Nouvelles démarches de réduction de modèles pour le traitement des problèmes à très grand nombre de paramètres

Abstract : Numerical simulation is nowadays a major tool in a large number of engineering fields. Nevertheless, even the recent incredible improvements of the computational power can hardly compensate the increasing complexity of the models used by engineers. In this context, Reduced Order Models (ROM) can be major decision-maker tools because, once they have been computed, they can be used to evaluate a very large number of test cases in a duration close to real time. The PGD (Proper Generalized Decomposition) in particular, is a method introduced at the LMT which has been adapted to many cases (non-linear problems, multiscale, multiphysics) and leads to savings of CPU time reaching several orders of magnitude.Unfortunately, it is currently difficult to build ROM with an increasing number of parameters. All the actual model reduction technics (including the PGD) can hardly solve problems with a high number of parameters (the current limit is about twenty parameters). It is a major barrier to a larger development of these methods. This PhD thesis presents a new methodology based on the PGD able to take into account high numbers of parameters.This goal has been achived thanks to three major contributions. First, a new data structure faithfull to mecanical properties of the problem has been developed. To that end, two different scales are introduced in the parametric space, giving its name to our method : Parameter-Multiscale PGD. Furthermore, the WTDG (Weak Treffz Discontinuous Galerkin) method has been inpemented. It is a discontinuous spatial discretisation adapted to our resolution techniques. Finally, new algorithms have been developed to built reduced order models of problems taking into account up to one thousand parameters.
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Charles Paillet. Nouvelles démarches de réduction de modèles pour le traitement des problèmes à très grand nombre de paramètres. Mécanique des solides [physics.class-ph]. Université Paris-Saclay, 2019. Français. ⟨NNT : 2019SACLN015⟩. ⟨tel-02276305⟩

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