réduction de modèles à l’issue de la théorie cinétique
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
Réduction de Modèles à l’Issue de la Réduction de Modèles à l’Issue de la
Théorie CinétiqueThéorie Cinétique
Francisco CHINESTA Francisco CHINESTA
LMSP – ENSAM ParisLMSP – ENSAM Paris
Amine AMMARAmine AMMAR
Laboratoire de Rhéologie, INPG Laboratoire de Rhéologie, INPG GrenobleGrenoble
The different scalesThe different scales
r1 r2
rN+1
q1 q2
qN
RR
tzyx ,,,
AtomisticAtomistic
Brownian dynamicsBrownian dynamics
Kinetic theory:Kinetic theory:
• Fokker-PlanckFokker-Planck
• StochasticStochastic
AtomisticAtomistic
1( , , , , ( ), , ( ))NU x y z t x t x t
i i iF F GradU
i i i i iiF m A A v x i
The 3 constitutive blocks:The 3 constitutive blocks:
Brownian dynamicsBrownian dynamicsr1 r2
rN+1
q1 q2
qN
usually modeled from a random motionusually modeled from a random motion
Beads equilibriumBeads equilibrium
r1 r2
rN+1
q1 q2
qN
Kinetic theory:Kinetic theory:
• Fokker-PlanckFokker-Planck
• StochasticStochastic),,,,,,(ψ1 N
qqtzyx
(3 1 3 )N D
1( )
4q A
t q q q
The Fokker-Planck formalismThe Fokker-Planck formalism
Coming back to the macroscopic scale:Coming back to the macroscopic scale:
Stress evaluationStress evaluation
qqFF( ) ( ) ( )
C
F q q F q q q dq
With With F F & & RR collinear collinear:: T
FF
Solving the deterministic Solving the deterministic Fokker-Planck equationFokker-Planck equation
Two new model Two new model reduction approachesreduction approaches
Model Reduction based on the Model Reduction based on the Karhunen-Loève decompositionKarhunen-Loève decomposition
, ,PDE u x t
( , ) 1, , , 1, , ppiu x t i N p P U
1 pp FUA
n N
Continuous:Continuous:
Discretization:Discretization:
1
1
, ,n
Pi i
i
U U
Karhunen-Loève:Karhunen-Loève:
Application in Computational Application in Computational RheologyRheology
1 1 p p p pM K M M
Fokker-Planck discretisation Fokker-Planck discretisation
010
(0)2
0
p p p
N
B
(0) (0) (0) (0) 1 T T
p pB M B B B
1 dof !1 dof !
First assumption:First assumption:
Initial reduced Initial reduced approximation approximation basisbasis
Fast simulation BUT bad results expectedFast simulation BUT bad results expected
Enrichment based on the use of the Krylov’s Enrichment based on the use of the Krylov’s subspaces: an “a priori” strategysubspaces: an “a priori” strategy
controlt
1mKSm M R
IFIF R IFIF R continuecontinue
1 p pR M B B
1 T p T pB M B B B
* , 1, 2, 3B B KS KS KS
(0)B B
*B B
The enrichment increases the number of approximation The enrichment increases the number of approximation functions BUT the Karhunen-Loève decomposition reduces it functions BUT the Karhunen-Loève decomposition reduces it
FENE 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)
It is time for dreamingIt is time for dreaming!!
qA
qt 4
1).(
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 pM
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 elemental particles in the universe !!elemental particles in the universe !!
Advanced deterministic approaches of Advanced deterministic approaches of Multidimensional Fokker-Planck equationMultidimensional Fokker-Planck equation
Separated representation and Tensor product Separated representation and Tensor product approximation bases approximation bases
q1 q2 q9
FEMFEM
GRIDGRID10 301000 10DIMDOF N
1 9 1 1 9 9 101
( , , , ) ( ) ( ) ( )n
j j j jj
q q t F q F q F t
Our Our proposalproposal
9 1DIM
41000 10 10DOF N DIM
Computing availabilityComputing availability ~10 ~109 9
ExamplExamplee
1
( , ) , ,
( ) 0
( , ) ( ) ( )h m
h hh
T f x y in L L x L L
T x
f x y a x b y
1
( , ) ( ) ( )j j jj
T x y F x G y
1
( , ) ( ) ( )n
j j jj
T x y F x G y
1
1
( ) ( ) ( )
( ) ( ) ( )
NT
ii k i kk
MT
ii k i kk
F x N x F x N F
G y M y G y M G
I - Projection:I - Projection:* * ( , ) T T d f x y T d
1
1 21 2 2
1 21 2
....
...
T T T T T Tn n
T T T T T Tn n
n
TdN F M G dN F M G dN F M Gx
T N F dM G N F dM G N F dM Gy
n
j T
1
( , ) ( ) ( )n
j j jj
T x y F x G y
1
1
n
n
RF
R
SG
S
* * ( , ) T T d f x y T d
(1, )
1 (1, )
. . 0
. 0 .
T T T Tnj j q
j T T T Tj j j p
TdN F M G M S dN Rx
T SN F dM G N R dMy
***
1
)()()()(),(
RSSRT
ySxRyGxFyxTn
jjjj
1
1
( ) ( ) ( )
( ) ( ) ( )
NT
ii k i kk
MT
ii k i kk
F x N x F x N F
G y M y G y M G
Only 1D interpolations and 1D integrations!
II - Enrichment:II - Enrichment:
q1 q2
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
Solving the Stochastic Solving the Stochastic representation of the representation of the
Fokker-Planck equationFokker-Planck equation
New efficient solversNew efficient solvers
Stochastic approaches …Stochastic approaches …
A way for solving the Fokker-Planck equation:A way for solving the Fokker-Planck equation:
(Ottinger & Laso)(Ottinger & Laso)
d
A Ddt q q q
dq A dt B dW WW : Wiener random process : Wiener random process
We need tracking a large ensemble of particles We need tracking a large ensemble of particles and control the statistical noise!and control the statistical noise!
0
1
( , 0) ( )j N
j jj
q t q q
TD BB
Fokker-Planck:Fokker-Planck:
, ,
( , , ) ( , , )( , , ) r
x t
d x t d x tx t D
dt dt
Stochastique:Stochastique:
(0, 2 )
ii i
r
Wd t W t
t
W N D t
Jeffery
BCF
1
1 BCF
ii
N
iiBCF
d
dt
N
Brownian Brownian Configuration Configuration
FieldsFields
SFS in a simple shear flowSFS in a simple shear flow
Rouge: MDF
1000 ddl / pdt
Bleu: BCF
100 BCF
1000 ddl / pdt
Vert: Reduced BCF
100 BCF
4 ddl / pdt
a11
t
The reduced approximation basis is constructed from some The reduced approximation basis is constructed from some snapshots computed on the averaged BFC distributionssnapshots computed on the averaged BFC distributions
Perspectives Perspectives (réduction de deuxième génération)(réduction de deuxième génération)
1
1 BCF
ii
N
iiBCF
d
dt
N
( )
i
f t
W
t
Séparation de variables ?Séparation de variables ?
Base commune pour les différents « configuration fields »?Base commune pour les différents « configuration fields »?