JIANG Haolin, XIE Yongjun, WU Peiyu, ZHANG Jianfeng, and NIU Liqiang,*

1. School of Infomation Science and Engineering, Southeast University, Nanjing 210096, China;2. School of Electronic and Information Engineering, Beihang University, Beijing 100191, China

Abstract: An unsplit-field higher order nearly perfectly matched layer (NPML) based on the auxiliary differential equation approach is introduced in three-dimensional finite-difference timedomain lattices. The proposed scheme has the advantage of both the NPML scheme and the higher order concept in terms of the improved absorbing performance and considerable computational efficiency. By incorporating with the generalized material independent concept, the proposed implementation is indepen dent of the material’s type. Thus, it has the advantages of terminating arbitrary media without changing the updated equations in the PML regions. Its effectiveness and efficiency is further demonstrated through numerical examples.

Keywords: finite-difference time-domain (FDTD), electromagnetic (EM) simulation, nearly perfectly matched layer (NPML), higher order perfectly matched layer (HO-PML).

1. Introduction

As one of the most powerful numerical methodologies in full-wave simulations, the finite-difference time-domain(FDTD) method has been widely employed in the analyzing of wave propagation, designing microwave devices,and so on [1].

To cope with the electromagnetism (EM) problems and model the object in the open regions, the perfectly matched layer (PML) with the advantages of the high calculation accuracy and the high simulation efficiency has gained considerable attention [2]. The original PML, proposed by Berenger, is implemented with the split-field equations resulting in a lower computational efficiency.Thus, the unsplit-field stretched coordinate PML (SCPML) [3,4] and uniaxial anisotropic PML (UPML) [5]are carried out to alleviate such phenomenon.

However, the update equations in the PML regions must be modified by employing the above-mentioned schemes directly to the Maxwell’s equations. To not only simplify the updated equations in the general media but also improve the computational efficiency, the nearly PML(NPML) is proposed [6]. It has been demonstrated that the low-frequency evanescent waves and the late-time reflections cannot be decreased by employing the original NPML scheme [7,8]. In order to alleviate such problem,the PML with the complex frequency-shifted (CFS) factor is regarded as one of the most efficient ways [9].

Within the problems in low frequency ranges with a large amount of propagation waves, the absorbing performance will be grievously influenced. Thus, the introduction of the higher order PML (HO-PML) is the most powerful way. The HO-PML is implemented by multiplying the stretched coordinate variables in a single term. The higher order scheme can be divided into second order PML, third order PML and so on. Generally, the HO-PML refers to the second order PML. The reason is that the computational efficiency will be increased with the improvement of the order. The second order stretched coordinate variable can satisfy the practical engineering.In [10], the second order PML is given not only to absorb the low-frequency waves but also to further enhance the absorbing performance both in the time domain and the frequency domain. Such conclusion has been further testified in [11,12].

Here, by incorporating the CFS scheme, the generalized material independent concept and the higher order concept, the higher order CFS-NPML is proposed in the three-dimensional FDTD computational domain. The efficiency and the effectiveness are testified through the comparison between different PML algorithms with the numerical examples. The results demonstrate that the proposed scheme can be employed to terminate arbitrary media without changing the updated equations in the PML regions.

2. Formulations

To enable the proposed scheme can terminate arbitrary media, the generalized material independent concept is introduced [13,14]. During the derivation, the x-projection of the Ampere’s law is employed as an example. In the higher order CFS-NPML regions, the Maxwell’s equations with the generalized material independent concept can be given as

where Sη(η=x,y,z) is the higher order stretched coordinate metric with the CFS factor which can be given as

where αηnand σηnare real and κηn≥1 is positive real.can be obtained by employing the partial fraction method,given as

where κη=1/(κη1κη2), aηn=αηn/ε0and bηn=aηn+aηn/κηn.By substituting (3) into (1), it can be obtained that

where Hzyand Hyzare the auxiliary variables. It can be noticed that the auxiliary variables almost hold the same updated forms. Here, Hzyis chosen as an example to demonstrate the proposed algorithm, given as

By rearranging (5), the equation can be rewritten as

According to the auxiliary differential equation (ADE)approach, one obtains

By transforming the resultants into the FDTD domain,one obtains

where

and

Thus, the auxiliary variables can be updated as

After updating the auxiliary variables, (4) can be updated by employing the following forms:

where the operator represents the finite-difference form,defined as

It should be noticed that the occupied memory increases by employing the higher order concept. In order to decrease the computational resource, the memory-minimized method is introduced during the updating. The temporary variable is introduced into (9), given as

where

By employing such procedure, the computational resource can be reduced significantly.

3. Numerical example

In this section, the effectiveness and the efficiency of the proposed scheme, donated as HO-NPML, are testified through the half-space soil problem and the inverted F antenna model. Here, the PC with Core TM i7-6820HQ CPU of 2.7 GHz and 16 GB RAM (DDR4 2 133 MHz) is employed to implement different PML formulations.

3.1 Half-space soil radiation problem

The half space soil radiation problem is considered in the first numerical example. The sketch picture is shown in Fig. 1.

Fig. 1 Sketch picture of half-space soil radiation problem

The whole computational domain is with the size of 100∆x×60∆y×40∆z. The source which is a Gaussian pulse with the maximum frequency of 1 GHz is located at the position of (50, 30, 30). The observation point which is located at (1, 1, 1) with the distance of 1 cell from three sides of the PML regions is employed to observe the wave reflections and evaluate the absorbing performance.

The half of the vertical height along z-direction is filled with the soil which can be expressed as the dispersive and lossy medium. The dispersive relationship can be expressed by the second order Debye model as

where ε∞=4.5 represents the permittivity at the infinite frequency, the pole amplitudes are chosen as A1=1.8,A2=1.6, the relaxation time is chosen as τ1=3.79 ns,τ2=0.151 ns, and the electrical conductivity is chosen as σbg=1.11 ms/m. At the boundaries, all sides are terminated by 8-cell-PML. To make comparison between the proposed HO-PML and previous works, CFS-PML [15],conventional NPML [6] and higher order CFS-PML [10]are selected.

The uniform mesh is employed as∆x=∆y=∆z=∆=0.015 m. The time step can be obtained by the Courant condition as ∆t=28.87 ps. Inside the PML regions,the parameters are chosen by the try and error method to obtain the best absorbing performance both in the time domain and the frequency domain. The parameters of HONPML and HO-PML are chosen as κη1=9 , αη1=0.75,mη1=3, ση1max=0.5ση1opt, κη2=15, αη2=0.07, mη2=3 and ση2max=0.4ση2opt, where

The parameters of CFS-PML are chosen as κη=10,αη=1.24 , mη=2, σηmax=1.2σηopt. The original NPML is implemented with the stretched coordinate (SC) variables,which has the parameters of [8, 2, 0.000 1%].

To obtain the absorbing performance in the time domain,the relative reflection error versus time is employed as

Fig. 2 Relative reflection error versus time with different PML algorithms

It can be observed that the maximum relative reflection errors of the NPML, CFS-PML, HO-PML and HONPML are -48.1 dB, -83.7 dB, -97.0 dB and -103.5 dB,respectively. Compared with the NPML, the absorbing performance of the HO-NPML is improved by 48.9 dB.Thus, it can be proved that the absorbing performance can be significantly improved by employing the higher order concept. Meanwhile, from Table 1, it can be observed that the efficiency and resources can be improved by the HO-NPML compared with the HO-PML indicating the effectiveness of the proposed scheme. The performance of the NPML is inferior compared with the CFSPML especially at the late-time. The reason is that the late-time reflections can also be reduced obviously by employing the CFS factor.

Table 1 CPU and memory employed by different PML algorithms

The absorbing performance can also be reflected by the reflection coefficient in the frequency domain, defined as

where FFT[·] is the fast Fourier transform. Fig. 3 shows the reflection coefficient obtained by different PML algorithms. It can be observed that the performance can be further enhanced by employing the higher order concept.Especially, the performance at low-frequency can be significantly improved, indicating that the low-frequency propagation waves can be efficiently absorbed.

Fig. 3 Reflection coefficient with different PMLs

3.2 Radiation problem in inverted-F antenna model

The inverted-F antenna model is introduced to demonstrate the effectiveness of the proposed HO-NPML algorithm in the second numerical example. The model of the inverted-F antenna is shown in Fig. 4.

Fig. 4 Sketch picture of the inverted-F antenna model

The whole FDTD computational domain which has the size of 200∆x×200∆y×50∆z is employed. To obtain a higher computational accuracy, the mesh sizes∆x=∆y=4 mm and ∆ z=0.262 mm are employed in each direction. The time step can be obtained as 0.504 ps. All sides of the boundaries are terminated by 8-cell-PML.The Gaussian voltage source with the maximum frequency of 10 GHz is excited at the bottom of the antenna.To evaluate the performance of the PML schemes, the observation point is located at the corner of the domain with one cell from all sides of the domain, which has been testified that it holds the worst absorbing performance. Within the PML regions, the parameters are selected to obtain the best absorption. The parameters of the HO-NPML and HO-PML are chosen as κη1=30 ,αη1=1.1, mη1=3, ση1max=2.6ση1opt, κη2=1, αη2=1.9 ,mη2=2 and ση2max=0.01ση2opt. The parameters of CFS-PML are selected as κη=45, αη=2.4 , mη=3, σηmax=0.7σηopt.Fig. 5 shows the relative reflection error versus time obtained by different PML algorithms.

Fig. 5 Relative reflection error versus time in the inverted-F antenna model with different algorithms

It can be observed that the maximum values of the relative reflection error with the NPML, CFS-PML, HOPML and HO-NPML are -38 dB, -57 dB, -101 dB and-101 dB, respectively. The absorbing performance can be improved significantly by employing the higher order concept. Compared with the NPML, the HO-NPML with the CFS factor can not only reduce the late-time reflections but also enhance the absorbing performance. As shown in Table 2, although the absorbing performance of the HO-NPML is inferior compared with that of the HOPML, the computational efficiency can be improved significantly. Thus, the NPML is a compromise between the computational efficiency and the absorbing performance.

Table 2 CPU and memory in the inverted-F antenna model employed by different PML algorithms

The absorbing performance of the PML algorithms in the frequency domain can be reflected by the return loss(S11parameters). Fig. 6 shows S11parameters of the inverted-F antenna model obtained by different PML algorithms. It can be observed that the S11parameters are almost overlapped, indicating that the proposed scheme can obtain the considerable performance.

Fig. 6 S11 parameters of the inverted-F antenna model obtained by different PML algorithms

4. Conclusions

An unsplit-field HO-NPML is proposed for modelling and simulating arbitrary media. As shown from the formulations, without changing the updated equations in the PML regions, the proposed implementation can terminate the arbitrary media directly. From the numerical examples, it has been demonstrated that the proposed scheme takes the advantages of the HO-PML and CFS-NPML in terms of enhancing the absorbing performance, reducing the late-time reflections and improving the computational efficiency.