International Journal of Antennas and Propagation

Volume 2016, Article ID 6893915, 9 pages

http://dx.doi.org/10.1155/2016/6893915

## Implementation of Associated Hermite FDTD Method in Handling INBCs for Shielding Analysis

National Key Laboratory on Electromagnetic Environmental Effects and Electro-optical Engineering, PLA University of Science and Technology, Nanjing 210007, China

Received 17 December 2015; Accepted 6 April 2016

Academic Editor: Rodolfo Araneo

Copyright © 2016 Lihua Shi 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

For modeling of electrically thin conductive shields, the unconditionally stable Associated Hermite (AH) FDTD scheme is combined with the impedance network boundary conditions (INBCs) in this paper. The two-port network equations of INBCs in frequency domain are transformed into AH domain to represent the relationship of tangential components of the electric and magnetic fields at faces of the shield. The established AH-INBCs shielding boundaries are incorporated into a set of implicit equations to calculate the expansion coefficients vectors of electromagnetic fields in the computational domain. The method is free of CFL condition and no convolution integral operation for solving the conventional INBCs-FDTD is involved. Numerical example shows that, compared with analytical solutions and conventional FDTD method, the proposed algorithm is efficient and accurate.

#### 1. Introduction

The finite-difference time-domain method [1, 2] has been widely used for many transient electromagnetic (EM) problems. In modeling of the electrically thin conductive layers (TCLs) by FDTD method [3, 4], it is needed to use sufficiently fine spatial grid size to achieve good accuracy and very small time step is required to meet the well-known Courant-Friedrich-Levy (CFL) stability condition [2]. To solve this problem, many approaches such as spatial filtering [5], subgridding technologies [6, 7], and subcell methods [8–14] have been developed. One of efficient subcell methods is based on the impedance network boundary conditions (INBCs) [10] and developed in recent years [11–16]. This method allows the shield region to be eliminated from the FDTD computational domain and to be reduced to a sheet of INBCs shielding boundary to model the coupling EM field tangential components on the shield surfaces. The self- and mutual impendences are used to represent the relationship of field components on both sides of the thin layer in frequency domain analytically. The time-domain transient expressions can be obtained by using inverse Fourier transform and recursive convolution approaches [4, 9–13]. The recursive process can be achieved by establishing a parametric model of the transient impedances, such as using a series of rational functions with vector fitting method [17], or incorporated with delay extraction techniques [18, 19] to reduce the complexity of the rational model to approximate the frequency-dependent impedances. Under these cases, the FDTD methods need to be modified to handle frequency-dependent parameters and it is still required to meet the CFL stability condition.

In this paper, the unconditionally stable Associated Hermite (AH) FDTD proposed recently [20] is incorporated with INBC two-port network equations to analyze the shielding problems. The main idea for AH-FDTD method is to use AH orthogonal functions as temporal basis and testing functions to expand Maxwell’s equations. By using Galerkin’s principle, the time variable is eliminated from the calculation, and the time step is not limited by CFL stability condition. Incorporated with this unconditionally stable method, INBCs technologies are rendered much more efficient than previous method. Frequency-domain impendences can be directly transformed into AH transformation matrixes [21]. By using the unique isomorphism of the AH function with its Fourier transform [22, 23], the INBCs boundary equations are then realized by multiplying an AH transformation matrix. A set of implicit equations incorporated with the AH-INBCs equations is derived to calculate the electromagnetic field expanding coefficients. The time-domain waveform or frequency-dependent results such as shielding effectiveness can be reconstructed directly from these expanding coefficients.

This paper is organized as follows. In Section 2, the INBCs in AH domain (AH-INBCs) are derived by using the AH transformation matrix and the unconditionally stable scheme based on the AH-INBCs (AH-INBCs-FDTD) is implemented in two-dimensional (2D) code. In Section 3, we adopted the same case as in [9] with the field penetration into thin multilayered conductive shells to calculate the shielding effectiveness and a comparison of our numerical results with the conventional FDTD method and the analytical solution is carried out. Finally, some conclusions are given in Section 4.

#### 2. Formulation of INBC in AH Domain

##### 2.1. AH Transformation Matrix for Frequency-Dependent Function

Given an input signal and output signal , the time-domain convolution integral can be represented by an -dimensional AH transformation matrix [21], where and ( or ) is the th Associated Hermite (AH) functions [18]. is transformed time variables, where is a time-translating parameter and is a time-scaling parameter. And then we can obtain the relationship between and in AH domain as follows:where the -tuple representation and , which both consisted of AH expanding coefficients. One can also calculate from the frequency-domain data by using isomorphism property of AH function [21]. For a proper approximation with AH expansion coefficients, the parameters selection can be selected according to [20, 24]: where and are the time and frequency support of the AH basis , respectively. One should note that the bigger could achieve more accuracy as well as more memory storage. Therefore, (2) means the minimum number of AH functions for a given time and frequency support.

Here, we give an example to illustrate the effectiveness of the AH transformation matrix for reconstructing the output response from an input signal as shown in Figure 1. Given the time and frequency support s and Hz, the parameters for AH basis can be obtained as and from formulations (2)-(3), and time-translating parameter is set as 0.5 to guarantee the causal responses. Figure 1(a) shows an input signal, which is a Gaussian-differential pulse. Figure 1(b) is the AH functions with the parameters selected above. The -tuple representation can be calculated based on these functions. If the frequency-dependent system function is , then its AH transformation matrix can be calculated. Figure 1(c) shows the absolute value for . Finally, the output can be reconstructed from -tuple representation , which can be calculated from (1). The reconstructed waveform fits well with the result from IFFT method as shown in Figure 1(d).