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

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.

Evaluation of the E‐polarization focusing ability in Thz range for microsize cylindrical parabolic reflector made of thin dielectric layer sandwiched between graphene

“…Here, the SIE kernel is separated to two parts, most singular (usually static) and the remainder. The most singular part is analytically inverted using special techniques like the Riemann-Hilbert problem (RHP) method [8][9][10] or the Abel integral equation method [11,12], or Galerkin MoM with weighted Chebyshev polynomials which are orthogonal eigenfunctions of the static part [13][14][15]. The remainder produces a Fredholm second-kind matrix equation, and in this case, the numerical solution is convergent.…”

Analysis of the Scattering from a Two Stacked Thin Resistive Disks Resonator by Means of the Helmholtz–Galerkin Regularizing Technique

“…This is what happens when: (1) the Galerkin scheme is adopted, and (2) the selected expansion functions are orthonormal eigenfunctions of a suitable operator containing the most singular part of the integral operator at hand. Such an approach, appropriately called method of analytical preconditioning, is very effective, as clearly shown in the literature devoted to the study of the scattering, propagation, and radiation problems [23][24][25][26][27][28][29][30][31][32][33]. Another way to obtain a guaranteed-convergence consists in solving numerically the singular integral equation by means of a Nyström-type discretization scheme taking into account the singularity of the integral equation and the behavior of the unknowns at the edges [34][35][36][37][38].…”

Asymptotic regularisation of the solution to the problem of electromagnetic field scattering from a set of small impedance particles

“…It turns out that the MAR is a promising tool for solving diffraction problems for both the acoustic and EM scattering from the objects of complicated geometry [21][22][23][24][25]. The modification of the MAR was applied efficiently for solving the spectral-domain integral equations related to the problem of EM wave scattering from a thick, perfectly conducting disk [26]. An in-depth review on the MAR applied to the different kinds of diffraction and scattering problems in electromagnetics is found in article [27].…”