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

2. The different scales The different scales r1r1 r2r2 r N+1 q1q1 q2q2 qNqN R Atomistic Brownian dynamics Kinetic theory: Fokker-PlanckFokker-Planck StochasticStochastic

3. Atomistic The 3 constitutive blocks:

4. Brownian dynamics r1r1 r2r2 r N+1 q1q1 q2q2 qNqN usually modeled from a random motion Beads equilibrium

5. r1r1 r2r2 r N+1 q1q1 q2q2 qNqN Kinetic theory: Fokker-PlanckFokker-Planck StochasticStochastic The Fokker-Planck formalism

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

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 1 dof ! First assumption: Initial reduced approximation basis Fast simulation BUT bad results expected

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

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

12. It is time for dreaming! For N springs, the model is defined in a 3N+3+1 dimensional space !! ~ 10 approximation functions are enough r1r1 r2r2 r N+1 q1q1 q2q2 qNqN

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

14. 1D 2D >1000D r1r1 r2r2 r N+1 q1q1 q2q2 qNqN 80D 10 dof 10x10 dof 10 80 dof No function can be defined in a such space from a computational point of view !! F.E.M. 10 80 ~ presumed number of elemental particles in the universe !! ~ presumed number of elemental particles in the universe !!

15. Advanced deterministic approaches of Multidimensional Fokker-Planck equation Separated representation and Tensor product approximation bases q1q1 q2q2 q9q9 FEM GRID Our proposal Computing availability ~10 9

16. Example I - Projection:

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

18. q1q1 q2q2

19. q1q1 q2q2 q9q9 80 9 ~ 10 16 FEM dof 80x9 RM dof 10 40 FEM dof 100.000 RM dof 1D/9D 2D/10D

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. Fokker-Planck: Stochastique: BCF Brownian Configuration Fields

23. SFS in a simple shear flow Rouge: MDF 1000 ddl / pdt Bleu: BCF 100 BCF 1000 ddl / pdt Vert: Reduced BCF 100 BCF 4 ddl / pdt a11 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 »?