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Réduction de Modèles à l’Issue de la Théorie Cinétique

Réduction de Modèles à l’Issue de la Théorie Cinétique. Francisco CHINESTA LMSP – ENSAM Paris Amine AMMAR Laboratoire de Rhéologie, INPG Grenoble. q 1. q 2. r 1. r 2. r N+1. q N. The different scales. R. Atomistic. Brownian dynamics. Kinetic theory: Fokker-Planck Stochastic.

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Réduction de Modèles à l’Issue de la Théorie Cinétique

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  1. Réduction de Modèles à l’Issue de la Théorie Cinétique Francisco CHINESTA LMSP – ENSAM Paris Amine AMMAR Laboratoire de Rhéologie, INPG Grenoble

  2. q1 q2 r1 r2 rN+1 qN The different scales R Atomistic Brownian dynamics • Kinetic theory: • Fokker-Planck • Stochastic

  3. Atomistic The 3 constitutive blocks:

  4. q1 q2 r1 r2 rN+1 qN Brownian dynamics Beads equilibrium usually modeled from a random motion

  5. q1 q2 r1 r2 rN+1 qN • Kinetic theory: • Fokker-Planck • Stochastic The Fokker-Planck formalism

  6. Coming back to the macroscopic scale: Stress evaluation q F q F With F & R collinear:

  7. Solving the deterministic Fokker-Planck equation Two new model reduction approaches

  8. Model Reduction based on the Karhunen-Loève decomposition Continuous: Discretization: Karhunen-Loève:

  9. Application in Computational Rheology Fokker-Planck discretisation Initial reduced approximation basis First assumption: 1 dof ! Fast simulation BUT bad results expected

  10. Enrichment based on the use of the Krylov’s subspaces: an “a priori” strategy IF IF continue The enrichment increases the number of approximation functions BUT the Karhunen-Loève decomposition reduces it

  11. 1D 300.000FEM dof ~10dof FENE Model 3D ~10 functions (1D, 2D or 3D)

  12. q1 q2 r1 r2 rN+1 qN • It is time for dreaming! For N springs, the model is defined in a 3N+3+1 dimensional space !! ~ 10 approximation functions are enough

  13. BUT How defining those high-dimensional functions ? Natural answer: with a nodal description 10 nodes = 10 function values 1D

  14. q1 q2 r1 r2 rN+1 1080 ~ presumed number ofelemental particles in the universe !! qN 1D 10 dof 10x10 dof 2D 1080 dof 80D >1000D No function can be defined in a such space from a computational point of view !! F.E.M.

  15. Advanced deterministic approaches of Multidimensional Fokker-Planck equation Separated representation and Tensor product approximation bases q1 q2 q9 Our proposal FEM GRID Computing availability ~109

  16. Example I - Projection:

  17. II - Enrichment: Only 1D interpolations and 1D integrations!

  18. q2 q1

  19. 1D/9D q1 q2 q9 809 ~ 1016 FEM dof 80x9 RM dof 2D/10D 1040 FEM dof 100.000 RM dof

  20. Solving the Stochastic representation of the Fokker-Planck equation New efficient solvers

  21. Stochastic approaches … A way for solving the Fokker-Planck equation: (Ottinger & Laso) W : Wiener random process We need tracking a large ensemble of particles and control the statistical noise!

  22. BCF Stochastique: Fokker-Planck: Brownian Configuration Fields

  23. SFS in a simple shear flow Rouge: MDF 1000 ddl / pdt a11 Bleu: BCF 100 BCF 1000 ddl / pdt Vert: Reduced BCF 100 BCF 4 ddl / pdt t The reduced approximation basis is constructed from some snapshots computed on the averaged BFC distributions

  24. Perspectives (réduction de deuxième génération) Séparation de variables ? Base commune pour les différents « configuration fields »?

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