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
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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
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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
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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:
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Brownian dynamicsBrownian dynamicsr1 r2
rN+1
q1 q2
qN
usually modeled from a random motionusually modeled from a random motion
Beads equilibriumBeads equilibrium
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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
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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
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Solving the deterministic Solving the deterministic Fokker-Planck equationFokker-Planck equation
Two new model Two new model reduction approachesreduction approaches
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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:
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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
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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
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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)
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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
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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
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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 !!
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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
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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
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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:
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q1 q2
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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
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Solving the Stochastic Solving the Stochastic representation of the representation of the
Fokker-Planck equationFokker-Planck equation
New efficient solversNew efficient solvers
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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
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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
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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
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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 »?