Réduction de modèles pour des applications en temps réel
description
Transcript of Réduction de modèles pour des applications en temps réel
Réduction de modèles 18 mai 2006ENSAM -
Réduction de modèles pour des applications en temps réel
F. Druesne, J-L Dulong, P. Villon, A. Ouahsine
Laboratoire Roberval – UTC Compiègne
Réduction de modèles 18 mai 2006ENSAM -
Contexte industriel : Savoir-faire & besoin
Approche : Apprentissage – Temps réel
Méthode a posteriori
Méthode a priori
Application sur une durit automobile
Réduction de modèles 18 mai 2006ENSAM -
Context
Virtual prototype
Decision aid for project review
3D immersive visualization of a product
Tool for mechanical design
Simulation of manual operations on rigid parts ( assembly simulation )
as early as design phase
Automotive industry & aeronautics
With haptic feedback
Tool for operators training
PSA EADS
3
Réduction de modèles 18 mai 2006ENSAM -
Example :Access to a motor unit by pushing an hose back
Tool for operators training
Tool for mechanical design
Simulation of manual operations on flexible parts
Virtual prototype
Context
Automotive industry
4
Problematic:Part deformation in real time, if non linearity
Réduction de modèles 18 mai 2006ENSAM -
Contexte industriel : Savoir-faire & besoin
Approche : Apprentissage – Temps réel
Méthode a posteriori
Méthode a priori
Application sur une durit automobile
Réduction de modèles 18 mai 2006ENSAM -
Approach
Real timeVirtual model
1000 Hz
30 Hz
Haptic device
LearningCAD model
FEM code
Response surface
Calculation campaign
Finite Element Model
Reduced response surface
Model reduction
How to build it ?
6
Réduction de modèles 18 mai 2006ENSAM -
Contexte industriel : Savoir-faire & besoin
Approche : Apprentissage – Temps réel
Méthode a posteriori
Méthode a priori
Application sur une durit automobile
Réduction de modèles 18 mai 2006ENSAM -
Problem geometrySlender structure in rubber• embedded at one extremity• handled at the other
Mechanical modelmeshed with H8 finite elementsn = 3408 degrees of freedomFinite deformationHyperelastic material (neo-hookean)Quasi-static problemFEAP code
Load casesS = 100 load cases following a regular grid
Test structure
yyxx eUeUU
xe
ye
ze
8
Réduction de modèles 18 mai 2006ENSAM -
)()()()( Ss21 tutututu
),()(),( sss tuFtutuK
Quasi-static campaign by solving u (ts ) on each point ts of the load cases grid
iiiT RduK Newton-Raphson scheme on u (size n) :
A posteriori methodology
S
n
n = 3408S = 100
9
Réduction de modèles 18 mai 2006ENSAM -
S
1s
Tss tutuC )()(
Model reduction by the Karhunen-Loeve Expansion (KLE) 1,2
1. Centered displacements by subtracting the average
ututu ss )()(
2. Covariance matrix
3. Eigenvectors of and selection of the m first (highest eigenvalues)C m21 ,...,,
A posteriori methodology
1 Krysl, Lall, Marsden 2000
2 Barbič, James 2005
10
Réduction de modèles 18 mai 2006ENSAM -
utaututu s
m
1ii
Tsis
)())(()(~
m21 ,...,,
Model reduction by the Karhunen-Loeve Expansion (KLE)
)( staA
4. Approached displacement
S
m
n = 3408S = 100m ~ 20
m
n
A posteriori methodology
11
Réduction de modèles 18 mai 2006ENSAM -
1
2
4
10
average
initial
A posteriori methodology
12
Réduction de modèles 18 mai 2006ENSAM -
0
5
10
15
20
25
30
35
40
0 2 4 6 8 10 12
number of retained modes
a po
ster
iori
rela
tive
erro
r
structure with slenderness ratio
2Ls
2Lss
S1s
KL
tu
tutue
)(
)(~)(max
Error induced by the KLE
A posteriori methodology
13
Réduction de modèles 18 mai 2006ENSAM -
Contexte industriel : Savoir-faire & besoin
Approche : Apprentissage – Temps réel
Méthode a posteriori
Méthode a priori
Application sur une durit automobile
Réduction de modèles 18 mai 2006ENSAM -
)()(),( sss tFutatuK iii
T RdaK
Quasi-static campaign by solving a (ts ) on each point ts of the loading cases grid
Newton-Raphson scheme on a (size m) :
iTiiT
T RdaK
iiiT RdaK ˆˆ
A priori methodology
• Convergence on a, with fixed
• Cost of (size m x m) is low1
TK ˆ
But a can converge, even if is large !Ris too ‘poor’ to describe solution
have to be enriched
15
Réduction de modèles 18 mai 2006ENSAM -
A priori methodology
RKu Te 1 ee1e u
,
Adaptative strategy by R-enrichment
orthonormalize
Algorithm :
enrichment loop
iterative loop (Newton Raphson)
until convergence on a
until convergence on R
else enrichment
Reduction by KLE if size( ) becomes too large 11 Ryckelynck 2005
16
Réduction de modèles 18 mai 2006ENSAM -
0
5
10
15
20
25
30
35
40
45
50
1
loading cases
base
dim
ensi
on
0
10
20
30
40
50
60
70
cpu
time
(s)
cpu timebase dimension
A priori methodology
17
load cases
base size base
siz
e m
Réduction de modèles 18 mai 2006ENSAM -
0
10
20
30
40
50
60
70
0 20 40 60 80 100
loading cases
base
dim
ensi
onenrichmentenrichment with reduction
A priori methodology
18
base
siz
e m
load cases
Réduction de modèles 18 mai 2006ENSAM -
0
100
200
300
400
500
600
700
800
900
0 20 40 60 80 100
loading cases
cum
uled
cpu
tim
e (s
)a posterioria priori enrichmenta priori enrichment with reduction
A priori vs A posteriori
19
1.61.8
Réduction de modèles 18 mai 2006ENSAM -
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
0 5000 10000 15000 20000
number of dof
cpu
time
(s)
a posterioria priori enrichmenta priori enrichment with reduction
20
A priori vs A posteriori
2.2 2.6
Réduction de modèles 18 mai 2006ENSAM -
1
2average
initial
1
2
average
initial
a posteriori
a priori
without reduction
21
A priori vs A posteriori
Réduction de modèles 18 mai 2006ENSAM -
Contexte industriel : Savoir-faire & besoin
Approche : Apprentissage – Temps réel
Méthode a posteriori
Méthode a priori
Application sur une durit automobile
Réduction de modèles 18 mai 2006ENSAM -
Application
Problem geometryAutomotive hose in rubber• embedded at its 2 extremities• handled in a point
Mechanical modelmeshed with H8 finite elementsn = 18720 degrees of freedomLarge deformationHyperelastic material (neo-hookean)Quasi-static problemFEAP code
Load casesS = 100 load cases following a regular grid
yyxx eUeUU
23
Réduction de modèles 18 mai 2006ENSAM -
0
5
10
15
20
25
30
35
40
45
0 5 10 15 20
number of retained modes
a po
ster
iori
rela
tive
erro
rstructure with slenderness ratio
automotive hose
Application : results
Error induced by the a posteriori KLE
24
Réduction de modèles 18 mai 2006ENSAM -
0
20
40
60
80
100
120
140
160
180
0 20 40 60 80 100
loading cases
base
dim
ensi
on
structure with slenderness ratioautomotive hose
Réduction de modèles 18 mai 2006ENSAM -
0
20000
40000
60000
80000
100000
0 20 40 60 80 100
loading cases
cum
uled
cpu
tim
e (s
)
a posterioria priori enrichmenta priori enrichment with reduction
Application : results
25
Réduction de modèles 18 mai 2006ENSAM -
Interpolation on data from training phase
Réduction de modèles 18 mai 2006ENSAM -
Conclusion
Conclusion
• Feasibility of large deformation in real time, with non linear hyperelastic material.
• New tool for mechanical design.
• The classical a posteriori methodology is possible but slower than the a priori one.
Perspectives
• Hyperreduction methodology (Ryckelynck 2005).
• Introduce material history in the reduced surface response.
• Introduce boundary conditions non linearity.
27