I. Bardi

Plane wave scattering from frequency-selective surfaces by the finite-element method

This paper presents the application of the finite-element method (FEM) to the analysis of reflection and transmission coefficients, effective permittivity, and permeability of periodic structures to incident plane waves. The problem domain is reduced to a unit cell by using linked boundary conditions (LBCs) and the accuracy and efficiency of the procedure is improved by using the scattered field formulation and perfectly matched layers (PMLs). Examples of the analysis of frequency-selective surfaces (FSSs) derived from photonic bandgap materials (PBGs) and composite materials with negative effective permeability and permittivity are presented.

Numerical analysis of 3D magnetostatic fields

Formulations of three-dimensional magnetostatic fields are reviewed for their finite-element analysis. Partial differential equations and boundary conditions are set up for various kinds of potentials. Besides the method using two scalar potentials, several vector potential formulations are also discussed. Galerkin techniques combined with the finite element method are applied for the numerical solution of the boundary value problems. The effect of gauging the vector potential upon the numerical performance is investigated. Solutions by different formulations to a simple test problem and a benchmark problem involving relatively thin saturated iron plates are presented. The latter is compared to measured results. >

Different finite element formulations of 3D magnetostatic fields

Several finite-element formulations of three-dimensional magnetostatic fields are reviewed. Both nodal and edge elements are considered. The aim is to suggest remedies to some shortcomings of widely used methods. Various formulations are compared based on results for Problem No. 13 of the TEAM Workshops, a nonlinear magnetostatic problem involving thin iron plates. >

Parameters of lossy cavity resonators calculated by the finite element method

The calculation of the resonance frequency and quality factor of closed or aperture coupled cavity resonators with volume and wall losses by an edge finite element method is discussed. An efficient solver is developed to solve the complex nonlinear eigenvalue problem. The effect of the roughness of the walls on the quality factor is taken approximately into account.

An efficient finite-element formulation without spurious modes for anisotropic waveguides

A numerically efficient finite-element formulation is presented for the analysis of lossless, inhomogeneously loaded, anisotropic waveguides of arbitrary shape. The electromagnetic field is described either by the three components of a magnetic vector potential and an electric scalar potential or by the three components of an electric vector potential and a magnetic scalar potential. The uniqueness of the potentials is ensured by the incorporation of the Coulomb gauge and by proper boundary conditions. Owing to the implementation of the solenoidality condition for the vector potential even in the case of zero wavenumber, no spurious modes appear. Variation expressions suited to the finite-element method are formulated in terms of the potentials. Standard finite-element techniques are employed for the numerical solution, leading to a generalized eigenvalue problem with symmetric, sparse matrices. This is solved by means of the bisection method with the sparsity of the matrices fully utilized. Dielectric- and ferrite-loaded waveguides with closed and open boundaries and including both isotropic and anisotropic materials are presented as examples. >

Finite element scheme for 3D cavities without spurious modes

A numerically efficient finite-element formulation is presented for the analysis of inhomogeneously loaded three-dimensional cavities of arbitrary shape. The electromagnetic field is described either by the three components of a magnetic vector potential and by an electric scalar potential, or by the three components of an electric vector potential and by a magnetic scalar potential. The uniqueness of the potentials is ensured by the incorporation of the Coulomb gauge and by proper boundary conditions. Owing to the correct description of the electromagnetic field, no spurious modes appear. The Galerkin equations are formulated for the finite element method leading to a generalized eigenvalue problem with symmetric, sparse matrices. This is solved by means of the bisection method with the sparsity of the matrices fully utilized. Several 3-D cavity problems are solved to illustrate the method. >

Nodal and edge element analysis of inhomogeneously loaded 3D cavities

The behavior of the nodal and edge finite-element approximation is investigated in the case of inhomogeneously loaded 3-D cavity resonators. The discussion is aimed at the origin and the avoidance of the spurious solutions. In the nodal element case, a joint vector and scalar potential description is presented while the field components are used in the case of the edge elements. In both cases, the number of zero eigenvalues can be predicted beforehand; isoparametric brick elements are used for the discretization, leading to a generalized eigenvalue problem with symmetric, sparse matrices. >

Perfectly matched layers in static fields

A perfectly matched layer technique is proposed in the paper for taking unbounded regions in static fields into account by means of the finite element method. The absorbing layer is an anisotropic medium having a nonphysical relative permittivity below one in the direction normal to the boundary with the relative permittivity in the tangential direction equal to the reciprocal of this value. A 2D problem with an analytical solution is solved to demonstrate the technique.

Nodal and edge element analysis of inhomogeneously loaded waveguides

Comparisons of nodal and edge finite element analyses of inhomogeneously loaded waveguides are made. For nodal elements, the vector potential together with a scalar potential description is applied. For edge elements, the field description is used. Both descriptions are free of spurious solutions. The case of the treatment of the propagation constant as an eigenvalue is also discussed. A modified version of the bisection method is used when the matrices of the eigenvalue problem are not positive definite. Examples including waveguides with sharp metal edges are presented to compare the different methods. For problems with sharp edges, the edge element approximation is favorable. >

Parameter estimation for PMLs used with 3D finite element codes

Perfectly matched layers (PMLs) are used to truncate the mesh in a 3D edge finite element code. A new interpretation of the PMLs helps to solve the problem of joining PMLs of different directions and to estimate the proper setting of the PML parameters. The finite element code uses an A,V description to obtain a better convergence rate of the iterative solver. The program is used to model the electromagnetic field of a linear dipole antenna in the vicinity of lossy objects.