1. 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. American Nuclear Society Topical Meeting in Mathematics & Computations, Avignon, France, 2005 2 OUTLINES General considerations and motivations Basic equations MINOS Solver The component mode synthesis method Numerical results Conclusions and perspectives

3. American Nuclear Society Topical Meeting in Mathematics & Computations, Avignon, France, 2005 3 General considerations and motivations General considerations and motivations Basic equations Basic equations MINOS Solver MINOS Solver Numerical results Numerical results Conclusions and perspectives Conclusions and perspectives The component mode synthesis method The component mode synthesis method

4. American Nuclear Society Topical Meeting in Mathematics & Computations, Avignon, France, 2005 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. American Nuclear Society Topical Meeting in Mathematics & Computations, Avignon, France, 2005 5 INTRODUCTION MINOS solver : omain core solver of the DESCARTES system, developed by CEA, EDF and Framatome omixed dual finite element method for the resolution of the SPn equations in 3D cartesian homogenized geometries o3D cell by cell homogenized calculations too expensive Standard reconstruction techniques to obtain the local pin power can be improved for MOX reloaded cores ointerface between UOX and MOX assemblies

6. American Nuclear Society Topical Meeting in Mathematics & Computations, Avignon, France, 2005 6 MOTIVATIONS Find a numerical method that takes in account the heterogeneity of the core Domain decomposition and two scale method : oCore decomposed in multiple subdomains oProblem solved with a fine mesh on each subdomain oGlobal calculation done with a basis that takes in account the local fine mesh results Perform calculations on parallel computers

7. American Nuclear Society Topical Meeting in Mathematics & Computations, Avignon, France, 2005 7 General considerations and motivations General considerations and motivations Basic equations Basic equations MINOS Solver MINOS Solver Numerical results Numerical results Conclusions and perspectives Conclusions and perspectives The component mode synthesis method The component mode synthesis method

8. American Nuclear Society Topical Meeting in Mathematics & Computations, Avignon, France, 2005 8 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 Strong formulation of SPN equations SPN one group equation written in the mixed form (odd – even) with albedo boundary condition reads : Coefficients : 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 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

9. American Nuclear Society Topical Meeting in Mathematics & Computations, Avignon, France, 2005 9 Mixed dual variational SPN formulation By projection and using the Green formula on the odd equations : Functional spaces : Even flux : discontinuous Odd flux : normal trace continuous

10. American Nuclear Society Topical Meeting in Mathematics & Computations, Avignon, France, 2005 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. American Nuclear Society Topical Meeting in Mathematics & Computations, Avignon, France, 2005 11 General considerations and motivations General considerations and motivations Basic equations Basic equations MINOS Solver MINOS Solver Numerical results Numerical results Conclusions and perspectives Conclusions and perspectives The component mode synthesis method The component mode synthesis method

12. American Nuclear Society Topical Meeting in Mathematics & Computations, Avignon, France, 2005 12 Discretized spaces RTk basis with : –Even basis => Orthogonallagrangian basis associated to nodes located at Gauss points of order 2k+1 –Even basis => Orthogonal lagrangian basis associated to nodes located at Gauss points of order 2k+1 –Odd flux basis such that : Finite Element basis on rectangle : Raviart Thomas Nedelec element (RTk) Even nodes X-odd nodes Y-odd nodes

13. American Nuclear Society Topical Meeting in Mathematics & Computations, Avignon, France, 2005 13 The matrix system Matrix Symmetric but not Positive Definite, elimination of the even flux : Linear system on the odd flux to solve : 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. American Nuclear Society Topical Meeting in Mathematics & Computations, Avignon, France, 2005 14 General considerations and motivations General considerations and motivations Basic equations Basic equations MINOS Solver MINOS Solver Numerical results Numerical results Conclusions and perspectives Conclusions and perspectives The component mode synthesis method The component mode synthesis method

15. American Nuclear Society Topical Meeting in Mathematics & Computations, Avignon, France, 2005 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. 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 : 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. American Nuclear Society Topical Meeting in Mathematics & Computations, Avignon, France, 2005 16 Diffusion model Monocinetic diffusion problem with homogeneous Dirichlet boundary condition. Mixed dual weak formulation :  Eigenvalue problem

17. American Nuclear Society Topical Meeting in Mathematics & Computations, Avignon, France, 2005 17 Local eigenmodes Overlapping domain decomposition : Computation on each of the first local eigenmodes with the global boundary condition on, and p=0 on \ : for all with

18. American Nuclear Society Topical Meeting in Mathematics & Computations, Avignon, France, 2005 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. American Nuclear Society Topical Meeting in Mathematics & Computations, Avignon, France, 2005 19 Linear system Unknowns : If all the integrals over vanish  sparse matrices with : Linear system associated :

20. American Nuclear Society Topical Meeting in Mathematics & Computations, Avignon, France, 2005 20 Global problem Global problem : H symmetric but not positive definite Not always well posed because of the inf-sup condition inf-sup  increase the number of odd modes

21. American Nuclear Society Topical Meeting in Mathematics & Computations, Avignon, France, 2005 21 General considerations and motivations General considerations and motivations Basic equations Basic equations MINOS Solver MINOS Solver Numerical results Numerical results Conclusions and perspectives Conclusions and perspectives The component mode synthesis method The component mode synthesis method

22. American Nuclear Society Topical Meeting in Mathematics & Computations, Avignon, France, 2005 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. American Nuclear Society Topical Meeting in Mathematics & Computations, Avignon, France, 2005 23 Power and scalar flux representation Power in the core Fast flux Thermal flux diffusion calculation two energy groups cell by cell mesh RTo element

24. American Nuclear Society Topical Meeting in Mathematics & Computations, Avignon, France, 2005 24 Comparison between our method and MINOS : 2D Keff difference, and norm of the power difference between CMS method and MINOS solution 4 modes 9 modes keff (pcm) keff (pcm)4.41.4 5 % 0.92 % More odd modes than even modes  inf-sup condition Two CMS method cases : –4 even and 6 odd modes on each subdomain –9 even and 11 odd modes on each subdomain

25. American Nuclear Society Topical Meeting in Mathematics & Computations, Avignon, France, 2005 25 Comparison between our method and MINOS : 2D Power gap between CMS method and MINOS in the two cases. Normalization factor : 4 even modes, 6 odd modes 9 even modes, 11 odd modes Positive Null Negative

26. American Nuclear Society Topical Meeting in Mathematics & Computations, Avignon, France, 2005 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. 4 even modes, 6 odd modes 95% of the cells : power gap < 1% 9 even modes, 11 odd modes 95% of the cells : power gap < 0,1%

27. American Nuclear Society Topical Meeting in Mathematics & Computations, Avignon, France, 2005 27 Same domain decomposition than in 2D. Keff difference, and norm of the power difference between CMS method and MINOS solution : 3D results 4 modes 8 modes keff (pcm) keff (pcm)7.32.5 5.1 % 1 % The core is split into 20 planes in the Z-axis : Reflector 18 planes with the same assemblies as in 2D Two CMS method cases : –4 even and 6 odd modes on each subdomain –8 even and 10 odd modes on each subdomain

28. American Nuclear Society Topical Meeting in Mathematics & Computations, Avignon, France, 2005 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. 4 even modes, 6 odd modes 95% of the cells : power gap < 1% 8 even modes, 10 odd modes 90% of the cells : power gap < 0,1%

29. American Nuclear Society Topical Meeting in Mathematics & Computations, Avignon, France, 2005 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. American Nuclear Society Topical Meeting in Mathematics & Computations, Avignon, France, 2005 30 General considerations and motivations General considerations and motivations Basic equations Basic equations MINOS Solver MINOS Solver Numerical results Numerical results Conclusions and perspectives Conclusions and perspectives The component mode synthesis method The component mode synthesis method

31. American Nuclear Society Topical Meeting in Mathematics & Computations, Avignon, France, 2005 31 Conclusions and perspectives Modal synthesis method : oGood accuracy for the keff and the local cell power oWell fitted for parallel calculation :  the local calculations are independent  the local calculations are independent  they need no communication  they need no communication Future developments : oParallelization of the code oExtension to 3D cell by cell SPn calculations oPin by pin calculation oComplete transport calculations