UPML‐ABC and TF/SF boundary for unconditionally stable AH‐FDTD method in conductive medium

Abstract

For analysing the scattering problem in conductive medium with the new unconditionally stable associated Hermite (AH) finite-difference time-domain (FDTD) method, the formulation of uniaxial anisotropic perfectly matched layer absorbing boundary condition (UPML-ABC) and the total-field/scattered-field (TF/SF) implementation scheme are deduced. By virtue of AH differential transformation matrix and the eigenvalue transformation, frequency-dependent parameters of the lossy medium can be well treated in AH-UPML, without using any additional auxiliary variables. Numerical verification for the scattering of buried conductor in lossy half-space is performed. Compared with Mur's ABC, UPML increases the absorbing performance by more than 80 dB in AH-FDTD analysis.Introduction: The finite-difference time-domain (FDTD) method is applied to analyse many electromagnetic problems. However, for the fine structures, a small time step must be chosen to meet the CourantFriedrich-Levy condition [1]. To eliminate the stability condition constraint, a new unconditionally stable FDTD method based on associated Hermite (AH) functions was proposed [2] recently and developed in [3]. This method employs the similar approach of space discretisation like the conventional FDTD scheme but uses AH expansion and testing functions in the time domain under the framework of the method of moment. In previous reports [2,3], only the Mur's first-order absorbing boundary condition was derived and the volume of electric and magnetic current densities is enforced as the electromagnetic sources. Here, the uniaxial anisotropic perfectly matched layer (PML) absorbing boundary condition (UPML-ABC) [4] and total-field/scattered-field (TF/SF) formulations [5] are developed for application in AH-FDTD. Unlike the conventional implementation of UPML, no additional auxiliary variable is used in AH-UPML due to the introduction of AH differential transformation matrix, which can directly deal with the frequency dependence of the lossy term. Also it can be readily implemented like paralleling-in-order solution scheme by using eigenvalue transformation [3]. For the treatment of total field/scattered field in free space domain, the coefficients of the incident field can be added to the source item on the right side of the original five-diagonal banded equations [2, 3]. Numerical example with terminating inhomogeneous lossy media shows that the time-domain results for the proposed method are consistent with the FDTD method very well and the reflection error deceased by almost 80 dB compared with that of Mur's first-order ABC.

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

“…In Figure 3, it is noted that the maximum relative error is −43 dB, −58 dB, and −70 dB for the BPML, UPML, and CPML, respectively. And it is obvious that, with the CFS factor, the CPML is superior to BPML and UPML [10]. In Figures 2 and 3, the computation time is much higher than the time-support of source wave.…”

“…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.…”

New hybrid FDTD algorithm for electromagnetic problem analysis*

“…[3] Among them, ADI [4][5][6] and LOD, [7,8] orthogonal expansions are implicit methods. [9][10][11][12] Namely, the inverse of the matrix is necessary to be solved in the process of iteration. Although these implicit methods are unconditionally stable, they also have some disadvantages.…”

An Unconditionally Stable Cylindrical FDTD Method to Analyze the EM Ground Wave Propagation

“…So for the problems with fine structures, it should choose a small time step and then a large number of marching-on-in-time steps for simulation, which is a time-consuming process. To eliminate the stability condition for fine structure, many unconditionally stable (US) methods were proposed, such as alternating direction implicit (ADI) method [3] and orthogonal function-based methods [4][5][6][7][8][9]. The orthogonal function-based methods can be classified into two main categories: the marchingon-in-order scheme and the paralleling-in-order scheme.…”