International Journal of Antennas and Propagation
Volume 2016 (2016), Article ID 8402697, 8 pages
http://dx.doi.org/10.1155/2016/8402697

Research Article

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

Yi-Gang Wang,¹ Yun Yi,¹ Bin Chen,¹ Hai-Lin Chen,¹ Kang Luo,¹ and Run Xiong²

¹PLA University of Science and Technology, Nanjing, Jiangsu 210007, China
²Engineering 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 and

for 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 fields

where 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 and

for 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.

Table 1: The floating point operations at the right-hand sides of the equations for two algorithms.

Mode
Conventional LOD-BOR-FDTD
One-step leapfrog LOD-BOR-FDTD

M/D
20
26
12
16

A/S
32
38
22
26

M/D + A/S
52
64
34
42

In the proposed algorithm, tridiagonal implicit equations for the magnetic field components are involved, so one should pay special attention to handling the perfectly electrically conducting (PEC) boundary, since there will be out-of-domain magnetic field components. To address this issue, one can resort to the image theory [17]. Such treatments are often used in handling the PEC resonator, PEC scatterer, PEC wall of the perfectly matched layer (PML), and so on.

3. CPML Implementation for the Proposed Algorithm

To solve open region problems efficiently, the CPML for the proposed one-step leapfrog LOD-BOR-FDTD algorithm is developed. The coordinate-stretching variables for the CPML are defined as

One can also define

where and represents the interface between FDTD and PML domains [18].

For simplicity, the CPML equations for the field component are given here. In fact, the CPML equations for the other field components can be obtained in a similar manner. According to the definition in (10) and (11), one can write the CPML equations of the conventional LOD-BOR-FDTD algorithm in the following form:

for the first subtime step and

for the second subtime step.

In the first subtime step, one can define

Then (12a) and (12b) can be rewritten as

In the second subtime step, one can define

Then (13a) can be rewritten as

The former time steps of (16a) and (17) are

Substituting (19) into (18), one can obtain

Substituting (15b) into (15a), one can obtain

Substituting (20) into (21), the following equation can be obtained:

Some of the auxiliary variables in (22) are approximated as

Ignoring the high order terms of in (23a), (23b), and (23c), then (22) can be approximated as

The following equation can be obtained from (14a) and (20):

To simplify the equations further, one can introduce an auxiliary variable . As a result, the final equations for can be reformulated as

Some of the auxiliary variables in (26a), (26b), and (26c) can be solved by

where and , .

In addition, the auxiliary variable in (26b) has been solved in updating , so it can be used directly here. It is obvious that the number of the convolutional auxiliary variables for the proposed CPML is the same as that of the CPML for the conventional BOR-FDTD algorithm. Consequently, one can say that the equations of proposed CPML are concise.

When and , the equations of the proposed CPML can be simplified to be the normal equations of the proposed LOD-BOR-FDTD algorithm. Similarly, since the coordinate stretching along the -direction only exists in the radial outer boundary, that is, and on (or near) the axis, the CPML equations for and are still (8a), (8b), (9a), (9b), and (9c).

In the Cartesian coordinate system, the coordinate stretching along each axis is similar to that along the -direction in the cylindrical coordinate system. So it can be concluded that the CPML for the LOD-FDTD algorithm in the Cartesian coordinate system [13] can be implemented by applying the similar procedure.

4. Numerical Results and Discussions

A resonant cavity and a scattering example are provided to validate the one-step leapfrog LOD-BOR-FDTD algorithm and its CPML.

4.1. Cavity Resonator

The mode of a PEC cylindrical cavity resonator is computed, which is frequently used as a benchmark [3, 4]. The radius and height of the cavity are 3.995 cm and 7.910 cm, respectively. Its theoretical resonant frequency is 4.950 GHz [3, 4]. The computational domain is divided into cells, with and . The cavity is computed with the conventional BOR-FDTD algorithm, the conventional LOD-BOR-FDTD algorithm, the one-step leapfrog ADI-BOR-FDTD algorithm, and the proposed algorithm, respectively. A time history of 52.73 ns is simulated.

Table 2 shows the computational results with different algorithms. Note that and is the maximum time step size that satisfies the CFL condition for the conventional BOR-FDTD algorithm [1], which is here. It can be seen from Table 2 that the one-step leapfrog LOD-BOR-FDTD algorithm obtains the same numerical results as the conventional LOD-BOR-FDTD algorithm. Moreover, the numerical errors of two algorithms increase as the time step sizes increase. In terms of the computational time, the proposed algorithm consumes about 26.5%, 25.8%, and 25.0% less time than the conventional LOD-BOR-FDTD algorithm when CFLN is chosen to be 1, 4, and 8, respectively. The two unconditionally stable algorithms use the same amount of memory, which is 1.06 MB. Therefore, one can say that the proposed algorithm preserves the properties of the conventional LOD-BOR-FDTD algorithm and obtains a higher computational efficiency. The proposed algorithm also shows the same accuracy as that of the one-step leapfrog ADI-BOR-FDTD algorithm but obtains a little higher computational efficiency, as shown in Table 2. Note that the comparison among the conventional LOD-BOR-FDTD algorithm, the one-step leapfrog ADI-BOR-FDTD algorithm, and the proposed algorithm was performed only in the case of the resonant cavity and that further analysis could be performed on other situations to rule out the possibility that the results could be linked to the particular case considered. Since the TE₀₁₁ mode is computed, the terms including multiplicative factor (mode number) in the equations are all zero. In the simulation, these terms are not computed. All of the calculations are performed on an Intel® Core™ 2 Quad CPU Q6600 @ 2.40 GHz machine and the Fortran codes are used.

Table 2: Comparison of the computational results with different algorithms.

BOR-FDTD
Conventional LOD-BOR-FDTD
One-step leapfrog ADI-BOR-FDTD
One-step leapfrog LOD-BOR-FDTD

CFLN
1
1
4
8
1
4
8
1
4
8

Iterations
40000
40000
10000
5000
40000
10000
5000
40000
10000
5000

Results, GHz
4.954
4.954
4.943
4.907
4.954
4.943
4.907
4.954
4.943
4.907

Relative error
0.08%
0.08%
−0.14%
−0.87%
0.08%
−0.14%
−0.87%
0.08%
−0.14%
−0.87%

Time, s
13.2
52.8
13.2
6.4
43.2
10.9
5.5
38.8
9.8
4.8

4.2. Scattering Example

To validate the proposed CPML for the one-step leapfrog LOD-BOR-FDTD algorithm, the scattered field from a PEC cylinder is computed. The radius and height of the cylinder are 20 cm and 40 cm, respectively. The oblique incident plane wave with incident angle is introduced by the TF/SF boundary conditions, which are placed at and . The incident plane wave is parallel polarized, as shown in Figure 1. The gray domain in Figure 1 denotes the PEC cylinder. The Gaussian pulse with , , , and is used as the incident electric field . The small computational domain is composed of cells, containing the 10 PML layers in the outer boundary, which are denoted by the oblique-line-domain as shown in Figure 1. According to the characteristic of the problem, the mode number ranges from 0 to 11. The uniform grids are chosen to be . Therefore, the maximum time step size for the conventional BOR-FDTD [1] is in the example. The observation point is placed at grid point , which is near the corner of the scattered field domain.

Figure 1: Illustration of the scattering example with oblique incidence.

The scattered field is computed with the conventional BOR-FDTD algorithm and the proposed algorithm, respectively, both with the CPML absorbing boundary conditions. The parameters of the CPML are chosen according to the following expressions [18]:

where is the thickness of the PML, is the interface between the FDTD and PML grids, and is the order of the polynomial. In the computing, the optimized parameters are chosen to be , , and .

To validate the one-step leapfrog LOD-BOR-FDTD algorithm further, the scattered field is computed via large computational domain first. In this case, the scattered field at the observation point is without reflection, so one can make a clear comparison. Figure 2 shows the fields obtained by the two algorithms via large computational domains. For the proposed algorithm, different CFLNs are used, where . It can be seen that the results of the proposed algorithm agree well with those of the conventional BOR-FDTD algorithm. Therefore, the proposed algorithm is validated further. Meanwhile, one can see that the numerical error of the proposed algorithm increases as the time step size grows.

Figure 2: fields at the observation point with the conventional BOR-FDTD and the proposed algorithm.

Figure 3 shows the reflection errors of the CPMLs for the conventional BOR-FDTD algorithm and the proposed algorithm, which are obtained by the fields at the observation point . The reflection error is defined as

where is the result obtained by the small computational domain as shown in Figure 1. is the numerical result via large computational domain. For the proposed algorithm, is computed separately for each CFLN. It can be seen from Figure 3 that the CPML for the one-step leapfrog LOD-BOR-FDTD algorithm shows good performance. Moreover, the reflection error of the one-step leapfrog CPML increases as the time step size grows. When , the maximum reflection error is about −40 dB.

Figure 3: Relative reflection errors of the CPML for the conventional BOR-FDTD and the proposed CPML for the one-step leapfrog LOD-BOR-FDTD.

5. Conclusion

In this paper, the one-step leapfrog LOD-BOR-FDTD algorithm is presented. It obtains a higher computational efficiency while keeping the properties of the conventional LOD-BOR-FDTD algorithm. In addition, the CPML for the proposed algorithm is implemented. The numerical results show the performances of the one-step leapfrog LOD-BOR-FDTD algorithm and its CPML. It can be concluded that, by applying the similar procedure, the CPML for the LOD-FDTD algorithm in the Cartesian coordinate system can be implemented.

Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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