In this paper, a novel algorithm based on the alternating direction implicit (ADI) multiresolution time-domain (MRTD) method for periodic structure simulation is proposed. By applying the multiresolution analysis in accordance with wavelet theory, the spatial sampling rate of the conventional finite-difference time-domain (FDTD) is significantly reduced by the MRTD method. The ADI method is then used to remove the Courant-Friedrich-Levy (CFL) limit that the MRTD method experiences. The periodic boundary condition (PBC) is directly implemented in the time domain using a constant transverse wave-number (CTW) wave. Numerical results are presented to confirm the efficiency and accuracy of the proposed method.

Over the past decades, notable progress on the engineering applications of periodic structures has been achieved, especially on frequency selective surfaces (FSS) [

Normally, in the analysis of periodic structures using the FDTD method, only a unit cell is simulated and computed (instead of the whole structure) by incorporating the appropriate periodic boundary condition (PBC). However, unlike the situation of a normal incident wave, the implementation of the PBC for an oblique incident is complicated due to the time delay in the transverse plane. The split-field technique [

However, the time step is constrained by the Courant-Friedrich-Levy condition due to the explicit time-marching technique. To overcome this problem, the weakly conditionally stable finite-difference time-domain (WCS-FDTD) [

In recent years, the multiresolution time-domain (MRTD) technique has been successfully developed due to its highly linear dispersion performance and ability to simulate complex electromagnetic structure [

In this paper, the spectral-FDTD (SFDTD) technique is applied to the conventional MRTD and ADI-MRTD methods, resulting in the PS-ADI-MRTD algorithms. The MRTD method and the ADI method are, respectively, used to reduce the spatial sampling rate and remove the KCL limit. The application, the SFDTD, is mainly using a constant transverse wave-number (CTW) wave to directly implement the PBC in the time domain. To verify the efficiency and accuracy of the PS-ADI-MRTD method, the numerical example is presented later.

Considering that the CTW travels in the

As shown in equation (

Therefore, if

As shown in Figure

Schematic diagram of the CTW incident wave.

Then,

By defining that the unit direction vector

The unit direction vector

The incident wave in the frequency domain is shown as follows:

Then, we obtain the magnetic field component of the incident wave (the transverse magnetic field):

By applying the inverse Fourier transform on (

For a better understanding of this paper, the conventional MRTD is briefly introduced here. The topology of the square patches periodic structure is depicted in Figure

Geometric diagram of the rectangle patches periodic structure: (a) Top view: _{x}_{y}

The equations of the proposed PS-ADI-MRTD method are similar to the ADI-MRTD method [

Matrix

We define that

According to Sherman-Morrison-Woodbury formula [

Finally,

The element

For the given

The TF/SF boundary and absorb boundary condition of the proposed PS-ADI-MRTD is similar to the conventional MRTD.

The detail equations for the open loop case are omitted for saving the space.

The structure of the thin metal rectangle array is shown in Figure

The TE wave of the CTW source is set as the excitation source. The wave vector is in the x-z plane and is represented by

Figure

Frequency characteristics at different angles of incidence.

As refer to the simulation about SFDTD algorithm, the implementation of SFDTD is mainly based on [

A novel algorithm referred to as the PS-AID-MRTD is presented to simulate the periodic structure. By using the CTW wave to handle the incident wave, one can directly implement the PBC. The PS-AID-MRTD algorithm has adopted each advantage of the MRTD method and the ADI technique. As a result, the spatial sampling rate is significantly reduced, and the time stability is guaranteed. Based on the numerical example, in the frequency range 2~12 GHz, the computing accuracy of the proposed method is basically the same with the SFDTD method and HFSS, and the computing effectiveness of the proposed method is superior to them.

The experimental data used to support the findings of this study are available from the corresponding author upon request.

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

This work was supported by China Postdoctoral Science Foundation (2016T90995) and the National Natural Science Foundation of Jiangsu Province (BK20150715).