International Journal of Antennas and Propagation

Volume 2019, Article ID 9542976, 14 pages

https://doi.org/10.1155/2019/9542976

## An Efficient Laguerre-Based FDTD Iterative Algorithm in 3D Cylindrical Coordinate System

Correspondence should be addressed to Hai-Lin Chen; moc.621@nehcnilyh

Received 12 April 2019; Accepted 17 June 2019; Published 25 September 2019

Academic Editor: Shiwen Yang

Copyright © 2019 Da-Wei Zhu 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

Here an efficient Laguerre-based finite-difference time-domain iterative algorithm is proposed. Different from the previously developed iterative procedure used in the efficient FDTD algorithm, a new perturbation term combined with the Gauss–Seidel iterative procedure is introduced to form the new Laguerre-based FDTD algorithm in the 3D cylindrical coordinate system. Such a treatment scheme can reduce the splitting error to a low level and obtain a good convergence; in other words, it can improve the efficiency and accuracy than other algorithms. To verify the performance of the proposed algorithm, two scattering numerical examples are given. The computation results show that the proposed algorithm can be better than the ADI-FDTD algorithm in terms of efficiency and accuracy. Meanwhile, the proposed algorithm is extremely useful for the problems with fine structures in the 3D cylindrical coordinate system.

#### 1. Introduction

In recent years, the unconditionally stable Laguerre-based finite-difference time-domain (FDTD) algorithm has been applied to simulate transient electromagnetic problems in the Cartesian coordinate. By transforming the time-domain problem to the Laguerre domain using the temporal Galerkin’s testing procedure, the transient solution is independent of time discretization. Thus, Laguerre-based FDTD formulation has the advantage of less numerical error when a larger time step is used.

The main drawback of the conventional Laguerre-based FDTD algorithm is that it requires solving a very large sparse matrix. To overcome this problem, a factorization-splitting scheme [1] was used to resolve the huge sparse matrix into six tridiagonal matrixes, and then the chasing method was used to solve them [2]. This provided excellent computational accuracy and can be efficiently parallel-processed on a computing cluster. To further reduce the splitting error, the modified perturbation term was introduced in [3, 4]; meanwhile, the Gauss–Seidel iterative procedure was applied, which made the new Laguerre-based FDTD scheme more efficient and accurate. Obviously, the good feasibility of the efficient Laguerre scheme had been conformed in the rectangular coordinate system.

However, in many applications, we have to deal with 3D cylindrical structures such as in optical fiber communication, integrated optics, and defense industry. Moreover, the geometry of interest may consist of fine structures. If we adopt the conventional FDTD method to discretize the cylindrical structure with the Cartesian grid, a significant staircasing error appears. In fact, due to the existence of the term in the 3D cylindrical coordinate system, the decomposition of the fields along the direction will lead the splitting error. How to reduce this kind of error in the iterative FDTD calculation has become a difficult problem. In addition, how to conduct direction transformation of the fields on the *z*-axis is also a research difficult in the 3D cylindrical coordinate system. Perhaps this is why, since 2000, the FDTD method has done little applications in the 3D cylindrical coordinate system. There is not even an article on the Laguerre technology.

Therefore, in order to expand the research field of the FDTD method, we propose an efficient Laguerre-based finite-difference time-domain iterative algorithm in the 3D cylindrical coordinate system. Firstly, according to the ideology of the weighted Laguerre polynomial (WLP) FDTD scheme, the WLP-FDTD equations of the proposed algorithm in the 3D cylindrical coordinate system are deduced by introducing the new perturbation term and the nonphysical intermediate variables. Secondly, the field components on the *z*-axis are amended with special treatment. Finally, to verify the proposed algorithm, two scattering numerical examples are given. Numerical results show that the proposed algorithm can be better than the ADI-FDTD algorithm [5] in terms of efficiency and accuracy.

#### 2. Iterative Theory and Formulations

##### 2.1. WLP-FDTD Equations in 3D Cylindrical Coordinate System

Introducing the WLP technology [6] into Maxwell’s equations of the 3D cylindrical coordinate [7], one can obtain the equations of the conventional WLP-FDTD in the 3D cylindrical coordinate system, directly:where is a time-scale factor, is the order of weighted Laguerre polynomials, and and are the electric and magnetic permeability, respectively.

##### 2.2. Iterative Equations for the Off-Axis Fields

Equations (1)–(6) can be written in the following matrix form:where and are the general form of WLP fields and the summation of order WLP fields, respectively, and and are the coefficient matrices.

Obviously, solving equation (7) must involve large sparse matrix problems, and it is impossible to solve it. Therefore, the efficient algorithm must be adopted to avoid the direct solution of the large sparse matrix.

Referring to the brief form of the 3-D FDTD equation in [4], when the excitation sources are added, the equations of the efficient Laguerre-based FDTD in the 3D cylindrical coordinate can be rewritten aswhere is the iteration and equations (8) and (9) are the initial and iterative computation, respectively. Clearly, (9) have the same form with (8). So, one can combine them as follows:where

In addition, the and in (10b) should bewhere and . , , and are the first-order central difference operators along , , and axes, respectively.

In fact, equation (10a) can be implemented by simple programming, and we only need to implement the design of the iterative calculation. Therefore, we actually only need to introduce the perturbation term in [4] into equation (7) to obtain (10b), and the advantage of this is that it greatly simplifies the programming process and improves the computational efficiency.

In accordance with the design theory of the proposed algorithm in this paper, we introduce the nonphysical intermediate variables , and it can divide (10b) into

Expanding (13a) and (13b), one can obtain

Substituting (14b) into (14a), one can obtain

Expanding (16), one can obtain three implicit equations , , and . Here, the Gauss–Seidel iterative procedure should be used, and the equations after the Gauss–Seidel procedure are

Obviously, the necessary nonphysical variables , , and for solving (15a) have been obtained through solving (17a)–(17c).

Next is how to solve the fields of the nonphysical variables for solving (15b). Here, it is not necessary because substituting (14b) into (15b), equation (15b) can be rewritten as

Clearly, the nonphysical variables are eliminated already. Meaning that, the nonphysical variables do not have to be solved in the whole iteration.

Substituting (18) into (15a), one can obtain a new equation which is similar with (16):

Expanding (18) and (19), one can obtain the computational equations of the electric and magnetic fields of the proposed algorithm:where

At this point, the implementation scheme of the proposed algorithm has been elaborated, and the overall execution process is described in Figure 1.