Download Component mode synthesis methods applied to 3D heterogeneous

Component mode synthesis methods
applied to 3D heterogeneous core calculations,
using the mixed dual finite element solver
MINOS
P. Guérin, A.-M. Baudron, J.-J. Lautard
[email protected]
CEA SACLAY
DEN/DANS/DM2S/SERMA/LENR
91191 Gif sur Yvette Cedex
France
1
OUTLINES
General considerations and motivations
The component mode synthesis method
A factorized component mode synthesis method
Parallelization
Conclusions and perspectives
2
General considerations and motivations
The component mode synthesis method
A factorized component mode synthesis method
Parallelization
Conclusions and perspectives
3
Geometry and mesh of a PWR 900 MWe core
Pin
Pin by pin geometry
assembly
Core
Cell by cell mesh
Whole core mesh
4
Introduction and motivations
MINOS solver :
–
–
–
main core solver of the DESCARTES project, developed by
CEA, EDF and AREVA
mixed dual finite element method for the resolution of the
SPN equations in 3D cartesian homogenized geometries
3D cell by cell homogenized calculations currently expensive
Standard reconstruction techniques to obtain the local pin
power can be improved for MOX reloaded cores
–
interface between UOX and MOX assemblies
Motivations:
–
–
Find a numerical method that takes in account the
heterogeneity of the core
Perform calculations on parallel computers
5
General considerations and motivations
The component mode synthesis method
A factorized component mode synthesis method
Parallelization
Conclusions and perspectives
6
The CMS method
CMS method for the computation of the eigenmodes
of partial differential equations has been used for a
long time in structural analysis.
The steps of the CMS method :
–
–
–
Decomposition of the domain in K subdomains
Calculation of the first eigenfunctions of the local
problem on each subdomain
All these local eigenfunctions span a discrete space
used for the global solve by a Galerkin technique
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Monocinetic diffusion model
Monocinetic diffusion eigenvalue problem with homogeneous
Dirichlet boundary condition:


 p  D  0 sur 

1
 

.
p




 f

a
keff

  0 sur 

p: Current
: Flux
sur  .
Fundamental eigenvalue
Mixed dual weak formulation :

find ( p,  )  H (div, R)  L2 ( R) and keff such that
 
1 

  D p.q   .q  0

R
R
2
,

(
q
,

)

H
(
div
,
R
)

L
( R).
 
 . p       1   
f
R a
R
keff R
S  

2
H (div , R)  q  L ( R) ; .q  L2 ( R) with S the space dimension


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Local eigenmodes
Overlapping domain decomposition : R 
K
k
R

k 1
A domain decomposition
in 9 subdomains for the
JHR research reactor
Computation on each R k of the first N klocal eigenmodes with the
global boundary condition on R, and p.n  0 on R k\ R
k k
 find ( pi , i ) solutions of the monocinetic diffusion problem,
1  k  K , 1  i  N k .
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Global Galerkin method
Extension on
 
k
~
W  span pi ,d
R by 0 of the local eigenmodes on each R k:
1 k  K
1i  N k , d
 
 H (div , R) and V  span ~ik
 global functional spaces on
1 k  K
1i  N k
 L2 ( R) .
R
Global eigenvalue problem: find the fundamental solution

( p ,  ) W  V and  of the discretized monocinetic diffusion
problem.
K Nk


k
pik,d and    f i k~ik
Unknowns in W and V : p   ci ,d ~
K
Nk
k 1 i 1
d
k 1 i 1
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Linear system
Linear system associated (2D): find ( p x , p y ,  ) and  such that
 Ax
0 0 0   p x 
0
Bx   p x 

  1 
 
0

A
B
p

0
0
0
y
y  y 


 py 

T
 BxT
0 0 T f    
B
Ta    
y





H
with :
 Ad11
 21
A
Ad   d
 .
 AK1
 d
A 
kl
d i, j

Ad12
Ad22
Adkl
AdK 1

R k Rl
If
 T11 T12
Ad1K 

 21
2K
.. Ad 
T22
 T
T 
kl
.
.    .
T


KK 
T K1 T K 2
.. Ad 

 
..
1 ~ k ~ l
pi ,d . p j ,d
D
T 
kl

i, j
 Bd11 Bd12 .. Bd1K 
.. T1K 
 21


22
2K
2K
Bd .. Bd 
 Bd
.. T 
B

d
 .

kl
B
.
.
.
. 
d


 B K 1 B K 2 .. B KK 
KK 
.. T 
d
d 
 d
    ~ik~ jl
R k Rl
    a or  f
B 
kl
d i, j
 ~ k ~ l
  . pi ,d  j
R k Rl
R k  R l   all the integrals over R k  R lvanish  sparse matrices
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PWR 900 Mwe: domain decomposition
Domain decomposition in 201 subdomains for a PWR 900 MWe
loaded with UOX and MOX assemblies :
Internal subdomains boundaries :
– on the middle of the assemblies

– condition p.n  0 is close to the real value
Interface problem between UOX and MOX is avoided
12
Power and scalar flux representation
diffusion calculation
two energy groups
cell by cell mesh
Power in the core
Thermal flux
Fast flux
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Comparison between CMS method and MINOS

keff difference, L and L norm of the power difference
between CMS method and MINOS solution
2
Two CMS method cases :
– 4 flux and 6 current modes on each subdomain
– 9 flux and 11 current modes on each subdomain
 keff (105 )
 P 2 (%)
 P  (%)
4 modes
9 modes
4.4
1.4
0,38
0,052
5
0.92
More current modes than flux modes
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Comparison between CMS method and MINOS
Power gap between CMS method and MINOS in the two cases.
5%
1%
0%
0%
-5%
-1%
4 flux modes, 6 current modes
9 flux modes, 11 current modes
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General considerations and motivations
The component mode synthesis method
A factorized component mode synthesis method
Parallelization
Conclusions and perspectives
16
Factorization principle
Goal: decrease CPU time and memory storage
 only the fundamental mode calculation
 replace the higher order modes by suitably chosen functions
Factorization principle on a periodic core:
–
–
i  ui  .
ui is a smooth function solution of a homogenized diffusion problem:

1
 


.(
D

ui )   ui

i

u  0 on R
 i
on R
,
 is the local fundamental solution of the problem on an assembly with
infinite medium boundary conditions
We adapt this principle on a non periodic core in order to replace
the higher order modes:
–
–
We use the solutions u k of homogenized diffusion problems on each
i
subdomain
We replace the higher order modes by:
   u , p
k
i
k
k
i
k
i ,d
uik

d
on R k , 2  i  N k , 1  k  K .
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Comparison between FCMS method and MINOS
Same domain decomposition
6 flux modes and 11 current modes
2.5E  2
FCMS
 keff (105 )
0
P
P
 2.5E  2
2.2
2
(%)
0,28

(%)
2,4
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General considerations and motivations
The component mode synthesis method
A factorized component mode synthesis method
Parallelization
Conclusions and perspectives
19
Parallelization of our methods
Most of the calculation time: local solves and matrix calculations
Local solves are independent, no communication
Matrix calculations are parallelized with communications between the
close subdomains
Global resolution: very fast, sequential
Proc 1
Subdomain 1: MINOS

i1 , pi1 , 1  i  N 1
MPI
Proc k
Matrices calculations
Global solve
Ad1l , T1l , Bd1l , l if 1  l  

 glob , pglob , keff
if 1  k  
MPI
Subdomain k: MINOS
Matrices calculations
k
 , pi , 1  i  N k
Adkl , Tkl , Bdkl , l if  k  l  
k
i
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CPU times and efficiency in parallel in 3D
1200
100
CPU time (s.)
1100
Efficiency (%)
90
1000
80
900
70
800
700
60
600
50
500
40
400
30
300
20
200
10
100
0
0
MINOS
1
2
4
7
8
16
24
Number of processors
25
26
32
49
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General considerations and motivations
The component mode synthesis method
A factorized component mode synthesis method
Parallelization
Conclusions and perspectives
22
Conclusions and perspectives
Modal synthesis method :
–
–
Good accuracy for the keff and the local cell power
Well fitted for parallel calculation:
local calculations are independent
they need no communication
Future developments :
–
–
–
–
–
–
Extension to 3D cell by cell SPN calculations
Another geometries (EPR, HTR…)
Pin by pin calculation
Time dependent calculations
Coupling local SPN calculation and global diffusion resolution
Complete transport calculations
23