On the efficient numerical simulation On the efficient numerical simulation of kinetic theory models of complex of kinetic theory models of complex

fluids and flowsfluids and flows

Francisco (Paco) Chinesta & Amine AmmarFrancisco (Paco) Chinesta & Amine AmmarLMSP UMR CNRS – ENSAM LMSP UMR CNRS – ENSAM

PARIS, FrancePARIS, Francefrancisco.chinesta@paris.ensam.frfrancisco.chinesta@paris.ensam.fr

Laboratoire de Rhéologie Laboratoire de Rhéologie

GRENOBLE, FranceGRENOBLE, France

In collaboration with:In collaboration with:

R. Keunings R. Keunings Polymer solutions and meltsPolymer solutions and melts

M. Laso M. Laso LCPLCP

M. Mackley & A. MaM. Mackley & A. Ma Suspensions of CNTSuspensions of CNT

r1 r2

rN+1

q1 q2

qN

RR

tzyx ,,,

Molecular dynamicsMolecular dynamics

Brownian dynamicsBrownian dynamics

Kinetic theory:Kinetic theory:

Fokker-Planck Eq.Fokker-Planck Eq.

Deterministic, Deterministic, Stochastic & Stochastic & BCF solversBCF solvers

Constitutive Eq.Constitutive Eq.

),,,,,,(ψ1 N

qqtzyx

(3 1 3 )N D

The The different different scales:scales:

General Micro-Macro approachGeneral Micro-Macro approach

d

A Ddt q q q

( ) ( ) C

g q q dq

2 NpId D

X ADiv 0 vDiv

Solving the deterministic Solving the deterministic Fokker-Planck equationFokker-Planck equation

New efficient solvers for:New efficient solvers for:

I.I. Reducing the simulation time of grid Reducing the simulation time of grid discretizations.discretizations.

II.II. Computing multidimensional solutions Computing multidimensional solutions where grid methods don’t run.where grid methods don’t run.

I. Reducing the simulation timeI. Reducing the simulation time

The idea …The idea …

1 P PR A U F 1

np p

j jj

U

Model: PDEModel: PDE1 p pa f

1

1

n

n n

R

n N

Model: PDEModel: PDE1 pp FUA

N N

( , )p piU u x t

1, , , 1, ,i N p P

n n+ Karhunen-Loève decomposition+ Karhunen-Loève decomposition

1. FENE 1. FENE ModelModel

300.000300.000 FEM dofFEM dof ~10~10 dofdof~10 functions (1D, 2D or 3D)~10 functions (1D, 2D or 3D)

3D3D

2

2

1

1

H( q )q

b

2

2

1 1

H(q)qb

1D1D

q

H(q)

Larson & Ottinger Larson & Ottinger

(Macromolecules, 1991)(Macromolecules, 1991)

( , , ) (0,0, )u v w x

2. Non-Linear 2. Non-Linear Models: Doi LCPModels: Doi LCP

With only 6 d.o.f. !!With only 6 d.o.f. !!

It is time for dreamingIt is time for dreaming!!

1( )

4q A

t q q q

For N springs, the model is defined For N springs, the model is defined in a 3in a 3NN+3+1 dimensional space !! +3+1 dimensional space !!

~ 10 approximation functions are ~ 10 approximation functions are enoughenough

),,,,,,,(21

tzyxqqqN

r1 r2

rN+1

q1 q2

qN

1

~10 10 ~10 1 ~10 1

p pa

1 2( , , , )

Nq q q q

II. Computing multidimensional solutionsII. Computing multidimensional solutions

BUTBUT ~10

1 2 3 3 1 2 3 31

( , , , , ) ( ) ( , , , )N i Nii

x x x t t x x x

How defining those How defining those high-dimensional functions ?high-dimensional functions ?

Natural answerNatural answer: with a nodal description: with a nodal description

1D1D

10 nodes = 10 function values10 nodes = 10 function values

1D

2D2D

>1000D>1000D

r1 r2

rN+1

q1 q2

qN

80D80D

10 dof10 dof

10x10 dof10x10 dof

10108080 dof dof

No function can be defined in a such space from No function can be defined in a such space from a computational point of view !!a computational point of view !!

F.E.M.

1080 ~ presumed number of~ presumed number of elementary particles in the universe !!elementary particles in the universe !!

The idea …The idea …

Model: PDEModel: PDE1

( , ) ( ) ( )n

j j jj

u x y F x G y

1

1

1

1

( )

( )n

n

n

n n

F x

G y

FEMFEM

GRIDGRID10 301000 10DIMDOF N

1 10 1 1 10 101

( , , ) ( ) ( )n

j j jj

x x F x F x

41000 10 10DOF N DIM

Computing multidimensional solutions Computing multidimensional solutions

q1

FG

q2

Solution EFSolution EFq1

q2

1 1 11 1 22( , ) ( ) ( )F q G qq q q1 q2

1. MBS-FENE1. MBS-FENE

q1

FG

q2

Solution EFSolution EFq1

q2

21 1 1 1 2 1 221 2 2( ) () )( ) (, () FF q Gq q qq G q

q1

FG

q2

Solution EFSolution EFq1

q2

3 3 1 31 1 1 1 2 2 1 21 2 2 22( )( , ) ( ) ( )( ) ( ) ( )FF q G F qq G q Gqq q q

q1

FG

q2

Solution EFSolution EFq1

q2

2 2 411 1 2 21 1 4 1 4 22 1 2 (( ( ( ), ) (( )) ) ) ( )Fq q F q GF q qqq Gq G

q1

FG

q2

Solution EFSolution EFq1

q2

2 2 511 1 2 21 1 5 1 5 22 1 2 (( ( ( ), ) (( )) ) ) ( )Fq q F q GF q qqq Gq G

q1

FG

q2

Solution EFSolution EFq1

q2

2 2 611 1 2 21 1 6 1 6 22 1 2 (( ( ( ), ) (( )) ) ) ( )Fq q F q GF q qqq Gq G

q1

FG

q2

Solution EFSolution EFq1

q2

2 2 711 1 2 21 1 7 1 7 22 1 2 (( ( ( ), ) (( )) ) ) ( )Fq q F q GF q qqq Gq G

q1

FG

q2

Solution EFSolution EFq1

q2

2 21 2 81 2 21 1 1 8 1 8 21 2 (( (, ) ( ) (( )) ) ) ( )Fq q F q GF q qqq Gq G

q1

FG

q2

Solution EFSolution EFq1

q2

2 2 911 1 2 21 1 9 1 9 22 1 2 (( ( ( ), ) (( )) ) ) ( )Fq q F q GF q qqq Gq G

q1

FG

q2

Solution EFSolution EFq1

q2

1 1 1 1 2 10 10 1 11 2 2 2 22 1 02 (( ) ( )( () ( (, )) )) F q G q F F q Gq q qGq q

q1

FG

q2

Solution EFSolution EFq1

q2

1 1 1 1 2 11 11 1 11 2 2 2 22 1 12 (( ) ( )( () ( (, )) )) F q G q F F q Gq q qGq q

q1 q2 q9

80809 9 ~ 10~ 1016 16 FEM dof FEM dof 80x9 RM dof80x9 RM dof

101040 40 FEM dof FEM dof 100.000 RM dof100.000 RM dof

1D/9D1D/9D

2D/10D2D/10D

2. Complex Flows2. Complex Flows

1

( , , ) ( ) ( ) ( )n

j j j jj

x q t F x G q H t

Example: Flow involving short fiber suspensionsExample: Flow involving short fiber suspensions

Kinematics:Kinematics:

FEM-DVESSFEM-DVESS

2

2

srDdt

uduDt

D

u

s = 0s = 1

Doi-Edwards ModelDoi-Edwards Model Ottinger Model: double Ottinger Model: double reptation, CCR, chain reptation, CCR, chain stretching, …stretching, …

1

, ( )n

j j jj

u s F u G s

s

3. Entangled polymer models based on 3. Entangled polymer models based on reptation motionreptation motion

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

y=1 1

Reduced Brownian Configurations FieldsReduced Brownian Configurations Fields

DiscretizationDiscretization

1.1. Solve i=1 and computed the Solve i=1 and computed the reduced approximation basisreduced approximation basis

2.2. Solve for all i>1 the reduced Solve for all i>1 the reduced problem:problem:

1( )T n T nB I G Ba B F

1000x10001000x1000

4x44x4

Ongoing works: Ongoing works: (II) (II) Suspensions of CNT: Suspensions of CNT: Aggregation/OrientatiAggregation/Orientati

on modelon model1

10

100

1000

10000

0,1 1 10 100 1000

Shear rate [s-1]

Appar

ent Vis

cosi

ty [Pa-

s]

Epoxy-A

0.025% MWNT in epoxy-A

0.05% MWNT in epoxy-A

0.1% MWNT in epoxy-A

0.25% MWNT in epoxy-A

0.5% MWNT in epoxy-A

Enhanced modeling:Enhanced modeling:

ψ( , , , , , )x y z t n p

+ The associated Fokker-Planck equation+ The associated Fokker-Planck equation

PerspectivesPerspectives

• Enhanced kinetic model for CNT suspensions Enhanced kinetic model for CNT suspensions taking into account orientation and aggregation taking into account orientation and aggregation effects: FP & BD simulations. Collaboration with M. effects: FP & BD simulations. Collaboration with M. Mackley Mackley

• Reduction of Stochastic, Brownian and molecular Reduction of Stochastic, Brownian and molecular dynamics simulations.dynamics simulations.

• Fast micro-macro simulations of complex flows: Fast micro-macro simulations of complex flows: Lattice-Boltzmann & Reduced-FP; and many others Lattice-Boltzmann & Reduced-FP; and many others mathematical topics (stabilization, wavelet bases, mathematical topics (stabilization, wavelet bases, mixed formulations, enhanced particles methods, mixed formulations, enhanced particles methods, …). Collaboration with T. Phillips.…). Collaboration with T. Phillips.