Réduction de modèles pour des applications en temps réel

1. Réduction de modèles pour des applications en temps réel F. Druesne, J-L Dulong, P. Villon, A. Ouahsine Laboratoire Roberval – UTC Compiègne

2. Contexte industriel : Savoir-faire & besoin Approche : Apprentissage – Temps réel Méthode a posteriori Méthode a priori Application sur une durit automobile

3. Context Simulation of manual operations on rigid parts( assembly simulation ) as early as design phase Tool for mechanical design Tool for operators training Automotive industry & aeronautics PSA EADS With haptic feedback Virtual prototype 3D immersive visualization of a product Decision aid for project review 3

4. Context Automotive industry Virtual prototype Simulation of manual operations on flexible parts Tool for mechanical design Tool for operators training Problematic: Part deformation in real time, if non linearity Example : Access to a motor unit by pushing an hose back 4

5. Contexte industriel : Savoir-faire & besoin Approche : Apprentissage – Temps réel Méthode a posteriori Méthode a priori Application sur une durit automobile

6. Approach Learning CAD model Finite Element Model Calculation campaign FEM code Response surface Model reduction Real time Virtual model 30 Hz Reduced response surface 1000 Hz Haptic device How to build it ? 6

7. Contexte industriel : Savoir-faire & besoin Approche : Apprentissage – Temps réel Méthode a posteriori Méthode a priori Application sur une durit automobile

8. Test structure • Problem geometry • Slender structure in rubber • embedded at one extremity • handled at the other • Mechanical model • meshed with H8 finite elements • n = 3408 degrees of freedom • Finite deformation • Hyperelastic material (neo-hookean) • Quasi-static problem • FEAP code • Load cases • S = 100 load cases following a regular grid 8

9. A posteriori methodology Quasi-static campaign by solving u (ts ) on each point ts of the load cases grid Newton-Raphson scheme on u (size n) : n = 3408 S = 100 9

10. A posteriori methodology Model reduction by the Karhunen-Loeve Expansion (KLE) 1,2 • Centered displacements by subtracting the average • Covariance matrix • Eigenvectors of and selection of the m first (highest eigenvalues) 1 Krysl, Lall, Marsden 2000 2 Barbič, James 2005 10

11. A posteriori methodology Model reduction by the Karhunen-Loeve Expansion (KLE) • Approached displacement n = 3408 S = 100 m ~ 20 11

12. average initial A posteriori methodology 12

13. A posteriori methodology Error induced by the KLE 13

14. Contexte industriel : Savoir-faire & besoin Approche : Apprentissage – Temps réel Méthode a posteriori Méthode a priori Application sur une durit automobile

15. Cost of (size m x m) is low • Convergence on a, with fixed But a can converge, even if is large ! is too ‘poor’ to describe solution have to be enriched A priori methodology Quasi-static campaign by solving a (ts ) on each point ts of the loading cases grid Newton-Raphson scheme on a (size m) : 15

16. 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 1 1 Ryckelynck 2005 A priori methodology Adaptative strategy by R-enrichment orthonormalize 16

17. A priori methodology base size m base size load cases 17

18. A priori methodology base size m load cases 18

19. A priori vs A posteriori 1.6 1.8 19

20. A priori vs A posteriori 2.2 2.6 20

21. average initial A priori vs A posteriori a posteriori a priori without reduction average initial 21

22. Contexte industriel : Savoir-faire & besoin Approche : Apprentissage – Temps réel Méthode a posteriori Méthode a priori Application sur une durit automobile

23. Application • Problem geometry • Automotive hose in rubber • embedded at its 2 extremities • handled in a point • Mechanical model • meshed with H8 finite elements • n = 18720 degrees of freedom • Large deformation • Hyperelastic material (neo-hookean) • Quasi-static problem • FEAP code • Load cases • S = 100 load cases following a regular grid 23

24. Application : results Error induced by the a posteriori KLE 24

26. Application : results 25

27. Interpolation on data from training phase

28. 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