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A robust preconditioner for the conjugate gradient method

A robust preconditioner for the conjugate gradient method. Pierre Verpeaux CEA DEN/DM2S/SEMT Stéphane Gounand CEA DEN/DM2S/SFME/LTMF Work context Direct solver Krylov solver New preconditionner Conclusion Further development. General context.

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A robust preconditioner for the conjugate gradient method

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  1. A robust preconditioner for the conjugate gradient method • Pierre Verpeaux CEA DEN/DM2S/SEMT • Stéphane Gounand CEA DEN/DM2S/SFME/LTMF • Work context • Direct solver • Krylov solver • New preconditionner • Conclusion • Further development ECCM 2010

  2. General context • General purpose Cast3m computer code developed at CEA for mechanical problem solving by FEM method. • Used in Nuclear Industry and others. • Toolbox for researcher • Blackbox for end user • Important need for qualification and validation. ECCM 2010

  3. Quality criteria for industrial code • No need for tuning • Correct solution if correct problem • Incorrect problem detection • Predictable time • No failure ECCM 2010

  4. Context • Cement paste studies – EHPOC project • Heterogeneous material with inclusions • High mechanical properties contrast • REV analysis • Need for realistic representation of inclusions including sharp edge • Need for accuracy. Reference calculation • CAO meshes with the Salome framework ECCM 2010

  5. Spherical inclusions ECCM 2010

  6. Sharp inclusions ECCM 2010

  7. 23.5 million tetrahedrons ECCM 2010

  8. Solver challenge • Large number of degrees of freedom • Very bad matrix conditioning due to • Properties contrast • Flattened elements • Various boundary condition including periodic condition of different kind. Use of cinematic constraints with Lagrange multiplier. • Multiple problem solving on the same matrix. ECCM 2010

  9. Direct solver • Robust and precise • Nested dissection ordering and sparse Crout solver • Spatial complexity in O(n4/3) • Temporal complexity in O(n5/3) for preconditioning (factorization). Parallelizable. • Temporal complexity in O(n4/3) for solving. ECCM 2010

  10. Direct solver - 2 • Out of core storage • Around 100Go of matrix storage for 10.000.000 dof. Practical limit on desktop PC. • 1d-12 accuracy • Unilateral constraints available ECCM 2010

  11. Krylov method • Constraint: maintain O(n) spatial complexity • In core preconditioner storage • No convergence with ILU(n) or ILUT preconditioner • No convergence with domain decomposition preconditioner • Some hope with algebraic multigrid preconditioner, but not yet implemented for mechanic. ECCM 2010

  12. New preconditioner • ILU(0) far better than ILU(n) for n small when it converges • Non convergence of Krylov iterations is related to small diagonal terms in the preconditioner • Idea: control the diagonal terms of the preconditioner by augmentation. • Risk of numerical instability due to the augmentation. ECCM 2010

  13. New preconditioner - 2 • Converge in all tested cases! • Most efficient with conjugate gradient method and RCM ordering • 1d-15 accuracy • O(n) space complexity • O(n4/3) temporal complexity approx • Poor parallelism • 16 000 000 dof on a 16gB desktop PC ECCM 2010

  14. Stresses in matrix ECCM 2010

  15. Convergence comparaison ECCM 2010

  16. CG versus BiCG ECCM 2010

  17. Iterations versus DOF ECCM 2010

  18. Shell mesh refinement • Convergence study • No refinement in the thickness of the element • Degradation of the stiffness matrix conditioning • No convergence of Krylov iteration at some point • In our case, loss of respect of cinematic constraints (Lagrange multiplier) ECCM 2010

  19. New preconditioner - 3 • Idea: exact factorization on constraint unknowns. Incomplete factorization on others • Applies well in solid mechanic since few filling due to cinematic constraints. To be tested in fluid mechanic with pressure constraint • Penalization of Lagrange multiplier • Convergence maintained on highly refined meshes. ECCM 2010

  20. Conclusion • New preconditioner for CG or BiCG method • RCM ordering • Augmentation of small diagonal terms • Penalization of Lagrange multipliers • Standard in Cast3M FEM code for mechanical analysis ECCM 2010

  21. Conclusion -2 • Robust – accurate • Low programming complexity (one instruction development, 6 months thinking) • Slower convergence than ILU(0) when ILU(0) works – factor 1.5-2 • No failure (yet) ECCM 2010

  22. Further development • Parallelization by block operations • Unilateral constraints on Lagrange multiplier • Automatic switch between ILU(0) and ILU(0)augmented? ECCM 2010

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