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Formulation mixte augmentée d’un modèle « Full-wave » tridimensionnel dans un plasma froid : analyse numérique d’une approximation ℙ₂-ℙ₁

Abstract : The object of this thesis is to study a numerical simulation method of a so-called “Full-wave” model of the propagation in a tokamak of an electromagnetic wave injected by a set of antennas installed in the boundary of the containment chamber. A Lagrange finite element approximation method is deployed. In the first chapter, we introduced the physical model considered to describe the propagation of an electrostatic wave of a frequency close to hybrid resonance in a so-called cold plasma, plasma confined by a magnetic field inside a tokamak. The propagation of electromagnetic waves is modeled by Maxwell's equations. This work contemplates an approximation of the time-harmonic solution. In the second chapter, we recall the mixed and mixed variational formulations already studied in the past. These formulations allow us to find solutions in (H¹(Ω))³ and therefore a conformal finite element approximation in this space. Chapter three is dedicated to the presentation of the discretization of the model's equations in three dimensions of space. In chapter four it is demonstrated the well-posed character of the discrete system of equations when one considers a Taylor-Hood type approximation ℙ₂-ℙ₁. In this section, we proved a result about the existence and the uniqueness of the solution in the case of a polyhedral “torus”. Chapter 5 is dedicated to numerical simulations. First are introduced the terms of the dielectric tensor K and its derivatives, needed to mount the stiffness matrix of the system. The first simulations concern the case where the density of electrons and ions is constant. Then results are presented in the case where densities have a parabolic profile. It is also considered the case where the wave vector is a function of the distance at the center of the tokamak.
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Submitted on : Thursday, March 17, 2022 - 4:45:03 PM
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Rihab Daadaa. Formulation mixte augmentée d’un modèle « Full-wave » tridimensionnel dans un plasma froid : analyse numérique d’une approximation ℙ₂-ℙ₁. Mathématiques [math]. Université de Lorraine, 2021. Français. ⟨NNT : 2021LORR0307⟩. ⟨tel-03612357⟩

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