International Journal of Antennas and Propagation

Volume 2016 (2016), Article ID 8402697, 8 pages

http://dx.doi.org/10.1155/2016/8402697

## One-Step Leapfrog LOD-BOR-FDTD Algorithm with CPML Implementation

^{1}PLA University of Science and Technology, Nanjing, Jiangsu 210007, China^{2}Engineering Academy of PLA, Xuzhou, Jiangsu 221004, China

Received 5 January 2016; Revised 12 March 2016; Accepted 14 April 2016

Academic Editor: Marta Cavagnaro

Copyright © 2016 Yi-Gang Wang 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 unconditionally stable one-step leapfrog locally one-dimensional finite-difference time-domain (LOD-FDTD) algorithm towards body of revolution (BOR) is presented. The equations of the proposed algorithm are obtained by the algebraic manipulation of those used in the conventional LOD-BOR-FDTD algorithm. The equations for -direction electric and magnetic fields in the proposed algorithm should be treated specially. The new algorithm obtains a higher computational efficiency while preserving the properties of the conventional LOD-BOR-FDTD algorithm. Moreover, the convolutional perfectly matched layer (CPML) is introduced into the one-step leapfrog LOD-BOR-FDTD algorithm. The equation of the one-step leapfrog CPML is concise. Numerical results show that its reflection error is small. It can be concluded that the similar CPML scheme can also be easily applied to the one-step leapfrog LOD-FDTD algorithm in the Cartesian coordinate system.

#### 1. Introduction

The body of revolution finite-difference time-domain (BOR-FDTD) algorithm is very efficient in analyzing electromagnetic problems towards rotationally symmetric structures [1, 2]. It has been widely used in modeling electromagnetic pulse effects, electromagnetic wave scattering, subsurface interface radar, optical lenses, guided waves, and so on [1]. However, the time step size of the conventional BOR-FDTD algorithm is strictly limited by the Courant-Friedrichs-Lewy (CFL) condition [1]. To remove the stability limit on the time step size of the BOR-FDTD algorithm and improve the efficiency, some unconditionally stable schemes such as the alternating-direction implicit (ADI) BOR-FDTD [3], the locally one-dimensional (LOD) BOR-FDTD [4], and the weighted Laguerre polynomials (WLP) BOR-FDTD [5] algorithms have been proposed. The WLP-BOR-FDTD algorithm needs to solve a large sparse matrix, so it is not so applicable for large computational domain [5]. The LOD-BOR-FDTD algorithm and the ADI-BOR-FDTD algorithm show comparable accuracy, and the LOD-BOR-FDTD algorithm shows a little higher computational efficiency [4]. In the conventional ADI-BOR-FDTD and LOD-BOR-FDTD algorithms, the equations for one full time step are split into two subtime steps; as a result, their computational expenditures are increased [3, 4]. Recently, the one-step leapfrog ADI-FDTD algorithm which eliminates the midtime step successfully has been proposed and developed [6–11]. It makes the simulation with the ADI-FDTD algorithm more efficient. The application of the one-step leapfrog ADI-FDTD algorithm to BOR has also been proposed [12]. In fact, the parallel improvement can also be made for the conventional LOD-BOR-FDTD algorithm.

Recently, an unconditionally stable one-step leapfrog LOD-FDTD algorithm was proposed [13]. In the algorithm, the equations are obtained by the manipulation of those used in the conventional LOD-FDTD algorithm. The resultant electric and magnetic field equations are interlaced half a time step apart and no subtime steps are involved. The new algorithm obtains a higher computational efficiency while preserving the properties of the conventional LOD-FDTD algorithm [13].

In this work, the one-step leapfrog LOD-FDTD algorithm for BOR is developed, called one-step leapfrog LOD-BOR-FDTD algorithm. In the proposed algorithm, the -direction electric and magnetic field components are dealt with differently. Moreover, the convolutional perfectly matched layer (CPML) [14] is introduced to the one-step leapfrog LOD-BOR-FDTD algorithm. The equations of the one-step leapfrog CPML are concise.

The remainder of the paper is organized as follows. In Section 2, the equations of the one-step leapfrog LOD-BOR-FDTD algorithm are presented and some discussions about the algorithm are made. In Section 3, the CPML is developed for the proposed algorithm. To assess the proposed algorithm and its CPML, numerical examples are given in Section 4. Finally, conclusions are made in Section 5.

#### 2. Formulations and Discussions

##### 2.1. Equations for Off-Axis Cells

The equations for the conventional LOD-BOR-FDTD algorithm can be expressed as [4]for the first subtime step andfor the second subtime step.

Here,are the matrices that contain the spatial differential operators, and are the electric and magnetic field vectors, is the mode number, is the permittivity, and is the permeability.

Following the similar procedure used in [13, 15, 16], one can obtain the following electric field equations for the one-step leapfrog LOD-BOR-FDTD algorithm with only algebraic manipulations of (1a), (1b), (2a), and (2b):where is a unit matrix, and are auxiliary variables, and .

Similarly, one can obtain the equations for the magnetic fieldswhere and are auxiliary variables and .

##### 2.2. Equations for On-Axis Field

The on-axis field component cannot be obtained by using (4a), (4b), and (4c) directly. Note that the on-axis field component is zero for , so one should only deal with it for . The equation for the on-axis field component in the conventional LOD-BOR-FDTD algorithm is [4]for the first subtime step andfor the second subtime step. Moreover, one can obtain the equation for in the conventional LOD-BOR-FDTD algorithm from (1b)

Applying the similar procedure used in Section 2.1, one can obtain the following equations for the on-axis field component by using (6a), (6b), and (7):Unlike the equations for the off-axis field components, no auxiliary variable is involved here.

##### 2.3. Equations for

The equations for should also be treated specially. With the algebraic manipulation of the relative difference equations used in the conventional LOD-BOR-FDTD algorithm [4], one can obtain the following equations for

##### 2.4. Some Discussions about the Proposed Algorithm

It can be seen that all the equations used in the proposed algorithm are obtained from those used in the conventional LOD-BOR-FDTD algorithm and only algebraic manipulations are made. Therefore, one can conclude that the proposed algorithm preserves the properties of the conventional LOD-BOR-FDTD algorithm.

In terms of memory, the variables used in the conventional LOD-BOR-FDTD algorithm are , , and in the first subtime step, which can also be reused in the second subtime step. In the proposed algorithm, , , and can occupy the same memory space. There are similar situations to , , and and and . As a result, the proposed algorithm consumes the same amount of memory as the conventional LOD-BOR-FDTD algorithm.

There are four tridiagonal equations in one full time step, for both the conventional LOD-BOR-FDTD algorithm and the proposed algorithm. So the floating point operations at the left-hand sides of the equations are the same for the two algorithms. However, the numbers of the multiplications/divisions (M/D) and the additions/subtractions (A/S) at the right-hand sides of the equations are different. Table 1 shows the count of the M/D and A/S at the right-hand sides of the equations in two algorithms for modes and , respectively. Obviously, the proposed algorithm needs less floating point operations, so one can conclude that it has a higher computational efficiency. Note that all the coefficients of the equations are precomputed and stored here, and special treatments for the components that lie on or near the axis are not considered.