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# Regularity Theory and Superalgebraic Solvers for Wire Antenna Problems

Bruno, Oscar P. and Haslam, Michael C. (2007) Regularity Theory and Superalgebraic Solvers for Wire Antenna Problems. SIAM Journal on Scientific Computing, 29 (4). pp. 1375-1402. ISSN 1064-8275. https://resolver.caltech.edu/CaltechAUTHORS:BRUsiamjsc07

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## Abstract

We consider the problem of evaluating the current distribution $J(z)$ that is induced on a straight wire antenna by a time-harmonic incident electromagnetic field. The scope of this paper is twofold. One of its main contributions is a regularity proof for a straight wire occupying the interval $[-1,1]$. In particular, for a smooth time-harmonic incident field this theorem implies that $J(z) = I(z)/\sqrt{1-z^2}$, where $I(z)$ is an infinitely differentiable function—the previous state of the art in this regard placed $I$ in the Sobolev space $W^{1,p}$, $p>1$. The second focus of this work is on numerics: we present three superalgebraically convergent algorithms for the solution of wire problems, two based on Hallén's integral equation and one based on the Pocklington integrodifferential equation. Both our proof and our algorithms are based on two main elements: (1) a new decomposition of the kernel of the form $G(z) = F_1(z) \ln\! |z| + F_2(z)$, where $F_1(z)$ and $F_2(z)$ are analytic functions on the real line; and (2) removal of the end-point square root singularities by means of a coordinate transformation. The Hallén- and Pocklington-based algorithms we propose converge superalgebraically: faster than $\mathcal{O}(N^{-m})$ and $\mathcal{O}(M^{-m})$ for any positive integer $m$, where $N$ and $M$ are the numbers of unknowns and the number of integration points required for construction of the discretized operator, respectively. In previous studies, at most the leading-order contribution to the logarithmic singular term was extracted from the kernel and treated analytically, the higher-order singular derivatives were left untreated, and the resulting integration methods for the kernel exhibit $\mathcal{O}(M^{-3})$ convergence at best. A rather comprehensive set of tests we consider shows that, in many cases, to achieve a given accuracy, the numbers $N$ of unknowns required by our codes are up to a factor of five times smaller than those required by the best solvers previously available; the required number $M$ of integration points, in turn, can be several orders of magnitude smaller than those required in previous methods. In particular, four-digit solutions were found in computational times of the order of four seconds and, in most cases, of the order of a fraction of a second on a contemporary personal computer; much higher accuracies result in very small additional computing times.

Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1137/050648262DOIUNSPECIFIED
ORCID:
AuthorORCID
Bruno, Oscar P.0000-0001-8369-3014
Additional Information:©2007 Society for Industrial and Applied Mathematics. Received by the editors December 22, 2005; accepted for publication (in revised form) December 27, 2006; published electronically June 12, 2007. This work was supported in part by the Air Force Office of Scientific Research, the National Science Foundation, and the National Aeronautics and Space Administration. The second author [M.C.H.] was supported by the Natural Sciences and Engineering Research Council of Canada.
Subject Keywords:electromagnetic scattering; wire antenna; Pocklington; Hallén
Issue or Number:4
Record Number:CaltechAUTHORS:BRUsiamjsc07
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:BRUsiamjsc07
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:9149
Collection:CaltechAUTHORS