Solution and Stability Analysis of a New Integral Equation for the Transient Scattering by a Flat, Rectangular Conducting Plate

In this contribution, we present a new Hallén-type formulation for scattering by a two-dimensional, perfectly conducting rectangular plate. We derive two coupled integral equations for the components of the induced surface current, which are solved by marching on in time. The numerical results may exhibit instabilities, but the solution can be stabilized in various ways. To arrive at a more detailed understanding of the stability problem, we reformulate the computation in a combination of a finite-difference time-domain computation and the solution of two scalar integral relations which no longer contain space and time differentiations. Both procedures are coupled via the boundary conditions at the edges of the plate. As a reference result, we consider the transient excitation of an infinite flat plate by a pulsed, vertical electric dipole, for which the unknown vector potential and the induced surface currents are available in closed form. INTRODUCTION The electric-field integral equation (EFIE) is a common tool for determining scattered or radiating fields emanating from a perfectly conducting, open surface. In this type of approach the induced surface current and/or charge density are solved from the integral equations, and, subsequently, the scattered electromagnetic field is determined at any given point by evaluating the integrals in a conventional integral representation. Advantages of this approach are that the radiation condition is accounted for inherently, and that only two-dimensional surface currents and/or charges need to be computed over the area of the scattering surface, instead of threedimensional electromagnetic fields over a volume surrounding that surface. When the EFIE is solved directly in the time domain, instabilities are often observed in the computed results. Numerical experiments for a number of two-dimensional transient-scattering problems [1] as well as for perfectly conducting surfaces indicate that the occurrence of such unstable solutions is enhanced by the presence of space derivatives in these equations. In particular, the manner in which these derivatives are approximated in the space discretization has a considerable influence on the stability of the computational solutions. To investigate this observation in more detail, we consider the “canonical” problem of a rectangular, flat plate excited by a pulsed incident field. The role of the space derivatives is isolated from the complete computation by reformulating the procedure in three steps. First, we reduce the original EFIE to a Hallén-type integral equation, which consists of two coupled, scalar integral equations for the transverse currents Jx(x, y, t) and Jy(x, y, t). This step was inspired by the success of using Hallén’s integral equation for the total current I(z, t) along a straight conducting thin wire in time-domain computations [2]. Marching-on-in-time schemes based on our new integral equation are still unstable, but can be stabilized with the aid of time averaging. Therefore, the second modification in our approach is to reformulate the integral equations in terms of two coupled differential equations that can be solved by a finite-difference time-domain scheme, and two uncoupled scalar integral equations, where the current densities Jx(x, y, t) and Jy(x, y, t) are resolved from the corresponding components of a vector potential A(x, y, t). Here, we found our inspiration in a similar treatment of the EFIE in [3]. A straightforward discretization of the differential equations results in a scheme that satisfies the well-known Courant stability criterion for two-dimensional wave propagation. The scalar integral equation is then solved by marching on in time, as described in [3]. For the case where the flat plate is infinite, both parts of the computation can be validated independently, by comparing numerical results with closed-form expressions for the field excited by a horizontally or vertically polarized, pulsed electric point dipole located above the plate. When the plate has finite dimensions, coupling occurs via the boundary conditions at its edges. This means that the two parts of the computation must be combined, and that the stability of the entire scheme depends on the implementation of these boundary conditions. FORMULATION OF THE PROBLEM A pulsed plane wave is incident on a two-dimensional, perfectly conducting rectangular plate as shown in Fig. 1. The Cartesian coordinate system is chosen such that the plate is located at 0 < x < a, 0 < y < b, and z = 0. The surrounding homogeneous, linearly and instantaneously reacting, isotropic medium has permittivity ε and permeability μ. We start from the standard form of the electric-field integral equation ( ∇T∇T · − 1 c2 ∂ t ) A(rT , t) = −ε∂tET (rT , t), (1) which holds for rT on the plate. In (1), the subscript T stands for transverse, and the vector potential A(rT , t) is related to the unknown surface current density JS(rT , t) in the plane z = 0 according to