OUTLINES

1. 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 pierre.guerin@cea.fr CEA SACLAY DEN/DANS/DM2S/SERMA/LENR 91191 Gif sur Yvette Cedex France

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 The component mode synthesis method A factorized component mode synthesis method Parallelization Conclusions and perspectives

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

5. 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 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 The component mode synthesis method A factorized component mode synthesis method Parallelization Conclusions and perspectives

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

9. Local eigenmodes • Overlapping domain decomposition : A domain decomposition in 9 subdomains for the JHR research reactor Computation on each of the first local eigenmodes with the global boundary condition on , and on \  solutions of the monocinetic diffusion problem,

10. Global Galerkin method • Extension on R by 0 of the local eigenmodes on each :  global functional spaces on R • Global eigenvalue problem: find the fundamental solution of the discretized monocinetic diffusion problem. and • Unknowns in W and V :

11. Linear system • Linear system associated (2D): find such that with : If all the integrals over vanish  sparse matrices

12. 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 is close to the real value • Interface problem between UOX and MOX is avoided

13. Power and scalar flux representation • diffusion calculation • two energy groups • cell by cell mesh Thermal flux Fast flux Power in the core

14. Comparison between CMS method and MINOS • keff difference, and norm of the power difference between CMS method and MINOS solution Two CMS method cases : • 4 flux and 6 current modes on each subdomain • 9 flux and 11 current modes on each subdomain • More current modes than flux modes

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

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

17. 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 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 of homogenized diffusion problems on each subdomain • We replace the higher order modes by:

18. Comparison between FCMS method and MINOS 0 • Same domain decomposition • 6 flux modes and 11 current modes

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

20. Subdomain 1: MINOS Subdomain k: MINOS Matrices calculations Global solve Matrices calculations 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 MPI MPI Proc k

21. CPU times and efficiency in parallel in 3D

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

23. 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 calculations • Another geometries (EPR, HTR…) • Pin by pin calculation • Time dependent calculations • Coupling local calculation and global diffusion resolution • Complete transport calculations