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Modal methods for 3D heterogeneous neutronics core calculations using the mixed dual solver MINOS. Application to complex

geometries and parallel processing.

Pierre Guérin, Jean-Jacques LautardPierre Guérin, Jean-Jacques [email protected]@cea.fr

CEA SACLAYCEA SACLAYDEN/DANS/DM2S/SERMA/LENRDEN/DANS/DM2S/SERMA/LENR91191 Gif sur Yvette Cedex91191 Gif sur Yvette Cedex

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OUTLINESOUTLINES

General considerations and motivationsGeneral considerations and motivations

The component mode synthesis methodThe component mode synthesis method

A factorized component mode synthesis A factorized component mode synthesis

methodmethod

ParallelizationParallelization

Conclusions and perspectivesConclusions and perspectives

33





A factorized component mode synthesis methodA factorized component mode synthesis method

44

Geometry and mesh of a PWR 900 MWe coreGeometry 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

55

Introduction and motivationsIntroduction and motivations

MINOS solver :MINOS solver :– main core solver of the DESCARTES system, developed by main core solver of the DESCARTES system, developed by

CEA, EDF and AREVACEA, EDF and AREVA– mixed dual finite element method for the resolution of the mixed dual finite element method for the resolution of the

equations in 3D cartesian homogenized equations in 3D cartesian homogenized geometriesgeometries

– 3D cell by cell homogenized calculations currently 3D cell by cell homogenized calculations currently expensiveexpensive

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

Motivations: Motivations: – Find a numerical method that takes in account the Find a numerical method that takes in account the

heterogeneity of the coreheterogeneity of the core– Perform calculations on parallel computersPerform calculations on parallel computers

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77

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 the CMS method : The steps of the CMS method : – Decomposition of the domain in K subdomainsDecomposition of the domain in K subdomains– Calculation of the first eigenfunctions of the local Calculation of the first eigenfunctions of the local

problem on each subdomain problem on each 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

88

Monocinetic diffusion modelMonocinetic diffusion model

Monocinetic diffusion eigenvalue problem with homogeneous Monocinetic diffusion eigenvalue problem with homogeneous Dirichlet boundary condition:Dirichlet boundary condition:

).(),(),(,1

.

0..1

2 RLRdivHq

keffp

qqpD

R

f

R

a

R

RR

.

0

1.

0

sur

surkeff

p

surDp

fa

Mixed dual weak formulation :Mixed dual weak formulation :

find such thatfind such thatkeffRLRdivHp and)(),(),( 2

Fundamental eigenvalueFundamental eigenvalue

: Current: Current

: Flux: Flux

p

dimension space the Swith)(.;)(),( 22 RLqRLqRdivHS

99

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 on \ :the global boundary condition on , and on \ :

kK

k

RR 1

kR kNkR RR

,1,1,1

.

),(),(),(,0..1 2

,0

k

R

kifk

iR

kia

R

ki

kk

R\RR

ki

R

ki

NiKkp

RLRdivHqqqpD

kkk

k

kk

:)(),(),( 2

,0thatsuchandfind

R\R kki

kkki

ki RLRdivHp

0. np

.0.),,(),(,0

RRonnqRdivHqRdivH kkk

RRk

\\

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Global Galerkin methodGlobal Galerkin method

Extension on Extension on RR by 0 of the local eigenmodes on each : by 0 of the local eigenmodes on each : kR

global functional spaces on global functional spaces on RR

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

.)(~),(~ 21

1

1

,1, RLspanVandRdivHpspanW Kk

Ni

ki

Kk

dNi

kdi kk

.),(,1

.

0..1

VWq

p

qqpD

R

f

R

a

R

RR

:thatsuchandFind VWp ),(

1111

Linear systemLinear system

Unknowns :Unknowns :

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

lk RR lk RR

with :

fa or

KKd

Kd

Kd

kld

Kddd

Kddd

d

AAA

A

AAA

AAA

A

..

...

..

..

11

22221

11211

KKd

Kd

Kd

kld

Kddd

Kddd

d

BBB

B

BBB

BBB

B

..

...

..

..

21

22221

11211

KKKK

kl

K

K

TTT

T

TTT

TTT

T

..

...

..

..

21

22221

11211

lk RR

ldj

kdiji

kld pp

DA ,,,

~.~1

lk RR

lj

kdiji

kld pB ~.~. ,,

lk RR

lj

kiji

klT ~~

,

Linear system associated :Linear system associated :

K

k

N

i d

kdi

kdi

k

pcp1 1

,,~

andand

K

k

N

i

ki

ki

k

f1 1

~

kif

kia

d

kdi

Td

kid

kdid

fTfTcB

dfBcA

1

0

,

,

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Global problemGlobal problem

Global problem :Global problem :

H symmetric but not positive definiteH symmetric but not positive definite

, , :

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

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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 is close to the real valuecondition is close to the real value

Interface problem between UOX and MOX is avoidedInterface problem between UOX and MOX is avoided0. np

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Power and scalar flux representationPower and scalar flux representation

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

diffusion calculationdiffusion calculation

two energy groupstwo energy groups

cell by cell meshcell by cell mesh

RTo elementRTo element

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Comparison between CMS method and MINOSComparison between CMS method and MINOS

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 4.44.4 1.41.4

(%)(%) 0,380,38 0,0520,052

(%)(%) 55 0.920.92

More current modes than flux modesMore current modes than flux modes

Two CMS method cases : Two CMS method cases : – 4 flux and 6 curent modes on each subdomain4 flux and 6 curent modes on each subdomain– 9 flux and 11 current modes on each subdomain9 flux and 11 current modes on each subdomain

P

2P

L2L

)10( 5

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

4 flux modes, 6 current modes4 flux modes, 6 current modes 9 flux modes, 11 current modes9 flux modes, 11 current modes

5%

0%

-5%

1%

0%

-1%

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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 flux modes, 6 current modes4 flux modes, 6 current modes95% of the cells : power gap < 95% of the cells : power gap <

1%1%

9 flux modes, 11 current modes9 flux modes, 11 current modes95% of the cells : power gap < 95% of the cells : power gap <

0,1%0,1%

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Factorization principleFactorization principle

Goal: decrease CPU time and memory storage Goal: decrease CPU time and memory storage only the fundamental mode calculationonly the fundamental mode calculation replace the higher order modes by suitably chosen replace the higher order modes by suitably chosen

functionsfunctions

Factorization principle on a periodic core :Factorization principle on a periodic core :– iis a smooth function solution of a homogenized diffusion s a smooth function solution of a homogenized diffusion

problem:problem:

– is the local fundamental solution on an assembly of the is the local fundamental solution on an assembly of the problem with infinite medium boundary conditionsproblem with infinite medium boundary conditions

We adapt this principle on a non periodic core in order to We adapt this principle on a non periodic core in order to replace the higher order modesreplace the higher order modes

. ii uiu

,

0

1).(

Ru

RuuD

i

ii

i

sur

sur

2020

The factorized CMS method : FCMSThe factorized CMS method : FCMS

solution of the problem:solution of the problem:

analytical solution analytical solution sines or cosinessines or cosines

the fundamental eigenmode on each subdomain.the fundamental eigenmode on each subdomain.

New current basis functions:New current basis functions:

New flux basis functions:New flux basis functions:

kki

kki

kki

ki

RRsuru

RRsurn

u

RsuruuD

0

,0 \

k

iu

.1,2 KkNi k

),( kk p

:),(~ RdivHp ki

.

1,20~

)0(~

0~

~

,

,

,1

,1

KkNiRRsurp

d

uifRsur

d

up

RRsurp

Rsurpp

kkkdi

kik

kik

di

kkd

kkd

kd

\and

\

:)(~ 2 RLki

.1,20~

~

0~

~

1

1 KkNiRRsur

Rsuru

RRsur

Rsur k

kki

kki

kki

kk

kkk

\and

\

2121

Same domain decompositionSame domain decomposition6 flux modes and 11 current modes6 flux modes and 11 current modesDifferences between FCMS and MINOS in 2D :Differences between FCMS and MINOS in 2D :97% of the cells 97% of the cells power gap < 1% power gap < 1%

FCMSFCMS

keffkeff 2.22.2

(%)(%) 0,280,28

(%)(%) 2,42,4

2P

P

25.2 E

25.2 E

0

Comparison between FCMS method and MINOSComparison between FCMS method and MINOS

0,0E+005,0E+051,0E+061,5E+062,0E+062,5E+063,0E+063,5E+064,0E+064,5E+065,0E+06

Power difference

Nu

mb

er

of

ce

lls

)10( 5

2222

JHR research reactor: first resultJHR research reactor: first result

9 subdomains9 subdomains : not yet a satisfactory result: not yet a satisfactory result improve the domain decompositionimprove the domain decomposition

321.1keff

2323

JHR: power distributionJHR: power distribution

2424

JHR: flux for the 6 energy groupsJHR: flux for the 6 energy groups

Thermal fluxThermal flux

Fast fluxFast flux

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Parallelization of our methods in 3DParallelization of our methods in 3D

Most of the calculation time: local solves and matrix calculationMost of the calculation time: local solves and matrix calculation

Local solves are independent, no communicationLocal solves are independent, no communication

Matrix calculations are parallelized with communications between the Matrix calculations are parallelized with communications between the close subdomainsclose subdomains

Global resolution: very fast, sequential Global resolution: very fast, sequential

0 500 1000 1500 2000 2500

Local Solve

Matrices

Global Solve

0,20

0,30

0,40

0,50

0,60

0,70

0,80

0,90

1,00

2 4 8 16 24 26 32

Total

Local Solve

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2828


Modal synthesis method :Modal synthesis method :– Good accuracy for the keff and the local cell powerGood accuracy for the keff and the local cell power– Well fitted for parallel calculation: Well fitted for parallel calculation:

the local calculations are independentthe local calculations are independentthey need no communicationthey need no communication

Future developments : Future developments : – Extension to 3D cell by cell calculationsExtension to 3D cell by cell calculations– Another geometries (EPR, HTR…) Another geometries (EPR, HTR…) – Pin by pin calculationPin by pin calculation– Time dependent calculations Time dependent calculations – Coupling local calculation and global diffusion Coupling local calculation and global diffusion

resolutionresolution– Complete transport calculationsComplete transport calculations

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