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Towards an Integrated NEM-Model Reduction Lagrangian Strategy in Fluid Structure Interaction

Towards an Integrated NEM-Model Reduction Lagrangian Strategy in Fluid Structure Interaction. F. Chinesta, D. Ryckelynck, E. Cueto, D. Gonzalez P. Villon. LMSP UMR CNRS-ENSAM Paris (France) I3A Universidad de Zaragoza (Spain) Laboratoire Roberval, UTC Compiegne (France). r. h.

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Towards an Integrated NEM-Model Reduction Lagrangian Strategy in Fluid Structure Interaction

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  1. Towards an Integrated NEM-Model Reduction Lagrangian Strategy in Fluid Structure Interaction F. Chinesta, D. Ryckelynck, E. Cueto, D. Gonzalez P. Villon LMSP UMR CNRS-ENSAM Paris (France) I3A Universidad de Zaragoza (Spain) Laboratoire Roberval, UTC Compiegne (France)

  2. r h When the element is too distorted:

  3. The Natural Element Method Delaunay triangulation and Voronoï Diagram Empty circumcircle criterion

  4. Properties: • Linear interpolation on the boundary (a-NEM / C-NEM) • Linear consistency • Partition of Unity • Better accuracy than linear FEM • Meshless character: no sensibility to the distorsion of the Delaunay triangles • The shape functions are C1 everywhere except at the nodes positions. • If one proceeds in an updated Lagrangian framework and the internal variables are defined at the nodes, remeshing, stabilization and field projections will be no more required.

  5. Generalized NewtonianFlows Power law LBB? No, but no locking ! The LBB problematic will be addressed later

  6. Vicoplasticity with thermal coupling

  7. Themo-Elasto-Plasticity in Large Transformations involving localization and refinements Z-Z

  8. Interface Tracking

  9. Vicsoelastic and two-phases flows

  10. D Y N A M I C S

  11. NEM-Hermite or NEM-Bubbles: A proper way to verify general LBB conditions in mixed formulations.

  12. Hermite approximation + NEM-Hermite: Mixed formulations Diffuse -MLS NEM shape function

  13. NEM-Bubbles: Mixed formulations b1 - NEM b2 - NEM

  14. NEM or Bubble-NEM shape function b2 – NEM+ b1 – NEM+

  15. About Model Reduction « To represent a linear function 1000000 degrees of freedom seem to be too much !! »

  16. The Karhunen-Loève Decomposition max Computethe best representation of nxn nx1 Nx1 NxN nx1

  17. 1 t 0 10 30 A Numerical Example

  18. h 1 i-1 i i+1 N Taking into account the boundary condition at x=1

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

  20. 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

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

  22. Enrichment Based on the use of the Krylov’s Subspaces: an “a priori” Strategy IF IF continue 1.000.000 dof ~10 rdof

  23. Perpectives • Reduction and LBB ???

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