OUTLINES

1. A component mode synthesis method for 3D cell by cell calculation using the mixed dual finite element solver MINOS P. Guérin, A.M. Baudron, J.J. Lautard Commissariat à l’Energie Atomique DEN/DM2S/SERMA CEA SACLAY 91191 Gif sur Yvette Cedex France pierre.guerin@cea.fr

2. OUTLINES • General considerations and motivations • Basic equations • MINOS Solver • The component mode synthesis method • Numerical results • Conclusions and perspectives

3. General considerations and motivations Basic equations MINOS Solver The component mode synthesis method Numerical results 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 • MINOS solver : • main core solver of the DESCARTES system, developed by CEA, EDF and Framatome • mixed dual finite element method for the resolution of the SPn equations in 3D cartesian homogenized geometries • 3D cell by cell homogenized calculations too expensive • Standard reconstruction techniques to obtain the local pin power can be improved for MOX reloaded cores • interface between UOX and MOX assemblies

6. MOTIVATIONS • Find a numerical method that takes in account the heterogeneity of the core • Domain decomposition and two scale method : • Core decomposed in multiple subdomains • Problem solved with a fine mesh on each subdomain • Global calculation done with a basis that takes in account the local fine mesh results • Perform calculations on parallel computers

7. General considerations and motivations Basic equations MINOS Solver The component mode synthesis method Numerical results Conclusions and perspectives

8. is a tridiagonal matrix coupling the harmonics, a full matrix which depends on the albedo coefficients, and respectively the even and odd removal diagonal matrices Strong formulation of SPN equations • Derived from 1D transport Pn equation • N+1 harmonics : The (N+1)/2 even components are scalar The (N+1)/2 odds components are vectors • Coefficients : • SPN one group equation written in the mixed form (odd – even) with albedo boundary condition reads :

9. Functional spaces : Mixed dual variational SPN formulation By projection and using the Green formula on the odd equations : Even flux : discontinuous Odd flux : normal trace continuous

10. Existence and unicity of the solution • Mixed dual variational SPN equations are a particular case of the more abstract problem : • The ellipticity of the bilinear continuous form a and the inf-sup condition on the continuous form b insure existence and unicity of the solution of this problem : 

11. General considerations and motivations Basic equations MINOS Solver The component mode synthesis method Numerical results Conclusions and perspectives

12. Even nodes X-odd nodes Y-odd nodes Discretized spaces Finite Element basis on rectangle : Raviart Thomas Nedelec element (RTk) • RTk basis with : • Even basis => Orthogonallagrangian basis associated to nodes located at Gauss points of order 2k+1 • Odd flux basis such that :

13. Matrix Symmetric but not Positive Definite, elimination of the even flux : Linear system on the odd flux to solve : The matrix system The matrix of the discretized system is : Block Gauss Seidel iteration (1 block corresponds to the set of nodes of one odd flux component) Eigenvalue problem solved by power iterations

14. General considerations and motivations Basic equations MINOS Solver The component mode synthesis method Numerical results Conclusions and perspectives

15. 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 our method : • Decomposition of the core in K small domains • Calculation with the MINOS solver 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

16. Diffusion model • Monocinetic diffusion problem with homogeneous Dirichlet boundary condition. • Mixed dual weak formulation :  Eigenvalue problem

17. for all with Local eigenmodes • Overlapping domain decomposition : • Computation on each of the first local eigenmodes with the global boundary condition on , and p=0 on \ :

18. Global Galerkin method • Extension on E by 0 of the local eigenmodes on each :  global functional spaces on E • Global eigenvalue problem on these spaces :

19. If all the integrals over vanish  sparse matrices Linear system • Unknowns : • Linear system associated : with :

20. Global problem • Global problem : • H symmetric but not positive definite • Not always well posed because of the inf-sup condition • increase the number of odd modes

21. General considerations and motivations Basic equations MINOS Solver The component mode synthesis method Numerical results Conclusions and perspectives

22. 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=0 is close to the real value • Interface problem between UOX and MOX is avoided

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

24. Comparison between our method and MINOS : 2D • Keff difference, and norm of the power difference between CMS method and MINOS solution Two CMS method cases : • 4 even and 6 odd modes on each subdomain • 9 even and 11 odd modes on each subdomain • More odd modes than even modes  inf-sup condition

25. Positive Null Negative Comparison between our method and MINOS : 2D • Power gap between CMS method and MINOS in the two cases. Normalization factor : 9 even modes, 11 odd modes 4 even modes, 6 odd modes

26. Comparison between our method and MINOS : 2D • Power cell difference between CMS method and MINOS solution in the two cases. Total number of cells : 334084. 9 even modes, 11 odd modes 95% of the cells : power gap < 0,1% 4 even modes, 6 odd modes 95% of the cells : power gap < 1%

27. Reflector 18 planes with the same assemblies as in 2D Reflector 3D results • The core is split into 20 planes in the Z-axis : • Same domain decomposition than in 2D. • Keff difference, and norm of the power difference between CMS method and MINOS solution : Two CMS method cases : • 4 even and 6 odd modes on each subdomain • 8 even and 10 odd modes on each subdomain

28. Comparison between our method and MINOS : 3D • Power cell difference between CMS method and MINOS solution in the two cases. Total number of cells : 6681680. 8 even modes, 10 odd modes 90% of the cells : power gap < 0,1% 4 even modes, 6 odd modes 95% of the cells : power gap < 1%

29. CPU time and parallelization • So far MINOS solver is faster than CMS method, BUT : • The code is not optimized • The deflation method used by the local eigenmodes calculations in MINOS can be improved • CMS method  most of the time spent in local calculations • Independent calculations, need no communication on parallel computers • Matrix calculations are easy to parallelize too. • Global solve time is very small • With N processors, we expect to divide the time by almost N  On parallel computer, the CMS method will be faster than a direct heterogeneous calculation

30. General considerations and motivations Basic equations MINOS Solver The component mode synthesis method Numerical results Conclusions and perspectives

31. 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 : • Parallelization of the code • Extension to 3D cell by cell SPn calculations • Pin by pin calculation • Complete transport calculations