Accurate and Efficient Algorithms for Frequency Domain Scattering from a Thin Wire

Davies, Penny J.; Duncan, Dugald B.; Funken, Stefan A.

doi:10.1006/jcph.2000.6688

Cited by 21 publications

(32 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(a) The normalization of the kernel (1.4) coincides with that of Jones [7] and Rynne [13]; the normalization employed by Davies, Duncan, and Funken [4], on the other hand, includes an additional multiple of 1/(4π), which, here, is incorporated on the right-hand side of (1.1) instead. (b) Another kernel for the equations above, which results from use of approximations additional to those implicit in (1.1), (1.3), and (1.4), is the so-called reduced kernel…”

Section: E(t) Sin[k(z − T)]dtmentioning

confidence: 99%

“…We will not utilize this kernel: as pointed out in [4] and [5], it is not possible to obtain reliable results using the reduced kernel G red (z) in place of G(z) in (1.1) or (1.3). (c) In a concise paper [16], Wang derives the expansion…”

Section: E(t) Sin[k(z − T)]dtmentioning

confidence: 99%

“…Equations (1.8) and (1.9) have been used as the basis for many numerical solvers for Pocklington and Hallén problems [4,9]. Numerical integration schemes for integrands containing such singularities which, based on polynomial interpolation, do not explicitly account for the singularity of the second derivative of the integrand typically exhibit loworder convergence.…”

Section: E(t) Sin[k(z − T)]dtmentioning

confidence: 99%

“…It is well known that the solution J of (1.1) and (1.3) with the end-point conditions (1.2) tends to zero like √ 1 − z 2 as z → ±1 [4,7,13]. Jones [7] established that if e(z) has a bounded derivative, then, under the conditions (1.2), equations (1.1) and (1.3) admit unique integrable solutions J(z).…”

Section: Regularity Of Solutionsmentioning

confidence: 99%

“…In addition to several efficient novel algorithms for wire problems, the contribution [4] presents an excellent survey of modern methods in the area; we therefore use the results in that reference to demonstrate the improvements that can be offered by our algorithms. We note that only the Pocklington equation is treated in [4], and, further, that our methods differ substantially from those presented in that contribution. In fact, several methods are discussed in [4], some which exhibit O(N −1 ) or O(N −2 ) convergence in the solution, and some methods which exhibit much faster convergence.…”

mentioning

confidence: 99%

See 4 more Smart Citations

Regularity Theory and Superalgebraic Solvers for Wire Antenna Problems

Bruno¹,

Haslam²

2007

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

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.

show abstract

Section: E(t) Sin[k(z − T)]dtmentioning

confidence: 99%

Section: E(t) Sin[k(z − T)]dtmentioning

confidence: 99%

Section: E(t) Sin[k(z − T)]dtmentioning

confidence: 99%

Section: Regularity Of Solutionsmentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

Regularity Theory and Superalgebraic Solvers for Wire Antenna Problems

Bruno¹,

Haslam²

2007

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

show abstract

A Posteriori Error Control for Finite Element Approximations of the Integral Equation for Thin Wire Antennas

Carstensen

Rynne²

2002

Zeitschrift für Angewandte Mathematik und Mechanik

View full text Add to dashboard Cite

In this paper we discuss a finite element approximation method for solving the Pocklington integro‐differential equation for the current induced on a straight, thin wire by an incident harmonic electromagnetic field. We obtain an a posteriori error estimate for finite element approximations of the equation, and we prove the reliability of this estimate. The theoretical results are then used to motivate an adaptive mesh‐refining algorithm which generates very efficient meshes and yields optimal convergence rates in numerical experiments.

show abstract

Some new problems in numerical integration

Monegato

2007

Journal of Computational and Applied Mathematics

View full text Add to dashboard Cite

show abstract

Accurate and Efficient Algorithms for Frequency Domain Scattering from a Thin Wire

Cited by 21 publications

References 17 publications

Regularity Theory and Superalgebraic Solvers for Wire Antenna Problems

Regularity Theory and Superalgebraic Solvers for Wire Antenna Problems

A Posteriori Error Control for Finite Element Approximations of the Integral Equation for Thin Wire Antennas

Some new problems in numerical integration

Contact Info

Product

Resources

About