# Well-Designed Termination Wall of Perfectly Matched Layers for ATS-FDTD Method

Authors: Ju Ge; Liping Gao; Rengang Shi
Journal: International Journal of Antennas and Propagation
 (2019)

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## 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.


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## Body

## 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 θm, 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, Exi,j+1/2n,Eyi+1/2,jn and Hzi,jn, 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 by(1)ε∂Ex∂t+σyEx=∂Hzx+Hzy∂yε∂Ey∂t+σxEy=-∂Hzx+Hzy∂xμ∂Hzx∂t+σmxHzx=-∂Ey∂xμ∂Hzy∂t+σmyHzy=∂Ex∂ywhere (and in what follows) Hz=Hzx+Hzy in the PML. σx,σy and σmx,σmy are the electric and magnetic conductivities of the PML, which satisfy the impedance matching condition(2)σxε=σmxμ,σyε=σmyμ.The boundaries of PML are given by Mur’s ABC:(3)∂Hzx∂x-1c∂Hzx∂t=0,on  left  boundary∂Hzx∂x+1c∂Hzx∂t=0,on  right  boundary∂Hzy∂y-1c∂Hzy∂t=0,on  below  boundary∂Hzy∂y+1c∂Hzy∂t=0,on  upper  boundarywhich c=1/εμ is the speed of light in vacuum.This paper uses the staggered grid technique:(4)xi=ih,xi+1/2=xi+h2,yj+1/2=yj+h2,yj=jh,tn=nΔt,tn+1/2=tn+Δt2,where h and Δt are the size of grid. By ATS-FDTD method, the approximations can be expressed as(5)Exi,j+1/2n=∑k=0KCxi,j+1/2kΔtk(6)Eyi+1/2,jn=∑k=0KCyi+1/2,jkΔtk(7)Hzi,jn=∑k=0KCzi,jkΔtk(8)Hzxi,jn=∑k=0KCzxi,jkΔtk(9)Hzyi,jn=∑k=0KCzyi,jkΔtkwhere Cxi,j+1/20=Exi,j+1/2n-1, Cyi+1/2,j0=Eyi+1/2,jn-1, Czi,j0=Hzi,j+1/2n-1 are the value at the former time level and K is positive integer and should be determined by(10)∑k=0K-4Δt/hkk!<1.The unknown coefficients Cxi,j+1/2k, Cyi+1/2,jk, Czi,jk, Czxi,jk, and Czyi,jk are determined by recursion formulas (11)-(21).In vacuum, the unknown coefficients are obtained by(11)εCxi,j+1/2k+1=δyCzi,j+1/2kk+1i=d+1,⋯,I-1-dj=d,⋯,J-1-d(12)εCyi+1/2,jk+1=-δxCzi+1/2,jkk+1i=d,⋯,I-1-dj=d+1,⋯,J-1-d(13)μCzi,jk+1=δyCxi,jk-δxCyi,jkk+1i=d+1,⋯,I-1-dj=d+1,⋯,J-1-dwhere I, J, and d are positive integer, I and J are the grid number in x and y direction, separately, d is the thickness of PML, and δx and δy are differential operator, such as δxCyi,jk=(Cyi+1/2,jk-Cyi-1/2,jk)/h; the other can be similarly defined.The coefficientsCxi,j+1/2k in PML are given by(14)εCxi,j+1/2k+1=-σyCxi,j+1/2k+δyCzxi,jk+Czyi,jkk+1i=0,⋯,I,j=0,⋯,d-1;J-d,⋯,J-1i=0,⋯,d;I-d,⋯,I,j=d,⋯,J-d-1The coefficients of Eyi+1/2,jn in PML are given by(15)Cyi+1/2,jk+1=-σxCyi+1/2,jk-δxCzxi,jk+Czyi,jkk+1i=0,⋯,d-1;I-d,⋯,I-1,j=0,⋯,Ji=d,⋯,I-d-1,j=0,⋯,d;J-d,⋯,JFrom the equation in (1), the schemes for coefficients of Hzxi,jn and Hzyi,jn in PML are written as(16)μCzxi,jk+1=-σmxCzxi,jk-δxCyi+1/2,jkk+1i=1,⋯,d;I-d,⋯,I-1,j=0,⋯,Ji=d+1,⋯,I-1-d,j=0,⋯,d;J-d,⋯,J(17)μCzyi,jk+1=-σmyCzyi,jk+δyCxi,j+1/2kk+1j=1,⋯,d;J-d,⋯,J-1,i=0,⋯,Ij=d+1,⋯,J-1-d,i=0,⋯,d;I-d,⋯,IThe schemes for coefficients of Hzx and Hzy at the Mur’s ABC are given by(18)μCzx0,jk+1=-3Czx0,jk+4Czx1,jk-Czx2,jk2hk+1j=0,⋯,J(19)μCzxI,jk+1=-3CzxI,jk+4CzxI-1,jk-CzxI-2,jk2hk+1j=0,⋯,J(20)μCzyi,0k+1=-3Czyi,0k+4Czyi,1k-Czyi,2k2hk+1i=0,⋯,I(21)μCzyi,Jk+1=-3Czyi,Jk+4Czyi,J-1k-Czyi,J-2k2hk+1i=0,⋯,Iwhere 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 conductivityσ in PML is defined as [1](22)σ=σmaxρdmwhere ρ(≤d) is the distance of PML layer from the inside PML interface. The reflection error at observation point in the time domain is defined as(23)En=20log⁡HzPML-HzRef.max⁡HzRef.nwhere HzPML denotes the numerical results and HzRef. denotes the reference results, free of reflection errors and computed with a large enough domain. K 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 101×101 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.Figures1–3 show the reflection errors with varying variables where HzP.A. denotes the numerical results computed with the PML-ABC boundary and HzP.P. denotes the numerical results computed with the PML-PEC boundary. Figure 1 illustrates the reflection error of PML-ABC with different thick d. From Figure 1 we can find that the reflection error decline along with increasing d. Figure 2 shows comparison between PML-ABC and PML-PEC with different σmax, where d=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 σmax, while the performance of PML-PEC is distinctly affected by σmax. Figure 3 shows the reflection errors of PML-ABC are slightly affected by power m.Figure 1
Reflection error of results of PML-ABC as a function of time step with m=2,σmax=2.6e-8, f= 18.83(Hz), h=2000(m), Δt=3.35e-6(s), K=5, and varying d.Figure 2
Reflection error of results of PML-ABC and PML-PEC as a function of time step with d=2, m=2,f=1.88e+11(Hz), h=0.01(m), Δt=2.37e-10(s), K=94, and varying σmax.Figure 3
Reflection error of results of PML-ABC as a function of time step with d=2,σmax=2.6e-7, f= 1.88e+17(Hz), h=2.0e-5(m), Δt= 4.34e-13(s), K=84, and varying m.Figures1–3 illustrate the good performance of PML-ABC, reflection error around -60[dB], with only two PML layers.Figures4–7 show that the ATS-FDTD method works well with the PML-ABC.Figure 4
μ H z i , j n at time step n=32 for Example II.Figure 5
Reflection error as a function of time step for Example II.Figure 6
μ H z i , j n at time step n=20 for Example III.Figure 7
Reflection error as a function of time step for Example III.Example II. The second example shows the good performance with a complex wave where time-harmonic source with frequency f= 1.43e+11(Hz) moves along x51 with speed 3m/ns. And the vacuum domain contains 101×101 cells with d=2, σmax=0.083, h=1.0e-2(m), dt= 1.04e-10(s), K=86, and m=3.Figure4 shows the numerical magnetic-field component at time steps n=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 A1(x2,y2) and A2(x15,y15).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 (x51,y51) with isogonal location and 10 cells away from center. With initial phases 0,2π/3,4π/3, these three sources circle around (x51,y51) with angular speed of π/18rad/ns. Figure 6 presents the numerical magnetic-field component at time steps n=20, 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 150×150 cell. Here, time-harmonic source is placed in the center. A circle with conductivity σ=10πμ/ε and 40 cells away from center is placed at (x75,y35). And a circle with electric permittivity 10.25ε and 40 cells away from center is placed at (x35,y75). Figure 8 presents the numerical magnetic-field component at time steps n=100, 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 (x2,y75) and A4 (x75,y2).Figure 8
μ H z i , j n at time step n=100 for Example IV.Figure 9
Reflection error as a function of time step for Example IV.


## 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.



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*Source: 6343641-2019-06-02.xml*