Etude d'une méthode volumes finis pour la résolution d'un modèle non linéaire d'un couplage maxwell/plasma dans le domaine temporel

Cette thèse présente l'étude d'une méthode numérique efficace pour résoudre les équations de Maxwell couplées à un modèle de plasma fluide. Le travail est organisé en cinq chapitres dans lesquels nous présentons la formulation du modèle physique, l'étude mathématique pour démontrer l'existence et l'unicité d'une solution, l'approximation numérique du problème, des résultats de validation et enfin, dans un cas simplifié, la mise en œuvre et l'étude numérique d'une stratégie de maillage auto-adaptatif en 1D. Dans ces travaux de recherche, nous nous sommes plus focalisés sur le choix d'une approximation numérique qui soit la plus performante pour résoudre notre problème couplé. En particulier, après avoir donné une approximation différences finies actuellement utilisée en 2D dans ce contexte, nous avons proposé une solution parallèle d'un outil FDTD et traité en 3D un couplage micro-onde/plasma en espace libre. Ensuite, dans le cas de problèmes de blindage, nous avons mis en avant les inconvénients de la méthode FDTD et proposé une approche basée sur un schéma volumes finis qui offre les avantages du raffinement local. Pour améliorer cette méthode, nous avons mis en œuvre une stratégie de pas de temps local et montré les gains obtenus dans le cas de plasma confinés.This thesis presents the study of an efficient numerical method to solve the Maxwell equations coupled with a fluid plasma model. The document is split into five chapters where we introduce the formulation of the physical model, a mathematical study to demonstrate the existence and uniqueness of a solution for the problem, numerical approximations of the equations, simulations and validations on 3D and 2D examples and a prospective work on a finite volume method with adaptative mesh for the 1D case. The accent is continuously put on the choice of the most efficient numerical approximation to solve the coupled problem. In this work, we exhibit the drawbacks of the finite difference method usually employed in this context. To overcome these drawbacks, we propose a method based upon a finite volume scheme which allows the capability to use local refinements. Then, to increase the gain in time CPU and memory storage, we introduce a local time-stepping scheme

The semi-Lagrangian method on curvilinear grids

International audienceWe study the semi-Lagrangian method on curvilinear grids. The classical backward semi-Lagrangian method [1] preserves constant states but is not mass conservative. Natural reconstruction of the field permits nevertheless to have at least first order in time conservation of mass, even if the spatial error is large. Interpolation is performed with classical cubic splines and also cubic Hermite interpolation with arbitrary reconstruction order of the derivatives. High odd order reconstruction of the derivatives is shown to be a good ersatz of cubic splines which do not behave very well as time step tends to zero. A conservative semi-Lagrangian scheme along the lines of [2] is then described; here conservation of mass is automatically satisfied and constant states are shown to be preserved up to first order in time

Optimality conditions for fractional differential inclusions with nonsingular Mittag–Leffler kernel

Abstract In this paper, by using the Dubovitskii–Milyutin theorem, we consider a differential inclusions problem with fractional-time derivative with nonsingular Mittag–Leffler kernel in Hilbert spaces. The Atangana–Baleanu fractional derivative of order α in the sense of Caputo with respect to time t, is considered. Existence and uniqueness of solution are proved by means of the Lions–Stampacchia theorem. The existence of solution is obtained for all values of the fractional parameter α∈(0,1) $\alpha\in(0, 1)$. Moreover, by applying control theory to the fractional differential inclusions problem, we obtain an optimality system which has also a unique solution. The controllability of the fractional Dirichlet problem is studied. Some examples are analyzed in detail

Existence and exact asymptotic behaviour of positive solutions for fractional boundary value problem with P-Laplacian operator

This paper deals with existence, uniqueness and global behaviour of a positive solution for the fractional boundary value problem $D^{\beta }(\psi (x)\Phi _{p}(D^{\alpha }u))=a(x)u^{\sigma }$ in $(0,1)$ with the condition $$\begin{equation*} \underset{x\rightarrow 0}\lim D^{\beta-1}(\psi(x)\Phi_{p}(D^{\alpha}u(x) ))=\underset{x\rightarrow 1}\lim \psi(x)\Phi_{p}(D^{\alpha}u(x))=0 \quad {\rm and} \quad \underset{x\rightarrow 0}\lim D^{\alpha-1}u(x)= u(1)=0, \end{equation*}$$ where $\beta ,\alpha \in (1,2]$ , $\Phi _{p}(t)=t|t|^{p-2}$ , p>1, $\sigma \in (1-p,p-1)$ , the differential operator is taken in the Riemann–Liouville sense and $\psi , a\ : (0,1)\longrightarrow \mathbb {R}$ are non-negative and continuous functions that may are singular at x=0 or x=1 and satisfies some appropriate conditions

Étude d'une méthode volumes finis pour la résolution d'un modèle non linéaire d'un couplage Maxwell/plasma dans le domaine temporel

TOULOUSE3-BU Sciences (315552104) / SudocSudocFranceF

Efficient numerical algorithm to simulate a 3D coupled Maxwell–plasma problem

International audienceThis paper proposes an improved algorithm, based upon an explicit finite difference scheme, in order to simulate the plasma breakdown induced by a monochromatic High Power Micro-Wave (HPM). The 3D coupled Maxwell–plasma equations are to be solved. We want to study with this model the geometry of the discharge and plasma formation at high pressure which may contribute to shield microwave sensors or circuits. Generally, the simulation of this kind of problem is very time-consuming, but by using the fact that the plasma evolution in time is slow relatively to the monochromatic source period, we can drastically reduce the simulation time. By considering this assumption, we describe in the paper a process which allows to obtain this important reduction. Finally, an example where we show the gain obtained in terms of computation time with our process is given to validate and illustrate the global work

FVTD Modeling of a Localized Microwave Plasma Discharge in Microstrip Wave Guide

International audienceA plasma hole in the substrate of a microstrip waveguide can be initiated by either high power RF signal or DC voltage. A 2D Finite Volume Time Domain modeling of such a configuration is proposed in this paper. Achieved results in terms of plasma raising time or microwave reflection are depicted to illustrate this topic

Guiding center simulations on curvilinear grids*

Semi-Lagrangian guiding center simulations are performed on sinusoidal perturbations of cartesian grids, and on deformed polar grid with different boundary conditions. Key ingredients are: the use of a B-spline finite element solver for the Poisson equation and the classical backward semi-Lagrangian method (BSL) for the advection. We are able to reproduce standard Kelvin-Helmholtz and diocotron instability tests on such grids. When the perturbation leads to a strong distorted mesh, we observe that the solution differs if one takes standard numerical parameters that are used in the cartesian reference case. We can recover good results together with correct mass conservation, by diminishing the time step

Development of 3D finite volume solver for plasma/microwave simulation

International audienceThe purpose of this work is to propose a method for solving a three-dimensional nonlinear model coupling Maxwell equations with plasma fluid equations. Cases where the plasma is generated or affected by an electromagnetic wave will be described, and we will see how the plasma will modify the propagation conditions

Modelization of plasma breakdown by using Finite Volume Time Domain Method

International audienceThe theoretical and experimental knowledge on high power microwave breakdown at near-atmospheric pressure is used to study the shielding of a metallic cavity with a plasma located in its aperture. In this paper we give a mathematical formulation for the shielding problem and we describe a FVTD approximation to solve it