1. Modèles réduits et interfaces
Francisco (Paco) Chinesta

2. Some numerical experiments on transient 3D models

3. A simple numerical examplet 10 30 1

5. Proper Orthogonal Decomposition

7. A significant reduction !!
Nx1
A significant reduction !!
4x1

9. Solving « a similar » problem
with the reduced order approximation basis computed from the solution of the previous problem 1 t 10 20 30 t 10 30 1

11. 1 t 10 20 30 -2

13. How quantify the accuracy without the knowledge of the reference solution?
How to enrich if the accuracy is not enough?

14. David Ryckelynk
Control
Enrichment

15. CPU reduction of some orders of magnitude
Time integration If If
CPU reduction of some orders of magnitude

16. Applications Control; Optimization and inverse identification;
Simulation in real time;

17. MAN + Grassman manifolds

18. Interfaces … can be reduced?
4 dof
Eigenfunctions:

19. * *

20. BUT interfaces can move:
Is it possible reducing its “tracking” description?
Non, in a direct manner !!
Is it possible reducing its “capturing” description?
Sometimes !!

21. The evolution of a characteristic function cannot be reduced in a POD sense !
Number of modes = Number of nodes !!!

22. BUT the evolution of the level set function can be also represented in a reduced approximation basis
Number of modes = 2
The number of modes increase with the geometrical complexity of interfaces

23. MEF or X-FEM / POD 1 (smooth evolution) & 2 (localization: X-FEM, …)
Each node belongs to one of these domains: 1 or 2 2 1 

24. M dX/dt + G X = F

25. Example

26. Domain decomposition

27. POD computation in W1

28. FEM calculation in W2
t = 0.01
t = 0.2
t = 0.4
t = 0.6
t stationnaire

29. Global solution

30. Drawbacks Convergence; Optimality (orthogonality, …);
Moving meshes (Lagrangian, MD, BD, …);
Hyperbolic models (Krylov enrichment fails);
Incremental time integration; …

31. BUT in fact the solution of many models can be approximated from:

32. A separated representation: We looks for the space and time functions for approximating the PDE solution

33. One possible approach …
Iter. n
R S
S R

34. What about CPU time ?
Incremental
Non-Incremental

35. On the separated representations

36. Separated representation
MEF, MDF, MVF, …
Modèles multidimensionnels
Maillage
Separated representation
Remark: can be a a group of coordinates.

37. Iter. n
R S
S R

39. Subdomains and Interfaces

48. Perspectives