Extrapolation methods for Sommerfeld integral tails

Abstract

A review is presented of the extrapolation methods for accelerating the convergence of Sommerfeld-type integrals (i.e., semi-infinite range integrals with Bessel function kernels), which arise in problems involving antennas or scatterers embedded in planar multilayered media. Attention is limited to partition-extrapolation procedures in which the Sommerfeld integral is evaluated as a sum of a series of partial integrals over finite subintervals and is accelerated by an extrapolation method applied over the real-axis tail segment (a a a, 1 1 1) of the integration path, where a a a > > > 0 is selected to ensure that the integrand is well behaved. An analytical form of the asymptotic truncation error (or the remainder), which characterizes the convergence properties of the sequence of partial sums and serves as a basis for some of the most efficient extrapolation methods, is derived. Several extrapolation algorithms deemed to be the most suitable for the Sommerfeld integrals are described and their performance is compared. It is demonstrated that the performance of these methods is strongly affected by the horizontal displacement of the source and field points and by the choice of the subinterval break points. Furthermore, it is found that some well-known extrapolation techniques may fail for a number of values of and ways to remedy this are suggested. Finally, the most effective extrapolation methods for accelerating Sommerfeld integral tails are recommended.

Lightning return stroke current radiation in presence of a conducting ground: 1. Theory and numerical evaluation of the electromagnetic fields

“…The presence in the integrand of the highly oscillating Bessel function and of an integrable singularity are the main causes of numerical troubles. Such troubles led some authors, involved in antenna modeling, to address their research to the investigation of some sort of approximate approaches in order to find simplified formulas and methods [Sarkar, 1975;Bannister, 1984;Michalski and Mosig, 1997;Michalski, 1998;Wait, 1999;Baumann and Sampaio, 1999].…”

Efficient Algorithms for Computing Sommerfeld Integral Tails

“…Since , (8) can be written in a following form: (9) Obviously, the optimal solution would come from the annihilation of the remainders of the linearly transformed sequence by imposing an appropriate ratio of the weights (10) In this point the WA method could be considered complete if the remainders were explicitly known, which is, unfortunately, not the case for the sequences of our interest. Instead, we use their asymptotic expansion (11) where are remainder estimates and are the associated coefficients, as given in [13]. Based on the asymptotic behavior of the spectral domain GFs (12) and using equidistant break points separated by asymptotic halfperiods of Bessel function, , as limits of partial integrals , the following analytical expression for the remainder estimates is obtained (13) for , where .…”

“…Exact zero crossings, extremum points and asymptotic half-periods of Bessel functions are common choices of break points [13]. Both integrals, and , are computed by an adaptive quadrature based on the Paterson rule [31].…”

“…The main assets of these sophisticated numerical methods are highly accurate results and error controllable behavior. The 0018-926X/$31.00 © 2012 IEEE most known representative is the integration-then-summation method combined with one of the numerous extrapolation techniques, such as weighted averages (WA) method, Shanks or generalized Levin transformation, among others [12], [13]. Also, several techniques for the direct integration of SI tails exist [14]- [18].…”

“…It was later further extended and improved in a very complete paper by Michalski [13], who found WA to be a sophisticated version of the Euler transformation. Since then it has been widely used in the EM community.…”