Approximate Boundary Conditions in Electromagnetics
Non-metallic materials and composites are now commonplace in modern vehicle construction, and the need to compute scattering and other electromagnetic phenomena in the presence of material structures has led to the development of new simulation techniques. This book describes a variety of methods for the approximate simulation of material surfaces, and provides the first comprehensive treatment of boundary conditions in electromagnetics. The genesis and properties of impedance, resistive sheet, conductive sheet, generalised (or higher order) and absorbing (or non-reflecting) boundary conditions are discussed. Applications to diffraction by numerous canonical geometries and impedance (coated) structures are presented, and accuracy and uniqueness issues are also addressed, high frequency techniques such as the physical and geometrical theories of diffraction are introduced, and more than130 figures illustrate the results, many of which have not appeared previously in the literature. Written by two of the authorities m the field, this graduate-level text should be of interest to all scientists and engineers concerned with the analytical and numerical solution of electromagnetic problems.
Inspec keywords: electromagnetic waves
Other keywords: cylindrical bodies; electromagnetics; second-order conditions; higher-order conditions; planar structures; GIBC conditions; steepest descent method; first-order conditions; generalized Rytov analysis; approximate boundary conditions; absorbing boundary conditions; impedance wedges
Subjects: Electromagnetic waves: theory
The book familiarizes engineers and scientists with approximate boundary conditions and their applications in electromagnetics. Both the development of approximate boundary conditions and their application to electromagnetic problems of practical interest are discussed.
The authors consider only time harmonic electromagnetic fields with a time dependence specified by the factor ejωt which is omitted. In a stationary, linear, isotropic, homogeneous medium which is free of sources, the field is described by Maxwell's equations.
The first order conditions have proved useful for simulating the properties of imperfectly conducting and coated geometries in propagation, scattering and antenna studies. Analytical and numerical solutions have been developed for a variety of configurations, and in the next two chapters we concentrate on the application of the impedance boundary and transition conditions to problems amenable to analytical solution. The coated half-plane and junction, the material half-plane and junction, and the coated wedge are particular examples of these. In this chapter we consider the analysis of half-planes and junctions that are modelled using the equivalent impedance or sheet geometries.
The half-plane which was the subject of the preceding chapter is the special case of a wedge whose interior angle is zero, but for a complex target such as an aircraft, a high frequency simulation requires the knowledge of the diffraction coefficient for a wedge of arbitrary angle. Indeed, the diffraction coefficient for a perfectly conducting wedge is at the heart of such well-known techniques as the uniform geometrical theory of diffraction (UTD), the physical theory of diffraction (PTD), and to extend these to impedance surfaces, it is essential to consider diffraction by the impedance wedge.
For a thin dielectric layer, second order transition conditions were developed by Weinstein (1969) and used (Leppington, 1983) to determine the field diffracted by an abrupt change in layer thickness. Since then there have been numerous applications of second (and higher) order boundary conditions in electromagnetics, but some of the solutions are either incomplete or in error through a failure to address the uniqueness. A second order condition is the simplest one that has this property and, because of the improved accuracy that can be achieved on going to a second order condition, it is convenient to start with this.
By increasing the order of the boundary condition it is possible to improve the accuracy with which the surface properties are simulated, but the penalty is an increase in the complication of an analytical or numerical solution of the problem. This chapter addresses the problem of choosing the appropriate form of these higher order conditions thus simplifying this task.
The GIBC applications discussed so far have been to planar surfaces but, as evident from the general forms presented in the last chapter, the conditions are also applicable to curved surfaces. To illustrate this fact, we now consider their application to the singly curved surface of a cylinder. In the special case of a homogeneous circular cylinder, the exact solution of the scattering problem is known, and this can be used to construct approximate boundary conditions as well as to determine their accuracy. An important practical application is to coated metallic cylinders. This is examined in the second section, where we demonstrate the accuracy that can be achieved with a second order GIBC, and we then develop a high frequency solution.
The GIBCs discussed in the preceding chapters were designed to simulate the surface properties of a scatterer, thereby eliminating the need to consider fields interior to the body. There is, however, another purpose for approximate boundary conditions, and this is to create a boundary which does not perturb a field incident upon it-in effect, to simulate a surface which is actually not there. The resulting conditions can be regarded as GIBCs for non-reflecting surfaces, and are generally referred to as absorbing boundary conditions (ABCs). They are of growing importance in numerical work where they are used to terminate the computational domain in a finite element (Silvester and Ferrari, 1990) or finite difference (Kunz and Luebbers, 1993) solution of the wave equation. In considering their two- and three-dimensional forms, emphasis will be placed on second order ABCs be cause of their extensive use in scattering and radiation problems.
One of the first rigorous derivations of an approximate boundary condition was by Rytov (1940), who developed conditions through the second order applicable at the curved surface of a highly conducting body. The results are important in showing the effect of surface curvature and material variations, and we will now apply the same method to the formulation used by Leontovich (1948). The boundary conditions obtained (Senior, 1990) are more general than those given by either author, and reveal some errors in the expressions quoted by Leontovich.
This appendix discusses numerical Wiener-Hoft factorisation used in approximate boundary conditions in electromagnetics.
The path CSDP defined by (C.2) is the steepest descent path (or SDP) through the point (ξs, ηs) and, because of the topology illustrated, this point is referred to as the saddle point. The fact that the exponential portion of the integrand in (C.4) decays most rapidly on the SDP simplifies the evaluation of I(κ), and is the primary reason for the path deformation from C to CSDP. We now consider the approximate closed form evaluation of the integral and show how the results depend on the algebraic properties of g(a) near to the SDP.
The physical optics (PO) currents are discontinuous at any edge or other line discontinuity in the surface slope, and, since they are non-zero only over the illuminated portion of a body, they may also be discontinuous at a shadow boundary. The discontinuity at an edge yields the PO diffracted field. This is part of the “true” edge-diffracted field.
The constants α1, α2, α3, A0 and A1 in this section are related by (5.213) with α = 3π/4 - θm, π/4 = θm (m = 1,2), and, since there are only four equations, it is clear that one constant is undetermined.
