Parameter-Multiscale PGD Methods for High Dimensional Parametric Spaces

International audienceModel reduction techniques such as Proper Generalized Decomposition (PGD) are decision-making tools which are about to revolutionize many domains. Unfortunately, their calculation remains problematic for problems involving many parameters, for which one can invoke the " curse of dimensionality ". This works proposes a tentative answer to this challenge in solid mechanics by the so-called " parameter-multiscale PGD ". This work is based on the classical PGD, a model reduction technique using separated variable representations to approximate high dimensional spaces. The method, introduced in [1], uses the physics of the problem to built a more structured representation. It is based on the Saint-Venant's Principle which highlights two different levels of parametric influence, which leads us to introduce a multiscale description of the parameters to separate a " macro " and a " micro " scale. To implement this " parameter-multiscale " vision, a completely discontinuous spacial approximation is needed. Thus, we use the Weak-Trefftz Discontinuous Method used in [2] for the calculation of " medium frequency " phenomena. Discontinuous spatial methods are rarely implemented in industrial solid mechanics software, thus, a non-intrusive version of the algorithm, compatible with classical finite element discretization, has been introduced. On different academic examples, we can show that the computation of the algorithm on a 3D linear elastic problem up to the second iteration leads to very small errors. That is done for cases with more than a thousand parameters [3]. REFERENCES [1] Ladevèze, P and Paillet, Ch and Néron, D, Extended-PGD Model Reduction for Nonlinear [3] Paillet, Ch, and Néron, D, and Ladevèze, P, A door to model reduction in high-dimensional parameter space, Comptes Rendus de l'Académie des Sciences, Mécanique, in publication (2018