1 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 Lautard CEA SACLAY DEN/DANS/DM2S/SERMA/LENR Gif sur Yvette Cedex

2 OUTLINES General considerations and motivations The component mode synthesis method A factorized component mode synthesis method Parallelization Conclusions and perspectives

3 General considerations and motivations General considerations and motivations The component mode synthesis method The component mode synthesis method Parallelization Parallelization Conclusions and perspectives Conclusions and perspectives A factorized component mode synthesis method A factorized component mode synthesis method

4 Geometry and mesh of a PWR 900 MWe core Pin assembly Core Pin by pin geometry Cell by cell mesh Whole core mesh

5 Introduction and motivations MINOS solver : –main core solver of the DESCARTES system, developed by CEA, EDF and AREVA –mixed dual finite element method for the resolution of the 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

6 General considerations and motivations General considerations and motivations The component mode synthesis method The component mode synthesis method Parallelization Parallelization Conclusions and perspectives Conclusions and perspectives A factorized component mode synthesis method A factorized component mode synthesis method

7 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. 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 : 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

8 Monocinetic diffusion model Monocinetic diffusion eigenvalue problem with homogeneous Dirichlet boundary condition: Mixed dual weak formulation : find such that Fundamental eigenvalue : Current : Flux

9 Local eigenmodes Overlapping domain decomposition : Computation on each of the first local eigenmodes with the global boundary condition on, and on \ :

10 Global Galerkin method Extension on R by 0 of the local eigenmodes on each :  global functional spaces on R Global eigenvalue problem on these spaces :

11 Linear system Unknowns : If all the integrals over vanish  sparse matrices with : Linear system associated : and

12 Global problem Global problem : H symmetric but not positive definite

13 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 is close to the real value Interface problem between UOX and MOX is avoided

14 Power and scalar flux representation Power in the core Thermal flux Fast flux diffusion calculation two energy groups cell by cell mesh RTo element

15 Comparison between CMS method and MINOS keff difference, and norm of the power difference between CMS method and MINOS solution 4 modes 9 modes keff keff (%)0,380,052 (%)50.92 More current modes than flux modes Two CMS method cases : –4 flux and 6 curent modes on each subdomain –9 flux and 11 current modes on each subdomain

16 Comparison between CMS method and MINOS Power gap between CMS method and MINOS in the two cases. 4 flux modes, 6 current modes 9 flux modes, 11 current modes 5% 0% -5% 1% 0% -1%

17 Comparison between CMS method and MINOS Power cell difference between CMS method and MINOS solution in the two cases. Total number of cells : flux modes, 6 current modes 95% of the cells : power gap < 1% 9 flux modes, 11 current modes 95% of the cells : power gap < 0,1%

18 General considerations and motivations General considerations and motivations The component mode synthesis method The component mode synthesis method Parallelization Parallelization Conclusions and perspectives Conclusions and perspectives A factorized component mode synthesis method A factorized component mode synthesis method

19 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 : – is a smooth function solution of a homogenized diffusion problem: – is the local fundamental solution on an assembly of the problem with infinite medium boundary conditions We adapt this principle on a non periodic core in order to replace the higher order modes

20 The factorized CMS method : FCMS solution of the problem: solution of the problem:  analytical solution  sines or cosines the fundamental eigenmode on each subdomain. the fundamental eigenmode on each subdomain. New current basis functions: New flux basis functions:

21 Same domain decomposition 6 flux modes and 11 current modes Differences between FCMS and MINOS in 2D : 97% of the cells  power gap < 1% FCMS keff keff2.2 (%)0,28 (%)2,4 0 Comparison between FCMS method and MINOS

22 JHR research reactor: first result 9 subdomains : not yet a satisfactory result : not yet a satisfactory result  improve the domain decomposition

23 JHR: power distribution

24 JHR: flux for the 6 energy groups Thermal flux Fast flux

25 General considerations and motivations General considerations and motivations The component mode synthesis method The component mode synthesis method Parallelization Parallelization Conclusions and perspectives Conclusions and perspectives A factorized component mode synthesis method A factorized component mode synthesis method

26 Parallelization of our methods in 3D Most of the calculation time: local solves and matrix calculation Local solves are independent, no communication Matrix calculations are parallelized with communications between the close subdomains Global resolution: very fast, sequential

27 General considerations and motivations General considerations and motivations The component mode synthesis method The component mode synthesis method Parallelization Parallelization Conclusions and perspectives Conclusions and perspectives A factorized component mode synthesis method A factorized component mode synthesis method

28 Conclusions and perspectives Modal synthesis method : –Good accuracy for the keff and the local cell power –Well fitted for parallel calculation: the local calculations are independent they need no communication Future developments : –Extension to 3D cell by cell calculations –Another geometries (EPR, HTR…) –Pin by pin calculation –Time dependent calculations –Coupling local calculation and global diffusion resolution –Complete transport calculations