The Plane-Wave Time-Domain Algorithm for the Fast Analysis of Transient Wave Phenomena zyx A. Arif Ergin, Balasubramaniam Shanker, and Eric Michielssen zyxwvutsrq Center for Computational Electromagnetics Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign 1406 W. Green Street, Urbana, IL 61801 E-mail: aergin@decwa.ece.uiuc.edu zyxwvutsr shanker@decwa.ece.uiuc.edu michiels@decwa.ece.uiuc.edu Keywords: numerical analysis; computational electromagnetics; time-domain analysis; electromagnetic transient scattering; marching-on-in-time method; plane-wave time-domain method; fast methods 1. Abstract This article describes a plane-wave time-domain (PWTD) algorithm that facilitates the fast evaluation of transient wave fields produced by surface scattering. The algorithm presented relies on a Whittaker-type expansion of transient fields in terms of propagat- ing plane waves. The incorporation of the PWTD scheme into existing marching-on-in-time- (MOT-) based integral-equation solvers is elucidated. It is shown that the computational cost of per- forming a surface-scattering analysis, using two-level and multi- level PWTD-enhanced MOT schemes, scales as zyxwvutsrq O( NtNi.5 log zyxwvutsr N,) and O(N,N, log2 N,), respectively, when the surface source den- sity is represented by N, spatial and Nf temporal samples. Hence, the computational cost of the proposed algorithms scales much more favorably than that of classical MOT schemes, which scale as 0 (N,N;). Therefore, PWTD-enhanced MOT schemes make pos- sible the analysis of broadband scattering from structures of unprecedented dimensions. 2. Introduction zyxwvutsrqpo s the need to model large-scale broadband and nonlinear A systems expands, efficient techniques for analyzing transient wave phenomena are becoming indispensable. The modem-day characterization of broadband antennas, the analysis of nonlinear circuits, and the design of well-logging tools all hinge upon the availability of transient-wave analysis schemes. Therefore, the development of fast and accurate time-domain wave simulators is of paramount importance to fields as diverse as electromagnetics, geophysics, and acoustics. Numerically rigorous techniques for analyzing transient- wave phenomena are based on either differential or integral equa- tions. While differential-equation techniques-e.g., the Finite- Difference and Finite-Element Time-Domain algorithms-are pre- ferred for studying scattering from inhomogeneous structures, integral-equation-based techniques offer two notable advantages when analyzing surface-scattering phenomena: /€€€Antennas and Propagation Magazine, Vol. 41, No. 4, August 1999 i) ii) Integral-equation techniques only require a discretiza- tion of the scatterer’s surface. In contrast, differential- equation formalisms are based on a volumetric discreti- zation of the scatterer and/or its sumounding, which results in a sharp increase in the number of unknowns when compared to integral-equation methods. Integral-equation techniques implicitly impose the radiation condition, which obfuscates the need for local absorbing boundary conditions. In spite of these advantages, integral-equation methods-often referred to as MOT schemes-have historically received only scant attention, as they have often been found unstable and computation- ally expensive when compared to their differential-equation coun- terparts. Recently, a variety of MOT implementations have been suggested [l-31 that are “for all practical purposes stable” [4]. As zy a result, the computational complexity of MOT schemes appears to be the stumbling block preventing their widespread use. Recently, we introduced a PWTD algorithm [5-81 that considerably reduces the cost of analyzing surface-scattering phenomena when used in conjunction with classical MOT schemes. The computational costs of the PWTD-enhanced MOT algorithms presented in [5-81 scale as 0 ( NtNs4‘3 log N,) and 0 (NI N, log N,) , where NI and N, denote the number of temporal and spatial degrees of freedom of the discretized surface quantity, as opposed to 0 zy (NfN:) for clas- sical MOT schemes. These algorithms can be considered the time- domain analogues of the windowed Fast Multipole methods [9-111 and Steepest-Descent-Path algorithms [ 121 that have been so remarkably successful in accelerating the solution of frequency- domain scattering problems. The purpose of this article is two-fold: i) To introduce a PWTD algorithm that is based on a Whittaker-typefield expansion [ 13-15]. The algorithm presented here relies on a field representation via a complete spherical expansion of the far field, instead of the finite-cone representations used in [5]. This renders the description of the scheme more transparent and its implementation easier. To some extent, the current scheme can be considered the direct time-domain counterpart of the frequency-domain Fast Multipole Method [16]. 1045-9243/99/$10.0001999 IEEE 39