209 research outputs found

    Regularity Theory and Superalgebraic Solvers for Wire Antenna Problems

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    We consider the problem of evaluating the current distribution J(z)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][-1,1]. In particular, for a smooth time-harmonic incident field this theorem implies that J(z)=I(z)/1z2J(z) = I(z)/\sqrt{1-z^2}, where I(z)I(z) is an infinitely differentiable function—the previous state of the art in this regard placed II in the Sobolev space W1,pW^{1,p}, p>1p>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)=F1(z)ln ⁣z+F2(z)G(z) = F_1(z) \ln\! |z| + F_2(z), where F1(z)F_1(z) and F2(z)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 O(Nm)\mathcal{O}(N^{-m}) and O(Mm)\mathcal{O}(M^{-m}) for any positive integer mm, where NN and MM 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 O(M3)\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 NN 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 MM 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

    Efficient high-order evaluation of scattering by periodic surfaces: deep gratings, high frequencies, and glancing incidences

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    We present a superalgebraically convergent integral equation algorithm for evaluation of TE and TM electromagnetic scattering by smooth perfectly conducting periodic surfaces z=f(x). For grating-diffraction problems in the resonance regime (heights and periods up to a few wavelengths) the proposed algorithm produces solutions with full double-precision accuracy in single-processor computing times of the order of a few seconds. The algorithm can also produce, in reasonable computing times, highly accurate solutions for very challenging problems, such as (a) a problem of diffraction by a grating for which the peak-to-trough distance equals 40 times its period that, in turn, equals 20 times the wavelength; and (b) a high-frequency problem with very small incidence, up to 0.01° from glancing. The algorithm is based on the concurrent use of Floquet and Chebyshev expansions together with certain integration weights that are computed accurately by means of an asymptotic expansion as the number of integration points tends to infinity