International Journal of Antennas and Propagation

Volume 2018, Article ID 3453495, 7 pages

https://doi.org/10.1155/2018/3453495

## Matrix Generation by First-Order Taylor Expansion in a Localized Manner

Ministerial Key Laboratory of JGMT, Nanjing University of Science and Technology, Nanjing, China

Correspondence should be addressed to Jun Hu; nc.ude.tsujn@uhnuj

Received 6 July 2018; Revised 8 October 2018; Accepted 31 October 2018; Published 27 December 2018

Academic Editor: Renato Cicchetti

Copyright © 2018 Jun Hu 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 method of moments is widely used, but its matrix generation is time-consuming. In the present paper, a localized multifrequency matrix-filling method is proposed. The method is based on the retarded first-order Taylor expansion of Green’s functions on each field point, which can reduce the number of callback Green’s functions and hence can solve double-surface integrals quickly. It is also based on the extraction of the common factors of different frequencies, and hence can sweep the frequency points quickly. Numerical examples are provided to validate the efficiencies of the proposed method.

#### 1. Introduction

The method of moments (MoM) has been widely used, and its solutions for most problems are still time-consuming. The matrix of MoM may come from electric-field integral equation (EFIE), magnetic-field integral equation, combined field integral equation [1–3], or current and charge integral equation [4] discretized by pulse basis functions, rooftop basis functions, RWG basis functions [5], pyramid-shaped functions [6], or other higher-order basis functions [7]. Among the above, solving the discretized EFIE with RWG functions is preferred. In most cases, the matrix-filling (both for single-frequency and wide-band problems) in the EFIE with RWG functions is time-consuming [8].

Many research activities have been finalized to accelerate the filling of MoM matrices. In particular, to save time in single-frequency problems, some researchers have provided exact closed-form expressions useful to analyze thin cylindrical structures [9], while others have accelerated the RWG-RWG cycle in primary MoM code using parallel technologies [7, 10, 11]. Some replaced the RWG-RWG loop by the optimized triangle-triangle loop [7, 12]. In order to save matrix-filling time for a wideband sweeping, some researchers developed matrix interpolation methods [8, 13] including Lagrange’s interpolation method, Chebyshev interpolation method, rational polynomial approximation, and Hermite interpolation method. Some developed result interpolation/extrapolation methods, like the MBPE [14], the AWE [15], and the ANN [16] methods.

Although these methods are very efficient, the matrix-filling problems are still serious. In both matrix interpolation and result interpolation/extrapolation methods, matrices of sampling frequencies should be filled independently. Meanwhile, in single-frequency problems, double-surface integrals are usually solved by many callback Green’s functions. It would be amazing if a method can alleviate double-surface integrals and also generate multimatrices of different frequencies quickly. Recently, several amazing methods based on the higher-order retarded Taylor expansion are developed, where one real matrix depending on geometry parameters is introduced [17–19]. They are applied into partial equivalent element method as in [20–24]. In these works, the Taylor expansions are usually global and of higher order [17].

In this paper, we will propose an amazing method where the 1-ordered Taylor expansion approximates Green’s functions locally. Like methods already published, it reduces the number of callback Green’s functions because of the ultra-wideband approximation. It also extracts the common factors in double-surface integrals of different frequencies and improves the matrix generation of multifrequency.

#### 2. Principle of the Proposed Method

The proposed method can be integrated into numerical solvers involving surface integral equations. In order to explain the principle clearly, the following description refers to the EFIE-based Galerkin’s method.

The discretized EFIE is usually written as where denotes the support of surface current, is the distance between source point and field point , is the th basis function, is the number of basis functions, is the coefficient, is the incident electrical field, is the wave number, is the angular frequency, and is the magnetic permeability. After being tested and transformed, the discretized EFIE is rewritten as follows: where denotes the vector consisting of unknown , is the vector consisting of tested incident electrical field , and is the impedance matrix.

In order to avoid singularities and in Eq. (1), the above matrix is filled by the following transformed double-surface integral: element by element, where denotes the th testing function. In Eq. (3), testing functions and basis functions are in the same completed function system. The preferred system of functions employed to describe the unknowns on arbitrary surfaces is that of based on linear functions, including the RWG and the rooftop basis functions. The traditional matrix filling and its repeated operations are very time-consuming. An improvement based on retarded Taylor expansion is as follows: where denotes the distance between the testing function and the basis function [17] and is the maximum order of expansion terms. In the above, the retarded Taylor expansion method is globally used for every element, and we call it as the global method. In the present paper, a different method based on first-order Taylor expansion is proposed, which is called as the localized method.

The localized method focuses on the following two important primary integrals, and , on the source regions for every field

Although the above two integrals can be solved by exact closed-form expression [9] for thin cylindrical structures, fast methods with primary functions for surface structures are necessary. After the first-order Taylor expansion, Eqs. (5) and (6) are approximated by concise expressions. where denotes the distance between the field point and the center of source region. Their relative errors are the following: respectively, where is the diameter of the source region as shown in Figure 1. In Figure 1, points and denote the source points in the source region and points , , and denote the field points in the field region, respectively. The accuracy () of the above approximation can be assured if the condition holds. This condition is usually compatible with the meshing requirement for linear basis functions. Compared with the global method in [17], the approximation based on Taylor expansion is localized for each field point in test function. The localized operation makes it easier to achieve the required accuracy. In the global method, the accuracy is assured by high order or by both and ; in the localized method, the accuracy is assured by .