Advances in Mathematical Physics

Volume 2019, Article ID 7902836, 9 pages

https://doi.org/10.1155/2019/7902836

## TM Electromagnetic Scattering from PEC Polygonal Cross-Section Cylinders: A New Analytical Approach for the Efficient Evaluation of Improper Integrals Involving Oscillating and Slowly Decaying Functions

D.I.E.I. and ELEDIA Research Center (ELEDIA@UniCAS), University of Cassino and Southern Lazio, 03043, Cassino, Italy

Correspondence should be addressed to Mario Lucido; ti.sacinu@odicul

Received 30 July 2018; Accepted 15 November 2018; Published 10 January 2019

Academic Editor: Ping Li

Copyright © 2019 Mario Lucido et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The analysis of the TM electromagnetic scattering from perfectly electrically conducting polygonal cross-section cylinders is successfully carried out by means of an electric field integral equation formulation in the spectral domain and the method of analytical preconditioning which leads to a matrix equation at which Fredholm’s theory can be applied. Hence, the convergence of the discretization scheme is guaranteed. Unfortunately, the matrix coefficients are improper integrals involving oscillating and, in the worst cases, slowly decaying functions. Moreover, the classical analytical asymptotic acceleration technique leads to faster decaying integrands without overcoming the most important problem of their oscillating nature. Thus, the computation time rapidly increases as higher is the accuracy required for the solution. The aim of this paper is to show a new analytical technique for the efficient evaluation of such kind of integrals even when high accuracy is required for the solution.

#### 1. Introduction

Spectral domain formulations are particularly suitable for the analysis of a wide class of electromagnetic problems ranging from the propagation in planar guides and waveguides or the radiation by planar antennas to the scattering from cylindrical structures or planar surfaces involving homogeneous or stratified media, just to give some examples. In general, the obtained integral equation in the spectral domain does not admit a closed form solution; hence, numerical schemes have to be adopted. The fast convergence of such methods is a key point. When dealing with polygonal cross-section cylindrical structures or canonical shape planar surfaces, just for examples, a well-posed matrix operator equation can be obtained by means of the method of analytical preconditioning [1]. It consists of the discretization of the integral equation by means of Galerkin’s method with a suitable set of expansion functions leading to a matrix equation at which Fredholm’s or Steinberg’s theorems can be applied [2, 3]. In the literature, it has been widely shown that this goal can be fully reached by selecting expansion functions reconstructing the physical behaviour of the fields on the involved objects with a closed-form spectral domain counterpart [4–15]. With such a choice, few expansion functions are needed to achieve highly accurate results and the convolution integrals are reduced to algebraic products. However, the obtained matrix coefficients are improper integrals of oscillating and, in the worst cases, slowly decaying functions to be numerically evaluated. The classical analytical asymptotic acceleration technique (CAAAT), consisting of the extraction from the kernels of such kind of integrals of their asymptotic behaviour while the slowly converging integrals of the extracted parts are expressed in closed form, allows us to obtain faster decaying integrands without overcoming the most important problem of their oscillating nature. Consequently, the convergence of the accelerated integrals becomes slower and slower as higher is the accuracy required for the solution.

In order to overcome this problem, a novel technique has been proposed for the analysis of the propagation in single and multiple coupled microstrip lines in planarly layered media [16], the propagation in multilayered single and coplanar coupled striplines [17, 18], the scattering form a rectangular plate in homogeneous medium or buried in a lossy half-space [19, 20], the complex resonances of a rectangular patch in a multi-layered medium [21], the scattering from a tilted strip buried in a lossy half-space at oblique incidence [22], and recently the scattering from a circular plate in homogeneous medium [23]. By means of algebraic manipulations and a suitable integration procedure in the complex plane, the matrix coefficients, which are single/double integrals involving products of Bessel functions of the first kind, are expressed as linear combinations of proper integrals and/or improper integrals of nonoscillating functions which can be quickly evaluated. It is interesting to observe that, despite the same line of reasoning, the procedures developed in the quoted papers are, in general, different from problem to problem.

In the papers [13, 15], the analysis of the electromagnetic scattering from perfectly electrically conducting (PEC) polygonal cross-section cylinders when a TM polarized plane wave orthogonally impinges onto the scatterer surface has been successfully approached. The problem has been formulated in terms of electric field integral equation (EFIE) in the spectral domain and discretized by means of Galerkin’s method. Jacobi polynomials multiplied by their orthogonality weight have been used as expansion functions. By means of a suitable choice of the polynomials’ parameters, the physical behaviour of the surface current density on each side of the polygonal cross-section and even on the adjacent wedges has been reconstructed. Moreover, it has been shown that the spectral domain counterpart of the selected expansion functions can be expressed in closed form in terms of confluent hypergeometric functions of the first kind. Due to the reciprocity theorem, it has been demonstrated that it is always possible to reduce the convolution integrals to algebraic products; i.e., the matrix coefficients are single improper integrals involving products of confluent hypergeometric functions of the first kind and complicated arguments numerically evaluated by means of CAAAT.

The aim of this paper is the introduction of a new analytical technique for the efficient evaluation of such kind of integrals. Algebraic manipulations and a suitable integration procedure in the complex plane allow us to reduce each integral to a linear combination of proper integrals and improper integrals of nonoscillating functions involving confluent hypergeometric functions of the first kind and the second kind. As will be shown in the following, the proposed technique outperforms the CAAAT especially when a high accuracy is required for the solution.

#### 2. Formulation and Solution of the Problem

##### 2.1. Background

In Figure 1, a polygonal cross-section PEC cylinder is sketched. A coordinate system is introduced such that the axis coincides with the cylinder axis. The sides of the polygonal cross-section are numbered clockwise and a local coordinate system is introduced on the i- side with the origin at the centre of the side itself in the position and the axis oriented in the outward direction. The angle ]−*π*, *π*] denotes the orientation of the axis with respect to the axis and denotes the length of the -th side. A TM polarized plane wave impinges onto the cylinder surface with an angle *ϕ* with respect to the axis and orthogonally with respect to the cylinder axis, namely,where and , is the wavenumber, is the wavelength, is the dielectric permittivity and the magnetic permeability of the external medium, is the angular frequency. With such a choice, only TM solutions can be obtained; i.e., the induced current is longitudinal and the electromagnetic field is invariant along the axis.