On the efficient numerical simulation of kinetic theory models of complex fluids and flows Francisco (Paco) Chinesta & Amine Ammar LMSP UMR CNRS – ENSAM.

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Presentation transcript:

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 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

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:

General Micro-Macro approach

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.

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

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

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

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

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

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

The idea … Model: PDE FEM GRID Computing multidimensional solutions

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

q1q1 F G q2q2 Solution EF q1q1 q2q2 

q1q1 F G q2q2 q1q1 q2q2 

q1q1 F G q2q2 q1q1 q2q2 

q1q1 F G q2q2 q1q1 q2q2 

q1q1 F G q2q2 q1q1 q2q2 

q1q1 F G q2q2 q1q1 q2q2 

q1q1 F G q2q2 q1q1 q2q2 

q1q1 F G q2q2 q1q1 q2q2 

q1q1 F G q2q2 q1q1 q2q2 

q1q1 F G q2q2 q1q1 q2q2 

q1q1 q2q2 q9q ~ FEM dof 80x9 RM dof FEM dof RM dof 1D/9D 2D/10D

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

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

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

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

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

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.