International Journal of Antennas and Propagation

Volume 2014, Article ID 720947, 9 pages

http://dx.doi.org/10.1155/2014/720947

## VIE-FG-FFT for Analyzing EM Scattering from Inhomogeneous Nonmagnetic Dielectric Objects

State Key Laboratory of Millimeter Waves, Southeast University, Nanjing 210096, China

Received 5 September 2014; Accepted 9 November 2014; Published 24 December 2014

Academic Editor: Stefano Selleri

Copyright © 2014 Shu-Wen Chen 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

A new realization of the volume integral equation (VIE) in combination with the fast Fourier transform (FFT) is established by fitting Green’s function (FG) onto the nodes of a uniform Cartesian grid for analyzing EM scattering from inhomogeneous nonmagnetic dielectric objects. The accuracy of the proposed method is the same as that of the P-FFT and higher than that of the AIM and the IE-FFT especially when increasing the grid spacing size. Besides, the preprocessing time of the proposed method is obviously less than that of the P-FFT for inhomogeneous nonmagnetic dielectric objects. Numerical examples are provided to demonstrate the accuracy and efficiency of the proposed method.

#### 1. Introduction

The volume integral equation (VIE) method [1] based on the method of moments (MoM) [2] is one of the efficient methods to analyze electromagnetic (EM) scattering from inhomogeneous dielectric objects. As is well known, for the traditional VIE-MoM, both the storage requirement and the computational complexity of a matrix-vector multiplication when an iterative method is applied are proportional to , where denotes the number of unknowns. Therefore, the VIE-MoM is not suitable for the direct analysis of EM scattering from electrically large and inhomogeneous dielectric objects.

One of approaches for improving the efficiency of the VIE-MoM is the VIE in combination with the fast Fourier transform (FFT), and it already has several implementations, such as the VIE-AIM [3, 4], the VIE-P-FFT [5, 6], and the VIE-IE-FFT [7, 8], which are simply called the FFT-based methods. These implementations are all transplanted from the corresponding versions [9–12] for the surface integral equation (SIE) [13]. Not long ago, a new realization, the FG-FFT, of the SIE in combination with the FFT for the electric field integral equation (EFIE) was proposed [14] and soon extended to the combined field integral equation (CFIE) [15].

In this paper, the FG-FFT for the SIE will be extended to the VIE for analyzing EM scattering from inhomogeneous nonmagnetic dielectric objects, and resultant method is simply called the VIE-FG-FFT. The remainder of this paper is organized as follows. In Section 2, the VIE-FG-FFT is presented in detail. In Section 3, some numerical examples are provided to demonstrate the accuracy and efficiency of the VIE-FG-FFT. Finally, the conclusion is given in Section 4. In this paper, the time convention is assumed and suppressed.

#### 2. Formulation

##### 2.1. The Volume Integral Equation

The permittivity and permeability of the free space are denoted by and , respectively. Let denote the volumetric domain occupied by an inhomogeneous nonmagnetic dielectric object with relative permittivity and relative permeability (meaning nonmagnetic).

Let be the incident electric field and the scattered electric field; then the total electric field can be expressed as the sum of and : and the volume integral equation (VIE) on for the total electric field can be rigorously expressed as [16] where and , which implies that where is the electric flux density of and Therefore, we have where is the outer boundary surface of .

It should be pointed out that the second term in the right-hand side of (5) cannot be ignored in the strict sense (see the second paragraph of Section 2 in [17]). However, this term will force one to introduce “half” basis function, which will be seen in the following.

##### 2.2. Buiding the MoM Model

The electric flux density can be chosen as the unknown function because it is continuous along the normal direction of the medium interface. After is discretized by using tetrahedrons, can be expanded with the SWG functions [1]: where “” and “” mean “full SWG function” and “half SWG function,” respectively, and the total number of the basis functions is . A full SWG function is defined on the union of a pair of tetrahedrons and that share a common face as follows: whose geometry is shown in Figure 1, while a half SWG function is defined only on a single tetrahedron . For convenience, we use to denote the common face of and , as shown in Figure 1.