Hassan Fahs

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.

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

In this paper, we discuss the formulation, stability and validation of a high-order non-dissipative discontinuous Galerkin (DG) method for solving Maxwells 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.

Improving accuracy of high-order discontinuous Galerkin method for time-domain electromagnetics on curvilinear domains

The paper discusses high-order geometrical mapping for handling curvilinear geometries in high-accuracy discontinuous Galerkin simulations for time-domain Maxwell problems. The proposed geometrical mapping is based on a quadratic representation of the curved boundary and on the adaptation of the nodal points inside each curved element. With high-order mapping, numerical fluxes along curved boundaries are computed much more accurately due to the accurate representation of the computational domain. Numerical experiments for two-dimensional and three-dimensional propagation problems demonstrate the applicability and benefits of the proposed high-order geometrical mapping for simulations involving curved domains.

Preliminary Investigation of a Nonconforming Discontinuous Galerkin Method for Solving the Time-Domain Maxwell Equations

This paper is concerned with the design of a high-order discontinuous Galerkin (DG) method for solving the 2-D time-domain Maxwell equations on nonconforming triangular meshes. The proposed DG method allows for using nonconforming 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. Numerical experiments are presented which both validate the theoretical results and provide further insights regarding to the practical performance of the proposed DG method, particulary when nonconforming meshes are employed.

Calculation of the SAR Induced in Head Tissues Using a High-Order DGTD Method and Triangulated Geometrical Models

The great majority of numerical calculations of the Specific Absorption Rate (SAR) induced in human tissues exposed to microwaves are performed using the Finite Difference Time Domain (FDTD) method and voxel based geometrical models. The straightforward implementation of the method and its computational efficiency are among the main reasons for FDTD being currently the leading method for numerical assessment of human exposure to electromagnetic waves. However, the rather difficult departure from the commonly used cartesian grid and cell size limitations regarding the discretization of very detailed structures of human tissues are often recognized as the main weaknesses of the method in this application context. We present here an alternative numerical dosimetry methodology combining a high order Discontinuous Galerkin Time Domain (DGTD) method and adapted geometrical models based on unstructured triangulations, and discuss its application to the calculation of the SAR induced in head tissues.

Hybrid Explicit–Implicit Time Integration for Grid-Induced Stiffness in a DGTD Method for Time Domain Electromagnetics

In the recent years, there has been an increasing interest in discontinuous Galerkin time domain (DGTD) methods for the numerical modeling of electromagnetic wave propagation. Such methods most often rely on explicit time integration schemes which are constrained by a stability condition that can be very restrictive on highly refined meshes. In this paper, we report on some efforts to design a hybrid explicit–implicit DGTD method for solving the time domain Maxwell equations on locally refined simplicial meshes. The proposed method consists in applying an implicit time integration scheme locally in the refined regions of the mesh while preserving an explicit time scheme in the complementary part.

Calculation of the SAR induced in head tissues using a high order DGTD method and triangulated geometrical models

The great majority of numerical calculations of the specific absorption rate (SAR) induced in human tissues exposed to microwaves are performed using the finite difference time-domain (FDTD) method and voxel-based geometrical models. The straightforward implementation of the method and its computational efficiency are among the main reasons for FDTD being currently the leading method for numerical assessment of human exposure to electromagnetic waves. However, the rather difficult departure from the commonly used Cartesian grid and cell size limitations regarding the discretization of very detailed structures of human tissues are often recognized as the main weaknesses of the method in this application context. In particular, interfaces between tissues where sharp gradients of the electromagnetic field may occur are hardly modeled rigorously in these studies. We present here an alternative numerical dosimetry methodology which is based on a high order discontinuous Galerkin time-domain (DGTD) method and adapted geometrical models constructed from unstructured triangulations of tissue interfaces, and discuss its application to the calculation of the SAR induced in head tissues.

Recent Achievements on a DGTD Method for Time-Domain Electromagnetics

We report on results concerning a discontinuous Galerkin time domain (DGTD) method for the solution of Maxwell equations. This DGTD method is formulated on unstructured simplicial meshes (triangles in 2-D and tetrahedra in 3-D). Within each mesh element, the electromagnetic field components are approximated by an arbitrarily high order nodal polynomial while, in the original formulation of the method, time integration is achieved by a second order Leap-Frog scheme. Here, we discuss about several recent developments aiming at improving the accuracy and the computational efficiency of this DGTD method in view of the simulation of problems involving general domains and heterogeneous media.