Download ppt - 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

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


Contexte industriel : Savoir-faire & besoin

Approche : Apprentissage – Temps réel

Méthode a posteriori

Méthode a priori

Application sur une durit automobile


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


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

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Problematic:Part deformation in real time, if non linearity





Méthode a priori



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 ?

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Méthode a priori



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

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)()()()( 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

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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 ,...,,


1 Krysl, Lall, Marsden 2000

2 Barbič, James 2005

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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


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1

2

4

10

average

initial


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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


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Méthode a priori



)()(),( 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

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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

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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


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load cases

base size base

siz

e m


0

10

20

30

40

50

60

70

0 20 40 60 80 100

loading cases

base

dim

ensi

onenrichmentenrichment with reduction


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base

siz

e m

load cases


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

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1.61.8


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

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2.2 2.6


1

2average

initial

1

2

average

initial

a posteriori

a priori

without reduction

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Méthode a priori



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

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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

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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


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

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Interpolation on data from training phase


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.

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