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Approximation par éléments finis C1 des modèles magnétohydrodynamiques pour les plasmas de fusion

Abstract : This thesis participates in the development of advanced numerical methods to simulate plasma instabilities for fusion by magnetic confinement in tokamaks. These flows are described in a general framework by magnetohydrodynamic(MHD) fluid models and can be considered incompressible in some approximations known as reduced MHD models. In this work, the incompressibility constraint is dealt with by the introduction of stream functions. A consequence of this formulation is the appearence of differential terms of order 4 in the equations. The use of C 1 functions is then required to apply the conforming Galerkin finite element method. We have used the the so-called reduced Clough-Tocher(CT) finite element method on general triangulations. The method has been validated on simple problems and then extented to problems relevant for the study of fusion plasmas. First, plasma equilibrium described by the Grad-Shafranov equation, has been investigated. Then we have studied incompressible models in a pure streamfunction formulation. First, we introduced a discretization of the incompressible Navier-Stokes equations which constitute a sub-model of the incompressible MHD equations. We have shown the stability in energy of the method and demontrated its performance on some standard test cases. We have then extended this numerical scheme to the incompressible MHD equations. We have also proved the stability in energy of the numerical approach and applied the numerical scheme to the simulation of the well known "tilt instability". In view of the results obtained, the CT method appears to be suitable for the simulation of plasma instabilities described by MHD models. Due to its capability to represent complex geometry, it compares favorably to other numerical methods in term of accuracy, CPU time, memory cost and versatility. jfavorably to other numerical methods in term of accuracy, CPU time, memory cost and versatility.
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Submitted on : Thursday, May 6, 2021 - 12:00:11 PM
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  • HAL Id : tel-03219276, version 1

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Ali Aboudou Elarif. Approximation par éléments finis C1 des modèles magnétohydrodynamiques pour les plasmas de fusion. Physique mathématique [math-ph]. Université Côte d'Azur, 2020. Français. ⟨NNT : 2020COAZ4108⟩. ⟨tel-03219276⟩

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