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The CFS-PML for 2D Auxiliary Differential Equation FDTD Method Using Associated Hermite Orthogonal Fu 时间:2017-06-26 11:37   来源:未知   作者:admin   点击: 次 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.