Comparative study of acceleration techniques for integrals and series in electromagnetic problems

Abstract

Most electromagnetic problems can be reduced to either integrating oscillatory integrals or summing up complex series. However, limits of the integrals and the series usually extend to infinity, and, in addition, they may be slowly convergent. Therefore numerically efficient techniques for evaluating the integrals or for calculating the sum of an infinite series have to be used to make the numerical solution feasible and attractive. In the literature there are a wide range of applications of such methods to various EM problems. In this paper our main aim is to critically examine the popular series transformation (acceleration) methods which are used in electromagnetic problems and to introduce a new acceleration technique for integrals involving Bessel functions and sinusoidal functions. IntroductionNumerical techniques used in the solution of electromagnetic problems require, in general, either evaluating oscillatory integrals over infinite domain or calculating the sums of infinite complex series. For example, the method of moments (MoM) in the spectral domain for two-dimensional geometry requires double-infinite integration of complex highly oscillatory functions; the MoM in the spatial domain employs the spatial domain Green's functions, which are defined as the Hankel transform of the spectral domain Green's function; in the analysis of a periodic structure one needs to employ a periodic Green's function which has double infinite summations; or, in the analysis of a microstrip patch antenna via cavity model, the input impedance or field distribution are written in terms of an infinite sum of modes in the cavity.If the summations and integrals given in the examples above are evaluated by "brute force" as they appear in the problems, the corresponding methods could be computationally very inefficient, rendering these problems impractical. To overcome this computational burden, special acceleration techniques, also called transformation techniques, for both integrals and summations have been proposed and successfully employed. Since these techniques have been studied for specific problems and compared to only a few other techniques, the potentials of these Copyright 1995 by the American Geophysical Union. Paper number 95RS02060. 0048-6604/95/95RS-02060508.00 techniques with their advantages and disadvantages have not been examined entirely for electromagnetic problems. Hence the contributions of this paper are in (1) providing the complete set of acceleration techniques used in the electromagnetic problems, (2) comparatively studying the acceleration techniques for integrals and series, and (3) introducing a new acceleration technique for integrals involving Bessel functions and sinusoidal functions. The transformations given and compared in this paper are the Euler transformation [Hildebrand, 1974], Shanks' Transformation [Shanks, 1955; Singh and Singh, 1991a], Wynn's • algorithm, the method of averages [Mosig and Gardiol, 1979], the Chebyshev-Toeplitz algorithm [Wimp, 1974; Singh and Singh, 1992b], the O algo...

A fast MoM code for finite arrays

“…This expression has a poor [O(1/n)] convergence, hence some acceleration techniques must be exploited to ensure a proper convergence [15]- [17].…”

Comparative Analysis of Acceleration Techniques for 2-D and 3-D Green's Functions in Periodic Structures Along One and Two Directions

“…A more sophisticated approach is based on a Kummer's decomposition before applying the Poisson's formula (6) where (7) and in the third member of (6) we applied Poisson's formula to the last series. The function corresponds to a function , where is the same as in (5).…”

“…A similar method illustrated in [5], [34], the so-called algorithm, can be applied to non-oscillating series. Other numerical techniques are reported in [6].…”

“…Numerical acceleration techniques like the Shanks' transformations [4] or the algorithm [5] may also be necessary to further accelerate the result. An overview of these and other numerical acceleration techniques is given in [6]. Improved convergence can be obtained by applying the so-called Ewald's transformation [7].…”

Periodic Green's function for skewed 3‐D lattices using the Ewald transformation

“…Among different techniques used to speed up the computation of Green's functions [3], the Ewald transform [4] has clearly demonstrated its suitability for periodic problems. It has been advantageously used in the efficient evaluation of GFs of infinite periodic phased arrays of line sources (2-D structures with 1-D periodicity) [5,6], 2-D structures with 2-D periodicity [7], in 3-D problems with 2-D orthogonal [8,9] and skewed [10] lattices as well as for rectangular cavities (3-D problems with 3-D orthogonal lattices) [11,12].…”