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Un Framework de calcul pour la méthode des bases réduites : applications à des problèmes non-linéaire multi-physiques

Abstract : Today, in many fields of applications, more and more engineering problems require to have an accurate and efficient evaluation of quantities of interest.Often, these quantities depend on a partial differential equation (PDE) parameterized solution -- physical or geometrical -- are the model inputs and the quantities of interest -- average values ​​-- are the outputs.The order reduction techniques, including reduced basis method which is the method used throughout this work, can meet these demands.In this thesis, we focus on the establishment of a framework in C ++ supporting parallel computing, which applies the reduced basis method to nonlinear multiphysics problems such as problems with natural convection (fluid-thermal coupling) or the high field resistive magnet modeling (we limit ourselves to thermo-electric coupling) leading to a study on the uncertainty quantification.The reduced basis method naturally relies on an approximation obtained using the finite element discretization of the problem being treated. For this, we use the Feel ++ computation library specialized in PDE resolution.We are also interested by multiscale problems.The particularity of these problems is to manipulate a set of phenomena involving different scales, as this is the case for example when we consider a flow in porous media.The multiscale finite element method allows having a "global" behavior, linked with large scales, of the problem solution without solving it on small scales.We propose a new construction of multiscale finite element basis functions based on the reduced basis method.
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Stéphane Veys. Un Framework de calcul pour la méthode des bases réduites : applications à des problèmes non-linéaire multi-physiques. Analyse numérique [cs.NA]. Université de Grenoble, 2014. Français. ⟨NNT : 2014GRENM069⟩. ⟨tel-01079415v2⟩

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