International Journal of Antennas and Propagation

Volume 2017, Article ID 5817380, 6 pages

https://doi.org/10.1155/2017/5817380

## The CFS-PML for 2D Auxiliary Differential Equation FDTD Method Using Associated Hermite Orthogonal Functions

^{1}College of Defense Engineering, PLA University of Science and Technology, Nanjing 210007, China^{2}Jiangsu Regulatory Bureau of Nuclear and Radiation Safety, Nanjing 210019, China^{3}National Key Laboratory on Electromagnetic Environmental Effects and Electro-Optical Engineering, PLA University of Science and Technology, Nanjing 210007, China

Correspondence should be addressed to Feng Lu; moc.361@67ulgnef

Received 25 October 2016; Revised 18 February 2017; Accepted 26 March 2017; Published 18 May 2017

Academic Editor: Sotirios K. Goudos

Copyright © 2017 Feng Jiang 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

The complex frequency shifted (CFS) perfectly matched layer (PML) is proposed for the two-dimensional auxiliary differential equation (ADE) finite-difference time-domain (FDTD) method combined with Associated Hermite (AH) orthogonal functions. According to the property of constitutive parameters of CFS-PML (CPML) absorbing boundary conditions (ABCs), the auxiliary differential variables are introduced. And one relationship between field components and auxiliary differential variables is derived. Substituting auxiliary differential variables into CPML ABCs, the other relationship between field components and auxiliary differential variables is derived. Then the matrix equations are obtained, which can be unified with Berenger’s PML (BPML) and free space. The electric field expansion coefficients can thus be obtained, respectively. In order to validate the efficiency of the proposed method, one example of wave propagation in two-dimensional free space is calculated using BPML, UPML, and CPML. Moreover, the absorbing effectiveness of the BPML, UPML, and CPML is discussed in a two-dimensional (2D) case, and the numerical simulations verify the accuracy and efficiency of the proposed method.

#### 1. Introduction

In the conventional finite-difference time-domain (FDTD) method [1, 2], the time step is constrained by the Courant-Friedrichs-Lewy (CFL) stability condition. When fine structures such as thin material and slot are simulated, more computer memory and computation time are required. In order to eliminate the CFL stability condition, some unconditionally stable FDTD methods have been developed such as alternating-direction implicit (ADI) method [3], Crank-Nicolson method [4], locally one-dimensional method [5], and Weighted Laguerre Polynomials (WLP) FDTD method [6]. Recently, the Associated Hermite (AH) FDTD methods to model wave propagation have been introduced in [7, 8].

The perfectly matched layer (PML) absorbing boundary conditions (ABCs) introduced by Berenger [9] have been widely used for truncating FDTD domains. Some PML variants have been proposed to improve the absorbing effectiveness for various electromagnetic waves, like modified PML (MPML), uniaxial anisotropic PML (UPML), generalized PML (GPML), and so forth. Recently, UPML-ABC for AH-FDTD method [10] is used in conductive medium. According to Kuzuoglu and Mittra in [11], the complex frequency shifted (CFS) PML [12, 13] is the most accurate in all the PML. The frequency-domain coordinate-stretching variable of CFS-PML (CPML) has three adjustable variables, while there are two adjustable variables for UPML-ABC. The reflection error of CPML can be greatly reduced from adjusting variables, and it has been implemented in the Cartesian coordinate, periodic structures, cylindrical coordinates, dispersive materials, WLP-FDTD, and so on.

In this paper, a 2D CPML for auxiliary differential equation (ADE) FDTD method using AH orthogonal functions is proposed. It is shown that the constitutive parameters of CPML are complicated, and if the method in [7] is used directly here, the second derivative field components would be involved and the final matrix equations will also be complex. In order to get a simple and easy formula, the auxiliary differential variables are introduced. Based on the ADE technique [14, 15] and Galerkin temporal testing procedure, one relationship between field components and auxiliary differential variables is derived. According to auxiliary differential variables and Maxwell’s equations of CPML, the other relationship between field components and auxiliary differential variables is derived. Then the formulations of all orders of AH functions are obtained to calculate the magnetic field expansion coefficients. According to [7], the matrix equations of CPML, Berenger’s PML (BPML), and free space can be unified. At last, the electric field expansion coefficients can also be obtained, respectively. To validate the efficiency of the proposed method, a 2D case is calculated. And the efficiency of the proposed method is verified through the comparison with BPML and UPML ABCs.

#### 2. Formulation

With free space, the frequency-domain Maxwell’s equations of CPML for 2D model case arewhere is the electric permittivity and is the magnetic permeability of free space; , . For CPML, and are assumed to be positive real and is real and ≥1.

An orthonormal set of basis functions can be defined aswhere are Hermite polynomials, , is a time-translating parameter, and is a time-scaling parameter. Choosing a finite order of basis functions [7] and proper parameters for and , and then using these transformed basis functions, the causal field components, taking for example, can be expanded as

The time derivative of the th order AH function is

According to (4), the first derivative of field components with respect to is [7]where .

Using ADE scheme [13, 14], four auxiliary differential variables are introduced:With the transition relationship from frequency domain to time domain, (6) can be written asApplying (3) and (5) to (10), the field functions in (10) can be expanded byMultiplying both sides by and integrating over , we getApplying the same above procedure with (7)–(9), we haveSimilar to [7, 8], we can rewrite (12)-(13) in a matrix form:whereEquation (14) can be rewritten in a new matrix form:Applying (6)–(9) to (1) and transferring from frequency domain to time domain, we haveApplying the same above procedure with (17), we havewhere

Using central difference scheme, substituting (16) into (18), grafting (18), and using , we have where , , , , , and and are the lengths of the cell edge where the electric fields are located. and are the distances between the center nodes where the magnetic fields are located. For the above subscripts for and in (20)–(22), is not a real position but an array index of each field variable, as shown in Figure 1 of [16].

In order to form the matrix equations only containing magnetic field, we apply (20)-(21) to (22): where is unit matrix.

From (23), only magnetic field vector variables remained. And it can also be found that each magnetic field variable is related to the four adjacent magnetic fields. If in (1), the final matrix equations of CPML will degrade to the matrix equations of free space in AH-FDTD method. If and in (1), the final matrix equations of CPML will degrade to the matrix equations of BPML in AH-FDTD method. So the matrix equations of CPML and BPML and free space can be unified. And this greatly facilitates the programming. Finally, a banded sparse coefficient matrix is obtained like [7], which contains all the points in free space and CPML ABC. And the paralleling-in-order solution scheme in [8] is applied to indirectly but efficiently calculate all of the expansion coefficients. If all of the expansion coefficients of the magnetic field are calculated, the expansion coefficients of the electric field can be obtained from (20)-(21). Finally, , , and can be reconstructed by using (3).

#### 3. Numerical Demonstration

In order to validate the efficiency of the presented method, wave scatter from a PEC medium is simulated. The BPML, UPML, and CPML ABCs are used to truncate the FDTD domains. The PML constitutive parameters are scaled using th order polynomial scaling [17]:where indicates the distance from the dispersive medium-PML interface into the PML, is the depth of the PML, and is the order of polynomial.

As illustrated in Figure 1, the computational domain is truncated by 10 additional PML in -direction and -direction, respectively. And we choose = = = = 10 mm, = 12 mm, = 6 mm, = 10 mm, = 2 mm, = 2 mm, = 13 mm, and = 15 mm (in Figure 1), which results in a cells lattice with mm. For the PEC rectangle’s scatter, we choose = 16 mm, = 8 mm, and = 10 mm.