Low-order continuous finite element spaces on hybrid non-conforming hexahedral-tetrahedral meshes

This article deals with solving partial differential equations with the finite element method on hybrid non-conforming hexahedral-tetrahedral meshes. By non-conforming, we mean that a quadrangular face of a hexahedron can be connected to two triangular faces of tetrahedra. We introduce a set of low-order continuous (C0) finite element spaces defined on these meshes. They are built from standard tri-linear and quadratic Lagrange finite elements with an extra set of constraints at non-conforming hexahedra-tetrahedra junctions to recover continuity. We consider both the continuity of the geometry and the continuity of the function basis as follows: the continuity of the geometry is achieved by using quadratic mappings for tetrahedra connected to tri-affine hexahedra and the continuity of interpolating functions is enforced in a similar manner by using quadratic Lagrange basis on tetrahedra with constraints at non-conforming junctions to match tri-linear hexahedra. The so-defined function spaces are validated numerically on simple Poisson and linear elasticity problems for which an analytical solution is known. We observe that using a hybrid mesh with the proposed function spaces results in an accuracy significantly better than when using linear tetrahedra and slightly worse than when solely using tri-linear hexahedra. As a consequence, the proposed function spaces may be a promising alternative for complex geometries that are out of reach of existing full hexahedral meshing methods

High order non-conforming multi-element Discontinuous Galerkin method for time domain electromagnetics

International audienceWe study a high order Discontinuous Galerkin Time Domain (DGTD) method for solving the system of Maxwell equations. This method is formulated on a non-conforming multi-element mesh mixing an unstructured triangulation for the discretization of irregularly shaped objects with a structured (Cartesian) quadrangulation for the rest of the computational domain. Within each element, the electromagnetic field components are approximated by a high order nodal polynomial. The DG method proposed here makes use of a centered scheme for the definition of the numerical traces of the electric and magnetic fields at element interfaces, associated to a second order or fourth order leap-frog scheme for the time integration of the associated semi-discrete equations. We formulate this DGTD method in 3D and study its theoretical properties. In particular, an a priori convergence estimation is elaborated. Finally, we present numerical results for the application of the method to 2D test problems

High order non-conforming multi-element Discontinuous Galerkin method for time-domain electromagnetics

Cette thèse porte sur l’étude d’une méthode de type Galerkin discontinu en domaine temporel (GDDT), afin de résoudre numériquement les équations de Maxwell instationnaires sur des maillages hybrides tétraédriques/hexaédriques en 3D (triangulaires/quadrangulaires en 2D) et non-conformes, que l’on note méthode GDDT-PpQk. Comme dans différents travaux déjà réalisés sur plusieurs méthodes hybrides (par exemple des combinaisons entre des méthodes Volumes Finis et Différences Finies, Éléments Finis et Différences Finies, etc.), notre objectif principal est de mailler des objets ayant une géométrie complexe à l’aide de tétraèdres, pour obtenir une précision optimale, et de mailler le reste du domaine (le vide environnant) à l’aide d’hexaèdres impliquant un gain en terme de mémoire et de temps de calcul. Dans la méthode GDDT considérée, nous utilisons des schémas de discrétisation spatiale basés sur une interpolation polynomiale nodale, d’ordre arbitraire, pour approximer le champ électromagnétique. Nous utilisons un flux centré pour approcher les intégrales de surface et un schéma d’intégration en temps de type saute-mouton d’ordre deux ou d’ordre quatre. Après avoir introduit le contexte historique et physique des équations de Maxwell, nous présentons les étapes détaillées de la méthode GDDT-PpQk. Nous réalisons ensuite une analyse de stabilité L2 théorique, en montrant que cette méthode conserve une énergie discrète et en exhibant une condition suffisante de stabilité de type CFL sur le pas de temps, ainsi que l’analyse de convergence en h (théorique également), conduisant à un estimateur d’erreur a-priori. Ensuite, nous menons une étude numérique complète en 2D (ondes TMz), pour différents cas tests, des maillages hybrides et non-conformes, et pour des milieux de propagation homogènes ou hétérogènes. Nous faisons enfin de même pour la mise en oeuvre en 3D, avec des simulations réalistes, comme par exemple la propagation d’une onde électromagnétique dans un modèle hétérogène de tête humaine. Nous montrons alors la cohérence entre les résultats mathématiques et numériques de cette méthode GDDT-PpQk, ainsi que ses apports en termes de précision et de temps de calcul.This thesis is concerned with the study of a Discontinuous Galerkin Time-Domain method (DGTD), for the numerical resolution of the unsteady Maxwell equations on hybrid tetrahedral/hexahedral in 3D (triangular/quadrangular in 2D) and non-conforming meshes, denoted by DGTD-PpQk method. Like in several studies on various hybrid time domain methods (such as a combination of Finite Volume with Finite Difference methods, or Finite Element with Finite Difference, etc.), our general objective is to mesh objects with complex geometry by tetrahedra for high precision and mesh the surrounding space by square elements for simplicity and speed. In the discretization scheme of the DGTD method considered here, the electromagnetic field components are approximated by a high order nodal polynomial, using a centered approximation for the surface integrals. Time integration of the associated semi-discrete equations is achieved by a second or fourth order Leap-Frog scheme. After introducing the historical and physical context of Maxwell equations, we present the details of the DGTD-PpQk method. We prove the L2 stability of this method by establishing the conservation of a discrete analog of the electromagnetic energy and a sufficient CFL-like stability condition is exhibited. The theoritical convergence of the scheme is also studied, this leads to a-priori error estimate that takes into account the hybrid nature of the mesh. Afterward, we perform a complete numerical study in 2D (TMz waves), for several test problems, on hybrid and non-conforming meshes, and for homogeneous or heterogeneous media. We do the same for the 3D implementation, with more realistic simulations, for example the propagation in a heterogeneous human head model. We show the consistency between the mathematical and numerical results of this DGTD-PpQk method, and its contribution in terms of accuracy and CPU time