International Journal of Antennas and Propagation

Volume 2018, Article ID 1612498, 6 pages

https://doi.org/10.1155/2018/1612498

## Efficient and Memory Saving Method Based on Pseudoskeleton Approximation for Analysis of Finite Periodic Structures

State Key Laboratory of CAD&CG, Zhejiang University, Hangzhou 310058, China

Correspondence should be addressed to Hai Lin; nc.ude.ujz.dac@nil

Received 5 April 2018; Revised 11 June 2018; Accepted 24 June 2018; Published 22 July 2018

Academic Editor: Paolo Baccarelli

Copyright © 2018 Chunbei Luo 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

An efficient and memory saving method based on pseudoskeleton approximation (PSA) is presented for the effective and accurate analysis of finite periodic structures. Different from the macro basis function analysis model, our proposed method uses the formulations derived by the local Rao-Wilton-Glisson basis functions. PSA is not only used to accelerate the matrix-vector product (MVP) inside the single unit but also adopted to decrease the calculation burden of the coupling between the different cells. Moreover, the number of decomposed coupling matrices is minimized due to the displacement invariance of the periodic property. Consequently, even compared with the multilevel fast multipole algorithm (MLFMA), the new method saves much more memory resources and computation time, which is also demonstrated by the numerical examples.

#### 1. Introduction

Periodic structures have recently found wide applications in the electromagnetic engineering such as antenna arrays and metamaterials with negative permittivity and negative permeability. Hence, the accurate and efficient analysis of periodic structures becomes quite essential. If the periodic structure is an infinite array, simple methods can be applied based on Floquet’s theorem [1] or periodic Green’s function [2], where only a single cell of the periodic structure is the domain of interest.

However, all periodic structures have finite size in real-life problems, although the size may be very large. Therefore, the numerical algorithms which accurately consider the mutual coupling between all cells should be used if the accurate results are required. The method of moments (MoM) [3] and its fast algorithms such as fast multipole method (FMM) [4], adaptive cross approximation (ACA) [5], and FFT-based methods [6] are flexible approaches to study the surface problems. However, the efficiency of numerical methods is still limited since the periodic property is not used in the algorithm framework. Recently, a hybrid method combining the accurate MoM and periodic method of moment (PMM) [7] has been proposed which can gain the balance between the two methods. Moreover, some physically based entire-domain basis functions [8] have been developed to reduce the number of unknowns. Further, the FMM and FFT techniques are integrated to accelerate the calculation [9].

Compared to the ACA method, pseudoskeleton approximation (PSA) [10] is also an efficient low-rank-based algebraic fast algorithm which makes it a really competitive alternative. In this paper, we propose an efficient method with low-memory requirement based on PSA to perform the analysis for finite periodic structures effectively and accurately. In consideration of the accuracy of the mutual interactions [8] and the simplicity of implementation, our proposed method uses the formulations derived from the local basis functions instead of macro basis function (MBF) [11, 12]. In this paper, PSA is not only used to accelerate the matrix-vector product (MVP) inside the single unit but also adopted to decrease the calculation burden of the coupling between the different cells. Moreover, the number of decomposed coupling matrices is minimized due to the displacement invariance of the periodic property. With these improvements, an efficient method with low-memory usage of finite periodic objects can be achieved. Several numerical examples are given to show the priority of the proposed method compared to the conventional multilevel fast multipole algorithm (MLFMA) [13] for periodic structures.

#### 2. MoM and PSA Formulation

In this section, the basis principle of MoM and PSA is briefly introduced at first. Then, the choice of arguments in PSA is discussed.

##### 2.1. MoM Equations and Its Fast Algorithms

Consider a time-harmonic electromagnetic wave scattering or radiation problem of an arbitrary perfect electrically conducting (PEC) object. The object is excited by an incident electric field , then the electric field integral equation (EFIE) associated with the surface equivalent currents can be expressed by where is the tangential unit vector on the surface of the object and and stand for the angular frequency and permeability, respectively. is 3D scalar Green’s function. The linear system of MoM is obtained by discretizing the unknown vector with Rao-Wilton-Glisson (RWG) [14] basis functions and applying Galerkin’s testing method. Let represent the above EFIE matrix system. The MVP process can be accelerated by fast algorithms which can be written as follows: where is the matrix of near field interactions which are directly computed and stored and stands for couplings between the far-field interactions which will be accelerated together with in the iterative solving process.

##### 2.2. Basic PSA Frameworks

According to the low-rank decomposition, the far-field interaction matrix with rank-deficient property can be approximated by a product of two much smaller submatrices and : where is the effective rank and satisfies . While in the skeleton approximation (SA) theory [10], there is a nonsingular submatrix in . Denote the rectangular matrices as and which contain the columns and rows of , respectively, then is expressed as where and have dimensions with and , respectively. In the PSA method, is reevaluated as where is the pseudoinverse of with dimensions of , then columns and rows are chosen from to get the and . Three aspects need to be noted here: (i) is the inverse of and the computation of inverse is very expensive; (ii) the determination of the value of is a balance between accuracy and efficiency, where is a number large enough so that most important bases will be embedded; (iii) how to choose those working columns and rows of and .

For the (ii) and (iii) problems, we will discuss them in the next subsection. For the first problem, assuming that can be decomposed via singular value decomposition (SVD) as where and are unitary matrices, is a diagonal matrix with nonnegative real numbers, and is the complex conjugate transpose of . In the actual implementation, the dimension of , , and can be further decreased by a preset threshold [15]. Let , , and represent the reduced submatrix of , , and , respectively. Then, calculation of the pseudoinverse of (i.e., ) is straightforward:

By combining (5) and (7), the original far-field interaction matrix can be decomposed as

##### 2.3. Choice of Arguments in PSA

As mentioned in the previous subsection, the choice of the value of and the specific sampling rows and columns determine the performance of PSA. In the randomized PSA (RPSA) [15], is equal to . Then, the problem of (ii) transforms into how to estimate the rank of the original matrix. In [16], Chai and Jiao give the approximation of rank of the 3D EM problem by , where is the studying wave number. In this paper, the rank is approximated by where is the diameter of the bounding box corresponding to the octree structure and is a preset positive parameter. The larger the is, the more accurate the matrix decomposition is. In this paper, when is set as 3, the satisfactory accuracy can be guaranteed.

For the problem of (iii), instead of using random numbers in RPSA, we use a strategy analogous to ACA in this paper. Firstly, initialize from the 0th row as the first row index. Then, find the largest entry in this row, and the corresponding column value in which this entry is located is chosen as the next column index. Similarly, find the largest entry in the current column and get the next row index which should be different from all previous row indexes. This process is carried out iteratively until rows and columns are found and stored. Please refer to [17] for more information.

#### 3. Proposed Method for Periodic Structures

We consider a case of arbitrarily shaped PEC patch, for example, refer in Figure 1. The total impedance matrix contains two parts: self-coupling blocks and mutual coupling blocks. Therefore, both the blocks are analyzed and decomposed by PSA to gain better efficiency and low-memory usage.