Sébastien Pernet

A spatial high-order hexahedral discontinuous Galerkin method to solve Maxwell's equations in time domain

In this paper, we present a non-dissipative spatial high-order discontinuous Galerkin method to solve the Maxwell equations in the time domain. The non-intuitive choice of the space of approximation and the basis functions induce an important gain for mass, stiffness and jump matrices in terms of memory. This spatial approximation, combined with a leapfrog scheme in time, leads also to a fast explicit and accurate method. A study of the dispersive error is carried out and a stability condition for the proposed scheme is established. Some comparisons with other schemes are presented to validate the new scheme and to point out its advantages. Finally, in order to improve the efficiency of the method in terms of CPU time on general unstructured meshes, a strategy of local time-stepping is proposed.

Dissipative terms and local time-stepping improvements in a spatial high order Discontinuous Galerkin scheme for the time-domain Maxwell's equations

In this paper, we present some improvements, in terms of accuracy and speed-up, for a particular well adapted Discontinuous Galerkin method devoted to the time-domain Maxwell equations. First, to reduce spurious modes on very distorted meshes, the addition of dissipative terms as penalization in the numerical scheme is studied and compared on examples. Second, in order to increase the efficiency of the method, a multi-class local time-stepping strategy is presented and its validation and advantages are highlighted on different examples.

BOUNDARY-INTEGRAL METHODS FOR ITERATIVE SOLUTION OF SCATTERING PROBLEMS WITH VARIABLE IMPEDANCE SURFACE CONDITION

We present an efficient boundary element method to solve electromagnetic scattering problems relative to an impedance boundary condition on an obstacle of arbitrary shape in the frequency domain. In particular, the technique is based on a Combined Field Integral Equation (CFIE) and is well adapted to treat the partially coated objects. Some methods are then proposed in order to eliminate the magnetic current and to treat correctly the rotation operator n × · (where n is the unit outward normal). After discretization, the final system is solved by an iterative method coupled with the Fast Multipole Method (FMM). Finally, a numerical comparison with a well-tried method to solve this kind of problem proves that we have proposed an attractive technique in terms of memory storage and CPU time.

High spatial order finite element method to solve Maxwell's equations in time domain

This paper presents a finite element method with high spatial order for solving the Maxwell equations in the time domain. In the first part, we provide the mathematical background of the method. Then, we discuss the advantages of the new scheme compared to a classical finite-difference time-domain (FDTD) method. Several examples show the advantages of using the new method for different kinds of problems. Comparisons in terms of accuracy and CPU time between this method, the FDTD and the finite-volume time-domain methods are given as well.

Hp a-priori error estimates for a non-dissipative spectral discontinuous galerkin method to solve the maxwell equations in the time domain

. In this paper, we present the hp-convergence analysis of a non-dissipative high-order discontinuous Galerkin method on unstructured hexahedral meshes using a mass-lumping technique to solve the time-dependent Maxwell equations. In particular, we underline the spectral convergence of the method (in the sense that when the solutions and the data are very smooth, the discretization is of unlimited order). Moreover, we see that the choice of a non-standard approximate space (for a discontinuous formulation) with the absence of dissipation can imply a loss of spatial convergence. Finally we present a numerical result which seems to confirm this property.

Finite Element and Discontinuous Galerkin Methods for Transient Wave Equations

This monograph presents numerical methods for solving transient wave equations (i.e. in time domain). More precisely, it provides an overview of continuous and discontinuous finite element methods for these equations, including their implementation in physical models, an extensive description of 2D and 3D elements with different shapes, such as prisms or pyramids, an analysis of the accuracy of the methods and the study of the Maxwell’s system and the important problem of its spurious free approximations. After recalling the classical models, i.e. acoustics, linear elastodynamics and electromagnetism and their variational formulations, the authors present a wide variety of finite elements of different shapes useful for the numerical resolution of wave equations. Then, they focus on the construction of efficient continuous and discontinuous Galerkin methods and study their accuracy by plane wave techniques and a priori error estimates. A chapter is devoted to the Maxwell’s system and the important problem of its spurious-free approximations. Treatment of unbounded domains by Absorbing Boundary Conditions (ABC) and Perfectly Matched Layers (PML) is described and analyzed in a separate chapter. The two last chapters deal with time approximation including local time-stepping and with the study of some complex models, i.e. acoustics in flow, gravity waves and vibrating thin plates. Throughout, emphasis is put on the accuracy and computational efficiency of the methods, with attention brought to their practical aspects.This monograph also covers in details the theoretical foundations and numerical analysis of these methods. As a result, this monograph will be of interest to practitioners, researchers, engineers and graduate students involved in the numerical simulationof waves.

New Trends in the Preconditioning of Integral Equations of Electromagnetism

A new family of source integral equations is presented, dedicated to the solution of time-harmonic Maxwell scattering problems. Regardless of the composition of the obstacle – metallic, full dielectric or coated with an impedance layer – we show that a general methodology is able to guide the construction of some special equations whose the foremost feature is to be well-conditioned. Indeed, all of them are free of spurious modes and appear as some compact perturbations of positive operators (when it is not the identity), leading therefore to fast iterative solutions without the help of any preconditioner. These intrinsically well-conditioned equations open the way for interesting new developments in the field of boundary equation methods for Maxwell applications.

A low-Mach number model for time-harmonic acoustics in arbitrary flows

This paper concerns the finite element simulation of the diffraction of a time-harmonic acoustic wave in the presence of an arbitrary mean flow. Considering the equation for the perturbation of displacement (due to Galbrun), we derive a low-Mach number formulation of the problem which is proved to be of Fredholm type and is therefore well suited for discretization by classical Lagrange finite elements. Numerical experiments are done in the case of a potential flow for which an exact approach is available, and a good agreement is observed.

Accurate, Automatic and Compressed Visualization of Radiated Helmholtz Fields from Boundary Element Solutions.

We propose a methodology to generate an accurate and efficient reconstruction of radiated fields based on high order interpolation. As the solution is obtained with the convolution by a smooth but potentially high frequency oscillatory kernel, our basis functions therefore incorporate plane waves. Directional interpolation is shown to be efficient for smart directions. An adaptive subdivision of the domain is established to limit the oscillations of the kernel in each element. The new basis functions, combining high order polynomials and plane waves, provide much better accuracy than low order ones. Finally, as standard visualization softwares are generally unable to represent such fields, a method to have a well-suited visualization of high order functions is used. Several numerical results confirm the potential of the method.

Well-suited and adaptive post-processing for the visualization of hp simulation results

Abstract While high order methods became very popular as they allow to perform very accurate solutions with low computational time and memory cost, there is a lack of tools to visualize and post-treat the solutions given by these methods. Originally, visualization softwares were developed to post-process results from methods such that finite differences or usual finite elements and therefore process linear primitives. In this paper, we present a methodology to visualize results of high order methods. Our approach is based on the construction of an optimized affine approximation of the high order solution which can therefore be handled by any visualization software. A representation mesh is constructed and the process is guided by an a posteriori estimate which control the error between the numerical solution and its representation pointwise. This point by point control is crucial as under their picture form, data correspond to values mapped on elements where anyone can pick up a pointwise information. A strategy is established to ensure that discontinuities are well represented. These discontinuities come either from the physical problem (material change) or the numerical method (Discontinuous Galerkin method) and are pictured accurately. Several numerical examples are presented to demonstrate the potential of the method.