Transcript Analysis of discontinuities in the meshless Natural

Réduction de Modèles à l’Issue de la
Théorie Cinétique
Francisco CHINESTA
LMSP – ENSAM Paris
Amine AMMAR
Laboratoire de Rhéologie, INPG
Grenoble
R
q1
The different scales
q2
Atomistic
r1
r2
Brownian dynamics
Kinetic theory:
x, y , z , t
• Fokker-Planck
rN+1
qN
• Stochastic
Atomistic
The 3 constitutive blocks:
U ( x, y, z, t, x1 (t ),
, x N (t ))
F i  F i GradU i 
F i  mi Ai  Ai  vi  xi
i
q1
q2
Brownian dynamics
r1
r2
rN+1
Beads equilibrium
qN
usually modeled from a random motion
q1
r1
q2
Kinetic theory:
• Fokker-Planck
r2
•
Stochastic
ψ( x, y, z, t, q1,, qN )
(3  1  3N ) D
rN+1
qN
The Fokker-Planck formalism


1    
  (q ) 
 A

t
q
4 q  q 
Coming back to the macroscopic scale:
Stress evaluation
q
F
q
  FF( q)  q   F ( q)  q  ( q) d q
C
With F & R collinear:
 
T
Solving the deterministic
Fokker-Planck equation
Two new model
reduction approaches
Model Reduction based on the
Karhunen-Loève decomposition
Continuous:
Discretization:
PDE , u  x , t 
u( xi , t ) i  1,
p
, N  , p  1,
AU  F
p
1
Karhunen-Loève: U ,
, P  U
p
p 1
n
,U  i  i n  N
P
i 1
Application in Computational
Rheology
Fokker-Planck discretisation
 M K  
p
M
p 1
 M  
p
  10 
 0
2  p
p
(0) p

First assumption:       B 
 0
 N 
 
B
1 dof !
(0)
T
 
M B   B
(0)
p
(0)
T
p 1
Initial reduced
approximation
basis
B  p1
(0)
Fast simulation BUT bad results expected
Enrichment based on the use of the Krylov’s
subspaces: an “a priori” strategy
B
(0)
B
 B T M B  p   B T B 




p 1
tcontrol
B B
*
B   B, KS1, KS 2, KS 3
*
R  M B  B
p
IF R  
KSm   M 
p 1
IF R   continue
m 1
R
The enrichment increases the number of approximation
functions BUT the Karhunen-Loève decomposition reduces it
FENE
Model
1D
q
3D
1
H(q) 
q2
1 2
b
H(q)
H( q ) 
1
1
q
2
b2
300.000 FEM dof
~10 dof
~10 functions (1D, 2D or 3D)
It is time for dreaming!
q1
r1
q2


1    
  ( q.) 
A
t
q
4  q   q 
r2
  (q1 , q2 ,, q N , x, y, z, t )
rN+1
qN
For N springs, the model is defined
in a 3N+3+1 dimensional space !!
~ 10 approximation functions are
 p 
 p1 
enough   M 

 


~1010
~101
~101
BUT
~10
 ( x1, x2 , , x3N 3 , t )  i (t ) i ( x1, x2 , , x3N 3 )
i 1
How defining those
high-dimensional functions ?
Natural answer: with a nodal description
10 nodes = 10 function values
1D
q1
1D
q2
10 dof
10x10 dof
2D
80D
r1
r2
1080 dof
rN+1
1080 ~ presumed number of
elemental particles in the universe !!
qN
>1000D
No function can be defined in a such space from
a computational point of view !!
F.E.M.
Advanced deterministic approaches of
Multidimensional Fokker-Planck equation
Separated representation and Tensor product
approximation bases
q1
q2
DIM  9  1
q9
Our proposal
FEM
( q1,
GRID
DOF  N DIM  100010  1030
n
, q9 , t )    j Fj1 ( q1 )
j 1
Fj 9 ( q9 ) Fj10 (t )
DOF  N  DIM  1000  10  104
Computing availability ~109
Example
T  f ( x, y ) in    L,  L x  L,  L
T ( x  )  0

T ( x, y)    j Fj ( x)G j ( y)
h m
f ( x, y )   ah ( x )bh ( y )
h 1
j 1
n
T ( x, y )    j Fj ( x )G j ( y )
j 1
I - Projection:
 T 
 x   dN T F 1 M T G
1
  T
 T   N F 1 dM T G1
 y 
N

T
F
(
x
)

N
(
x
)
F
(
x
)

N
Fi

k
i
k
 i

k 1

M
G ( y )  M ( y )G ( y )  M T G

i
k
i
k
 i
k 1
  T T * d    f ( x, y ) T *d 

T

T
dN F 2 M G 2
T
T
N F 2 dM G 2
 1 
T
T
... dN F n M G n    2 
.
T
T
... N F n dM G n   
 
 n 
j Tn
II - Enrichment:
n
T ( x, y )    j Fj ( x )G j ( y )
j 1
N

T
 Fi ( x )   N k ( x ) Fi ( xk )  N F i

k 1

M
G ( y )  M ( y )G ( y )  M T G

i
k
i
k
 i
k 1

T ( x, y )    j F j ( x )G j ( y )  R( x ) S ( y ) 
j 1


T *  R*S  RS*

n
  T T * d    f ( x, y ) T *d 


 T 
 x  n
 dN T F j .M T G j   M T S . dN T
   j  T
 
T

T
  j 1  N F j .dM G j   0(1, p )
 y 
 R

T
T  
N R . dM   S 
0(1,q )
Only 1D interpolations and 1D integrations!
R
Fn 1 
R
Gn 1 
S
S
q1
q2
1D/9D
q1
q2
809 ~ 1016 FEM dof
2D/10D
1040 FEM dof
q9
80x9 RM dof
100.000 RM dof
Solving the Stochastic
representation of the
Fokker-Planck equation
New efficient solvers
Stochastic approaches …
(Ottinger & Laso)
A way for solving the Fokker-Planck equation:
d 
 
  

  A  
D

dt  q
q 
 q 

D  BB
T
jN
( q, t  0)    j ( q  q j )
0
j 1
dq  A dt  B dW
W : Wiener random process
We need tracking a large ensemble of particles
and control the statistical noise!
Fokker-Planck:
d ( x,  , t ) 

dt


d

 ( x,  , t )
dt



    ( x,  , t ) 

 Dr



x , ,t 
  
Stochastique:





W
d

i 
i

di  t  Wi   
t
t 

dt


BCF



 W  N (0, 2 Dr t )
Brownian  

Configuration
  Jeffery
Fields

  i 


 
1
N BCF
N BCF

i 1
i
SFS in a simple shear flow
Rouge: MDF
1000 ddl / pdt
a11
Bleu: BCF
100 BCF
1000 ddl / pdt
Vert: Reduced BCF
100 BCF
t
4 ddl / pdt
The reduced approximation basis is constructed from some
snapshots computed on the averaged BFC distributions
Perspectives
(réduction de deuxième génération)
d i

  i 


dt
 

1
N BCF
N BCF

i 1
i
Wi
 
t
f (t )
Séparation de variables ?
Base commune pour les différents « configuration fields »?