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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. Laboratoire de Rhéologie GRENOBLE, France. [email protected]. In collaboration with: - PowerPoint PPT Presentation
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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, [email protected]@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.