ADAPTIVE AND IMPLICIT HAAR-WAVELET-BASED TIME-DOMAIN INTEGRAL EQUATION ANALYSIS OF STRAIGHT THIN WIRE SCATTERER

An adaptive, implicit, multiresolution time-domain algorithm is applied to solve the integro-differential equation arising from the canonical straight thin wire scattering problem. The extended Haar wavelet system is adopted for the expansion of the induced current along the wire responding to a Gaussian input. Results are compared from implicit and explicit marching-on-time algorithms using Harr wavelet basis functions and pulse basis functions. The implicit algorithm using Haar wavelet system provides an adaptive solution to the transient problem, thus potentially reducing the requirement of computer memory and CPU time. Time-domain techniques have proven to be useful in electromagnetic analysis in transient applications such as short pulse antenna radiation, high-resolution radar scattering, electromagnetic compatibility and electromagnetic pulse interference. Time-domain (TD) methods have gained more and more attention in computational electromagnetics, as they have merits such as obtaining broadband response with a single analysis and good performance with time-varying or nonlinear systems. Most importantly, as in our transient applications in pulsed power, only the early time response is of interest, time domain methods can effectively be truncated, allowing us to compute solutions only for as long as necessary. We prefer integral equation methods when analyzing our electrically large structures, although we have also attempted differential equation methods. Integral equation (IE) methods need only discretize over a surface rather than a volume, and the reduced number of unknowns makes the computational cost both in time and in storage more economical. In transient applications, there is still room for improvement. Multiresolution analysis has found wide application recently in both TD and frequency domain (FD) methods. The wavelet-based multiresolution analysis (MRA) has proven to be an effective method in the analysis of electrically large structures. Wavelet based MRA can capture the local as well as the global properties of the solutions efficiently. In the vast literature of MRA applied to electromagnetic scattering problems, various wavelet systems have been utilized (1), (2), (3). The simplest Haar scaling and wavelet functions have finite temporal/spatial support (though sinc-function-type frequency support), and thus, the algorithms adopting Haar subdomain basis functions have wide flexibility when applied to arbitrary domain and are usually easier to implement than other wavelet systems. The Haar wavelet system is also directly analogous to a traditional pulse basis function approach. Most of the MRA research done in time-domain has been applied to differential equations (1), (2), while most of that done with integral equations has been in frequency-domain (3). In our pulsed power project, it is preferred to utilize multiresolution analysis based on time-domain integral equation method. Previously we used Haar wavelet basis functions in the derivation of the marching-on-time (MOT) algorithm and we have presented satisfactory results with explicit multiresolution formulation (4). In this paper, we provide an implicit and fully adaptive multiresolution formulation. After the introduction, we state the problem and the formulation in section 2. Numerical results are presented in section 3, followed by discussions and conclusions in section 4. 2. PROBLEM AND FORMULATION Consider a straight thin wire scatterer exposed to an input TM electric filed. The thin wire is of diameter of 1 centimeter and length of 2 meters. The transient input electric field is a Gaussian pulse