Efficient Evaluation of Slowly Converging Integrals Arising from MAP Application to a Spectral-Domain Integral Equation

Mario Lucido,Marco Donald Migliore,Alexander I Nosich,Gaetano Panariello,Daniele Pinchera,Fulvio Schettino

doi:10.3390/electronics8121500

Abstract

In this paper, we devised an analytical technique to efficiently evaluate the improper integrals of oscillating and slowly decaying functions arising from the application of the method of analytical preconditioning (MAP) to a spectral-domain integral equation. The reasoning behind the method’s application may consistently remain the same, but such a procedure can significantly differ from problem to problem. An exhaustive and understandable description of such a technique is provided in this paper, where we applied MAP for the first time to analysis of electromagnetic scattering from a zero-thickness perfectly electrically conducting (PEC) disk in a planarly layered medium. Our problem was formulated in the vector Hankel transform domain and discretized via the Galerkin method, with expansion functions reconstructing the physical behavior of the surface current density. This ensured fast convergence in terms of the truncation order, but involved numerical evaluation of slowly converging integrals to fill in the coefficient matrix. To overcome this problem, appropriate contributions were pulled out of the kernels of the integrals, which led to integrands transforming into exponentially decaying functions. Subsequently, integrals of the extracted contributions were expressed as linear combinations of fast-converging integrals via the Cauchy integral theorem. As shown in the numerical results section, the proposed technique drastically outperformed the classical analytical asymptotic-acceleration technique.

Highlights

The classical statement of a general electromagnetic propagation, radiation, and scattering problem requires that fields, being solutions of Maxwell equations, satisfy boundary, edge, and radiation conditions [1]
In a series of papers, the problem of the accurate and efficient numerical evaluation of these kinds of integrals was addressed under a different perspective [28,29,30,31,32,33,34,35,36]: (1) If needed, suitable contributions were pulled out of the kernels, making the integrands exponentially decaying functions; (2) an analytical technique was devised on the basis of the Cauchy integral theorem to express the integrals of extracted contributions as a linear combination of fast-converging integrals, or series
We presented an analytical technique for the efficient evaluation of improper integrals of oscillating and slowly decaying functions arising from method of analytical preconditioning (MAP) application to a spectraldomain integral equation

Summary

Introduction

The classical statement of a general electromagnetic propagation, radiation, and scattering problem requires that fields, being solutions of Maxwell equations, satisfy boundary, edge (i.e., local power boundedness conditions), and radiation conditions [1]. In a series of papers, the problem of the accurate and efficient numerical evaluation of these kinds of integrals was addressed under a different perspective [28,29,30,31,32,33,34,35,36]: (1) If needed, suitable contributions were pulled out of the kernels, making the integrands exponentially decaying functions; (2) an analytical technique was devised on the basis of the Cauchy integral theorem to express the integrals of extracted contributions as a linear combination of fast-converging integrals, or series In this way, this method drastically outperformed the classical analytical asymptotic acceleration technique (CAAAT).

Spectral-Domain Integral Equation

Procedure orthogonal cylindrical harmonics

Integration complex

Results and Discussion

Conclusions