Solution of the time-harmonic Maxwell equations using discontinuous Galerkin methods

We present numerical results concerning the solution of the time-harmonic Maxwell's equations discretized by discontinuous Galerkin methods. In particular, a numerical study of the convergence, which compares different strategies proposed in the literature for the elliptic Maxwell equations, is performed in the two-dimensional case.Comment: Preprint submitted for publication for the proceedings of ICCAM06 (11/04/2007

Locally implicit discontinuous Galerkin method for time domain electromagnetics

In the recent years, there has been an increasing interest in discontinuous Galerkin time domain (DGTD) methods for the solution of the unsteady Maxwell equations modeling electromagnetic wave propagation. One of the main features of DGTD methods is their ability to deal with unstructured meshes which are particularly well suited to the discretization of the geometrical details and heterogeneous media that characterize realistic propagation problems. Such DGTD methods most often rely on explicit time integration schemes and lead to block diagonal mass matrices. However, explicit DGTD methods are also constrained by a stability condition that can be very restrictive on highly refined meshes and when the local approximation relies on high order polynomial interpolation. An implicit time integration scheme is a natural way to obtain a time domain method which is unconditionally stable but at the expense of the inversion of a global linear system at each time step. A more viable approach consists of applying an implicit time integration scheme locally in the refined regions of the mesh while preserving an explicit time scheme in the complementary part, resulting in an hybrid explicit–implicit (or locally implicit) time integration strategy. In this paper, we report on our recent efforts towards the development of such a hybrid explicit–implicit DGTD method for solving the time domain Maxwell equations on unstructured simplicial meshes. Numerical experiments for 3D propagation problems in homogeneous and heterogeneous media illustrate the possibilities of the method for simulations involving locally refined meshes

An inconditionnally stable discontinuous Galerkin method for solving the 2D time-domain Maxwell equations on unstructured triangular meshes

Numerical methods for solving the time-domain Maxwell equations often rely on cartesian meshes and are variants of the finite difference time-domain (FDTD) method due to Yee. In the recent years, there has been an increasing interest in discontinuous Galerkin time-domain (DGTD) methods dealing with unstructured meshes since the latter are particularly well adapted to the discretization of geometrical details that characterize applications of practical relevance. However, similarly to Yee's finite difference time-domain method, existing DGTD methods generally rely on explicit time integration schemes and are therefore constrained by a stability condition that can be very restrictive on locally refined unstructured meshes. An implicit time integration scheme is a possible strategy to overcome this limitation. The present study aims at investigating such an implicit DGTD method for solving the 2D time-domain Maxwell equations on non-uniform triangular meshes

Etude de stabilité d'une méthode Galerkin discontinu pour la résolution numérique des équations de Maxwell 2D en domaine temporel sur des maillages triangulaires non-conformes

On étudie la stabilité d'une méthode Galerkin discontinu pour la résolution numérique des équations de Maxwell 2D en domaine temporel sur des maillages triangulaires non-conformes. Cette méthode combine l'utilisation d'une approximation centrée pour l'évaluation des flux aux interfaces entre éléments voisins du maillage, á un schéma d'intégration en temps de type saute-mouton. La méthode repose sur une base de fonctions polynomiales nodales Pk et on considère ici les schémas obtenus pour k=0,..3. L'objectif de cette étude est d'exhiber des conditions sous lesquelles les schémas correspondant sont stables, et de comparer ces conditionsá celles obtenues dans le cas de maillages conformes

Parallel Solutions of Three-Dimensional Compressible Flows

In this report, we present parallel solutions of realistic three-dimensional flows obtained on the {\tt Intel Paragon}, the {\tt Cray T3D} and the {\tt Ibm SP2} MPPs (Massively Parallel Processors). The solver under consideration is a representative subset of an existing industrial code, {\tt N3S-MUSCL} (a three-dimensional compressible Navier-Stokes solver, see Chargy\cite{Cha1}). It implements a mixed finite element/finite volume formulation on unstructured tetrahedral meshes. Defining a good strategy for the parallelisation of an unstructured mesh based solver is a challenge, particularly when one aims at reaching a high level of performance while maintaining portability of the source code between scalar, vector and parallel machines. The parallelisation strategy adopted in this study combines mesh partitioning techniques and a message-passing programming model. The mesh partitioning algorithms and the generation of the corresponding communication data-structures are gathered in a preprocessor in order to introduce a minimum change in the original serial code. The portability from one message passing parallel system to another may be enhanced with the use of a communication library such as {\tt PVM}

A hp-like discontinuous Galerkin method for solving the 2D time-domain Maxwell's equations on non-conforming locally refined triangular meshes

This work is concerned with the design of a hp-like discontinuous Galerkin (DG) method for solving the 2D time-domain Maxwell's equations on non-conforming locally refined triangular meshes. The proposed DG method allows non-conforming meshes with arbitrary-level hanging nodes. This method combines a centered approximation for the evaluation of fluxes at the interface between neighboring elements of the mesh, with a leap-frog time integration scheme

Méthodes de type Galerkin discontinu pour la résolution numérique des équations de Maxwell en régime fréquentiel

On s'intéresse à la résolution numérique des équations de Maxwell tridimensionnelles en régime harmonique par des méthodes de type Galerkin Discontinu (GD) en maillages tétraédriques non-structurés où l'on approche les inconnues du problème par des fonctions constantes par morceaux (méthode GD d'ordre 0 ou GDP0) ou linéaires par morceaux (méthode GD d'ordre 1 ou GDP1). De nombreux travaux ces dernières années ont démontré l'intérêt des formulations GD pour la modélisation numérique de phénomènes de propagation d'ondes en milieux hétérogènes. Dans cette étude, on adapte au domaine fréquentiel (ou régime harmonique) des méthodes GD centrées précédemment dévelopées pour les mêmes équations résolues en domaine temporel. Dans ce rapport, on traite essentiellement de la formulation des méthodes en question en 1D et 3D, et de leur analyse théorique (dispersion numérique et caractère bien posé du problème discret). La difficulté principale réside dans le fait que l'on traite d'un problème posé dans le domaine complexe pour lequel des résultats d'inversibilité sont difficiles à prouver en l'absence d'hypothèses supplémentaires. On conclut en présentant une série de résultats numériques préliminaires en 1D et 3D, ces derniers visant essentiellement à valider les méthodes GD proposées. La question de la résolution (itérative ou directe) des systèmes linéaires obtenus sera traitée dans un prochain rapport

Upwind stabilization of Navier-Stokes solvers

We present a study of the effect of upwinding on stabilisation of both advective and pressure terms in a family of primitive-variable Navier-Stokes solvers. We consider two MUSCL schemes, the first one applies to compressible flow, the second one to incompressible flow. We illustrate the fact that both numerical models suffer oscillations if a minimal (but not large) amount of upwinding is not associated with acoustics, while advection can be stabilized by the physical diffusion terms when the mesh Reynolds number is small enough

A Two-Level Parallelization Strategy for Genetic Algorithms Applied to Shape Optimum Design

This report presents a two-level strategy for the parallelization of a genetic algorithm coupled to a compressible flow solver designed on unstructured triangular meshes. The parallel implementation is based on MPI and makes use of the process group features of this environment. The resulting algorithm is used for the optimum shape design of aerodynamic configurations. Numerical and performance results are presented for the optimization of two-dimensional airfoils for calulations performed on the following systems : a SGI Origin 2000 and a IBM SP-2 MIMD systems; a Pentium Pro (P6/200 Mhz) cluster where the interconnection is realized through a FastEthernet (100 Mbits/s) switch

A high-order non-conforming discontinuous Galerkin method for time-domain electromagnetics

International audienceIn this paper, we discuss the formulation, stability and validation of a high-order non-dissipative discontinuous Galerkin (DG) method for solving Maxwell's equations on non-conforming simplex meshes. The proposed method combines a centered approximation for the numerical fluxes at inter element boundaries, with either a second-order or a fourth-order leap-frog time integration scheme. Moreover, the interpolation degree is defined at the element level and the mesh is refined locally in a non-conforming way resulting in arbitrary-level hanging nodes. The method is proved to be stable and conserves a discrete counterpart of the electromagnetic energy for metallic cavities. Numerical experiments with high-order elements show the potential of the method