International Journal of Antennas and Propagation

Volume 2018, Article ID 9562780, 6 pages

https://doi.org/10.1155/2018/9562780

## ADI-MRTD Algorithm for Periodic Structures Analysis

National Key Laboratory on Electromagnetic Environmental Effects and Electro-optical Engineering, Army Engineering University of PLA, Nanjing 210007, China

Correspondence should be addressed to Pin Zhang; moc.621@ruofgnahznip

Received 25 January 2018; Revised 10 April 2018; Accepted 12 June 2018; Published 10 October 2018

Academic Editor: Rodolfo Araneo

Copyright © 2018 Xiaohua Dong 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

In this paper, a novel algorithm based on the alternating direction implicit (ADI) multiresolution time-domain (MRTD) method for periodic structure simulation is proposed. By applying the multiresolution analysis in accordance with wavelet theory, the spatial sampling rate of the conventional finite-difference time-domain (FDTD) is significantly reduced by the MRTD method. The ADI method is then used to remove the Courant-Friedrich-Levy (CFL) limit that the MRTD method experiences. The periodic boundary condition (PBC) is directly implemented in the time domain using a constant transverse wave-number (CTW) wave. Numerical results are presented to confirm the efficiency and accuracy of the proposed method.

#### 1. Introduction

Over the past decades, notable progress on the engineering applications of periodic structures has been achieved, especially on frequency selective surfaces (FSS) [1] and metamaterials [2, 3]. For an accurate analysis of electromagnetic scatter characteristics, many numerical methods have been developed. The finite-difference time-domain (FDTD) method is a typical algorithm that plays an important role in dealing with electromagnetic problems.

Normally, in the analysis of periodic structures using the FDTD method, only a unit cell is simulated and computed (instead of the whole structure) by incorporating the appropriate periodic boundary condition (PBC). However, unlike the situation of a normal incident wave, the implementation of the PBC for an oblique incident is complicated due to the time delay in the transverse plane. The split-field technique [4] is proposed for periodic structures analysis under oblique incidence circumstance. However, one has to adopt a multipart algorithm to solve the associated extra terms due to the transformed field. The spectral FDTD (SFDTD) method is a novel technique in which the constant transverse wave-number (CTW) wave is applied [5]. It is more efficient than the split-field technique due to the angle-independent stability criterion.

However, the time step is constrained by the Courant-Friedrich-Levy condition due to the explicit time-marching technique. To overcome this problem, the weakly conditionally stable finite-difference time-domain (WCS-FDTD) [6] and the locally one-dimensional finite-difference time-domain (LOD-FDTD) [7] have been widely applied to the study of periodic structures. Furthermore, the alternating direction implicit finite-difference time-domain (ADI-FDTD) method incorporates the advantages of both the explicit and implicit formats, namely, relatively simple calculation and unconditional stability [8]. Although the weighted Laguerre polynomials based spectral finite-difference time-domain (WLP-SFDTD) scheme for periodic structure analysis is efficient, solving the correspondingly huge banded system matrix is time-consuming [9]. One has to use the sparse lower-upper (LU) factorization packages or add a perturbation term.

In recent years, the multiresolution time-domain (MRTD) technique has been successfully developed due to its highly linear dispersion performance and ability to simulate complex electromagnetic structure [10–12]. Moreover, with highly linear dispersion performance, the MRTD scheme implies that a low sampling rate in space can still provide a relatively small phase error in the numerical simulation of a wave propagation problem. Therefore, it becomes possible that larger targets can be simulated without sacrificing accuracy. However, the MRTD scheme has a major drawback that the time stability condition is more rigorous than that of the FDTD scheme, which limits the computational efficiency of the MRTD algorithm.

In this paper, the spectral-FDTD (SFDTD) technique is applied to the conventional MRTD and ADI-MRTD methods, resulting in the PS-ADI-MRTD algorithms. The MRTD method and the ADI method are, respectively, used to reduce the spatial sampling rate and remove the KCL limit. The application, the SFDTD, is mainly using a constant transverse wave-number (CTW) wave to directly implement the PBC in the time domain. To verify the efficiency and accuracy of the PS-ADI-MRTD method, the numerical example is presented later.

#### 2. Mathematical Formulation

##### 2.1. Formulation of the Incident Wave

Considering that the CTW travels in the plane, when the incident wave is oblique, the expression of the PBC can be shown in the frequency domain as follows:

As shown in equation (1), if and are constant numbers, then will be a constant complex number. By transforming equation (1) into the time domain, we obtain the following:

Therefore, if and are constant numbers, the PBC will have no time delay. Then, the implementation of the PBC is similar to that of the normal incidence case.

As shown in Figure 1, the constant transverse wave-number (CTW) wave is employed as the incident wave in this discussion [5], and a TE wave plane is used here. The polarization direction of the TE wave is along is the incident wave vector, and is the incident angle. From Figure 1, we obtain the following: