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Explore numerical experiments on transient 3D models, Proper Orthogonal Decomposition, and reduced order approximation for accurate and efficient solutions. Discover methods to quantify accuracy, enrich solutions, and control time integration while reducing CPU usage. Applications include control, optimization, real-time simulation, and more.
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Modèles réduits et interfaces Francisco (Paco) Chinesta
1 t 0 10 30 A simple numerical example
Nx1 4x1 A significant reduction !!
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
1 t 0 10 20 30 -2
How quantify the accuracy without the knowledge of the reference solution? How to enrich if the accuracy is not enough?
David Ryckelynk Control Enrichment
Time integration If If CPU reduction of some orders of magnitude
Applications • Control; • Optimization and inverse identification; • Simulation in real time;
Interfaces … can be reduced? 4 dof Eigenfunctions:
* *
BUT interfaces can move: Is it possible reducing its “tracking” description? Non, in a direct manner !! Is it possible reducing its “capturing” description? Sometimes !!
The evolution of a characteristic function cannot be reduced in a POD sense ! Number of modes = Number of nodes !!!
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
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
FEM calculation in W2 t = 0.01 t = 0.2 t = 0.4 t = 0.6 t stationnaire
Drawbacks • Convergence; • Optimality (orthogonality, …); • Moving meshes (Lagrangian, MD, BD, …); • Hyperbolic models (Krylov enrichment fails); • Incremental time integration; • …
BUT in fact the solution of many models can be approximated from:
A separated representation: We looks for the space and time functions for approximating the PDE solution
One possible approach … Iter. n R S S R
What about CPU time ? Non-Incremental Incremental
MEF, MDF, MVF, … Modèles multidimensionnels Maillage Separated representation Remark: can be a a group of coordinates.
Iter. n R S S R