1. On the efficient numerical simulation of kinetic theory models of complex fluids and flows Francisco (Paco) Chinesta & Amine Ammar LMSP UMR CNRS – ENSAM PARIS, France PARIS, France francisco.chinesta@paris.ensam.fr Laboratoire de Rhéologie GRENOBLE, France GRENOBLE, France In collaboration with: R. Keunings Polymer solutions and melts M. Laso LCP M. Mackley & A. MaSuspensions of CNT

2. r1r1 r2r2 r N+1 q1q1 q2q2 qNqN R Molecular dynamics Brownian dynamics Kinetic theory: Fokker-Planck Eq. Deterministic, Stochastic & BCF solvers Constitutive Eq. The different scales: The different scales:

3. General Micro-Macro approach

4. Solving the deterministic Fokker-Planck equation New efficient solvers for: I.Reducing the simulation time of grid discretizations. II.Computing multidimensional solutions where grid methods don’t run.

5. I. Reducing the simulation time The idea … Model: PDE + Karhunen-Loève decomposition

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

7. Larson & Ottinger (Macromolecules, 1991) 2. Non-Linear Models: Doi LCP With only 6 d.o.f. !!

8. 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 II. Computing multidimensional solutions

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

10. 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 elementary particles in the universe !! ~ presumed number of elementary particles in the universe !!

11. The idea … Model: PDE FEM GRID Computing multidimensional solutions

12. q1q1 F G q2q2 Solution EF q1q1 q2q2  q1q1 q2q2 1. MBS-FENE

13. q1q1 F G q2q2 Solution EF q1q1 q2q2 

14. q1q1 F G q2q2 q1q1 q2q2 

15. q1q1 F G q2q2 q1q1 q2q2 

16. q1q1 F G q2q2 q1q1 q2q2 

17. q1q1 F G q2q2 q1q1 q2q2 

18. q1q1 F G q2q2 q1q1 q2q2 

19. q1q1 F G q2q2 q1q1 q2q2 

20. q1q1 F G q2q2 q1q1 q2q2 

21. q1q1 F G q2q2 q1q1 q2q2 

22. q1q1 F G q2q2 q1q1 q2q2 

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

24. 2. Complex Flows Example: Flow involving short fiber suspensions Kinematics:FEM-DVESS

25. s = 0 s = 1 Doi-Edwards Model Ottinger Model: double reptation, CCR, chain stretching, … 3. Entangled polymer models based on reptation motion

26. Ongoing works : (I) Stochastic models can be also reduced ! y=1

27. Reduced Brownian Configurations Fields Discretization 1.Solve i=1 and computed the reduced approximation basis 2.Solve for all i>1 the reduced problem: 1000x1000 4x4

28. Ongoing works: (II) Suspensions of CNT: Aggregation/Orientati on model Enhanced modeling: + The associated Fokker-Planck equation

29. Perspectives Enhanced kinetic model for CNT suspensions taking into account orientation and aggregation effects: FP & BD simulations. Collaboration with M. MackleyEnhanced kinetic model for CNT suspensions taking into account orientation and aggregation effects: FP & BD simulations. Collaboration with M. Mackley Reduction of Stochastic, Brownian and molecular dynamics simulations.Reduction of Stochastic, Brownian and molecular dynamics simulations. Fast micro-macro simulations of complex flows: Lattice-Boltzmann & Reduced-FP; and many others mathematical topics (stabilization, wavelet bases, mixed formulations, enhanced particles methods, …). Collaboration with T. Phillips.Fast micro-macro simulations of complex flows: Lattice-Boltzmann & Reduced-FP; and many others mathematical topics (stabilization, wavelet bases, mixed formulations, enhanced particles methods, …). Collaboration with T. Phillips.