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Modèles réduits et interfaces 

Modèles réduits et interfaces . Francisco (Paco) Chinesta. Some numerical experiments on transient 3D models. 1. t. 0. 10. 30. A simple numerical example. P roper O rthogonal D ecomposition. Nx1. 4x1. A significant reduction !!. 1. t. 0. 10. 30. Solving « a similar » problem.

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Modèles réduits et interfaces 

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  1. Modèles réduits et interfaces  Francisco (Paco) Chinesta

  2. Some numerical experiments on transient 3D models

  3. 1 t 0 10 30 A simple numerical example

  4. Proper Orthogonal Decomposition

  5. Nx1 4x1 A significant reduction !!

  6. 1 t 0 10 30 Solving « a similar » problem with the reduced order approximation basis computed from the solution of the previous problem 1 t 0 10 20 30

  7. 1 t 0 10 20 30 -2

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

  9. David Ryckelynk Control Enrichment

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

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

  12. MAN + Grassman manifolds

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

  14. * *

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

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

  17. 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

  18. 2 1 MEF or X-FEM / POD 1 (smooth evolution) & 2 (localization: X-FEM, …) Each node belongs to one of these domains: 1 or 2

  19. M dX/dt + G X = F

  20. Example

  21. Domain decomposition

  22. POD computation in W1

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

  24. Global solution

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

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

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

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

  29. What about CPU time ? Non-Incremental Incremental

  30. On the separated representations

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

  32. Iter. n R S S R

  33. Subdomains and Interfaces

  34. Perspectives

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