission energy deposited, and this apparently has hitherto not been investigated. In this work we describe

the application of the theory of discrete-state continuous-time Markov processes to develop a backward

Master equation model for the probability distribution of the cummulative number of fissions as a function

of time. For a lumped model description, we use generating function techniques to obtain and examine

solutions for the probability distribution function under various conditions. We contrast the time

development of the stochastic fission distribution with the corresponding neutron population, develop

conditions under which the fission number in a supercritical system diverges, and examine the steady state

distribution of fissions under all conditions of criticality. Finally, a generalization of the approach to

account for the neutron phase space dependence is presented and equations derived from which the mean

fission density and its variance can in principle be obtained.

REFERENCES

[1] L. Pal, “ On the theory of stochastic processes in nuclear reactors,” Il Nuovo Cimento, SupplementVII, pp. 25-42 (1958)

[2] G. Bell, “Probability distribution of neutrons and precursors in a multiplying assembly,” Annals ofPhysics, 21, pp. 243-283 (1063)

[3] G. Bell, “On the stochastic theory of neutron transport,” Nucl. Sci. Eng, 21, pp. 390-401 (1965)

[4] I. Pazsit and L. Pal, Neutron Fluctuations, Elsevier, Oxford, UK (2008).


SP3 SOLUTION VERSUS DIFFUSION SOLUTION IN NODAL CODES —WHICH IMPROVEMENT CAN BE EXPECTED

Bruno MerkInstitute of Safety Research

Helmholtz-Zentrum Dresden-RossendorfP.O. Box 51 01 19

01314 DresdenGermany

[email protected]

Susan DuerigenInstitute of Safety Research

Helmholtz-Zentrum Dresden-RossendorfP.O. Box 51 01 19

01314 DresdenGermany

[email protected]

In the last years, the simplified PN method derived by Gelbard [1] and brought to application by

Larsen [2, 3] has attracted much attention in the reactor physics field. This method gives the possibility to

partly capture the neutron transport effect without a severe increase in computational burden. Various

computer programs based on the SPN method have been developed for thermal reactor applications to

date, e.g., [4–7]. The publication of Brantley and Larsen [3], with the advanced numeric procedure offering

the possibility to solve the SP3 equations on the basis of the already existing diffusion solver, was the basis

for the decision to implement an SP3 solver into the DYN3D code at the Helmholtz-Zentrum

Dresden-Rossendorf.

The implementation of an increased order spherical harmonics expansion (from P1 to SP3) to better

represent the neutron transport equation has the potential to improve the overall calculation result of the

codes significantly. Nevertheless, it has to be investigated if this potential can be released under the

specifics of the code. The very specific feature of nodal codes is the use of large calculation cells, so-called

nodes, which are homogenized in pre-calculations with higher order transport tools during the

cross-section preparation procedure. In contrast to finite difference methods, the neutron flux distribution

inside the large nodes is represented in a spatial resolution. This spatial resolution is classically derived

either from analytical solutions (analytic nodal method, analytic function expansion nodal method) or from

expansion solutions (nodal expansion methods). The spatial flux is used to determine the coupling between

the nodes via the neutron current at the boundary surfaces. The power production in the nodes is calculated

using the integral neutron fluxes inside the nodes.

In this work, full one-dimensional analytical solutions for the one and two energy-group diffusion, SP3,

and P3 equations are derived to investigate the effect of the SP3 approach in nodal calculations. The results

Bruno Merk and Susan Duerigen

are compared with SN reference solutions. A first test is performed on the one-group test case of Brantley

and Larsen [3] to verify the different analytical solutions. In this test case, tailored cross sections have been

used by Brantley and Larsen to highlight the transport effects. A second test is performed for a pin cell

with real reactor fuel-moderator configuration based on the material configuration of a German benchmark

definition.

Figure 1. Normalized fast and thermal neutron flux for the diffusion (red - green) and the P3 transportsolution (red - blue) for case 2 burnt UOX fuel next to fresh MOX fuel in the center.

For the evaluation of the differences between the diffusion and the SP3 solutions in the nodal code

configuration, fuel configurations from the OECD/NEA AND U.S. NRC PWR MOX/UO2 CORE

TRANSIENT BENCHMARK are investigated. Different combinations of challenging fuel assembly pairs

(like fresh UOX and burnt MOX, a rodded neighboring assembly, or an assembly next to the reflector),

representative for the benchmark core, are investigated. Fig. 1 shows the neutron flux distributions for a

case consisting of burnt UOX fuel with one fresh MOX fuel assembly in the center. The fast neutron flux is

increased significantly in the center where the MOX fuel assembly is located. This increase is expected

since more fission events occur in the fresh MOX fuel. The difference between the diffusion (green line in

the central zone) and the P3∗ solution (blue line in the central zone) is very limited. The thermal neutron

flux shows a strong dip in the zone with the MOX fuel, which is caused by the strong thermal absorption

resonances of the even Pu isotopes. Nevertheless, even in this case, the results of the diffusion solution and

the P3 solution are very close.

The analysis of the deviations for the fast (left) and the thermal (right) neutron flux in Fig. 2 shows the∗identical to SP3 in the 1D case

2/4

SP3 Solution Versus Diffusion Solution in Nodal Codes

Figure 2. Deviation of the fast and thermal neutron flux in the diffusion solution compared to the P3

solution for case 2 burnt UOX fuel next to fresh MOX fuel in the center (in percent).

typical behavior of a comparison of expansion solutions of different orders. Oscillations can be found at

the boundaries where the strong material gradients appear. The diffusion approximation is not valid exactly

at these positions. It is known that the results for the diffusion approximation are of reduced quality in the

vicinity of about one migration length from the material boundaries, i.e., about 7 cm. The deviation is

small apart from the area close to the material boundaries. The oscillations are smaller for the fast flux than

for the thermal flux which is caused by the mean free path of 1.85 cm of the fast neutrons being longer than

the mean free path of 0.72 cm of the thermal neutrons.

With the investigations of several cases given in the publication, it can be demonstrated that a major part of

the accuracy improvement due to the increased number of terms in the spherical harmonics expansion will

not be seen in a nodal solution approach. The traditional large, fuel assembly sized, homogenized nodes

tend to homogenize the result significantly and thus reduce the deviations of the diffusion solution. The

major differences between the diffusion and the SP3 solution appear only in the vicinity of the material

boundaries where the diffusion approximation is not fully valid. This differences partly level out due to the

integration over the full nodes.

Consequently, only a limited part of the gain achievable due to the use of the SP3 approximation will be

seen in nodal calculations.

3/4

Bruno Merk and Susan Duerigen

REFERENCES

[1] E. M. Gelbard, Application of Spherical Harmonics Method to Reactor Problems, WAPD-BT-20,Bettis Atomic Power Laboratory (1960).

[2] E. W. Morel, J. E. Larsen, and J.M. McGhee, “Asymptotic derivation of the multigroup P1 andsimplified PN equations with anisotropic scattering,” Nucl. Sci. Eng., 123, pp. 328-342 (1996).

[3] P. S. Brantley and E. W. Larsen, “The simplified P3 approximation,” Nucl. Sci. Eng., 134, pp. 1-21(2000).

[4] M. Tatsumi and A. Yamamoto, “Advanced PWR core calculation based on multi-group nodaltransport method in three-dimensional pin-by-pin geometry,” J. Nucl. Sci. Technol., 40, pp. 376-387(2003).

[5] C. Beckert and U. Grundmann, “Development and verification of a nodal approach for solving themultigroup SP3 equations,” Ann. Nucl. Energy, 35, pp. 75-86 (2007).

[6] A. M. Baudron and J. J. Lautard, “MINOS: a simplified Pn solver for core calculation,” Nucl. Sci.Eng., 155, pp. 250-263 (2007).

[7] K. Tada, A. Yamamoto, Y. Yamane, and Y. Kitamura, “Applicability of diffusion and simplified P3

theories for pin-by-pin geometry of BWR,” J. Nucl. Sci. Technol., 45, pp. 997-1008 (2008).

4/4

The 22nd International Conference on Transport Theory (ICTT-‐22) Portland, Oregon, September 11-‐15, 2011

CONSISTENT RECONDENSATION THEORY

Steven Douglass and Farzad Rahnema Georgia Institute of Technology

[email protected]

The size and complexity of many radiation transport problems, such as whole-‐core reactor neutronics, results in solution methods which attempt to coarsen the discretization of the phase space with a minimal effect on accuracy while improving the efficiency. While cross section homogenization and low-‐order angular approximations have been traditionally used to address the spatial and angular variables, improved computer technology has caused fine-‐mesh transport on large problems (e.g., whole reactor cores) to be much more competitive in terms of solution time. Despite this trend, the complex energy structure of neutron cross sections still requires the use of cross section condensation within the multigroup framework to coarsen the discretization of the energy variable. The use of cross section condensation to reduce the energy structure from continuous energy to a few coarse groups requires several approximations in the condensation procedure, and introduces known inaccuracies into the coarse-‐group solution. The method presented in this paper addresses these approximations by ensuring an exactly consistent preservation of reaction rates and fine-‐group flux spectrum during the condensation procedure. This allows the fine-‐group flux to be obtained in the coarse-‐group calculation and uses a recently developed recondensation scheme to eliminate the necessity of lattice-‐cell calculations. As a result, the coarse-‐group calculation is able to provide the accuracy of a fine-‐group calculation, fully accounting for core-‐environment and spectral effects. In this paper, the method is fully described, and demonstrated for 1D reactor benchmarks characteristic of LWRs and optically thin reactors.

ACKNOWLEDGMENT This work was supported by NEUP Award Number DE-‐AC07-‐O5ID14517. Any opinions, findings, conclusions, or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the Department of Energy Office of Nuclear Energy.


A NEW ANALYTIC SOLUTION OF THE ONE-SPEED NEUTRONTRANSPORT EQUATION FOR ADJACENT HALF-SPACES WITH

ISOTROPIC SCATTERING

Richard P. Smedley-StevensonAWE plc

Aldermaston, Reading, Berkshire

RG7 4PR, UK

and

Department of Earth Science and Engineering

Imperial College London

SW7 2AZ, UK

[email protected]

This paper derives a new analytic solution of the one-speed neutron transport equation for two isotropically

scattering half-spaces, where each half-space has its own isotropic internal source. The constant source

problem has been studied previously by Auerbach [1], and this analysis has been generalised to consider a

source whose strength varies linearly with position. This problem arises in connection with assessing the

accuracy of linear treatments of the thermal emission source in Monte Carlo thermal radiation transport

algorithms [2]. Simple expressions for the angular flux at the interface as well as its zero and first moments

are provided as a function of the scattering ratios in the two half-spaces.

We want to solve the following transport problem in the two half-spaces:

µdψ

dx(x, µ) + ψ(x, µ) =

ω10

2

∫ 1

−1ψ(x, µ′)dµ′ +

(1− ω10)

2φ11x, x > 0

µdψ

dx(x, µ) + ψ(x, µ) =

ω20

2

∫ 1

−1ψ(x, µ′)dµ′ +

(1− ω20)

2φ21x, x < 0 (1)

where −1 ≤ µ ≤ 1. Using the procedures in Auerbach’s report, the solution to this problem can be written

in terms of the Chandrasekhar’s H-functions in each of the half-spaces.

The interface angular flux values are given by:

ψ(0,+µ) =1

2

ζ(ω10, ω20)√1− ω10

√1− ω20

ω20H2(µ)

H1(µ)(φ11 − φ21)

− µ (f1(µ, ω10, ω20)φ11 + f2(µ, ω10, ω20)φ21) (2)

and:

ψ(0,−µ) =1

2

ζ(ω10, ω20)√1− ω10

√1− ω20

ω10H1(µ)

H2(µ)(φ11 − φ21)

AWE c© Crown Owned Copyright (2011)

Richard P. Smedley-Stevenson

+ µ (f1(µ, ω20, ω10)φ21 + f2(µ, ω20, ω10)φ11) (3)

where µ ≥ 0.

These expressions involve the function:

ζ(ω10, ω20) =1

2

(1− ω10)(1− ω20)

(ω20 − ω10)

ω20h21√1− ω20

− ω10h11√1− ω10

= η(ω10, ω20)(1− ω10) = η(ω20, ω10)(1− ω20) (4)

which depends on the first angular moment (h1) of the H-function in each half-space, and the functions:

f1(µ, ω10, ω20) =1

2

(1− ω10)ω20 − (√1− ω10

√1− ω20)ω20

H2(µ)H1(µ)

(ω20 − ω10)

=1

2− f2(µ, ω10, ω20) (5)

Note that the ζ function also appears in the expression for the current in the corresponding constant source

problem.

The interface scalar flux can be written as:

ψ0(0) =

∫ 1

−1ψ(x, µ′)dµ′

=1

2

√1− ω10

√1− ω20

(ω20 − ω10)

ω20h21√1− ω20

− ω10h11√1− ω10

(φ11 − φ21)

=ζ(ω10, ω20)√

1− ω10√1− ω20

(φ11 − φ21) (6)

and the current crossing the interface is given by:

ψ1(0) =

∫ 1

−1µ′ψ(x, µ′)dµ′

= −1

6(φ11 + φ21)

+1

8

(1− ω10)(1− ω20)

(ω20 − ω10)

ω20h21√1− ω20

− ω10h11√1− ω10

2

(φ11 − φ21)

= −1

3θ(ω10, ω20)φ11 −

1

3θ(ω20, ω10)φ21 (7)

where:

θ(ω10, ω20) =1

2− 3

8

(1− ω10)(1− ω20)

(ω20 − ω10)

ω20h21√1− ω20

− ω10h11√1− ω10

2

= 1− θ(ω20, ω10) (8)

The interface current is just a linear combination of the asymptotic values of the current deep within each

half-space.

Finally, we present some example results for the variation of the angular flux as a function of angle for

different scattering ratios in Figures 1 and 2.

2/5

A new analytic solution of the one-speed neutron transport equation

ACKNOWLEDGMENTS

The author would like to thank Professor Mike Williams for making the author aware of the Auerbach’s

work on the constant source variant of this problem and for many useful discussions at Imperial College.

REFERENCES

[1] T. Auerbach, ”Some applications of Chandrasekhar’s method to reactor theory,” Technical ReportBNL 676 (T-255), Brookhaven National Laboratory (1961).

[2] J. D. Densmore, ”Asymptotic analysis of the spatial discretization of radiation absorption andre-emission in Implicit Monte Carlo,” Journal of Computational Physics, 230, pp. 1116 – 1133(2011).

3/5

Richard P. Smedley-Stevenson

-1 -0.5 0 0.5 1µ

0

0.2

0.4

0.6

0.8

1

ψ(0

,µ)

ω20=0

ω20=0.5

ω20=0.9

ω20=0.95

ω20=0.99

ω20=0.999

ω10=0

-1 -0.5 0 0.5 1µ

0

0.2

0.4

0.6

0.8

1

ψ(0

,µ)

ω10=0.5

-1 -0.5 0 0.5 1µ

0

0.5

1

1.5

2

ψ(0

,µ)

ω10=0.9

-1 -0.5 0 0.5 1µ

0

0.5

1

1.5

2

ψ(0

,µ)

ω10=0.95

-1 -0.5 0 0.5 1µ

0

1

2

3

4

5

ψ(0

,µ)

ω10=0.99

-1 -0.5 0 0.5 1µ

0

1

2

3

4

5

ψ(0

,µ)

ω10=0.999

Figure 1. Plot of ψ(0, µ) verses µ for the linear source problem with φ11 = 1 and φ21 = 0, for a range ofω20 values and different fixed values of ω10.

4/5

A new analytic solution of the one-speed neutron transport equation

-1 -0.5 0 0.5 1µ

0

0.2

0.4

0.6

0.8

1

ψ(0

,µ)

ω10=0

ω10=0.5

ω10=0.9

ω10=0.95

ω10=0.99

ω10=0.999

ω20=0

-1 -0.5 0 0.5 1µ

0

0.2

0.4

0.6

0.8

1

ψ(0

,µ)

ω20=0.5

-1 -0.5 0 0.5 1µ

0

0.5

1

1.5

2

ψ(0

,µ)

ω20=0.9

-1 -0.5 0 0.5 1µ

0

0.5

1

1.5

2

ψ(0

,µ)

ω20=0.95

-1 -0.5 0 0.5 1µ

0

1

2

3

4

5

ψ(0

,µ)

ω20=0.99

-1 -0.5 0 0.5 1µ

0

1

2

3

4

5

ψ(0

,µ)

ω20=0.999

Figure 2. Plot of ψ(0, µ) verses µ for the linear source problem with φ11 = 1 and φ21 = 0, for a range ofω10 values and different fixed values of ω20.

5/5


Energy-dependent analytical solutions for the charged particle transportequation.

Tobias Geback and Mohammad AsadzadehDepartment of Mathematical SciencesChalmers University of Technology

SE-412 96 Goteborg, [email protected], [email protected]

In this work, we investigate the possibility to derive more accurate analytical approximate solutions to the

Boltzmann equation for charged particle transport, under the countinuous slowing down assumption

(CSDA). The method is based on the Fermi-Eyges equation and the corresponding analytical solution, but

additional factors are included to account for the energy dependence of the fluence of particles.

The Fermi-Eyges equation for the fluencef = f(r, θx, θy, E) of a beam of charged particles entering a

material in thez-direction, under the CSDA (including energy loss straggling), is given by

∂f

∂z+ θx

∂f

∂x+ θy

∂f

∂y+ σa(E)f −

∂

∂E(S(E)f) −

1

2

∂2

∂E2(ω(E)f) = T (z)

(

∂2f

∂θ2x

+∂2f

∂θ2y

)

(1)

Here,(θx, θy) is the projection onto the(x, y)-plane, perpendicular to the particle movement direction,σa

is the absorption cross section,S(E) andω(E) are the stopping power and straggling coefficient,

respectively, and

T (z) =

∫ 1

−1σs(E(z), µ) · µ2dµ (2)

with the elastic collision cross-sectionσs, and the average energyE at depthz.

Using separation of variables and by applying the narrow energy spectrum approximation (NESA), we are

able to derive analytical expressions that solve (1) approximately, both for the case when the energy loss

straggling term is included and when it is omitted. The analytical solutions are based on the Fermi-Eyges

solution for a Gaussian boundary condition, see [1], with correction factors to handle the energy

dependence. The correction factors are also used to computemore accurate values for the collision

coefficientT (z).

The accuracy of the analytical solutions are investigated using numerical solutions for the correction

factors obtained through finite element computations, as shown in Fig. 1. The expressions for the total

fluence are also compared to the results of Monte Carlo computations.

Tobias Geback and Mohammad Asadzadeh

z (cm)

E (

MeV

)

FEM

0 5 10 15 20 25 30 350

50

100

150

200

250

0.1

0.2

0.3

0.4

0.5

0.6

z (cm)

E (

MeV

)

Explicit (NESA)

0 5 10 15 20 25 30 350

50

100

150

200

250

300

0.1

0.2

0.3

0.4

0.5

0.6

Figure 1. Level curves for the numerical solution for the correction factor (top), and the analytical solutionunder the narrow energy spectrum approximation (NESA) (bottom). The cross-sections for electrons inwater, as in [2], were used. The dashed lines are the curvesE = E(z), and the red lines are the averageenergies at given depth for the respective solutions.

REFERENCES

[1] A. Brahme, “Investigations of the Application of Microtron Accelerator for Radiation Therapy”,Ph. D. Thesis, Institute of Radiation Physics (Stockholm University), Stockholm, 1975.

[2] L. Zhengming and A. Brahme. “High-energy electron transport”. Phys. Rev. B, 46(24), pp.15739-15751 (1992).

2/2


SPATIAL MOMENTS OF CONTINUOUS TRANSPORT PROBLEMSCOMPUTED ON GRIDS

Jeffery D. DensmoreComputational Physics and Methods Group

Los Alamos National LaboratoryP.O. Box 1663, MS D409Los Alamos, NM 87545

[email protected]

The method of moments is a technique for determining exact expressions for spatial and angular moments

of radiation distributions in infinite, homogeneous media [1]. These moments canfurther be used to

calculate other quantities of interest. For example, we consider the following transport problem in such a

medium,

µ∂ψ

∂x+ Σtψ =

Σs

2

∫ 1

−1ψ(x, µ′)dµ′ +

Q

2, (1)

where the notation is standard. If we define moments of the form

φm =

∫

∞

−∞

xmφ(x)dx , (2)

and

Qm =

∫

∞

−∞

xmQ(x)dx , (3)

with m ≥ 0, we have for the first few values ofm [2]

φ1

φ0=Q1

Q0, (4)

andφ2

φ0=Q2

Q0+ 2L2 , (5)

whereL = 1/√

3ΣtΣa is thediffusion length. Equation (4) is a statement that the flux-weighted average of

x is equal to the source-weighted average ofx, while Eq. (5) is a statement that the flux-weighted average

of x2 is greater than the source-weighted average ofx2 by 2L2.

Here, we examine how Eqs. (4) and (5) are altered when a uniform spatial grid is specified and Eqs. (2) and

(3) are replaced by

φm =∑

j

xmj φj∆x , (6)

and

Qm =∑

j

xmj Qj∆x . (7)

In these equations,∆x is the cell size,xj is the center of cellj, andφj andQj are the average scalar flux

and source in cellj, respectively. For simplicity, we will also assume that the source is piecewiseconstant

Jeffery D. Densmore

such thatQj is the value of the source in cellj, although the work that follows can certainly be adapted to

other spatial dependencies. Moments of the type of Eqs. (6) and (7) areof interest, for example, in the

Monte Carlo simulation of radiative transfer, where some quantities are treated as spatially continuous (i.e.,

the radiation intensity), while others are treated as spatially discrete (i.e., the material temperature) [3].

Under these conditions, it is not straightforward to generate expressions for moments of the type of Eqs. (2)

and (3).

We first develop versions of Eqs. (4) and (5) corresponding to Eqs.(6) and (7) using a singular

eigenfunction expansion [4]. In this case, the solution to Eq. (1) within cellj is of the form

ψ(x, µ) = ajφ+(µ)e−Σt(x−xj−1/2)/ν0 + bjφ−(µ)eΣt(x−xj−1/2)/ν0

+

∫ 1

−1Aj(ν)φν(µ)e−Σt(x−xj−1/2)/νdν +

Qj

2Σa. (8)

Here,φ+(µ) andφ−(µ) are the discrete eigenfunctions,φν(µ) is the continuum eigenfunction,ν0 is the

(positive) discrete eigenvalue,xj−1/2 is the left cell edge (xj+1/2 is the right cell edge), andaj , bj , and

Aj(ν) are coefficients yet to be determined. From Eq. (8), we have

φj =

1

Σt∆x

[

ajν0

(

1 − e−Σt∆x/ν0

)

− bjν0

(

1 − eΣt∆x/ν0

)

+

∫ 1

−1Aj(ν)ν

(

1 − e−Σt∆x/ν)

dν

]

+Qj

Σa, (9)

while continuity of the angular flux at cell edges and orthogonality of the eigenfunctions yield

ajN0e−Σt∆x/ν0 +

ν0Qj

2Σt= aj+1N0 +

ν0Qj+1

2Σt, (10)

bjN0eΣt∆x/ν0 +

ν0Qj

2Σt= bj+1N0 +

ν0Qj+1

2Σt, (11)

and

Aj(ν)N(ν)e−Σt∆x/ν +νQj

2Σt= Aj+1(ν)N(ν) +

νQj+1

2Σt. (12)

In Eqs. (10)–(12),N0 andN(ν) are the discrete and continuum full-range normalization constants,

respectively.

Equations (9)–(12) form a system of equations forφj along withaj , bj , andAj(ν). Multiplying these

expressions by1, xj , andx2j , summing the results over all cells, and applying the definitions in Eqs. (6) and

(7) allows us to write

φ0 =1

ΣaQ0 , (13)

φ1 =1

ΣaQ1 , (14)

and

φ2 =1

ΣaQ2 +

∆x

Σ2t

[

ν20

N0coth

(

Σt∆x

2ν0

)

+

∫ 1

0

ν2

N(ν)coth

(

Σt∆x

2ν

)

dν

]

Q0 . (15)

2/4

Spatial Moments of Continuous Transport Problems Computed on Grids

From these equations, we see that Eq. (4) still holds, but Eq. (5) is replaced by

φ2

φ0=Q2

Q0+ 2L2 + E , (16)

where the error termE is

E =1 − c

Σ2t

2ν30

N0

[

Σt∆x

2ν0coth

(

Σt∆x

2ν0

)

− 1

]

+

∫ 1

0

2ν3

N(ν)

[

Σt∆x

2νcoth

(

Σt∆x

2ν

)

− 1

]

dν

. (17)

Here,c = Σs/Σt is thescattering ratio, and we have employed the identity

1 = 3(1 − c)2[

ν30

N0+

∫ 1

0

ν3

N(ν)dν

]

, (18)

which follows from the singular eigenfunction expansions of1 andµ2.

As an alternative to Eq. (17), we can also develop an expression forE via a Fourier transform approach. To

proceed, we first represent the piecewise-constant source for allvalues ofx using

Q(x) =∑

j

H(x− xj−1/2)(Qj −Qj−1) , (19)

whereH(x− xj−1/2) is the Heaviside step function shifted to the cell edgexj−1/2. Then, through a

Fourier transform of Eqs. (1) and (19), we can show that the Fouriertransform of the scalar flux is

φ(k) =1

ik

tan−1 (k/Σt)

k − Σs tan−1 (k/Σt)

∑

j

(

e−ikxj−1/2 − e−ikxj+1/2

)

Qj . (20)

In addition, we have from the definition of the inverse Fourier transform

φj =1

2π∆x

∫

∞

−∞

eikxj+1/2 − eikxj−1/2

ikφ(k)dk . (21)

Substituting Eq. (20) into Eq. (21) gives

φj =∆x

2π

∫

∞

−∞

[

sin (k∆x/2)

k∆x/2

]2 tan−1 (k/Σt)

k − Σs tan−1 (k/Σt)

∑

j′

eik∆x(j−j′)Qj′dk . (22)

In a manner similar to the derivation of Eqs. (13)–(15), we multiply Eq. (22) by 1, xj , andx2j , sum the

results over all cells, and apply the definitions in Eqs. (6) and (7), but now also make use of the following

Fourier series and its derivatives,

∑

j

eik∆x(j−j′) = 1 + 2∞

∑

n=1

cos (nk∆x)

=2π

∆x

∞∑

n=−∞

δ

(

k −2πn

∆x

)

.

(23)

3/4

Jeffery D. Densmore

From this process, we again obtain Eqs. (13) and (14), while Eq. (15) becomes

φ2 =1

ΣaQ2 +

1

Σa

[

2L2 +

(

∆x

π

)2 ∞∑

n=1

1

n2

2πn/Σt∆x− tan−1 (2πn/Σt∆x)

2πn/Σt∆x− c tan−1 (2πn/Σt∆x)

]

Q0 . (24)

Constructing Eq. (16) with Eq. (24) instead of Eq. (15) yields

E =

(

∆x

π

)2 ∞∑

n=1

1

n2

2πn/Σt∆x− tan−1 (2πn/Σt∆x)

2πn/Σt∆x− c tan−1 (2πn/Σt∆x). (25)

ACKNOWLEDGMENTS

This work was performed under U.S. government contract DE–AC52–06NA25396 for Los Alamos

National Laboratory, which is operated by Los Alamos National Security, LLC, for the U.S. Department of

Energy.

REFERENCES

[1] H.W. Lewis, “Multiple Scattering in an Infinite Medium,”Phys. Rev., 78, 526 (1950).

[2] P.S. Brantley and E.W. Larsen, “Spatial and Angular Moment Analysisof Continuous andDiscretized Transport Problems,”Nucl. Sci. Eng., 135, 195 (2000).

[3] J.A. Fleck, Jr. and J.D. Cummings, “An Implicit Monte Carlo Scheme for Calculating Time andFrequency-Dependent Nonlinear Radiation Transport,”J. Comp. Phys., 8, 313 (1971).

[4] K.M. Case and P.F. Zweifel,Linear Transport Theory, Addison-Wesley Publishing Company,Reading, Massachusetts (1967).

4/4


Eigenvalues of the Anisotropic Transport Equation in a Slab

Esequia Sauter1, Fabio de S. Azevedo2, Mark Thompson3 and Marco Tulio B. M. Vilhena4PPGMAp

Universidade Federal do Rio Grande do SulBento Goncalves, P.O. Box 15080

Porto Alegre - RS, [email protected]@[email protected]

[email protected]

The critical slab problem is standard in transport theory, many works in the literature having been devoted

to developing methods to calculate the critical conditions and simulating them. These problems rely on

calculating the leading eigenvalue of the transport equation given by:

µd

dxI(y, µ) + I(y, µ) =

w

2

∫ 1

−1p(µ, µ′)I(y, µ′)dµ′, 0 < y < L (1)

where p(µ, µ′) is the scattering kernel and (1) is complemented with boundary conditions. In [1], the

leading (smallest) eigenvalue is calculated by the Case’s singular eigenfunction method applied to

semi-reflecting boundary condition and isotropic scattering. In that work, however, the reflecting

coefficient is equal on both boundaries and does not depend on the incident angle µ. In [3], benchmark

results were provided for isotropic case with non-reflecting boundary conditions. The leading eigenvalue is

obtained as well by the Case’s formalism. In [2], benchmark results were reported for the isotropic case

with non-reflecting boundary conditions by two different methodologies: Case’s formalism and the

singularity subtraction method.

The aim of this work is to derive a methodology for calculating the eigenvalues of the a transport equation

and present numerical simulations to illustrate and validate our method. In our work, we condider a

scattering kernel p(µ, µ′) in the following form:

p(µ, µ′) =M∑l=0

βlPl(µ)Pl(µ′), βM 6= 0 (2)

where Pl is the l-th order Legendre polynomial, βl are real constant and w is the eigenvalue. The boundary

is semireflective satisfying the following condition:

I(0, µ) = ρ0(µ)I(0,−µ), µ > 0 (3a)

I(L, µ) = ρL(µ)I(L,−µ), µ < 0. (3b)

We convert the problem of calculating the eigenvalues of the transport equation given by (1)-(3) in the

problem of calculating the eigenvalues of a integral operator in L2(0, L)M0 , where M0 is the number

Esequia Sauter, Fabio S. de Azevedo, Mark Thompson and Marco Tulio M. B. Vilhena

non-null coefficients βl. Defining l1, l2, . . . lM0 such that⋃M0j=1li = l : βl 6= 0, we have to solve the

following problem:βl1K

l1,l1g βl2K

l1,l2g · · · βlM0

Kl1,lM0g

βl1Kl2,l1g βl2K

l2,l2g · · · βlM0

Kl2,lM0g

......

. . ....

βl1KlM0

,l1g βl2K

LM0,l2

g · · · βlM0KlM0

,lM0g

Jl1(y)Jl2(y)

...JlM0

(y)

=1

w

Jl1(y)Jl2(y)

...JlM0

(y)

(4)

where K li,ljg are integral operators.

Therefore in our work we solve the eigenvalue problem given by equation (4). In order to solve

numerically this problem we construct a finite dimensional approximation for each of the integral operators

involved. Then we approximate L2(0, L) by a finite-dimensional spaceM. The projection of the integral

operator inM is a real matrix, whose spectrum is calculated by stardard linear algebra methods.

So letM be a N -dimensional subspace of L2(0, L) spanned by the vectors ψiNi=1 and define the

operators K lkg as

K lkg = PK lk

g

where P : L2(0, L)→M is the orthogonal projector from L2(0, L) intoM defined by

‖Pq − q‖L2(0,L) = infφ∈M

‖φ− q‖L2(0,L).

From the standard theory of Hilbert spaces, P is uniquely determined and is an bounded self-adjoint

operator in L2(0, L). (see [5] for futher details) Now denote ϕ the isomorphism between CN andM given

by:

ϕ :M ←→ CN

q =N∑j=1

qjψj ←→[q1, q2, · · · , qN

]T= q.

Using this terminology we define the matrices W lk by

W lk = ϕK lkg ϕ−1 = ϕPK lk

g ϕ−1.

In order to obtain the coefficients of W lk, we consider q =∑Nj=1 q

jψj and u = K lkg q =

∑Nj=1 u

jψj . The

coefficients uj are calculated taking the inner product with each element of the base ofM:

N∑j=1

uj 〈ψj , ψi〉 =N∑j=1

qj⟨K lkg ψj , ψi

⟩, ∀1 ≤ i ≤ N

which admits the matricial form Cu = Aq where q = ϕq and u = ϕu. The matrices A and C are given

by:

Aij =⟨K lkg ψj , ψi

⟩and Cij = 〈ψj , ψi〉 (5)

2/4

Eigenvalues of the Anisotropic Transport Equation in a Slab

we note that ⟨K lkg ψj , ψi

⟩=

⟨PK lk

g ψj , ψi⟩=⟨K lkg ψj , Pψi

⟩=⟨K lkg ψj , ψi

⟩Therefore W lk = C−1A.

We then consider the spaceM =Ms ⊕Mc where the spacesMs andMc are generated respectively by

the functions an sin(2πnL y

)Nn=1 and bn cos

(2πnL y

)Nn=0 with the coefficients:

an =

√2

Land bn =

√

1L , n = 0√2L , n > 0

Here we have chosen the coefficients an and bn so that the basis forM is orthonormal under the usual

inner product defined by

〈f(y), g(y)〉 =∫ L

0f(y)g(y)dx

where g(y) indicates the complex conjugate of g(y). The coefficients of the matrices W lk are obtained

using (5). Since we have chosen an orthonomal basis formM, we have only to calculate the inner products

related to the matrix A, the matrix C being equal to identity. Each coefficient of the matrix A is then given

by a triple integral of the form:⟨K lkg ψi, ψj

⟩=

∫ 1

−1

∫ L

0

∫ L

0klk(µ, s, y)ψi(s)ψj(y)dsdydµ

here klk(µ, s, y) is the kernel representing the integral operator K lkg . The complex conjugation was left out

here because we are dealing only with real funcions. Fortunately the integration in the variables y and s

involves only the product of trigonometric and exponential functions, which can be solved exactly, leading

to expressions in µ which need to be solved numerically.

Finally we report numerical results for the isotropic (β0 6= 0), linearly anisotropic (β0 6= 0 and β1 6= 0) and

Rayleigh (β0 6= 0 and β2 6= 0).

ACKNOWLEDGMENTS

This work was conducted as part of a project of the National Institute of Science and Technology on

Innovative Nuclear Reactors (INCT-Brazil). F. Azevedo was supported by a postdoctoral fellowship of the

CAPES(Brazil). E. Sauter was supported by a doctoral fellowship of CNPq(Brazil). Professor Vilhena

thanks the CNPq for the partial financial support of this work.

REFERENCES

[1] M. A. Atalay, “The critical slab problem for reflecting boundary conditions in one-speed neutrontransport theory”, Ann. Nucl. Energy, 23, pp. 183-193 (1996)

3/4

Esequia Sauter, Fabio S. de Azevedo, Mark Thompson and Marco Tulio M. B. Vilhena

[2] S. Naz, and S. Loyalka. “One speed criticality problems for a bare slab and sphere: Somebenchmark results”, Ann. Nucl. Energy, 35, pp. 2426-2431 (2008)

[3] A. Rawat, and N. Mohankumar “Benchmark results for the critical slab and sphere problem inone-speed neutron transport theory”, Ann. Nucl. Energy, In Press (2011)

[4] F. S. Azevedo, E. Sauter, M. Thompson, M. T. Vilhena Existence theory and simulations forone-dimensional radiative flows, Ann. Nucl. Energy , 38, pp. 1115-1124 (2011)

[5] W. Rudin Functional Analysis, Mcgraw-Hill, New York USA (1973).

4/4


IN-WATER OCEAN OPTICS INVERSION ALGORITHM

N. J. McCormickUniversity of Washington

Department of Mechanical EngineeringSeattle, WA 98195-2600

[email protected]

Eric RehmUniversity of WashingtonSchool of Oceanography

College of Ocean and Fishery SciencesSeattle, WA 98195-5350

[email protected]

We have derived and tested an analytic algorithm that enables the ratio of the backscattering to absorption

cross sections bb/a to be estimated from light field measurements of only the vertically upward radiance

(angular flux) Lu(z) and the downward directed planar irradiance (downward partial current) Ed(z) for a

given wavelength λ. Because of the strongly forward scattering of typical ocean waters,[1, 2] the ratio bb/a

is a natural parameter with which to characterize the optical properties of such waters.[3]

The algorithm is for use with in-water, depth-dependent measurements sufficiently beneath the surface that

the light at depth z is approaching the asymptotic regime, z → zas, which typically occurs within a few

optical depths. The algorithm complements an analogous algorithm applicable for near-surface

measurements.[4] To circumvent the lack of detailed information about the strongly anisotropic scattering

of all ocean waters at all wavelengths, the angular scattering (phase) function is assumed to be of the

delta-isotropic shape with one adjustable parameter F . That free parameter can be fit using data from a

limited portion of an experimental data set or by a priori computer simulations, for example with the

widely used Hydrolight c© program[5].

The inversion of light field measurements to obtain inherent optical properties of ocean waters has been

extensively investigated[6] and the classic difficulties of ill-posedness and nonuniqueness associated with

such problems has been illustrated.[7] Although our algorithm provides only a single estimate of bb/a for a

given wavelength, it can be used either a) to provide an approximate estimate of that inherent optical

property to help characterize the amount of chlorophyll in the water or b) to provide an initial starting guess

for algorithms employing a look up table or iterative use of forward computations with a computer

program.

Because of the azimuthal symmetry of the light field detectors, the algorithm is based on the simplest form

of the radiative transfer equation,

µ∂

∂zL(z, µ) + cL(z, µ) = b

∫ 1

−1f(µ′, µ)L(z, µ′)dµ′ , z ≥ 0 ,

N. J. McCormick and Eric Rehm

with L(z, µ) and f(µ′, µ) the azimuthally integrated radiance at depth z and the scattering phase function,

respectively. The parameters c and b are the total and scattering cross sections, respectively. Because it is

assumed that z → zas, the algorithm is valid for any surface illumination.

The efficacy of the algorithm is tested against such a composite data set acquired at eight stations in the

North Atlantic in which the light field measurements and separate, independent measurements of the

backscattering and absorption cross sections were obtained.[8]

REFERENCES

[1] T. J. Petzold, “Volume Scattering Functions for Selected Ocean Waters,” SIO Publ. 72-78, ScrippsInst. Oceanogr., La Jolla, Calif. 79 pp. (1972). Condensed as Chapter 12 in Light in the Sea, J. E.Tyler, ed., Dowden, Hutchinson & Ross, Stroudsberg, pp. 150–174 (1977).

[2] C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, New York, 1994).

[3] H. Gordon, O. B. Brown, and M. M. Jacobs, “Computed relationshps between the inherent andapparent optical properties of a flat homogeneous ocean,” Appl. Opt. 14, 417–427 (1975).

[4] A. Kaskas, M. C. Gulecyuz, C. Tezcan, and N. J. McCormick, “Analytic algorithms for determiningradiative transfer optical properties of ocean waters,” Appl. Opt. 45, 7698–7705 (2006).

[5] C. D. Mobley and L. K. Sundman, Hydrolight 5.0, Ecolight 5.0 Technical Documentation (SequoiaScientific, Inc., Redmond, WA, 2008).

[6] H. R. Gordon, “Inverse methods in hydrologic optics,” Oceanologia 44, 9–58 (2002).

[7] M. Defoin-Platel and M. Chami, “How ambiguous is the inverse problem of ocean color in coastalwaters?,” J. Geophys. Res. 112, C03004, doi: 10.1029/2006JC003847, 2007.

[8] E. Rehm, University of Washington Ph.D. dissertation (in preparation).

2/2


On Boundedness of Higher Velocity Moments for theLinear Boltzmann Equation with Diffuse Boundary Conditions

Rolf PetterssonDepartment of Mathematics

Chalmers University of TechnologySE-412 96 Goteborg, SWEDEN

[email protected]

Abstract. The linear Boltzmann equation is frequently usedfor mathematical modelling in physics, e.g.

for describing the neutron distribution in reactor physics. In a long range of our earlier papers we have

studied the existence and uniqueness problems for the space- and time-dependent equation with general

”nonheating” boundary, (e.g. with specular reflections), both in the elastic and also in the inelastic (granular)

collision case, cf. ref. [1] - [5]. Here we will study the casewith diffuse (Maxwellian) boundary; cf. also ref.

[6] for the stationary equation case. Furthermore we discuss the question on convergence to equilibrium,

when time goes to infinity, using a general H-theorem.

REFERENCES

[1] R.Pettersson,On solutions and higher moments for the linear Boltzmann equation with infinite rangeforces. IMA J.Appl.Math.38, 151-166 (1987).

[2] R.Pettersson,On solutions to the linear Boltzmann equation with general boundary conditions andinfinite range forces. J.Stat.Phys.59, 403-440 (1990).

[3] R.Pettersson,On weak and strong convergence to equilibrium for solutionsto the linear Boltzmannequation.J.Stat.Phys.72, 355-380 (1993).

[4] R.Pettersson, On solutions to the linear Boltzmann equation for granular gases.Transp.Th.Stat.Phys.33, 527-543 (2004).

[5] R.Pettersson,n global boundedness of higher velocity moments for solutions to the linear Boltzmannequation with hard sphere collisions.Il Nuovo Cimento 33, 189-197 (2010).

[6] L. Falk Existence of solutions to the stationary linear Boltzmann equation.Transp.Th. Stat.Phys.32,34-62 (2003).


ON THE SPEED OF HEATH WAVE

Mih aly MakaiDepartment of Nuclear Techniques

Budapest Technical University3 Muegyetem rkp. R-317

Budapest, Hungary, [email protected]

We revisit the problem of heat conductance and diffusion, two remarkable transport processes

characterized by instantaneous actions. We show that the assumption of local thermal equilibrium sets a

limit to the speed of change in the distribution function of a statistical systemS. A statistical system

consists of a large number of components, and its state is changed through a large number if interactions

among ita components. A macroscopic phenomenon is obtained by averaging thus it would be rather

unexpected if any macroscopic phenomenon would exhibit a speed faster than the change rate of the

distribution function itself. Using Onsager’s approximation, we show that the balance equations of the

extensive parameters also have solutions with finite velocities involved. At the same time the infinite speed

is obtainable when second order terms are neglected. We show how the presented technique is applied in

plasma physics to determine the speeds of the physical processes in a plasma.

1 The problem

Wigner [10][p. 235], speaking about the derivation of the neutron diffusion equation remarks ”The correct

P1 approximation to the Boltzmann equation is not the first order diffusion equation but the second-order

equation (9.18), which is called the ”telegrapher’s equation”. The general solution of the telegrapher’s

equation shows the phenomenon of retardation; that is, the solution has a well-defined wave front, in

addition to a residual disturbance which persists at all points traversed by the wave front. The telegrapher’s

equation thus lies between the simple wave equation, whose solutions have a wave front but no residual

disturbance, and the diffusion equation, whose solutions have a residual disturbance but no wave front.”

Speaking of heat waves, Joseph and Preciosi [1] discusses the problem of diffusion and heat conductance.

They refer to a stream of contributors, and say: ”Two problems are the source of this stream: the problem

of infinite wave speed and the problem of second sound.” Joseph and Preciosi [1] remarks that the infinite

speed of propagation in diffusion theory appeared as early as 1948 in the work of Cattaneo [11].

The present work starts from the non-equilibrium statistical physics setting. We discuss a statistical system

S, its volume isV , its mass isM . S is in local thermal equilibrium (LTE). In LTE, the first law of

thermodynamics holds.

We assume that sufficiently small elementary volumes exist inS such that the elementary volume can be

Mihaly Makai

regarded as infinitesimal but is large enough the intensive parameters (temperature, pressure, chemical

potential) are meaningful. Starting out from the balance equations of the extensive parameters inS, it is

shown that the general form of an equation for a disturbance is the telegrapher’s equation thus a

disturbance propagates with finite speed. The heat conductance is explicitly discussed.

It is shown that in local thermal equilibrium (LTE) the distribution function of a statistical system may

change only with limited speed.

The internal energyU and the entropyS are expressed with the help of the distribution functionf(r,v, t)

as

U =∫

12mv2f(r,v, t)d3v (1)

S =∫

ln f(r,v, t)f(r,v, t)d3v. (2)

From all these, we obtain the following condition:

∇U

∂tU=

∇S

∂tS. (3)

That condition imposes the following restriction on the particle density distribution functionf(r,v, t):

C(r, t)∇f(r,v, t) = ∂tf(r,v, t). (4)

Equation (4) is a hyperbolic equation with one characteristic speedC.

Below we use the following notation.X,Y are the vectors of extensive and intensive parameters.Je,Jo

are the conductive and convective currents. Finally,Λ is defined byJe = Λ∇Y.

Assume the validity of Onsager’s approximation: in which the conductive current changes as

Je(X) = Λ∇Y(X) = ΛN∇X.

Here elements of matrixN are

Nij =∂Yi

∂Xj, i, j = 1, . . . , n; (5)

and∇X stands for three derivatives ofX. Furthermore

Jo(X) = vρX (6)

and

∂Xρ

∂t+ vρ∇X + ΛN∆X = q (7)

2/??

On the Speed of Heath Wave

Equation (7) is the telegrapher’s equation, its solution travels with a finite speed and has a wave front.

Analyze now the internal energy:

ρ∂u

∂t+ ∇Jq = 0. (8)

Here it is assumed that thermal expansion may be neglected. The phenomenology gives the following form

for the heat fluxJq:

Jq = Lqq∇(

1T

)+

n−1∑

k=1

Jk∇µk − µn

T. (9)

In a one component, homogeneous system, the second term is zero. WhenV is kept fixed, the internal

energy is

u = cvT (10)

wherecv is heat capacity at constant volume.

Now let us return to equation (8). We have seen that

∇(

1T

)= − cv

u2∇u,

hence

Jq = Lqqcv

u2∇u, (11)

assuming that in a homogeneous systemLqq, cv are position independent, we get

∇Jq = −Lqqcv

(1u3

(∇u)2 − ∇∇u

u2

)(12)

Substituting thatJq into (8), we get

ρ∂u

∂t= −Lqqcv

(1u3

(∇u)2 − ∇∇u

u2

)(13)

At the same time [6][p. 41]

ρ∂u

∂t= λ∇∇T. (14)

That expression is obtained from (13) if the term(∇u)2 is neglected in accordance with Onsager’s concept

of near equilibrium:

ρcv∂T

∂t= −Lqq

(−∇∇T

T 2

)(15)

In other words, when the second order(∇u)2 is neglected, we obtain the traditional heat equation.

Finally, it will be shown that the above outlined thought has been applied in plasma physics [14], [13] to

assess plasma stability, and the possible speeds of physical processes.

3/??

Mihaly Makai

REFERENCES

[1] D. D. Joseph, L. Preziosi : Heat Waves,Rev Mod Phys., 69, 41 (1989)

[2] D. D. Joseph, L. Preziosi: Addendum to the paper ”Heat waves”,Rev Mod Phys.62, 375 (1990)

[3] R. Zwanzig: Memory Effects in Irreversible Thermodynamics,Phys. Rev.124, 983 (1961)

[4] A. A. Le-Zakharov, A. M. Krivtsov: Molecular Dynamics Investigation of Heat Conduction in

Crystals with Defects,Doklady Physics, 53, 261 (2008)

[5] T. Tel, T., J. Vollmer, W. Breymann: Transient chaos: the origin of transport in drivren systems,

Europhys. Lett., 35, 659 (1996)

[6] S. R. de Groot, P. Mazur:Non-Equilibrium Thermodynamics, North Holland, Amsterdam, (1962)

[7] J. Keizer: A theory of spontaneous fluctuations in viscousous fluids far from equilibrium,Phys.

Fluids, 21, 198, (1978)

[8] T. Kato: Perturbation theory for linear operators, Springer, Berlin, (1966)

[9] E. T. Jaynes: Information Theory and Statistical Mechanics,Phys. Rev.106, No. 4, 620-630, (1957)

[10] A. Weiberg, E. P. Wigner:The physical Theory of Neutron Chain Reactors, The University of

Chicago Press, Chicago, (1958)

[11] C. Cattaneo: Sulla condizione de calore,Atti del Semin. Mat. e Fis. Univ. Modena, 3, 3, (1948)

[12] J. Stachel (Ed.):Einstein’s Miraculous Year, Princeton University Press, New Jersey, (1998)

[13] W. Dai, P. R. Woodward: A Simple Finite Difference Scheme for Multidimensional

Magnetohydrodynamical Equations,J. Comp. Phys, 142, 331-369 (1998)

[14] P. L. Roe, D. S. Balsara: Notes on the Eigensystem of Magnetohydrodynamics: SIAM,J. Appl.

Math.56, 57-67 (1996)

4/??


DENSITY DISTRIBUTION OF THE MOLECULES OF A LIQUID

IN A SEMINFINITE SPACE

V. Molinari, B. Ganapol1

(DIENCA) Energy Department and D. Mostacci

University of Bologna Bologna, Italy

[email protected]

1. INTRODUCTION The variation of the number density in a liquid, as a consequence of a boundary discontinuity, is often disregarded in many applications. Nevertheless, this subject is of growing in interest in many fields such as in the study of thin films or in biology, where investigations are becoming more and more detailed with decreasing dimensions [1,2,3]. The aim of this work is to obtain an estimate of the number density distribution in a liquid as a function of the distance from a discontinuity when the molecules of the system are in the state of thermodynamic equilibrium (the velocity distribution function is the Maxwellian distribution) and the average velocity is equal to zero. The physical situation refers to a liquid composed of certain molecules surrounded by a medium of different molecules that can be in various phases (solid, liquid or gas). We will also consider the case of a liquid neglecting the effects of the surrounded medium (in other words a liquid bordering vacuum). The problem of a liquid in contact with its vapour has been treated in [4]. 2. THEORY Kinetic theory is a microscopic theory of the processes in many body systems that can be used to analyze a great variety of phenomena. Here, our study is performed in the framework of this theory (Vlasov equation) to obtain some liquid properties as a consequence of discontinuities. The first equation of the of BBGKY hierarchy is [5]

221

2,12,1

1

11

1

11

1 vdrdvf

mF

vf

mF

rfv

tf

∂∂⋅−=

∂∂⋅+

∂∂⋅+

∂∂

∫ (1)

as obtained from the Liouville equation [2]. This equation describes the microscopic temporal behaviour of the particles of the system in terms of position and velocity. In Eq(1), ),,( 111 tvrf

denotes

the single particle, or simple, velocity distribution function, ( )tvvrrf ,,,, 21212,1

, is the two particle, or

double, distribution function, 1F

denotes the external force on a particle of mass m at ( tr ,1

) and 2,1F

is

the inter-particle force between two particles located at positions 1r

and 2r

. Since the collision integral

on the right hand side depends on 2,1f , the correlation in position and velocity between colliding

particles is explicitly considered. We observe that this equation is valid up to the order associated with the model used for the intermolecular potential as described below.

1 Visiting Fellow of the Institute of Advanced Studies, University of Bologna, from the Department of Aerospace and Mechanical Engineering, University of Arizona.



2

We consider a system in thermodynamic equilibrium and whose particles are not subject to an external

force, so that 01 =∂∂ tf . In addition, we assume that the inter-particle force 2,1F

is a central potential

independent of velocity In this case, the double distribution function, 2,1f , is given by the product of

two Maxwellian distributions ( )vfM

( ) ( ) ( )212,1212,1 , vfvfnvvf MM

= , (2)

where ( )212,1 ,rrn

denotes thetwo-particle number density

( ) ∫= 212,1212,1 , vdvdfrrn

. (3)

Observing that

( )111

vfvKTm

vf

MM

−=∂∂

, (4)

where K is the Boltzmann constant and T is the temperature, Eq(1) becomes

∫=∂∂

22,12,11

1 rdFnrnKT

(5)

where ( )11 rn

is the single particle density ( ) ∫= 22,111 rdnrn

. (6)

If the interaction between particles is negligible, 02,1 ≈F

, and the collision integral on theright hand

side of Eq (5) vanishes. Moreover, if the correlation between pairs of particles can be disregarded, then 212,1 nnn ≈ . (7)

In the case of a liquid phase, the effects of the interaction between molecules cannot be disregarded however. An estimate of this effect is obtained resorting to inclusion of a self-consistent field (Vlasov field). The self-consistent field is important to describe the dynamic behaviour of a system where every molecule interacts simultaneously with a large number of surrounding molecules and the correlation between pairs of molecules can be disregarded since its effect becomes negligible with respect to the collective interaction. This physical situation occurs when the range of the

intermolecular force, 2,1F

, is large in comparison to the average distance between molecules, as is the

case of a plasma or liquid state. From this point of view, interactions in a liquid system are treated somewhat like is done in plasma physics, where a specific parameter, the “plasma parameter”, establishes when there are enough particles in the so-called “Debye sphere” and simultaneous multiple collisions prevail.


3

Then, for the liquid state, we can assume Eq(7) to be valid and the collision integral becomes ∫ ∫ == '

122,12122,12,1 FnrdFnnrdFn

(8)

where 'F

is the unknown self-consistent field that must be determined on the bases of the pairwise

interaction force, 2,1F

and the assumed interaction potential. Now Eq(5) can be written as

( )r'Fr

n

n

1KT

1

1

1

=∂∂

(9)

2.1. Self-consistent field The first step is to find the self-consistent field, 'F

from

( ) ( ) 21

2,121 dr

rrnrF

V

∂∂

=′ ∫ϕ

, (10)

where 2,1ϕ is the pair wise interaction potential between two molecules located at positions r and 2r

respectively. To calculate the Vlasov self-consistent field, several phenomenological potential models have been proposed which, more or less, describe the real multiple force interaction mechanism [6,7]. In this work the Sutherland pair potential model is used because it is mathematically simple and the predictions on the liquid properties are expected to be qualitatively consistent with the behaviour of many real fluids. The phenomenological Sutherland pair potential model given by [6]

( )α

σε−=ϕ

r4r2,1 for σ≥r (11a)

( ) ∞=r2,1ϕ for σ<r (11b)

with the corresponding force 2,1F

rr

F ˆ412,12,1 +=∇−= α

ασεαϕ

. (12)

The Sutherland parameters ε and σ correspond, respectively, to the depth of the potential well (divided by 4) and to the distance of closest approach in the Sutherland approximation, essentially related to the molecular radius. Both parameters are tabulated for numerous molecules, see e.g. [6] To calculate F ′

, a molecule located at the point ( )z,0,0 will be considered, and the force from all the surrounding liquid will be determined using the Sutherland potential. We assuming slab symmetry, in which density depends only on the z – coordinate and consider an elementary volume dV at a location defined by the coordinates of a spherical reference system


4

centered on the molecule of interest. By finding the Cartesian force components and integrating the forces over all space, after some algebra, we find the self-consistent field for the semi-infinite medium

( )( )

( )( )

−+

−−= ∫ ∫

− +∞

+

σ

σ

ζζ

ζζζζπεσ

z

z

dz

ndznF

055

68' ; (13a)

whereas, for the case of a slab of thickness 2a

( )( )

( )( )

−+

−−= ∫ ∫

−

− +

σ

σ

ζζ

ζζζζπεσ

z

a

a

z

dz

ndznF 55

68' . (13b)

In deriving Eqs(13), a minimum approach in a Vander Waal’s potential was introduced. If the density variation is mild, ( )n z can be expanded in Taylor series retaining only the first few terms

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]54

3

33

3

32

2

2

zO!4

z

dz

znd

!3

z

dz

znd

2

z

dz

zndz

dz

zdnznn −ζ+

−ζ+

−ζ+

−ζ+−ζ+=ζ (14)

Neglecting terms of order 3 and higher, and substituting into Eqs (13) (after some algebra), the following equations are obtained:

+

−σ

+πεσ=22

2

3346

z4

1

dz

nd

z

12

3

1

dz

dn

z4

n8)z('F (15a)

for the semi-infinite case and

( ) ( ) ( ) ( )

( ) ( )

−−

++

+

−−

+−

σ+

−−

+πεσ=

222

2

333446

za

1

az

1

4

1

dz

nd

za

1

az

12

3

1

dz

dn

za4

1

az4

n

8)z('F (15b)

for the case of the slab. 3 THE SEMI-INFINITE LIQUID Consider a liquid occupying a semi-infinite space, 0z ≥ . The other part of the space, 0z < , can be empty or filled with a material in gas, liquid or solid phase. We will consider here only the first case. 3.1. z < 0 is empty The equation for the number density distribution, Eq. (9), for the one dimensional case becomes

)(' nFKTn

dzdn

= (16)


5

with nn ≡1 and, substituting the expression of 'F in Eq.(16), we obtain

+

−+= 22

2

334

6

4

112

3

1

4

81

zdznd

zdzdn

zn

KTdzdn

n σπεσ

,

which also may be written as

02

12

3

4'"

2

2

6332 =+

−

−+

znKT

znznnn

πεσσ (17a)

that is to say, a nonlinear (singular) differential equation of second order. This equation can be recast into self-similar form by defining the following dimensionless quantities 0; nnzx == χσ where

( )∞== znn0 to give

023

4

3

8'"

2

2

30

32 =+

−

−+

xnKT

xx χ

σπεχχχχ (17b)

For the value of ( )910 mσ −≥ , the parameter 62πεσ

KT can be disregarded and Eq.(17b) becomes

03

4

3

8 32

22 =+

−+ χχχ

dxdxx

dxdx (18)

With several variable transformations, the solution can be shown to be

( ) ( )303 3

1 1 1 0, ;B xkx C x e A k A B xχ −= Φ − (19)

where Φ is the confluent hypergeometric function [8] and k is the largest solution to

( ) 0102 =+−+ akAk

with

9

20 =A , 0

8

9B = ,

9

1=a , kAA 201 += .

To find C1, we can establish the following asymptotic behaviour

( ) ( )( )

( )

⋅⋅⋅⋅⋅++−

+−Γ

Γ=

30

1

10

11

11

xBkkA

kABACx kχ (20a)

and since ( ) 1=∞→xχ , we find


6

( )( )1

011 A

BkAC

k

Γ−Γ

= . (20b)

4. THE NUMERICAL SOLUTION A companion effort has been the consideration of the purely numerical solution to Eqs(17). In particular, a numerical algorithm, based on Taylor series, has been developed to solve Eq(17b). Since the ODE is singular, special attention must be given to the singularity at the origin, which comes about because of the discontinuity in material properties. The algorithm begins with the solutions determined from the method of Frobenius near the origin and continues with a conventional Taylor series solution thereafter to the initial value problem. On comparison with the analytical solution of Eq(19), agreement to 10 figures can easily be achieved. The same procedure will be attempted for Eq(20a), where the solution is first expanded in the neglected parameter 02 / LKT n Gπ . This will now involve

additional integration to evaluate a particular solution, since the resulting ODEs are no longer homogeneous. 5. A PHYSICAL INTERPRETATION OF RESULTS First recall the case considered in 3.1, that of an interface between a liquid and vacuum. Molecules in the bulk of the homogeneous liquid are subject to binding forces having no preferential direction, or in other words, isotropic forces. Hence density is to be expected constant in regions far enough from interfaces. However, as an interface is approached (i.e., at distances of the order of the tens of molecular diameters) binding forces become peaked toward the bulk of the liquid, and it can be expected that this prevailing inward force would lead to an increase in the density, which is the opposite to what is commonly assumed. The aim of the present work was to verify the existence of this phenomenon: figure 1 confirms this prediction, insofar as the attractive forces present are of the van der Waals type. This behaviour will be of particular importance in investigating surface tension. If an interface with a solid surface is considered, on the other hand, this behaviour can be significantly changed, or even reversed, depending upon the relative forces between the molecules belonging to the two phases. This is the subject of further work already in progress.


7

Fig. 1. Normalized liquid density distribution

Dimensionless Distance from Wall

0 20 40 60 80 100 120

Nor

mal

ized

Den

sity

0.0

0.2

0.4

0.6

0.8

1.0

Expanded View

0 20 40 60 80 100 120

0.830

0.835

0.840

0.845

0.850

0.855

ACKNOWLEDGMENT The second author thanks the University of Bologna Institute for Advanced Study for hosting his stay during the completion of this work.

REFERENCES [1] O. Sinanoglu, J. Chem. Phys. 75 (1981) 463-468.

[2] Plech1, U. Klemradt and J. Peisl, J. Phys.: Condens. Matter 13 (2001) 5563–5576.

[3] R. Tadmor, J. Phys.: Condens. Matter 13 (2001) L195–L202.

[4] A. Frezzotti, L. Gibelli, F. Lorenzani, Phys. Fl. 17 (2005) 012102-1.

[5] J.L. Delcroix, A. Bers, Physique des plasmas, Interéditions, Paris, 1994.

[6] J.O. Hirschfelder, C.F. Curtis and R.B. Bird, Molecular Theory of Gases and Liquids, Wiley, New York, 1954.

[7] R. K. Pathria, Statistical Mechanics, Pergamon Press, Oxford U.K., 1972.

[8] M. Abramowitz, I.A. Stegun, Handbook of mathematical functions, Dover Publications, New York 1964.


Friday, September 16, 2011

Deterministic Transport III

8:35 am Exploration of an Adaptive Angular Solver in Neutron Transport - D. Lathouwers, D.J. Koeze

9:00 am An Angular Multigrid Acceleration Method for SN Equations with Highly Forward-Peaked Scattering

- B. Turcksin, J.C. Ragusa, J.E. Morel

9:25 am Methodology for Decomposition into Transport and Diffusive Subdomains for the Linear Discontinu-

ous Method - N.D. Stehle, D.Y. Anistratov

9:50 am A Piecewise-Linear Discontinuous Spatial Discretization for Polyhedral Grids in 3D Cartesian Ge-

ometry -T.S. Bailey, W.D. Hawkins, M.L. Adams

Transport Applications I/Reactor Physics III

10:45 am Dosimetry Calculation of Two Commercially Available Iodine Brachytherapy Seeds Using Spencer-

Lewis 3D Multigroup SN Transport Code - N. Ayoobian

11:10 am Radiation Field Characterization, Shielding Assessment and Activation Calculations for the MYRRHA

Design - A. Ferrari, B. Merk, and J. Konheiser

11:35 am Two-Region Diffusion Model for Improved Analysis of ADS Experiments - V. Glivici-Cotruta, B. Merk

12:00 pm A Mesh-Free Approximation of Spatial Neutron Flux Distribution - D. Altiparmakov


Exploration of an Adaptive Angular Solver in Neutron Transport

D. Lathouwers & D.J. KoezeDepartment of Radiation, Radionuclides and Reactors

Delft University of TechnologyMekelweg 15

2629 JB Delft, [email protected], [email protected]

The paper investigates an angular adaptive solver of the neutron transport equation. By using an adaptive

solver, one can solve neutron transport problems at less computational expense. The computational power

is spent on the parts that contribute most to errors in the solution, e.g. getting rid of ray effects. The subject

of the present work is that of angular adaptivity, spatiallyadaptive refinement has been reported on

elsewhere [1] [2]. The angular and spatial adaptive methodscan in principle be combined into a combined

adaptive solver.

The discretization of both spatial and angular part is done using a Discontinuous Galerkin method. In the

present work we use piece wise constant basis functions on the unit sphere to represent the angular flux. In

the one dimensional case the patches are ring shaped. Local refinement is performed by locally cutting

these rings into new, smaller rings. The discrete transportequation with isotropic scatter for the fluxψ on a

single patch is[

Ω′pUψp

]

left+

[

Ω′pUψp

]

right+ (−Ω′

pK + σtM)ψp=σs4πMφ+ S, (1)

where the matricesK,M andU are 2 by 2 matrices and follow from the linear spatial basis functions, and

Ω′p is a weighted angle on the patch. Only upwind information is used in the boundary streaming terms.

Depending on the direction under consideration one takes either the upwind or current flux.

Since all angular basis functions are independent, this equation can still be solved by a sweeping algorithm,

which is important for maintaining Discrete Ordinates efficiency. The flux moments required for the scatter

operator are computed by quadrature, which follows naturally from the discretization using patches.

Figure 1 shows the error in the detector response for two testcases where the angular patches are uniformly

refined. (i) A homogeneous slab, with uniform source and detector for both optically thick and thin media

and (ii) the Reed test case using a coarse and a fine distribution of the spatial elements.

Refinement strategy is either traditional or goal oriented.Each patch in the angular variables can be refined

independent of other patches and independent of the spatialelement. When using a goal oriented

refinement strategy the local refinement is based on the dual weighted residual approach [3], which

requires both the computed forward and adjoint solutions corresponding to the problem. The adjoint flux is

estimated in a more refined space than the forward problem. Patches that contribute most to the error are

Danny Lathouwers and Dion J. Koeze

1e-14

1e-13

1e-12

1e-11

1e-10

1e-09

1e-08

1e-07

1e-06

1 2 3 4 5 6 7 8 9

resi

dual

level of angular refinement

Coarse Reed TestFine Reed Test

Thick AbsorptionThin Absorption

Figure 1. Error reduction with homogeneous refinement. In each level of refinement the number ofrings is doubled. The ratio of scatter and total cross section is .99 in the thick case and .5 in the thincase.

marked for refinement, after which a better forward solutioncan be computed. We present the

mathematical derivation of theSN like equations, we discuss some implementation issues required for

local refinement, and we investigate the efficiency of the method by applying the developed algorithms to

several test cases where angular refinement is important forerror reduction.

REFERENCES

[1] D. Lathouwers, “Spatially Adaptive Eigenvalue Estimation for theSN Equations on UnstructuredTriangular Meshes,”accepted for publication in Annals of Nuclear Energy.

[2] D. Lathouwers, “Goal-oriented Spatial Adaptivity For theSN Equations on Unstructured TriangularMeshes,”accepted for publication in Annals of Nuclear Energy, 38, pp. 1373-1381 (2011).

[3] W. Bangerth and R. Rannacher,Adaptive Finite Element Methods for Differential Equations,Birkhauser (2003).

2/2


An Angular Multigrid Acceleration Method for Sn Equations with HighlyForward-Peaked Scattering

Bruno TurcksinDepartment of Nuclear Engineering

Texas A&M UniversityCollege Station, Texas 77843-3133

[email protected]

Jean C. RagusaDepartment of Nuclear Engineering


[email protected]

Jim E. MorelDepartment of Nuclear Engineering


[email protected]

The angular multigrid method for one-dimensional slab geometry has proved to be very effective to solve

the Sn equations with highly forward-peaked scattering [1]. However, the extension of this method to

multidimensional geometries is unstable [2]. A diffusive filter to the angular multigrid corrections has to be

added as a stabilizer technique within the standard preconditioned Source Iteration. For instance, the

scheme for a Sn calculation inRd (d = 2, 3) works as follows :

1. Perform a transport sweep for the Sn equations with a Pn+a (a = 0 if d = 2 and 1 if d = 3)

expansion of the scattering of the cross sections.

2. Perform a transport sweep for the Sn/2 equations with a Pn/2+a expansion for the Sn residual as the

inhomogeneous source.

3. Repeat 2 up to perform a transport sweep for the S2 equations with a P2+a expansion for the S4residual as the inhomogeneous source.

4. Solve the diffusion equation with a P0 expansion for the S2 residual as the inhomogeneous source.

5. Apply a diffusive filter to the correction from steps 2 and 3.

6. Add the correction from steps 4 and 5 to the Legendre moments of the Sn iterate to obtain the

accelerated Sn+a moments.

However, here we propose to reformulate the problem within a Krylov solver (GMRES) and therefore

abandon SI. The previous successive sweeps are now different stages of a new preconditioner. For instance,

Bruno Turcksin, Jean C. Ragusa and Jim E. Morel

the system solved for the Sn equations using GMRES can be written as :

(I − Tn)Aφn = DnL−1n q (1)

Φn = Aφn (2)

where q is a volumetric source, Φn is the flux-moments for the Sn equations, I is the identity matrix, Tn is

DnL−1n MnΣn, Dn is the discrete-to-moment matrix for the Sn equations, Ln is the operator

Ωdir ·∇ •+Σt•, Mn is the moment-to-discrete matrix for the Sn equations, Σn is the scattering matrix

and A is the preconditioner :

A = I + Pn/2→nTn/2(I + Pn/4→n/2Tn/2(. . . (I + P0→2T0R2→0) . . .)Rn→n/2 (3)

where Pn/2→n is the projection matrix of φn/2 to φn, Rn→n/2 is the restriction matrix of φn to φn/2 and T0is a Diffusion Synthetic Acceleration operator.

All transport sweeps use Discontinuous Finite Element (DGFEM). For DSA, we use the DGFEM DSA of

[3]. To solve the DSA in the most efficient way, an algebraic multigrid method is used instead of the

conjugate gradient preconditioned with SSOR of [3].

REFERENCES

[1] J. E. Morel and T. A. Manteuffel, “An Angular Multigrid Acceleration Technique Sn Equations withHighly Forward-Peaked Scattering,” Nuclear Science and Engineering, 107, pp. 330-342 (1991).

[2] S. D. Pautz, J. E. Morel and M. L. Adams, “An Angular Multigrid Acceleration Method for SnEquations with Highly Forward-Peaked Scattering,” Proceedings of the International Conference onMathematics and Computation, Reactor Physics and Environmental Analyses in NuclearApplications, Madrid, Spain, September 27-30, 1999, Vol. 1, pp. 647-656 (1999)

[3] Y. Wang and J. C. Ragusa, “Diffusion Synthetic Acceleration for High-Order Discontinuous FiniteElement Sn Transport Schemes and Application to Locally Refined Unstructured Meshes,” NuclearScience and Engineering, 166, pp 145-166 (2010).

2/2


Methodology for Decomposition into Transport and Diffusive Subdomains forthe Linear Discontinuous Method

Nicholas D. Stehle and Dmitriy Y. AnistratovDepartment of Nuclear Engineering

North Carolina State UniversityRaleigh NC, 27695-7909

[email protected]; [email protected]

A large class of radiative transfer problems contain highly diffusive regions. It is possible to reduce

computational costs by solving a diffusion problem in diffusive subdomains instead of the transport

equation. Note that this enables one to decrease dimensionality of the transport problem. In this paper we

present a methodology for splitting a spatial domain of a transport problem into transport and diffusion

subregions for the linear discontinuous (LD) transport discretization method in 1D slab geometry. The

proposed methodology consists of the following main components:

• determining the spatial range of diffusive and transport subdomains,

• formulating methods for solving the given problem in subdomains of each kind,

• definition of interface conditions at boundaries of transport and diffusion subdomains.

To identify and locate diffusive regions, we apply metrics for measuring the transport effects that are based

on the Eddington factor [1]. The proposed methodology for the LD method uses the Second-Moment (SM)

method [2]. The low-order SM (LOSM) equations are discretized consistently with the LD transport

discretization. On each s-th transport iteration, the method is defined as follows:

1. Solve the transport equation in transport subdomains with special boundary conditions at interfaces

with diffusion regions specifying the angular flux for particles coming from the diffusion

subdomains.

2. Calculate the terms in the right-hand side of the LOSM equations in transport subdomains and set

them to the diffusion value in the interior of diffusive subdomains.

3. Solve the LOSM equations in the whole problem domain to compute the scalar flux φ(s) in both

diffusion and transport subdomains.

The transport boundary conditions at interfaces with diffusion subdomains are defined by means of a

linearly anisotropic angular flux the form of which is derived from an asymptotic diffusion analysis of the

LD discretization method.

Nicholas D. Stehle and Dmitriy Y. Anistratov

Table 1. Parameters of the Test

Region σt σs q ∆x BC’s

0 ≤ x ≤ 1 2 0 0 0.1 ψ|x=0=1

1 ≤ x ≤ 11 100 100 0 1 ψ|x=11=0

Table 2. Parameters of Spatial Domain Decomposition

Spatial Range

Type of Region Variant 1 Variant 2

Transport 0 ≤ x ≤ 3 0 ≤ x ≤ 4

Diffusion 3 ≤ x ≤ 10 4 ≤ x ≤ 9

Transport 10 ≤ x ≤ 11 9 ≤ x ≤ 11

To demonstrate the accuracy of the proposed method, we present the results of a well-known test problem

[3]. The parameters of this test are shown in Table 1. The calculations are performed with two variants of

domain decomposition presented in Table 2. The Eddington factor as a function of position is shown in

Figure 1. Figure 2 presents the relative error in the cell-average scalar flux compared to the LD solution

calculated by solving the transport problem in the whole domain on the given spatial grid. Note that if a

data point is not shown for some interval, then its value equals zero. The presented numerical results show

that the proposed method is very accurate. The maximum relative errors equal 3×10−7 and 3×10−11 for

two different cases of domain decompositions, respectively. Note that the analysis of metrics of transport

effects enables one to control the accuracy of the final numerical solution.

REFERENCES

[1] D. Anistratov, “Evaluation of Transport Effects and Spatial Domain Decomposition into Transportand Diffusive Subdomains in 1D Geometry,” Trans. Am. Nucl. Soc., 101, pp. 390-393 (2009).

[2] M. L. Adams and E. W. Larsen, “Fast Iterative Methods for Discrete-Ordinates Particle TransportCalculations,” Prog. Nucl. Energy, 40, pp. 3-159 (2002).

[3] E. W. Larsen and J. E. Morel, “Asymptotic Solutions of Numerical Transport Problems in OpticallyThick, Diffusive Regimes II,” J. Comp. Phys., 83, pp. 212-236 (1989)

2/3

Methodology for Decomposition into Transport and Diffusive Subdomains for the LD Method

0 1 2 3 4 5 6 7 8 9 10 110.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

E(x

)

x

Figure 1. The Eddington factor versus position.

0 1 2 3 4 5 6 7 8 9 10 111E-17

1E-16

1E-15

1E-14

1E-13

1E-12

1E-11

1E-10

1E-9

1E-8

1E-7

1E-6

1E-5

x

diffusion subdomain: 3< x<10 diffusion subdomain: 4< x< 9

Figure 2. The relative errors in the cell-average scalar flux.

3/3


A PIECEWISE LINEAR DISCONTINUOUS FINITE ELEMENT SPATIAL DISCRETIZATION OF THE SN TRANSPORT EQUATION

FOR POLYHEDRAL GRIDS IN 3D CARTESIAN GEOMETRY

Teresa S. Bailey Lawrence Livermore National Laboratory

P.O. Box 808, L‐095 Livermore, CA 94551 [email protected]

W. Daryl Hawkins and Marvin L. Adams

Texas A&M University Department of Nuclear Engineering College Station, TX 77843‐3133


Introduction We introduce a new spatial discretization of the linear Boltzmann transport equation for 3D Cartesian geometry. This spatial discretization, called the Piecewise Linear Discontinuous Finite Element method (PWLD), is a standard Galerkin DFEM method that utilizes Piecewise Linear basis functions, which were first used in 2D by Stone and Adams[1,2], and in 3D for a CFEM diffusion discretization[3]. The PWLD method is designed for accuracy in the thick diffusion limit on arbitrary polyhedral grids. Previously, only Corner Balance methods have been shown to be successful in the diffusion limit for these 3D grid types [4,5]. Other methods that may be extended to 3D for polyhedral grids include DFEMs using Wachspress basis functions [6,7], the CFEM‐based DFEM methods developed by Warsa [8], and characteristics methods that use PWL source approximations [9]. We will briefly derive the PWLD method for polyhedral grids and present a variety of numerical methods to demonstrate its numerical resiliency and correctness. Furthermore, we note that the PWLD method meets all of Adams’ requirements for acceptable diffusion limit behavior [10] on polyhedral grids, but will not show numerical results to this effect. The goal of this paper is to serve as an introduction to the application of PWLD for 3D transport. Development We write the time‐independent, monoenergetic Sn transport equation in 3D Cartesian geometry as

, , , , , , , ,m m m mx y z x y z x y z Q x y z , (1)

where the m subscript indicates the index of an element in the quadrature set used for the discrete‐ordinates approximation [11].


2

The application of a DFEM to this equation is straightforward, and described in many different references [1,2,6,8,10,12,13]. DFEMs expand all spatially dependent terms ( and Q in Eq. (1)) in terms of basis functions, u. For convenience when working with polyhedral cells, we divide our cells into tetrahedral sub‐cell units, called sides. We define a side by two adjacent vertices, a face center point, and the cell center point. Because a face can be determined by more than three vertices, the vertices on a face do not have to be co‐planar in general. As a result, we facet our face about the face center point. A side for a faceted face is shown in Figure 1.

Figure 1: A side in a hexahedral cell with a faceted face

We now define the basis functions as linear combinations of the standard linear functions on tetrahedral sides:

, ,( ) ( ) ( ) ( )j j f j f c j cfaces at j

u r t r t r t r , (2)

where the t functions are standard linear functions defined tetrahedron by tetrahedron. For example, tj equals one at the j‐th vertex and decreases linearly to zero on all other vertices of each side that touches point j. tc is unity at the cell midpoint and zero at each face midpoint and each cell vertex. tf is unity at the face midpoint and zero at the cell midpoint and at each of the face’s vertices. The c and f are weights that give the cell and face midpoints as weighted averages of their vertices:

,@

cell midpointc c j jj c

r r ; (3)


3

,@

face midpointf f j jj f

r r . (4)

In this paper we assume that ,

1c j J

and ,

1f j

fN , where J is the number of

vertices in the cell, and Nf is the number of vertices in a face. As a result, is the same for every basis function in the cell, and f is the same for every basis function

on a face. We note that this basis function definition meets Adams’ diffusion limit requirement of full‐resolution for DFEMs [10] because we can define a basis function supported at each node in a general polyhedral cell. The result of the DFEM discretization is a single‐cell matrix that determines the unknowns in each cell in terms of its source and its incident intensities (from upstream cells or boundary conditions). The size of the single cell matrix is JxJ where J is the number of basis functions used to approximate the flux in the cell. A row of a single‐cell matrix is determined by testing the equation with a weight function. The ith row of the single cell matrix is given by

, , ,1 1

@

,1

, ,1 1

, , , ,

, ,

, , , , , ,

s

s

s

J J

m i m f j j m j jsf face s f j jV

i

J

i m m j js cell jV

J J

i m j j i m j js cell j jV V

n v u x y z u x y z ds

v u x y z dV

v x y z u x y z dV v Q u x y z dV

ss cell

(5)

where m denotes an angular intensity unknown on the boundary of the cell. These surface quantities are determined by an upwinding condition.

,

,

0

0

m cell m

m

m upwind cell m

if n

if n

. (6)

The upwinding condition along with the weight and basis function definitions allows this method to retain the surface matching property requirement for the diffusion limit [10]. Again, for convenience when working with polyhedral cells, the integrals in Eq. (5) are divided into sums of integrals over sides. Eqs. (5) and (6) represent a general discontinuous finite element spatial discretization applied to the 3D Cartesian transport equation. We solve the system of equations local to a cell


4

generated by multiplying Eq. (5) by J distinct weight functions vi, which are the same set of equations as the basis functions, to produce an approximate solution to the transport equation in the cell. We have now fully defined our method except for boundary conditions, which are straightforward. We have also developed a lumped and lumping parameter version of the method, which we will not describe in detail in this paper [12]. Numerical Test Problems We have developed a variety of numerical test problems to further characterize our method and compare it against existing methods. We implemented our method in the Parallel Deterministic Transport Code (PDT) being developed at Texas A&M University. PDT is a massively parallel code that is designed to be a general methods test bed for deterministic transport. We have also implemented unlumped, lumped, and lumping parameter versions of the Tri‐Linear DFEM (TriLD) [13] in PDT, and use it as a standard by which to compare the accuracy of PWLD. The first test problem we will present tests the resiliency of the method on extremely distorted cells. In this test problem, we have a one cell spatial domain of 4 cm x 4 cm x 4 cm, ranging from the point (0,0,0) to the point (4,4,4). The vertex at the origin, (0,0,0), is moved incrementally toward the vertex located at (4,4,4). That is, we begin with a cube and incrementally make it closer to concave, slightly concave, and ultimately dramatically concave. As we do this, some of the faces become significantly non‐planar. Moreover, some of the “side” subcells take on negative volumes when the cell center point is outside the cell, which is true for some of tests. The next set of test problems are truncation error test problems. We ran a series of truncation error problems in the thin limit, and compared the results of PWLD and TriLD. For these truncation error problems we developed a problem with a quadratic solution using the method of manufactured solutions [12]. These quadratic solutions have spatial coordinate cross‐terms which are contained in the TriLD solution space, but not in the PWLD solution space. As a result, TriLD has an advantage for this test problem. We will show that all methods produce a second‐order truncation error rate and very similar accuracy. Finally, we will present PDT solutions to a selected set of the Kobayashi 3‐D Radiation Transport Benchmark problems[14]. We will also make some comments about PDT computational performance on these and other problems.

ACKNOWLEDGMENTS This work performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE‐AC52‐07NA27344.


5

REFERENCES

[1] H.G. Stone and M.L. Adams, “A Piecewise Linear Finite Element Basis with Application to Particle Transport,” in Proc. ANS Topical Meeting Nuclear Mathematical and Computational Sciences Meeting, April 6-11, 2003, Gatlinburg, TN (2003), CD-ROM.

[2] H.G. Stone and M.L. Adams, “New Spatial Discretization Methods for Transport on Unstructured Grids,” in Proc. ANS Topical Meeting Mathematics and Computation, Supercomputing, Reactor Physics and Biological Applications, September 12-15, 2005, Avignon, France (2005), CD-ROM.

[3] T.S. Bailey, M.L. Adams, B. Yang, and M.R. Zika, “A Piecewise Linear Finite Element Discretization of the Diffusion Equation for Arbitrary Polyhedral Grids,” J. Comput. Phys. 227, 3738‐3757 (2008).

[4] K. Thompson and M.L. Adams, “A Spatial Discretization for Solving the Transport Equation on Unstructured Grids of Polyhedra,” in Proc. ANS Topical Meeting Mathematics and Computation, Reactor Physics and Environmental Analysis in Nuclear Applications, September, 1999, Madrid, Spain (1999).

[5] M.L. Adams, “Subcell Balance Methods for Radiative Transfer on Arbitrary Grids,” Transport Theory Statist. Phys. 26, 385‐431, (1997).

[6] G.G. Davidson and T.S. Palmer, “Finite Element Transport Using Wachspress Rational Basis Functions on Quadrilaterals in Diffusive Regions,” in Proc. ANS Topical Meeting Mathematics and Computation, Supercomputing, Reactor Physics and Nuclear and Biological Applications, September 12‐15, 2005, Avignon, France (2005), CD‐ROM.

[7] E. Wachspress, “Generalized Finite Elements,” Proc. International Conference on Mathematics and Computational Methods & Reactor Physics, Saratoga Springs, New, NY, May 3‐7, 2009, CD‐ROM (2009).

[8] J. S. Warsa, “A Continuous Finite Element‐Based, Discontinuous Finite Element Method for SN Transport,” Nucl. Sci. and Eng. 160, 385‐400 (2008).

[9] T.M. Pandya, M.L. Adams, and W. Daryl Hawkins, “Long Characteristics with Piecewise Linear Sources Designed for Unstructured Grids, “ in Proc. ANS International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering, May 8‐12, 2011, Rio de Janeiro, Brazil (2011), CD‐ROM.

[10] M.L. Adams, “Discontinuous Finite Element Transport Solutions in Thick Diffusive Problems,” Nucl. Sci. and Eng. 137, 298-333 (2001).

[11] E.E. Lewis and W. F. Miller, Computational Methods of Neutron Transport, American Nuclear Society, La Grange Park, Il (1993).

[12] T.S. Bailey, “The Piecewise Linear Discontinuous Finite Element Method Applied to the RZ and XYZ Transport Equations,” Doctoral Dissertation, Texas A&M University (2008).

[13] T.A. Wareing, J.M. McGhee, J.E. Morel, and S.D.Pautz, “Discontinuous Finite Element SN Methods on Three-Dimensional Unstructured Grids,” Nucl. Sci. and Eng. 138, 256-268 (2001).


6

[14] K. Kobayashi, N. Sugimura, Y. Nagaya, “3-D Radiation Transport Benchmark Problems and Results for Simple Geometries with Void Regions,” OECD Proceedings, Nuclear Energy Agency (2000)


Dosimetry calculation of two commercially available Iodine brachytherapy seeds using Spencer-Lewis 3D multi-group SN transport code

Navid Ayoobian

Department of Nuclear Engineering Shiraz University; Shiraz; Iran

[email protected]

Kamal Hadad Department of Nuclear Engineering

Shiraz University; Shiraz; Iran

The Discrete Ordinates Method (DOM), SN, is a deterministic solution of the general Boltzmann/Spencer-Lewis equations governing particle transport. The method is well known and routinely used for reactor physics applications. However, DOM is rarely applied to medical radiation physics problems, where Monte Carlo solutions of the Boltzmann equation are almost universally used [1-4]. A general derivation of the multigroup energy approach and the spatial and angular discretizations can be found in the textbook by Lewis and Miller [5]. The solution of discrete ordinates equations approaches the exact solution of the Boltzmann equation as the space, energy, and angle discrete bin sizes approach differential size. SMARTEPANTS [6-7] (Simulating Many Accumulative Rutherford Trajectories Electron Photon and Neutral Transport Solver) is a diamond difference SN Boltzmann/Spencer-Lewis solver in x-y-z geometry for coupled electron/photon transport of up to four types of charged-neutral particles. The SMARTEPANTS approach is based on the Gousmit-Saunderson solution to the infinite medium Spencer-Lewis equation. Energy and charge deposition, particle fluxes and leakage currents are determined in x-y-z geometries. With the Spencer-Lewis approach, energy loss is modeled by means of stopping power, group to group transfer cross sections are not defined, and the energy grid need only be fine enough to adequately describe the energy variation of the angular flux. For a homogeneous medium, the Spencer-Lewis equation can be written as:

Ω. , Ω, , Ω Ω ,Ω , Ω , Ω, (1) in which , Ω, = electron flux at position r, direction and path length s, σt (s) = total scattering cross section , Ω Ω = differential scattering cross section , Ω, = fixed electron source. The path length variable s is substituted for the energy and is defined as the distance an electron travels in slowing down from some reference energy E0 (usually taken to be the highest energy in the system) to energy E, that is for a region the relationship between s and E is: (2)

The SMARTEPANTS capabilities include its speed, accuracy and benchmark results. Despite all appealing features, the code uses an old cross section library which is


2

prepared by the CEPXS [8]. In this work, we present the improved multigroup-Legendre photon cross section library by implementing the early cross section library, EPDL97 (Evaluated Photon Data Library, 1997 version) [9] into SMARTEPANTS code. The multigroup method involves a discretization of the particle energy domain into energy intervals or groups:

where E1> E2> E3>…> EG+1 and EG+1 is the cutoff energy. The multigroup approximation is realistic only if the cross sections do not vary greatly in energy within a group. Hence, the structure of the energy grid can impact the accuracy of a prediction. The multigroup angular flux, φ r, Ω , is defined as:

φ r, Ω φ r, Ω, EEE dE (3)

where φ r, Ω, E is the angular flux. The multigroup scalar flux is defined as:

φ r φ r, Ω dΩ (4) The differential cross section, σ(r,E→E’,μ), can be represented by a Legendre expansion:

σ r, E E’, µ ∑ L σL r, E E’ PL µL (5)

where: σL r, E E’ PL µ σ r, E E’, µ dΩ

2π PL µ σ r, E E’, µ dµ (6) For a differential cross section:

σ ,L rE E

EE L ,E E’ E

EE

E EEE

(7)

where w(E) is the multigroup weighting function. Since a weighting function must be arbitrarily chosen, a unique set of multigroup cross sections does not exist. In this work, w(E)=cg for Eg≥ E ≥ Eg+1 , where cg is a group-dependent constant. Hence the multigroup-Legendre expansion coefficients become:

σ ,L rE

EE L ,E E’ E

EE

∆E (8)

where ΔEg=Eg -Eg+1. The MATLAB software is used to generate the multigroup-Legendre cross section from EPDL97. The code starts by reading the atomic numbers, weight fractions, energy group number, energy group type (linear or logarithm), maximum energy, cut off energy, Legendre moment number and density for each materials. By running the code, two files (cpx.MG and cpx.CSD) are produced that are input file for SMARTEPANTS. These files include the multigroup total/ pair production/ triplet production/ self-scattering/ down-scattering cross sections, multigroup-Legendre scattering cross sections and energy deposition for different values of Legendre coefficients (L). The new code is used to calculate dosimetry parameters, such as dose rate constant, radial dose function and anisotropy functions of two commercially available iodine brachytherapy seeds (Models LS-1 and Intersource) [10-14]. The results are presented in Tables 1-2 and Figures 1-2.

EG+1 EG E3 E1 E2

group G group 2 group 1 …


3

Table 1. Dose rate constant, Λ, at 1 cm along the transverse axis of two Iodine sources

Model Reference Method Λ (cGy h−1 U−1)

LS-1 Wang et al. (10) Monte Carlo (EGS4) 0.967 LS-1 Williamson et al. (11) Monte Carlo 0.920 LS-1 Nath et al. (12) Measurement 1.02 LS-1 Faghihi et al. (13) Monte Carlo(MCNP4C) 0.953

LS-1 This work Discrete

Ordinates

S=4 L=5 0.760 L=7 0.761

S=8 L=5 1.106 L=7 1.105

S=12 L=5 0.971 L=7 0.970

S=16 L=5 0.971 L=7 0.970

Intersource Meigooni et al. (14) Monte Carlo(PTRAN) 0.981 Intersource Faghihi et al. (13) Monte Carlo(MCNP4C) 0.986

Intersource This work Discrete

Ordinates

S=4 L=5 0.751 L=7 0.753

S=8 L=5 1.110 L=7 1.111

S=12 L=5 0.988 L=7 0.989

S=16 L=5 0.988 L=7 0.989

Table 2. The values of the anisotropy function, F(r,θ) for LS1 source

θ (degrees) r (cm)

1 2 3 5 7

0 (this work) - - 0.769 0.763 0.772 0 (Wang et al. (10)) 0.814 0.763 0.771 0.760 0.773 10 (this work) - 0.780 0.770 0.782 0.782 10 (Wang et al. (10)) 0.816 0.777 0.773 0.780 0.781 20 (this work) - - 0.808 0.823 0.813 20 (Wang et al. (10)) 0.844 0.811 0.809 0.820 0.815 30 (this work) 0.921 0.855 0.851 0.862 0.861 30 (Wang et al. (10)) 0.872 0.849 0.850 0.860 0.863 40 (this work) - 0.899 0.999 0.989 0.903 40 (Wang et al. (10)) 0.900 0.895 0.896 0.900 0.903 50 (this work) - 0.933 0.938 0.941 0.939 50 (Wang et al. (10)) 0.930 0.930 0.937 0.940 0.938 60 (this work) - 0.960 0.966 0.969 0.968 60 (Wang et al. (10)) 0.960 0.958 0.965 0.970 0.968 70 (this work) 0.977 0.991 0.989 0.990 0.992 70 (Wang et al. (10)) 0.983 0.989 0.987 0.990 0.993 80 (this work) - 0.996 1.001 1 1.003 80 (Wang et al. (10)) 0.996 0.995 1 1 1.004 90 (this work) 1 1 1 1 1 90 (Wang et al. (10)) 1 1 1 1 1

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FERENCES

gers, J.F. Wite 192Ir sou10):2307–1Rogers, J.F. eed dose din”, Med Phys

4

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4r (cm)

el LS1 brach

rce Iodine b

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6

Faghihi et a

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Williamson

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6

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Faghihi et

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et al.

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5

[3]K.A. Gifford, J.L. Horton, T.A. Wareing, G. Failla, F. Mourtada, “Comparison of a finite‐element multi‐group discrete‐ordinates code with Monte Carlo for radiotherapy calculations”, Phys Med Biol., 51(9):2253–65, (2006).

[4]K.A. Gifford, T.A. Wareing, G. Failla, J.L. Horton, P.J. Eifel, F. Mourtada, “Comparison of a 3D multi‐group SN particle transport code with Monte Carlo for intercavitary brachytherapy of the cervix uteri”, J App Clin Medi Phys., 11(1):2‐9, (2010).

[5]E.E. Lewis, W.F. Miller, “Computational Methods of Neutron Transport”, Wiley, New York; (1984).

[6]K. Hadad, W.L. Filippone, “Coupled Electron/Photon SN Calculation in Lattice Geometry”, American Nuclear Society (ANS) winter meeting. San Francisco, CA, United States, 14‐18 Nov, (1993).

[7]W.L. Filippone, “The Theory and Application of SMART Electron Scattering Matrices,” Nucl. Sci. Eng.,99,232 (1988).

[8]L. Lorence, J. Morel, G. Valdez, “Physics Guide to CEPXS: A multi‐group coupled electron‐photon cross‐section generating code, version 1.0”, Albuquerque, NM: Sandia National Laboratory; (1989).

[9]D.E. Cullen, J.H. Hubbell, L. Kissel, “EPDL97: The Evaluated Photon Data Library, 97 Version”, University of California, UCRL‐LR‐50400, vol 6, Rev 5: Lawrence Livermore National Laboratory; (1997).

[10]R. Wang, R.S. Sloboda, “Monte Carlo dose parameters of brachy seed model LS‐1 I‐125 brachytherapy source”, Applied Radiation and Isotopes, 56: 805‐813, (2001).

[11]J.F. Williamson, “Dosimetric characterization of a new I‐125 interstitial brachytherapy source design: a Monte Carlo investigation”, Med Phys, 29(2), (2002).

[12]R.L. Nath, N. Yue, “Dosimetric characterization of a newly designed encapsulated interstitial brachytherapy source of iodine 125 model LS‐1 Brachy seed TM”, Appl Radiat Isotop, 55: 813‐821, (2001).

[13] R. Faghihi, M. Zehtabian, S. Sina, “Comparison of dosimetry parameters of two commercially available Iodine brachytherapy seeds using Monte Carlo calculations”, Iran. J. Radiat. Res., 7 (3), (2009).

[14]A.S. Meigooni, M.M. Yoe‐Sein, A.Y. Al‐Otoom, K. Sowards, “Determination of the dosimetric characteristics of InterSource125 Iodine brachytherapy source”, Med Phys, 27: 2168–2173, (2000).



RADIATION FIELD CHARACTERIZATION,

SHIELDING ASSESSMENT AND ACTIVATION CALCULATIONS

FOR THE MYRRHA DESIGN

Anna Ferrari 1,2 *

, Bruno Merk 1, Joerg Konheiser

1

1 Institute of Safety Research

Helmholtz-Zentrum Dresden-Rossendorf

PF 510119, 01314 Dresden, Germany

2 Institute of Radiation Physics

Helmholtz-Zentrum Dresden-Rossendorf

PF 510119, 01314 Dresden, Germany

Accelerator-driven systems (ADS) are one of the options studied for the transmutation of

nuclear waste in the European Community. The design of sub-critical ADS requires high

energy and high power proton accelerators, of the order of hundreds MeV and some MW

for the proposed demonstration experiments. The use of high energy Mega-Watt proton

beams, in combination with a nuclear reactor core operating in sub-critical or critical

mode, presents many challenges for various aspects of the design. Radiation shielding

and minimization of the induced activation are key points.

The present study has been done in the framework of the Central Design Team european

project (CDT), which has the goal to design the FAst Spectrum Transmutation

Experimental Facility (FASTEF), able to demonstrate efficient transmutation of high

level waste and associated ADS technology. The heart of the system is a 100 MW LBE

cooled reactor, working both in critical and sub-critical modes. A beamline aims to

transport a 600 MeV, 4 mA proton beam produced by a linear accelerator up to the

spallation target for the neutron production, which is located inside the reactor core

(fig. 1). Based on the FASTEF design, the MYRRHA facility [1], which should enter the

construction phase in 2015, will be built at SCK·CEN in Mol (Belgium). MYRRHA is

conceived as a multi purpose facility: as technology demonstrator for lead-bismuth

cooled fast reactor, as demonstrator for efficient transmutation, and as high flux

irradiation facility for material testing and medical isotope production.

An extensive simulation study has been done to assess the shielding of the reactor

building and the proton accelerator, as well as to fix the activation problems that have a

heavy influence on the beamline and building design. For this purpose, the shielding

calculations have been performed on both configurations, the ADS configuration

consisting of the accelerator, the beam line and the subcritical core, and the critical

configuration consisting of the reactor core only.

* Email address: [email protected]



2

Figure 1. A view of the MYRRHA beamline: in the last part the vertical proton

beam is hitting the spallation target inside the reactor core. The footprint of

the beam at the target is also drawn: the beam moves along a circular pattern,

allowing a uniform beam power density at the window surface in the rotation

direction.

The presented simulations are based on both MCNPX (version 2.6.0) [2] and FLUKA

(version 2011.2.2) [3, 4] Monte Carlo transport codes, with the aim to do a code-code

comparison and to cross check the results.

Starting from the MCNPX models of the reactor core both for the critical and the sub-

critical mode (including in the latter case the vertical part of the proton beamline with the

spallation target), the radiation source terms due to neutron and photon fields have been

fully characterized on suitable surfaces around the reactor core. To assess the shielding of

the reactor building, these source terms have been then used as input in a second row of

simulations performed with both FLUKA and MCNPX. The results, which are expressed

as neutron and photon fluences and then as ambient dose equivalent profiles, are used to

determine the dimensions of the concrete walls for the shielding of the complete nuclear

system and to tune dedicated technological solutions. All the results are presented,

together with the implications on the design.

A parallel work has been done to assess the shielding around the proton accelerator by

optimizing at the same time the elements of the beamline that are devoted to the partial or

total beam absorption (beam dump, collimators). This second study has been fully carried

out by using the FLUKA code, which has the unique feature to perform the transport of

the residual radiation via a full Monte Carlo method, allowing in addition modifications

in the geometry and material characterization from the prompt to the residual radiation

transport. It will be shown how a suitable material configuration, with the introduction of

low-activation materials, is a key issue: it will improve the accessibility and the long-term



3

treatment of the irradiated elements allowing to maintain - and sometimes to improve –

the shielding efficiency.

ACKNOWLEDGMENTS

This work has been carried out within the framework of the CDT EU FP7 project.

REFERENCES

[1] H. Aït Abderrahim, P. Baeten, D. De Bruyn, J. Heyse, P. Schuurmans and J.

Wagemans, “MYRRHA, a Multipurpose hYbrid Research Reactor for High-end

Applications”, Nuclear physics News, 20, 24-28 (2010).

[2] MCNPX User’s Manual Version 2.6.0, LA-CP-07-1473 (2008).

[3] G. Battistoni et al., “The FLUKA code: description and benchmarking”,

Proceedings of the hadronic shower simulation workshop 2006, Fermilab, 6-8

September 2006, M. Albrow, R. Raja (Eds.), AIP Conference Proceedings, 896, 31-

49 (2007).

[4] A. Fassò et al., “FLUKA: a multi-particle transport code”, CERN-2005-10 (2005),

INFN/TC_05/11, SLAC-R-773.


TWO-REGION DIFFUSION MODEL FOR IMPROVED ANALYSIS OFADS EXPERIMENTS

Varvara Glivici-Cotruta, Bruno MerkInstitut fur Sicherheitsforschung

Helmholtz-Zentrum Dresden-RossendorfBautzner Landstr. 400

01328 Dresden, [email protected]

Each year nuclear power generation facilities produce about 200,000 m3 of low- and intermediate-level

radioactive waste and 10,000 m3 of high-level waste (including spent fuel designated as waste) worldwide

[1]. One of the ways to manage the nuclear waste is a conversion of long-lived highly toxic fission products

into short-lived or stable fission products by transmutation. Accelerator Driven System (ADS) is a

developing alternative for the transmutation of the long-lived radioisotopes, energy production, as well as

an important link in the chain of a nuclear fuel cycle. A number of international experimental campaigns

(MUSE [2], YALINA [3]) have taken place and are planned (GUINEVERE [4]) in order to investigate the

operational procedures in an ADS, sub-criticality determination and on-line reactivity monitoring which is

a key point for the ADS safety.

The results form the YALINA-Booster experimental campaign have demonstrated the imprecision of the

traditional reactivity calculation methods based on the point kinetic equation for the analysis of the ADS

experiments [5]. The origin of this behaviour lies in the strong dependency of the neutron detector response

on the detector position in a highly heterogeneous core and at different subcritical states. It indicates a

much stronger space and time dependence of the neutron flux in the subcritical system in comparison with

a critical one. For instance, the area ratio technique [6] demonstrates a strong spacial dependence [7], and

the results for the prompt neutron decay constant method [8] vary over the core at deep subcritical levels. A

space and time dependent diffusion and P1 problems were solved aiming to handle this space-time

dependence. The space and time dependent solutions were developed by means of Green’s function

technique. Although the Fick’s law is only valid in the presence of external sources in regions sufficiently

far from the source (several mean free paths) and far from the boundary between different types of

materials [9], the obtained analytical results were in a very good agreement with experimental ones [10].

A fast lead core and an external neutron source create rapidly varying transients in which the spatial and

time effects are important, and, in combination with a thick lead reflector, impose a constraint on an

application of the traditional point reactor kinetics approximation. Thereby, for the foreseen GUINEVERE

experiments a two-region space and time dependent diffusion approximation was chosen to be solved and

analysed:

1

v1· ∂Φ1(x, t)

∂t− 1

3 · Σtr1· ∂

2Φ1(x, t)

∂x2+ Σa1 · Φ1(x, t) = S(x, t), 0 < x < L, t > 0 (1)

Varvara Glivici-Cotruta and Bruno Merk

1

v2· ∂Φ2(x, t)

∂t− 1

3 · Σtr2· ∂

2Φ2(x, t)

∂x2+ Σa2 · Φ2(x, t) = 0, L < x < b, t, L, b > 0 (2)

The appropriate boundary conditions are:

Φ1(0, t) = 0, t > 0

Φ2(b, t) = 0, t > 0(3)

The initial conditions are:

Φ1(x, 0) = 0, 0 < x < L

Φ2(x, 0) = 0, L < x < b(4)

The conditions at the interface between two media:

Φ1(L, t) = Φ2(L, t), t > 0

1

Σtr1· ∂Φ1(x, t)

∂x

∣∣∣x=L

=1

Σtr2· ∂Φ2(x, t)

∂x

∣∣∣x=L

, t > 0(5)

It is necessary to represent the solution for two-region core owing to a presence of an external neutron

source, sub-criticality of the system, and an increasing impact of the reflector in a small experimental

reactor, as well as due to the inaccuracy of the diffusion approximation around the core reflector, core

blanket interfaces, and throughout fast reactor blankets [11]. This two-region solution without separation of

space and time gives a significantly improved methodology for the analysis of the future experiments like

GUINEVERE. The efficiency of the derived solution over the accurate numerical solutions (like Monte

Carlo calculation, for example) lies in a comparatively short calculation time, which is of major importance

for the on-line monitoring the reactivity of a subcritical reactor system.

REFERENCES

[1] I. A. E. Agency, “Managing radioactive waste.” Factsheet.

[2] R. Soule and et al., “Neutronic studies in support of accelerator-driven systems: The museexperiments in the masurca facility,” Nuclear Science and Engineering, vol. 148, pp. 124–152, 2004.

[3] C.-M. Persson, P. Seltborg, A. Ahlander, W. Gudowski, T. Stummer, H. Kiyavitskaya, V. Bournos,Y. Fokov, I. Serafimovich, and S. Chigrinov, “Analysis of reactivity determination methods in thesubcritical experiment yalina,” Nuclear Instruments and Methods in Physics Research, vol. 554,pp. 374–383, 2005.

2/3

TWO-REGION DIFFUSION MODEL FOR THE IMPROVED ANALYSIS OF ADS EXPERIMENTS

[4] A. Billebaud, P. Baeten, and et al., “The guinevere project for accelerator driven system physics,” inInternational Conference GLOBAL 2009 ”The Nuclear Fuel Cycle: Sustainable Options & IndustrialPerspectives”, September 6-11 2009.

[5] V. Becares, D. Villamarın, M. Fernandez-Ordonez, and E. M. Gonzalez-Romero, “Analysis of pnsmeasurements at the yalina-booster subcritical facility,” in 3rd ECATS Progress Report Meeting,2nd-3rd March 2010.

[6] N. G. Sjostrand, “Measurements on a subcritical reactor using a pulsed neutron source,” Arkiv ForFysik, vol. 11 [13], pp. 233–246, 1956.

[7] C. Berglof, M. Fernandez-Ordonez, D. Villamarin, V. Becares, E. M. Gonzalez-Romero, V. Bournos,I. Serafimovich, S. Mazanik, and Y. Fokov, “Spatial and source multiplication effects on the area ratioreactivity determination method in a strongly heterogeneous subcritical system,” Nuclear Science andEngineering, vol. 166, pp. 134–144, 2010.

[8] B. E. Simmons and J. S. King, “A pulsed neutron technique for reactivity determination,” NuclearScience and Engineering, vol. 3, pp. 595–608, 1958.

[9] H. Nifenecker, O. Meplan, and S. David, Accelerator Driven Subcritical Reactors. Institute ofPhysics Publishing, 2003.

[10] B. Merk, V. Glivici-Cotruta, and F. P. Weiß, “A solution for telegrapher’s equation with externalsource: Development and first application,” IL NUOVO CIMENTO, vol. 125 B, N.12, pp. 1547–1559,2010.

[11] K. O. Ott and R. J. Neuhold, Introductory Nuclear Reactor Dynamics. American Nuclear Society,1985.

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A MESH-FREE APPROXIMATION

OF SPATIAL NEUTRON FLUX DISTRIBUTION

Dimitar V. Altiparmakov

Atomic Energy of Canada Limited

Chalk River Laboratories

Chalk River, ON, K0J 1J0, Canada

[email protected]

A common objective of the current deterministic transport methods is to approximate the

solution with a set of trial functions from a less restrictive class of functions than the class of

functions to which the exact solution belongs. This approach is mainly due to the inability of

conventional numerical mathematics to handle complex spatial domains on a whole. Instead,

such a domain needs to be subdivided into a number of subdomains of simple geometric shapes

(triangles, rectangles, etc.), to each of which a classical approximation (finite difference, finite

element, etc.) could be applied. By lessening the requirements on trial functions, however, the

number of degrees of freedom (unknown coefficients) of the approximate solution may increase

significantly. Despite the tremendous capabilities of today’s computers, this is still a severe

limitation in transport calculation of large heterogeneous systems.

This paper presents two levels of a mesh-free approximation that is expected to allow a

significant reduction of the number of unknowns. The first order approximation preserves the

neutron flux continuity by interpolating the neutron flux at interface boundaries. The second

order approximation, similar to the Hermite interpolation, is devised to conserve neutron current

continuity as well. However, unlike conventional interpolation, which deals with node (point)

influence functions, here the basis functions are surface specified. They are derived according to

the geometric shape of material regions using the mathematical apparatus of the R-function

theory [1]. Accordingly, the geometry of the problem is a priori and analytically implemented in

the approximate solution. The basis functions are continuous and differentiable everywhere, as

the exact solution is, except at material interfaces where the continuity of the neutron flux and

current is preserved by a proper choice of the unknown coefficients.

A CANDU reactor1 lattice cell is used as the model problem to illustrate the feasibility of mesh-

free approximation. A spatially continuous group-wise neutron flux distribution is generated as a

reference solution by the integral transport equation applied to a collision probability solution

obtained by the lattice code WIMS-AECL [2]. The collision probability calculation is carried

out in 89 energy groups on a very fine mesh represented by 631 unknowns per group. Figure 1

shows the geometric model of the lattice cell considered and the mesh subdivision applied. The

continuous flux is calculated on a rectangular mesh grid with 1 mm step size. For the sake of

simplicity, the 89-group flux is condensed into two groups: fast group that covers the neutron

energy range from 0.625 eV to 10 MeV, and thermal group below 0.625 eV. Figure 2 presents

the thermal neutron flux.

The infinite lattice model implies that the entire space consists of a periodically repeating

structure of material regions. To construct the approximation basis, each material region is

1 CANDU (CANada Deuterium Uranium) is a registered trademark of Atomic Energy of Canada Limited




considered as a distinct spatial domain represented mathematically by a domain function. The

latter is a real continuous and differentiable function that is greater than zero inside the domain,

equal to zero at its boundary, and smaller than zero everywhere else. In addition, it is normalized

to the first order, i.e. the normal derivative at the domain boundary is equal to unity, while the

second order normal derivative is equal to zero.

For a spatial domain bounded by an analytic surface, the domain function can be specified

according to the related analytic equation. In the case of an irregular boundary and/or multi-

connected domain (a union/intersection of several domains), the related domain function can be

constructed by the R-function apparatus. For a periodically repeating domain it is necessary to

apply translational symmetry transformation to the related spatial coordinates [1], [3]. As a

visual illustration, Figures 3 and 4 show the positive parts of the domain functions of the

moderator and coolant regions, respectively.

The next step in the construction of the basis functions is to approximate the neutron flux at the

interface boundaries. In conventional numerical mathematics, this is usually done by a complete

set of functions. Such an approach, however, neglects the environment, and increases

unnecessarily the number of degrees of freedom. Instead, the interface flux can be efficiently

approximated by a linear combination of a constant term (isolated region) and domain functions

of the neighbouring regions. This linear combination of functions is interpolated over the space

of the related material region by the gluing formula [1], which is in fact a generalization of the

Lagrange interpolation formula. Each domain function is also used as a basis function over the

domain interior. To get the second order approximation, the basis functions are further modified

and mutually correlated according to the normalization properties in order to preserve the current

continuity.

To investigate the ability of the basis functions to approximate the actual solution, the unknown

coefficients are determined by the least squares method applied to the discrepancy between the

known solution and the mesh-free approximation. The results show that the first order

approximation using 22 unknowns per group yields very good agreement with the reference

solution. The difference in neutron multiplication factor is 0.12 mk (12 pcm), while the relative

mean square error in spatial distribution is of the order of one percent in magnitude for each of

the 89 groups considered. As a visual illustration, Figure 5 compares the thermal flux (E < 0.625

eV) of the mesh-free approximation and the reference solution along three lines across the cell: a

horizontal line passing through the cell centre, a horizontal line at the cell edge, and a diagonal

line.

REFERENCES

[1] V.L. Rvachev, R-Function Theory and Its Applications, Izd. Naukova Dumka, Kiev,

1982 (in Russian).

[2] D. Altiparmakov, “New Capabilities of the Lattice Code WIMS-AECL”, Proceedings of

PHYSOR 2008, the International Conference on Reactor Physics, Nuclear Power: A

Sustainable Resource”, Interlaken, Switzerland, September 14-19, 2008, Paper 246.

[3] D. Altiparmakov and J. Pop-Jordanov, “Implementation of Certain Analytical

Information in Transport Theory Computation”, Transport Theory and Statistical

Physics, 15(6&7), 861-870 (1986).



Figure 1. Geometric model and mesh

subdivision of the CANDU lattice cell

Figure 2. Spatial distribution of thermal

neutron flux in the CANDU lattice cell

Figure 3. Domain function of the moderator region



Figure 4. Domain function of the coolant region

Figure 5. Thermal flux distribution along three lines across the lattice cell

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