1 a component mode synthesis method for 3d cell by cell calculation using the mixed dual finite...

11

A component mode synthesis method for A component mode synthesis method for 3D cell by cell calculation using the 3D cell by cell calculation using the

mixed dual finite element solver MINOSmixed dual finite element solver MINOS

P. Guérin, A.M. Baudron, J.J. LautardP. Guérin, A.M. Baudron, J.J. LautardCommissariat à l’Energie AtomiqueCommissariat à l’Energie Atomique

DEN/DM2S/SERMADEN/DM2S/SERMACEA SACLAYCEA SACLAY

91191 Gif sur Yvette Cedex France91191 Gif sur Yvette Cedex [email protected]@cea.fr

22American Nuclear Society Topical Meeting in Mathematics & Computations, Avignon, France, 2005

OUTLINESOUTLINES

General considerations and motivationsGeneral considerations and motivations

Basic equationsBasic equations

MINOS SolverMINOS Solver

The component mode synthesis methodThe component mode synthesis method

Numerical resultsNumerical results

Conclusions and perspectivesConclusions and perspectives


Geometry and mesh of a PWR 900 MWe core Geometry and mesh of a PWR 900 MWe core

PinPin assemblyassembly CoreCore

Pin by pin geometryPin by pin geometry Cell by cell meshCell by cell mesh Whole core meshWhole core mesh


INTRODUCTIONINTRODUCTION

MINOS solver :MINOS solver :o main core solver of the DESCARTES system, developed main core solver of the DESCARTES system, developed

by CEA, EDF and Framatomeby CEA, EDF and Framatome

o mixed dual finite element method for the resolution of mixed dual finite element method for the resolution of the SPn equations in 3D cartesian homogenized the SPn equations in 3D cartesian homogenized geometriesgeometries

o 3D cell by cell homogenized calculations too expensive3D cell by cell homogenized calculations too expensive

Standard reconstruction techniques to obtain the local Standard reconstruction techniques to obtain the local pin power can be improved for MOX reloaded corespin power can be improved for MOX reloaded coreso interface between UOX and MOX assembliesinterface between UOX and MOX assemblies


MOTIVATIONSMOTIVATIONS

Find a numerical method that takes in account Find a numerical method that takes in account the heterogeneity of the corethe heterogeneity of the core

Domain decomposition and two scale method : Domain decomposition and two scale method :

o Core decomposed in multiple subdomainsCore decomposed in multiple subdomains

o Problem solved with a fine mesh on each Problem solved with a fine mesh on each subdomainsubdomain

o Global calculation done with a basis that Global calculation done with a basis that takes in account the local fine mesh resultstakes in account the local fine mesh results

Perform calculations on parallel computersPerform calculations on parallel computers


Derived from 1D transport Pn equationDerived from 1D transport Pn equation

N+1 harmonics : N+1 harmonics :

The (N+1)/2 even components The (N+1)/2 even components are scalar are scalar

The (N+1)/2 odds components are The (N+1)/2 odds components are vectorsvectors

Strong formulation of SPN equations Strong formulation of SPN equations

)()12( st

continuous be tosupposed are andp

SPN one group equation written in SPN one group equation written in the mixed form (odd – even) with the mixed form (odd – even) with albedo boundary condition reads :albedo boundary condition reads :

oe

p

Ein

10)().( andEnpH eT on

oo

eeT

SpTH

STpH

.

Coefficients : Coefficients :

.

.5

43

21

H

.4

2

0

σ

σ

σ

eT

.5

3

1

oT

is a tridiagonal matrix is a tridiagonal matrix coupling the harmonics, coupling the harmonics, a full matrix which depends on a full matrix which depends on the albedo coefficients, and the albedo coefficients, and respectively the even and respectively the even and odd removal diagonal matricesodd removal diagonal matrices

H)(e

eT oT


Mixed dual variational SPN formulationMixed dual variational SPN formulation

By projection and using the Green formula on the odd equations :

),()(),( 2 EdivHELVp d

Functional spaces :

Even flux : discontinuous

Odd flux : normal trace continuous

E

oo

E

ooet

E

ooo

E

oe

E

ee

E

eee

E

eoT

SnnHHTH

STH

.).)(.)((.).(

).(

1


Existence and unicity of the solutionExistence and unicity of the solution

Mixed dual variational SPN equations are a particular case Mixed dual variational SPN equations are a particular case of the more abstract problem :of the more abstract problem :

The ellipticity of the bilinear continuous form a and the inf-The ellipticity of the bilinear continuous form a and the inf-sup condition on the continuous form b insure existence sup condition on the continuous form b insure existence and unicity of the solution of this problem :and unicity of the solution of this problem :

'

'

( , ) :

( , ) ,

( , ) ( , ) ,

( , ) ( , ) ,W W

V V

Find p u W V such that

q v W V

a p q b q u f q

b p v t u v g v

BKerqqqqaW

2),(,0

0),(

supinf,0

VW

WqVv vq

vqb


Discretized spacesDiscretized spaces

1

1.

.1

1

ˆ

ˆˆˆ~

R

xdqdivB

RTk basis with :RTk basis with :

– Even basis => OrthogonalEven basis => Orthogonal lagrangian basis associated to nodes lagrangian basis associated to nodes located at Gauss points of order 2k+1located at Gauss points of order 2k+1

– Odd flux basis such that :Odd flux basis such that :

Finite Element basis on rectangle : Raviart Thomas Nedelec element Finite Element basis on rectangle : Raviart Thomas Nedelec element (RTk)(RTk)

Even nodesX-odd nodesY-odd nodes

:1RT


The matrix systemThe matrix system

e

oT Q

Qp

TB

BR

)(1 pBQT Te

Matrix Symmetric but not Positive Definite, elimination of the even Matrix Symmetric but not Positive Definite, elimination of the even flux :flux :

Linear system on the odd flux to solve :Linear system on the odd flux to solve :

eoT QBTQpBBTR 11

The matrix of the discretized system is :The matrix of the discretized system is :

Block Gauss Seidel iteration (1 block corresponds to the set of Block Gauss Seidel iteration (1 block corresponds to the set of nodes of one odd flux component)nodes of one odd flux component)

Eigenvalue problem solved by power iterationsEigenvalue problem solved by power iterations


The CMS methodThe CMS method

CMS method for the computation of the CMS method for the computation of the eigenmodes of partial differential equations has been eigenmodes of partial differential equations has been used for a long time in structural analysis.used for a long time in structural analysis.

The steps of our method : The steps of our method : – Decomposition of the core in K small domainsDecomposition of the core in K small domains– Calculation with the MINOS solver of the first Calculation with the MINOS solver of the first

eigenfunctions of the local problem on each eigenfunctions of the local problem on each subdomain subdomain

– All these local eigenfunctions span a discrete All these local eigenfunctions span a discrete space used for the global solve by a Galerkin space used for the global solve by a Galerkin techniquetechnique


Diffusion modelDiffusion model

Monocinetic diffusion problem with homogeneous Dirichlet boundary Monocinetic diffusion problem with homogeneous Dirichlet boundary condition.condition.

Mixed dual weak formulation :Mixed dual weak formulation :

0)(.1

1)(

),(),(),(

:)(),(),,(2

2

EE

E

f

E

a

E

qdivqpD

pdiv

ELEdivHq

thatsuchRELEdivHpFind

Eigenvalue problemEigenvalue problem


Local eigenmodesLocal eigenmodes

Overlapping domain decomposition :Overlapping domain decomposition :

Computation on each of the first local eigenmodes with Computation on each of the first local eigenmodes with the global boundary condition on , and p=0 on \ :the global boundary condition on , and p=0 on \ :

k

K

k

EE 1

kE kNkE EE

0)(.1

1)(

),(),(),(

:)(),(),,(2

E\E ,0

2 E\E ,0

k

k

kk

kkk

E

ki

E

ki

E

kifk

iE

kia

E

ki

kk

kkki

ki

ki

qdivqpD

pdiv

ELEdivHq

thatsuchRELEdivHpFind

kNiKk 1,1for allfor all

EEonnqEdivHqEdivH kkkEEk \0.),,(),(\,0with


Global Galerkin methodGlobal Galerkin method

Extension on E by 0 of the local eigenmodes on each :Extension on E by 0 of the local eigenmodes on each : kE

)(),( 2

,1

,1

~,1

,1

~

ELVectVandEdivHpVectWKk

Ni

ki

Kk

Ni

ki

kk

0)(.1

1)(

,),(

:),,(

EE

E

f

E

a

E

qdivqpD

pdiv

VWq

thatsuchRVWpFind

global functional spaces on Eglobal functional spaces on E

Global eigenvalue problem on these spaces :Global eigenvalue problem on these spaces :


Linear systemLinear system

Unknowns :Unknowns :

K

k

N

i

K

k

N

i

ki

ki

ki

ki

k k

fandpcp1 1 1 1

~~

If all the integrals over vanish If all the integrals over vanish sparse sparse matricesmatrices

lk EE lk EE

with :

fa or

KKKK

kl

K

K

AAA

A

AAA

AAA

A

..

...

..

..

21

22221

11211

KKKK

kl

K

K

BBB

B

BBB

BBB

B

..

...

..

..

21

22221

11211

KKKK

kl

K

K

TTT

T

TTT

TTT

T

..

...

..

..

21

22221

11211

lk EE

lj

kiji

kl ppD

A~~

, .1

lk EE

lj

kiji

kl pdivB~~

, .)(

lk EE

lj

kiji

klT~~

,

0

1

ki

ki

kif

kia

ki

T

fBcA

fTfTcB

Linear system associated :Linear system associated :


Global problemGlobal problem

Global problem :Global problem :

H symmetric but not positive definiteH symmetric but not positive definite

Not always well posed because of the Not always well posed because of the inf-supinf-sup condition condition increase the number of odd modesincrease the number of odd modes

, , :

0 0 0 01

0 0 0 0

0 0

x y

x x x x

y y y y

T Tx y a f

H

Find p p and such that

A B p p

A B p p

B B T T


Domain decompositionDomain decomposition

Domain decomposition in 201 subdomains for a PWR 900 Domain decomposition in 201 subdomains for a PWR 900 MWe loaded with UOX and MOX assemblies :MWe loaded with UOX and MOX assemblies :

Internal subdomains boundaries :Internal subdomains boundaries :– on the middle of the assemblieson the middle of the assemblies– condition p=0 is close to the real valuecondition p=0 is close to the real value

Interface problem between UOX and MOX is avoidedInterface problem between UOX and MOX is avoided


Power and scalar flux representationPower and scalar flux representation

Power in the corePower in the core Fast fluxFast flux Thermal fluxThermal flux

diffusion calculationdiffusion calculation

two energy groupstwo energy groups

cell by cell meshcell by cell mesh

RTo elementRTo element


Comparison between our method and MINOS : Comparison between our method and MINOS : 2D2D

Keff difference, and norm of the power difference Keff difference, and norm of the power difference between CMS method and MINOS solutionbetween CMS method and MINOS solution

4 modes4 modes 9 modes9 modes

keff keff (pcm)(pcm)

4.44.4 1.41.4

5 %5 % 0.92 %0.92 %

More odd modes than even modesMore odd modes than even modes

inf-sup conditioninf-sup condition

Two CMS method cases : Two CMS method cases : – 4 even and 6 odd modes on each subdomain4 even and 6 odd modes on each subdomain– 9 even and 11 odd modes on each subdomain9 even and 11 odd modes on each subdomain

33.8 10 45.2 10

P

2P

L2L



Power gap between CMS method and MINOS in the two cases. Power gap between CMS method and MINOS in the two cases. Normalization factor : Normalization factor :

210

4 even modes, 6 odd modes4 even modes, 6 odd modes 9 even modes, 11 odd modes9 even modes, 11 odd modes

Positive

Null

Negative



Power cell difference between CMS method and MINOS solution in Power cell difference between CMS method and MINOS solution in the two cases. Total number of cells : 334084.the two cases. Total number of cells : 334084.

0

50000

100000

150000

200000

250000

Power difference

Nu

mb

er

of

ce

lls

0

50000

100000

150000

200000

250000

Power difference

Nu

mb

er

of

ce

lls

4 even modes, 6 odd modes4 even modes, 6 odd modes95% of the cells : power gap < 95% of the cells : power gap <

1%1%

9 even modes, 11 odd modes9 even modes, 11 odd modes95% of the cells : power gap < 95% of the cells : power gap <

0,1%0,1%


Same domain decomposition than in 2D.Same domain decomposition than in 2D.

Keff difference, and norm of the power difference between Keff difference, and norm of the power difference between CMS method and MINOS solution :CMS method and MINOS solution :

3D results3D results

4 modes4 modes 8 modes8 modes

keff keff (pcm)(pcm)

7.37.3 2.52.5

5.1 %5.1 % 1 %1 %

The core is split into 20 planes in the Z-axis :The core is split into 20 planes in the Z-axis :

Reflector

Reflector18 planes with the same 18 planes with the same assemblies as in 2Dassemblies as in 2D

2L L

Two CMS method cases : Two CMS method cases : – 4 even and 6 odd modes on each subdomain4 even and 6 odd modes on each subdomain– 8 even and 10 odd modes on each subdomain8 even and 10 odd modes on each subdomain3107.3 4109.7

2P

P



Power cell difference between CMS method and MINOS solution in Power cell difference between CMS method and MINOS solution in the two cases. Total number of cells : 6681680.the two cases. Total number of cells : 6681680.

0500000

100000015000002000000250000030000003500000400000045000005000000

-1,2

E-02

-8,2

E-03

-4,3

E-03

-3,4

E-04

3,6E

-03

7,5E

-03

1,1E

-02

Power difference

Nu

mb

er

of

ce

lls

0500000

100000015000002000000250000030000003500000400000045000005000000

Power difference

Nu

mb

er

of

ce

lls

4 even modes, 6 odd modes4 even modes, 6 odd modes95% of the cells : power gap < 1%95% of the cells : power gap < 1%

8 even modes, 10 odd modes8 even modes, 10 odd modes

90% of the cells : power gap < 0,1%90% of the cells : power gap < 0,1%


CPU time and parallelizationCPU time and parallelization

So far MINOS solver is faster than CMS method, BUT :So far MINOS solver is faster than CMS method, BUT :– The code is not optimizedThe code is not optimized– The deflation method used by the local eigenmodes calculations in The deflation method used by the local eigenmodes calculations in

MINOS can be improvedMINOS can be improved

CMS method CMS method most of the time spent in local calculations most of the time spent in local calculations– Independent calculations, need no communication on parallel Independent calculations, need no communication on parallel

computerscomputers– Matrix calculations are easy to parallelize too.Matrix calculations are easy to parallelize too.– Global solve time is very smallGlobal solve time is very small– With N processors, we expect to divide the time by almost NWith N processors, we expect to divide the time by almost N

On parallel computer, the CMS method will be faster than a direct On parallel computer, the CMS method will be faster than a direct heterogeneous calculationheterogeneous calculation



Modal synthesis method :Modal synthesis method :

o Good accuracy for the kefGood accuracy for the kefff and the local cell power and the local cell power

o Well fitted for parallel calculation : Well fitted for parallel calculation :

the local calculations are independentthe local calculations are independent

they need no communicationthey need no communication

Future developments : Future developments :

o Parallelization of the codeParallelization of the code

o Extension to 3D cell by cell SPn calculationsExtension to 3D cell by cell SPn calculations

o Pin by pin calculationPin by pin calculation

o Complete transport calculationsComplete transport calculations

1 a component mode synthesis method for 3d cell by cell calculation using the mixed dual finite...

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