A General Formulation for Time-Harmonic Maxwell Equations in 3D Cavities and Approximation by a Spectral Method (original) (raw)

Analysis of Time-Domain Maxwell's Equations for 3-D Cavities

Advances in Computational Mathematics, 2002

Time–domain Maxwell's equations are studied for the electromagnetic scattering of plane waves from an arbitrarily shaped cavity filled with nonhomogeneous medium. A transparent boundary condition is introduced to reduce the problem to the bounded cavity. Existence and uniqueness of the model problem are established by a variational approach and the Hodge decomposition. The analysis forms a basis for numerical solution

Numerical Solution of the Three-Dimensional Time-Harmonic Maxwell Equations by DG Method Coupled with an Integral Representation

International Journal for Research in Applied Science and Engineering Technology -IJRASET, 2020

This work is dedicated to the numerical results and the implementation of the method coupling a discontinuous Galerkin with an integral representation (CDGIR). The originality of this work lies in the choice of discretization by discontinuous Galerkin element and a mixed form for Maxwell's equations. The numerical tests justify the effectiveness of the proposed approach. Keywords: Finite element method, Maxwell equations, Discontinuous Galerkin method, fictitious domain, integral representation, time-harmonic. I. INTRODUCTION Mathematically, the phenomenon of the electromagnetic waves propagation is generally modeled by the system of equations known as the Maxwell equations. There are two modes of the Maxwell equations to be treated, a first mode that is known by the time domain Maxwell equations in which the evolution of electromagnetic fields is studied as a function of time and the second mode that is known by the frequency domain Maxwell equations where one studies the behavior of electromagnetic fields when the source term follows a harmonic dependence in time. Numerical modeling has become the most important and widely used tool in various fields such as scientific research. The finite-difference methods (FDM), the finite element methods (FEM) and the finite volume methods (FVM) are the three classes of methods known for the numerical resolution of the problems of electromagnetic waves propagation. In 1966, Yee cited the first efficient method in [42] which is the finite-difference methods in the time domain (FDMTD). When diffraction problems are posed in unbounded domain, the use of these methods induces a problem. In order to solve it, two techniques are used. The first consists in reducing to a bounded domain by truncating the computational domain, then it is necessary to impose an artificial condition on the boundary on the truncation boundary. The second technique consists in writing an equivalent problem posed on the boundary of the obstacle, it is therefore what is called the theory of integral equations. The numerical resolution can then be done by discretizing the problem by collocation (method of moments, method of singularity) or by a finite element discretization of the boundary. In 1980, Nedelec introduces the edge finite element method developed in [31] which is also available in [29, 30]. With the conservation of energy, this method also possesses several advantages; it allows to treat unstructured meshes (complex geometries) as it can be used with high orders (see [41, 24, 29]). In recent years, research has revealed a new technique known as Discontinuous Galerkin Methods (GDM); this strategy is based on combining the advantages of FEM and FVM methods since it approaches the field in each cell by a local basis of functions by treating the discontinuity between neighboring cells by approximation FVM on the flows. Initially, these methods have been proposed to treat the scalar equation of neutron transport (see [35]). In the field of wave propagation, precisely for the resolution of the Maxwell equations in the time domain, many schemes are based on two forms of formulations: a concentrated flux formulation (see [16, 34]) and an upwind flux formulation (see [22, 12]). Discontinuous Galerkin methods have shown their effectiveness in studying the problem with discrete eigenvalues (see [23]). In frequency domain, for the resolution of Maxwell equations, the majority consider the second order formulation (see [25, 32, 33]), as others study the formulation of the first order as in [6, 20]. This strategy of the CDGIR method allows us to write a problem in an unbounded domain into an equivalent problem in a domain bounded by a fictitious boundary where a transparent condition is imposed. This transparent condition is based on the use of the integral form of the electric and magnetic fields using the Stratton-chu formulas (see [7]). This process has been studied, in the

Discrete boundary finite element schemes for an exterior problem for the time-harmonic Maxwell’s equation

Applied Mathematics and Computation, 2014

A general boundary integral formulation using Galerkin procedure is applied to compute the scattering electric field produced by the diffraction of an incident electromagnetic wave by a perfectly conducting obstacle. This electric field satisfies the three-dimensional timeharmonic Maxwell's equations for which the skin currents and charges are to be approximated using boundary finite element method. With the help of linear and quadratic finite elements of Lagrange type, we introduce an approximate surface on which the discrete formulation is defined, and construct approximate surface currents and charges. In addition, we study the existence and the uniqueness of the solution of the discrete problem and develop some error estimates for the currents and charges. Numerical results are also presented in order to validate our numerical approach.

Debye Sources and the Numerical Solution of the Time Harmonic Maxwell Equations II

Communications on Pure and Applied Mathematics, 2012

In this paper, we develop a new integral representation for the solution of the time harmonic Maxwell equations in media with piecewise constant dielectric permittivity and magnetic permeability in R 3. This representation leads to a coupled system of Fredholm integral equations of the second kind for four scalar densities supported on the material interface. Like the classical Müller equation, it has no spurious resonances. Unlike the classical approach, however, the representation does not suffer from low frequency breakdown. We illustrate the performance of the method with numerical examples.

A 3D finite element method for the modelling of bounded and unbounded electromagnetic problems in the time domain

International Journal of Numerical Modelling Electronic Networks Devices and Fields

This paper presents a 3D body-conforming finite element solution of the time-dependent vector wave equation. The method uses edge elements on tetrahedra for the electric field interpolation. This kind of element is suited to model Maxwell's equations since it only enforces tangential continuity of vector fields. For the discretization of time derivatives we use the Newmark method, which allows obtaining an unconditionally stable scheme with second-order accuracy. The Silver–Müller absorbing boundary condition is employed for the domain truncation in unbounded problems. Numerical results for some examples are provided to validate the presented method. Copyright © 2000 John Wiley & Sons, Ltd.

A time-domain vector potential formulation for the solution of electromagnetic problems

IEEE Microwave and Guided Wave Letters, 1998

We present an alternative vector potential formulation of Maxwell's equations derived upon introduction of a quantity related to the Hertz potential. Once space and time are discretized, within this formulation the electric field and vector potential components are condensed in the same point in the elementary cell. In three dimensions the formulation offers an alternative to finite-difference time-domain (FDTD) method; when reduced to a two-dimensional (2-D) problem, only two variables, instead of three, are necessary, implying a net memory saving of 1/3 with respect to FDTD.

A novel integral formulation for the solution of maxwell equations

IEEE Transactions on Magnetics, 2003

This paper presents a novel integral formulation for the solution of the full Maxwell equations in the frequency domain. The solenoidality of the current density inside a homogeneous conductor is exploited to introduce a two-component electric vector potential, which is numerically expanded in terms of edge elements.