Solution of time domain electric field integral equation for arbitrarily shaped dielectric bodies using an unconditionally stable methodology

Abstract

In this work, we present a new and efficient numerical method to obtain an unconditionally stable solution for the time domain electric field integral equation (TD‐EFIE) for arbitrary homogeneous dielectric bodies, derived utilizing the surface equivalence principle. This novel method does not utilize the customary marching‐on in time solution method often used to solve a hyperbolic partial differential equation. Instead we solve the wave equation by expressing the transient behaviors in terms of weighted Laguerre polynomials. By using these orthonormal basis functions for the temporal variation, the time derivatives in the TD‐EFIE formulation can be handled analytically. Since these weighted Laguerre polynomials converge to zero as time progresses, the induced electric and magnetic currents when expanded in a series of weighted Laguerre polynomials also converge to zero. In order to solve the wave equation, we introduce two separate testing procedures, a spatial and temporal testing. By introducing first the temporal testing procedure, the marching‐on in time procedure is replaced by a recursive relation between the different orders of the weighted Laguerre polynomials. The other novelty of this approach is that through the use of the entire domain Laguerre polynomials for the expansion of the temporal variation of the currents, the spatial and the temporal variables can be separated. For convenience, we use the Hertz vector as the unknown variable instead of the equivalent electric current density. However, we use the equivalent magnetic current density as it is. To verify our method, we apply the proposed method to various dielectric scatterers and compare the results of an inverse Fourier transform of a frequency domain EFIE.