Regularity Theory and Superalgebraic Solvers for Wire Antenna Problems

Abstract

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 thatis 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 O(N −m ) and 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 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.

Method-of-Moments Analysis of Resonant Circular Arrays of Cylindrical Dipoles

“…There are no divergence problems when the improved kernel is used because the real part in (2c) is logarithmically singular at [18], [32], [33]. Therefore, the associated in [12,Eq.…”

A generalized Calderón formula for open-arc diffraction problems: theoretical considerations

“…where C is the Euler constant, and where si(ξ) and ci(ξ) are the sine and cosine integrals respectively, (both of which are bounded functions of ξ as |ξ| tends to infinity), it is easily verified that the second term in (166) behaves asymptotically as ln(ξ) ξ 2 as ξ tends to infinity. Clearly, the first term of (166) decays as O( 1 ξ ), and therefore [34,35,37] and embodied by equations (10), (11) and definition 1: the image of the operator S is not contained in the domain of definition of the operator N; see equations (10) and (11).…”

Second‐kind integral solvers for TE and TM problems of diffraction by open arcs

“…Introducing the changes of variables t = cos θ and t ′ = cos θ ′, and defining n θ = n r (cos θ ) , we obtain the periodic weighted single‐layer and hypersingular operators and Clearly then, the solutions to the periodic equations where ( θ ) = f ( r (cos θ )) and ( θ ) = f ( r (cos θ )), are related to the solutions of and by the relations In view of , the solutions to are smooth and periodic, and it is therefore natural to study the properties of and in the Sobolev spaces H e s (2 π ) of 2 π periodic and even functions [cf. Yan and Sloan , 1988; Bruno and Haslam , 2007].…”

“…In the particular case where the frequency vanishes ( k = 0) and the curve under consideration is the flat strip (Γ = [−1, 1]), the operator reduces to Symm's operator whose well‐known diagonal property in the cosine basis e n ( θ ) = cos nθ , [ Yan and Sloan , 1988; Bruno and Haslam , 2007], namely plays a central role in our analysis. Note that, in particular, establishes the bicontinuity of the operator from H e s (2 π ) into H e s +1 (2 π ).…”