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Model reduction: the best way to realize many simulation dreams

Model reduction: the best way to realize many simulation dreams. Francisco (Paco) Chinesta. LMSP UMR 8106 CNRS – ENSAM. http://lms-web.paris.ensam.fr/lms/ francisco.chinesta@paris.ensam.fr. An example motivating the use of advanced strategies. V. F. V. Q. Numerical difficulties :.

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Model reduction: the best way to realize many simulation dreams

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  1. Model reduction: the best way to realize many simulation dreams Francisco (Paco) Chinesta LMSP UMR 8106 CNRS – ENSAM http://lms-web.paris.ensam.fr/lms/ francisco.chinesta@paris.ensam.fr

  2. An example motivating the use of advanced strategies V F V Q Numerical difficulties: • Small thickness & moving thermal source implying very fine meshes. • Large simulation times induced by the low thermal conductivity of polymer. • Evolving surfaces and interfaces.

  3. To account for the difficulties related to: Model Reduction & PTI • Small thickness & moving thermal source implying very fine meshes, • Large simulation times induced by the low thermal conductivity of polymer, NxN nxn with Eventually

  4. 1 t 0 10 30 A numerical example

  5. Proper Orthogonal Decomposition

  6. Nx1 More than a significant reduction !! 4x1

  7. 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

  8. Evaluating accuracy and enriching the approximation basis 1 t 0 10 20 30 -2

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

  10. Control Enrichment

  11. Time integration If If

  12. Parallel time integration Linear case

  13. This strategy allows a real parallel time integration BUT it requires to solve N+1 linear problems in each time interval (N=dof) !!

  14. Parallel time integration in reduced basis or + MORTAR

  15. This strategy allows a real parallel time integration AND it requires to solve n+1 linear problems in each time interval (n~10) CPU reduction ~ 10.000

  16. Taking into account non-linearities … e.g. I. Linearization (Newton, fixed point, …) e.g. and the problem being linear … II. Space discretization & parallel time integration

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

  18. * *

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

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

  21. 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

  22. 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

  23. M dX/dt + G X = F

  24. Example

  25. Domain decomposition

  26. POD computation in W1

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

  28. Global solution

  29. Perspectives Iter. n R S S R

  30. Iter. n+1 Convergence ? A. Huerta & P. Diez

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