model reduction: the best way to realize many simulation dreams
DESCRIPTION
Model reduction: the best way to realize many simulation dreams. Francisco (Paco) Chinesta. LMSP UMR 8106 CNRS – ENSAM. http://lms-web.paris.ensam.fr/lms/ [email protected]. An example motivating the use of advanced strategies. V. F. V. Q. Numerical difficulties :. - PowerPoint PPT PresentationTRANSCRIPT
Model reduction: the best way Model reduction: the best way to realize many simulation to realize many simulation
dreamsdreams
Francisco (Paco) ChinestaFrancisco (Paco) Chinesta
LMSP UMR 8106 CNRS – ENSAM LMSP UMR 8106 CNRS – ENSAM http://lms-web.paris.ensam.fr/lms/ http://lms-web.paris.ensam.fr/lms/ [email protected]@paris.ensam.fr
An example motivating the use of An example motivating the use of advanced strategiesadvanced strategies
V
FQ
Numerical difficultiesNumerical difficulties::
1.1. Small thickness & moving thermal source implying Small thickness & moving thermal source implying very fine meshes.very fine meshes.
2.2. Large simulation times induced by the low thermal Large simulation times induced by the low thermal conductivity of polymer.conductivity of polymer.
3.3. Evolving surfaces and interfaces.Evolving surfaces and interfaces.
V
Model Reduction & PTIModel Reduction & PTI
1.1. Small thickness & moving thermal source implying Small thickness & moving thermal source implying very fine meshes,very fine meshes,
2.2. Large simulation times induced by the low thermal Large simulation times induced by the low thermal conductivity of polymer,conductivity of polymer,
1 p pA T F
NNxxNN1( )pA T
EventuallyEventually
To account for the difficulties related to:To account for the difficulties related to:
1 p pa f
nnxxnn n Nwithwith 6~ 10N
~ 10n
A numerical exampleA numerical example
)(tq
t
00 1010 3030
11
0
)(
1)0,(
10300
,1
,0
2
2
tx
tx
x
T
tqx
T
txT
, , x, tx
T
t
T01.0
( )q t
1 n n n M + K T M T Q
10 20, ,t t T T
st 1.0
1 n n n M + K T M T Q
10 20 3001 1 110 20 300
2 2 2
10 20 300
t t t
t t t
t t tN N N
T T T
T T T
T T T
Q
T QQ φ φ
1 1 , ... , N N φ φ
1
84
3
2
1
10
15.0
15.1
1790
1
2
30
( )
( )
( )
( )
x
x
x
T x
PProper roper OOrthogonal rthogonal DDecompositionecomposition
30
01
0 1 2 3
( ) ( ) ( ) with
( 0) 1& ( 0) ( 0) ( 0) 0
i
i ii
t t t
t t t t
T T
1
2
30
( )
( )
( )
( )
x
x
x
T x
k ( )x
01 1 1 2 1 3 1
02 1 2 2 2 3 2
01 2 3N N N N
T x x x x
T x x x x
T x x x x
B
T = B ζ
Nx1Nx1
4x14x1More than a significant reduction !!More than a significant reduction !!
1
1
1
n n n
n n n
T n T n T n
M + K T M T Q
M + K Bξ M Bξ Q
B M + K Bξ B M Bξ B Q
1 T n T n T n B M + K B ζ B MB ζ B Q
44 !!fo d ...4
Solving « a similar » problemSolving « a similar » problem)(tq
t
00 1010 2020
11
3030
with the reduced order with the reduced order approximation basis approximation basis computed from the computed from the
solution of the previous solution of the previous problemproblem
)(tq
t
00 1010 3030
11
01 1 1 2 1 3 1
02 1 2 2 2 3 2
01 2 3N N N N
T x x x x
T x x x x
T x x x x
B
dof 4
dof 101
1 T n T n T n B M + K B ζ B MB ζ B Q
1 n n n M + K T M T Q
Evaluating accuracy and Evaluating accuracy and enriching the approximation basisenriching the approximation basis
)(tq
t
00 1010 2020
11
3030
-2-2
1 n n n M + K T M T Q
dof 4
dof 101
How quantify the accuracy How quantify the accuracy without the knowledge of without the knowledge of the reference solution?the reference solution?
How to enrich if the accuracy is not enough? How to enrich if the accuracy is not enough?
1 T n T n T n B M + K B ζ B MB ζ B Q
1 n n n M + K T M T Q
52.6 10 R
R
RR
nT 1nT
ControlControl 1 T n T n T n B M + K B ζ B MB ζ B Q
1 n n n R M + K B ζ MB ζ Q
EnrichmentEnrichment B B, R
Time integrationTime integration
1 p pA T F
B
1 T p T pB A B B F
1 p pR A B F
,B B R
IfIf R
IfIf R
1 T p T pB A B B F
Parallel time integrationParallel time integration
dTA T F
dt
1
0
0
0
0
1
0
0
0
0dT
A Tdt
1 ( )pT t
11( )hT t
1 ( )hNT t
1 1 11
1
( ) ( ) ( )i N
p hiii
T t T t T t
2 ( )pT t
21( )hT t
2 ( )hNT t
2 2 22
1
( ) ( ) ( )i N
p hiii
T t T t T t
1
1 2
1 21 1
( 0),
( ) ( )
g
i i
T t T
T t T t
Linear caseLinear case
This strategy allows a real parallel time This strategy allows a real parallel time integrationintegration
BUT BUT
it requires to solve it requires to solve NN+1 linear problems in +1 linear problems in each time interval (each time interval (NN=dof) !!=dof) !!
Parallel time integration in reduced Parallel time integration in reduced basisbasis
1 p pA T F
B
1 T p T pB A B B F
1 11 1 p pR A B F
2 22 1 p pR A B F
1 i ii p pR A B F
1 2, , ,B B R R
oror 1
1, , , , ,i
iB B R B B R + MORTAR+ MORTAR
This strategy allows a real parallel time This strategy allows a real parallel time integrationintegration
AND AND
it requires to solve it requires to solve nn+1 linear problems in +1 linear problems in each time interval (n~10)each time interval (n~10)
CPU reduction ~ 10.000CPU reduction ~ 10.000
Taking into account non-linearities …Taking into account non-linearities …
e.g.e.g. ( ) T ( ) T
K T ft
x
I.I. Linearization (Newton, fixed point, …) Linearization (Newton, fixed point, …)
II.II. Space discretization & parallel time integration Space discretization & parallel time integration
( )
( 1) (n)( ) T ( ) n
nTK T f
t
x
and the problem being linear and the problem being linear ……
e.g.e.g.
Interfaces … can be reduced?Interfaces … can be reduced?
-1 -0.5 0 0.5 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
X
T
2 110K K
-1 -0.5 0 0.5 1-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
X
T
0t
1K
t
0 10 20 30 40 50 60-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
X
TEigenfunctions:Eigenfunctions:4 dof4 dof
ii i
TK T f
t
( ,4) (4,1)p p
NT B ξ
1p p T G T F
-1 -0.5 0 0.5 1-0.01
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
X
T
( ,4) (4,1)p p
NT B ξ
1p p T G T F
1p p ξ g ξ f
**
BUT interfaces can move: BUT interfaces can move:
Is it possible reducing its “tracking” description? Is it possible reducing its “tracking” description?
Non, in a direct manner !!Non, in a direct manner !!
Is it possible reducing its “capturing” description?Is it possible reducing its “capturing” description?
Sometimes !!Sometimes !!
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0t 1t 3t 4t 5t 6t 7t 9t8t2t
0 0.2 0.4 0.6 0.8 1-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
Number of modes = Number of nodes !!!Number of modes = Number of nodes !!!41 10N
The evolution of a characteristic function cannot be The evolution of a characteristic function cannot be reduced in a POD sense !reduced in a POD sense !
BUTBUT the evolution of the level set function can be the evolution of the level set function can be also represented in a reduced approximation basisalso represented in a reduced approximation basis
0 0.2 0.4 0.6 0.8 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0t 1t 3t 4t 5t 6t 7t 9t8t2t
0 0.2 0.4 0.6 0.8 1-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
Number of modes = 2Number of modes = 20, 2i i
The number of modes increase with the geometrical The number of modes increase with the geometrical complexity of interfacescomplexity of interfaces
2
1
MEF or X-FEM / PODMEF or X-FEM / POD1 (smooth evolution) & 1 (smooth evolution) & 2 (localization: X-FEM, …)2 (localization: X-FEM, …)
Each node belongs to one of these domains: Each node belongs to one of these domains: 1 or 1 or 22
M dX/dt + G X = FM dX/dt + G X = F
11 12 11 12 1 11
21 22 21 22 2 22
M M G G X FX
M M G G X FX
11 12 1
21 22 2
11 12 1 1
21 22 2 2
T T
T T T T
M M B IB B
M M B II I X
G G B I FB B B B
G G B I X FI I I I
ExampleExample
Domain decompositionDomain decomposition
POD computation in POD computation in 11
FEM FEM calculationcalculation in in 22
t = 0.01
t = 0.2
t = 0.4
t = 0.6
t stationnaire
Global solutionGlobal solution
PerspectivesPerspectives
1
* * *
( , ) ( ) ( ) ( ) ( )
( , ) ( ) ( ) ( ) ( )
n
j jj
T t X t R S t
T t R S t R S t
x x x
x x x
*
I
( ) 0T T dt d
xM A
Iter. n
R S
*
I
( ( , , , )) 0j jRS T R S X dt d S
xM A
S R
*
I
( ( , , , )) 0j jR S T R S X dt d R
xM A
Convergence ?Convergence ?
1
1
( ) ( )
( ) ( )n
n
X R
t S t
x x
1
1
* * *
( , ) ( ) ( ) ( ) ( )
( , ) ( ) ( ) ( ) ( )
n
j jj
T t X t R S t
T t R S t R S t
x x x
x x x
Iter. n+1Iter. n+1
A. Huerta & P. DiezA. Huerta & P. Diez