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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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R duction de mod les l issue de la th orie cin tique

Réduction de Modèles à l’Issue de la Théorie Cinétique

Francisco CHINESTA

LMSP – ENSAM Paris

Amine AMMAR

Laboratoire de Rhéologie, INPG Grenoble


R duction de mod les l issue de la th orie cin tique

q1

q2

r1

r2

rN+1

qN

The different scales

R

Atomistic

Brownian dynamics

  • Kinetic theory:

  • Fokker-Planck

  • Stochastic


R duction de mod les l issue de la th orie cin tique

Atomistic

The 3 constitutive blocks:


R duction de mod les l issue de la th orie cin tique

q1

q2

r1

r2

rN+1

qN

Brownian dynamics

Beads equilibrium

usually modeled from a random motion


R duction de mod les l issue de la th orie cin tique

q1

q2

r1

r2

rN+1

qN

  • Kinetic theory:

  • Fokker-Planck

  • Stochastic

The Fokker-Planck formalism


R duction de mod les l issue de la th orie cin tique

Coming back to the macroscopic scale:

Stress evaluation

q

F

q

F

With F & R collinear:


Solving the deterministic fokker planck equation

Solving the deterministic Fokker-Planck equation

Two new model reduction approaches


R duction de mod les l issue de la th orie cin tique

Model Reduction based on the Karhunen-Loève decomposition

Continuous:

Discretization:

Karhunen-Loève:


R duction de mod les l issue de la th orie cin tique

Application in Computational Rheology

Fokker-Planck discretisation

Initial reduced approximation basis

First assumption:

1 dof !

Fast simulation BUT bad results expected


R duction de mod les l issue de la th orie cin tique

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


R duction de mod les l issue de la th orie cin tique

1D

300.000FEM dof

~10dof

FENE Model

3D

~10 functions (1D, 2D or 3D)


R duction de mod les l issue de la th orie cin tique

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


R duction de mod les l issue de la th orie cin tique

BUT

How defining those

high-dimensional functions ?

Natural answer: with a nodal description

10 nodes = 10 function values

1D


R duction de mod les l issue de la th orie cin tique

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.


R duction de mod les l issue de la th orie cin tique

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


R duction de mod les l issue de la th orie cin tique

Example

I - Projection:


R duction de mod les l issue de la th orie cin tique

II - Enrichment:

Only 1D interpolations and 1D integrations!


R duction de mod les l issue de la th orie cin tique

q2

q1


R duction de mod les l issue de la th orie cin tique

1D/9D

q1

q2

q9

809 ~ 1016 FEM dof

80x9 RM dof

2D/10D

1040 FEM dof

100.000 RM dof


R duction de mod les l issue de la th orie cin tique

Solving the Stochastic representation of the Fokker-Planck equation

New efficient solvers


R duction de mod les l issue de la th orie cin tique

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!


Fokker planck

BCF

Stochastique:

Fokker-Planck:

Brownian Configuration Fields


Sfs in a simple shear flow

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


Perspectives r duction de deuxi me g n ration

Perspectives (réduction de deuxième génération)

Séparation de variables ?

Base commune pour les différents « configuration fields »?


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