Abstract
This paper presents a well-designed termination wall for the perfectly matched layers (PML). This termination wall is derived from Mur’s absorbing boundary condition (ABC) with special difference schemes. Numerical experiments illustrate that PML and the termination wall works well with ATS-FDTD(Shi et al. 2015). With the help of termination wall, perfectly matched layers can be decreased to two layers only; meanwhile, the reflection error still reaches -60[dB] when complex waveguide is simulated by ATS-FDTD.
1. Introduction
It is well known that the perfectly matched layers (PML) [1–4] are wildly used for simulating Maxwell’s equations in unbounded domain. The overall performance of PML depends on its all parameters, and the optimal parameters have been studied by many scientists [5–9].
To avoid causing much rebound at PEC, the traditional termination wall of PML, the PML are improved by some termination walls [8–11]. In [8], a lossy version of the one-way Engquist Majda ABC was used for the termination of magnetic-field component of PML and the electric-field component of PML was terminated by an according transverse impedance with specific direction , which lead to the inconvenience of this method. In [9–11], unified lossy version of Mur’s second-order ABC and dispersive boundary condition were developed to be the PML termination for wave equation which could be derive from Maxwell equations. The thickness of PML is still a restriction; none of the above improved PML can reach optimal thickness, two layers, with good performance.
In this paper, a special single-layer Mur’s ABC is designed as the PML termination for solving 2D Maxwell equations by ATS-FDTD [12] method. The staggered grids are used in this paper and the electromagnetic field components are placed on the boundary of each cell, and , which is different from the traditional Yee’s grid [13]. The magnetic-field component of PML is terminated by first-order Mur’s ABC where the difference schemes will be designed very differently. This termination wall, PML-ABC, is very convenient and effective to be implemented with the ATS-FDTD method to solve Maxwell equation.
2. The PML-ABC and ATS-FDTD
The equations of PML [8] are given bywhere (and in what follows) in the PML. and are the electric and magnetic conductivities of the PML, which satisfy the impedance matching condition
The boundaries of PML are given by Mur’s ABC:which is the speed of light in vacuum.
This paper uses the staggered grid technique:where and are the size of grid. By ATS-FDTD method, the approximations can be expressed aswhere =, =, = are the value at the former time level and is positive integer and should be determined byThe unknown coefficients , , , , and are determined by recursion formulas (11)-(21).
In vacuum, the unknown coefficients are obtained bywhere , , and are positive integer, and are the grid number in and direction, separately, is the thickness of PML, and and are differential operator, such as ; the other can be similarly defined.
The coefficients in PML are given byThe coefficients of in PML are given by
From the equation in (1), the schemes for coefficients of and in PML are written asThe schemes for coefficients of and at the Mur’s ABC are given bywhere two-order difference schemes are used for one-order Mur’s ABC.
3. Numerical Results
To show the good performance of the PML-ABC with implementation of the ATS-FDTD, three examples are given in this section. The conductivity in PML is defined as [1]where is the distance of PML layer from the inside PML interface. The reflection error at observation point in the time domain is defined aswhere denotes the numerical results and denotes the reference results, free of reflection errors and computed with a large enough domain. and size of grid are calculated by (10).
Example I. In the first example, comparison between PML-ABC and the traditional PML, PML-PEC, is presented. The vacuum domain contains cells and the time-harmonic source is located at the center of the vacuum. Observation point is located one cell diagonally away from a corner of the PML interface.
Figures 1–3 show the reflection errors with varying variables where denotes the numerical results computed with the PML-ABC boundary and denotes the numerical results computed with the PML-PEC boundary. Figure 1 illustrates the reflection error of PML-ABC with different thick . From Figure 1 we can find that the reflection error decline along with increasing . Figure 2 shows comparison between PML-ABC and PML-PEC with different , where =12 for PML-PEC. From Figure 2 we can find that the reflection errors of PML-ABC are less than that of PML-PEC. The performance of PML-ABC is slightly affected by , while the performance of PML-PEC is distinctly affected by . Figure 3 shows the reflection errors of PML-ABC are slightly affected by power .
Figures 1–3 illustrate the good performance of PML-ABC, reflection error around -60[dB], with only two PML layers.
Figures 4–7 show that the ATS-FDTD method works well with the PML-ABC.
Example II. The second example shows the good performance with a complex wave where time-harmonic source with frequency = 1.43e+11(Hz) moves along with speed . And the vacuum domain contains cells with , =0.083, h=1.0e-2(m), dt= 1.04e-10(s), K=86, and .
Figure 4 shows the numerical magnetic-field component at time steps =32 where the outgoing wave does not cause an obvious echo from the PML-ABC. Figure 5 presents the error reflection around -60[dB] at point and .
Example III. In the third example, variables and coefficients are the same as those in Example II. Here, three time-harmonic sources are placed around with isogonal location and 10 cells away from center. With initial phases , these three sources circle around with angular speed of . Figure 6 presents the numerical magnetic-field component at time steps , where the complex outgoing wave goes through the PML-ABC and does not cause an obvious echo. Figure 7 shows reflection errors around -60[dB] at points A1 and A2.
Example IV. In the fourth example, variables and coefficients are the same as those in Example II while the vacuum domain contains cell. Here, time-harmonic source is placed in the center. A circle with conductivity and 40 cells away from center is placed at . And a circle with electric permittivity and 40 cells away from center is placed at . Figure 8 presents the numerical magnetic-field component at time steps , where the complex outgoing wave goes through the PML-ABC and does not cause an obvious echo. Figure 9 shows reflection errors around -60[dB] at points A3 and A4 .
4. Conclusions
This paper presents a well-designed termination wall for PML on staggered grid where all electromagnetic field components are placed on the boundary of each cell. Numerical experimentations show that the PML-ABC works well with ATS-FDTD method to solve Maxwell equation in free-domain. The PML-ABC can be decreased to two layers with reflection error -60[dB] when simulate complex waveguide.
Data Availability
No data were used to support this study.
Ethical Approval
This article does not contain any studies with human participants or animals performed by any of the authors.
Conflicts of Interest
All authors declare that they have no conflicts of interest.
Acknowledgments
The work was supported by the National Natural Science Foundation of China (11571367), Shandong Provincial Natural Sciences Foundation (ZR2014AM029, ZR2015AL006), and China Scholarship Council Foundation (Grant no. 201806455012).