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International Journal of Antennas and Propagation
Volume 2014, Article ID 720947, 9 pages
http://dx.doi.org/10.1155/2014/720947
Research Article

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 [912] 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.

Figure 1: The geometry of the SWG function.

In fact, half SWG functions are applied only at the outer boundary of , and hence the number is just that of the triangular elements over . In this paper, we have a convention that a half SWG function is so defined on that when .

When (6) is substituted into (1) and after the Galerkin procedure is applied, the VIE-MoM matrix equation can be built as follows: where

Note that, in our VIE-MoM model, the restriction of on a tetrahedral region will be considered as a constant, and the following symbol abbreviations will be applied: Then the elements of matrix have the following expressions: where Note that in (11) when and are properly separated and that in (12) when and are properly separated: where Note that in (14) when and are properly separated and that in (15) when and are properly separated.

2.3. The Frame for the VIE-FG-FFT

In this section, the FG-FFT technology is introduced into the VIE-MoM for both reducing memory requirement and improving computational efficiency.

The entire MoM matrix can be split into two parts: the near-field matrix and the far-field matrix , and where the is a sparse matrix (the identification of a near-element will be presented in the second paragraph of Section 3), which is obtained by forcing all “far elements” of to be equal to zero, and can ultimately expressed in such a form as follows: where , , , and are all sparse matrices which will be constructed in Section 2.4 and where the head mark “” implies matrix elements being 3D vectors; is a triple Toeplitz matrix related to Green’s function; the superscript indicates matrix transpose.

When an iterative solver is applied, the matrix-vector product will be performed by means of where is directly calculated, while can be speeded up by means of the FFT through (18). In this way, the VIE-FG-FFT can reduce the memory requirement and the computational complexity to and theoretically (also see [9] for more detailed analysis).

2.4. Fitting Green’s Function

In this section, matrices , , , and in (18) will be constructed. First, let a uniform Cartesian grid enclose the given volumetric region . Use , , and to denote the three grid spacing sizes in the directions , , and , respectively. In this paper, however, the convention is always selected.

In a uniform Cartesian grid, an expansion box (or simply box) is defined as a cube-like collection composed of nodes. When , includes nodes and is called its expansion order (or simply order), written as . Figure 2 illustrates a box of order 2.

Figure 2: An expansion box of order 2.

Assume that is a box centered at of radius and is a fixed point within . Let Green’s function be represented into where the coefficients are to be determined and is an arbitrary point outside the box , as shown in Figure 3.

Figure 3: A 3D representation of the matching Green’s function.

Now we select a spherical surface with the center at and the radius slightly larger than , and select a group of sample points on the surface [18], where the number ought to be significantly greater than that of the nodes within . In this paper, wavelength, which was adopted in [15]. Then we match Green’s function values at these sample points. The resultant systems of matrix equations can be solved by the least-square method [19]. In this paper, the order of a box is selected to be 2, and at this time the number of sample points on the testing spherical surface is selected to be 120.

Without loss of generality, assume that and ( and ) are in box (). Calculations of MoM matrix elements ultimately come down to calculations on the Gaussian points. We specify using Gaussian points in a tetrahedral region and Gaussian points in a triangular region. For convenience, the correspondences between integral regions and Gaussian points and weights that will be used are all listed in Table 1. When the testing element and the source element are separated by a proper distance, all the first terms in the right-hand side of (11), (12), (14), and (15) will vanish. At this case, we can easily obtain the following approximate expressions: where the coefficients are calculated by using the following formulae:

Table 1: Correspondences between integral regions and Gaussian points and weights.

Substituting (21) into (11)–(16), we can easily obtain formula (18). It should be pointed out that performing matrix-vector product once based on (18) requires at least 8 FFTs.

2.5. On the Preprocessing Time

Here, the preprocessing time means the time taken by generating the coefficients in (21). Clearly, the preprocessing time is proportional to the corresponding computational complexity. For general cases, from the coefficient formulae in Section 2.4, it can be evaluated that the number of required float multiplications is about

In the VIE-P-FFT, the corresponding coefficients are generated by utilizing “projecting coefficients,” which requires solving the least-square problem with multiple right-hand terms. If the order of a box is selected to be 2, containing 27 grid points, and the number of the sample points on the testing spherical surface is selected to be , then the matrix of the least-square problem in whether the VIE-P-FFT or the VIE-FG-FFT is a complex matrix. Further, if the SVD process of the matrix is ignored, then the least-square solution for a right-hand term requires about complex multiplications or about real multiplications because a complex multiplication requires at least 3 real multiplications. It is easily known from the projection scheme introduced in [20] that, for generating the corresponding coefficients, the number of required float multiplications is about which is usually much larger than .

It can be concluded from the above analysis that the preprocessing time of the VIE-FG-FFT is obviously less than that of the VIE-P-FFT for inhomogeneous nonmagnetic dielectric objects.

3. Numerical Results

In this section, several examples are provided to demonstrate the validity, accuracy, and efficiency of the VIE-FG-FFT. In all the examples, the expansion order is always chosen as 2, and the Cartesian grid spacing sizes in different directions are always selected to be the same as each other; namely, . Besides, denotes the wavelength in free space. Our computing platform is a DELL T5400 workstation with 8 cores of clock frequency 3 GHz, and the FFT codes are from the FFTW [21].

Assume that the testing function and the source function are located within boxes and , respectively. Then the corresponding matrix element is identified as a near element if and only if the distance between the center of and that of is smaller than the sum of the radius of and that of .

3.1. Example A: Homogeneous Ball

A dielectric spherical shell with the relative permittivity 2.0 is considered, as shown in Figure 4. This object is discretized by using 23174 tetrahedrons with the average edge length , producing 51231 unknowns. The Cartesian grid spacing size is chosen as .

Figure 4: Bistatic RCS curves of the dielectric spherical shell in Example A.

The bistatic RCS curve obtained by the FG-FFT is compared with the Mie series solution in Figure 4. It can be seen that the two curves coincide very well, which demonstrates the validity of the FG-FFT.

3.2. Example B: Homogeneous Slab

A homogeneous hollow dielectric slab with the relative permittivity 2.5 is considered, as shown in Figure 5. The direction of the incident wave is . This object is discretized by using 19301 tetrahedrons with the average edge length , producing 42862 unknowns. The Cartesian grid spacing size is chosen as , , and , respectively.

Figure 5: The dimensions of the hollow dielectric slab in Example B.

The bistation RCS curves obtained by the direct MoM, the FG-FFT, and the other FFT-based methods (the P-FFT, the AIM, and the IE-FFT) are all shown in Figure 6, and the are recorded in Table 2, which are calculated by the following formula: where is the number of sampling theta azimuth angles. In this example, .

Table 2: RMSEs of the RCSs in Example B.
Figure 6: Bistatic RCS curves of the homogeneous dielectric slab in Example B.

It can be seen from Figure 6 and Table 2 that when the grid spacing size is , the four RCS curves obtained by the FG-FFT, P-FFT, IE-FFT, and AIM agree well with that by the direct MoM. However, when increasing the grid spacing size gradually, both the FG-FFT and P-FFT can keep consistent with the MoM, while both the P-FFT and IE-FFT do not.

It can be concluded from the above experiments that compared with both the AIM and IE-FFT, both the FG-FFT and P-FFT are more accurate and not sensitive to the Cartesian grid spacing size.

3.3. Example C: Partitioned Homogeneous Slab

A partitioned homogeneous hollow dielectric slab with 4 different relative permittivities is considered, as shown in Figure 7. The direction of the incident wave is . This object is discretized by using 17714 tetrahedrons with the average edge length , producing 39927 unknowns. The Cartesian grid spacing size is chosen as , , and , respectively.

Figure 7: The dimensions of the hollow dielectric slab in Example C.

The bistatic RCS curves obtained by the direct MoM, the FG-FFT, and other FFT-based methods are all shown in Figure 8, and the are recorded in Table 3. In this example, for the .

Table 3: RMSEs of the RCSs in Example C.
Figure 8: Bistatic RCS curves of the partitioned homogeneous dielectric slab in Example C.

It is again seen from Figure 8 and Table 3 that the accuracy of both the FG-FFT and P-FFT is higher than that of both the IE-FFT and AIM.

3.4. Example D: Inhomogeneous Slab

Here the dielectric slab in Figure 5 is again considered. The direction of the incident wave is kept unchanged. But at this time, the relative permittivity is defined by where . Now, the average edge length used in the discretization is selected within .

Different discretization granularities correspond to different numbers of unknowns, as shown in Table 4. The bistatic RCS curves obtained by the FG-FFT and P-FFT for different discretization granularities are all shown in Figure 9, and the preprocessing time for the FG-FFT and P-FFT is recorded in Table 4.

Table 4: The Preprocessing time in Example D.
Figure 9: Bistatic RCS curves of the inhomogeneous dielectric slab () in Example D.

It can be seen from Figure 9 that the RCS curves for different discretization granularities are almost the same as each other. It can be known from Table 4 that the preprocessing time of the FG-FFT is obviously less than that of the P-FFT for different discretization granularities.

4. Conclusions

In this paper, a new realization of the VIE combined with the fast Fourier transform (VIE-FG-FFT) has been established to solve EM scattering from inhomogeneous nonmagnetic dielectric objects. The proposed method has been compared with several existing popular FFT-based methods, including the VIE-P-FFT, VIE-AIM, and VIE-IE-FFT. The accuracy of the VIE-FG-FFT is almost the same as that of the VIE-P-FFT and higher than that of the VIE-AIM and VIE-IE-FFT especially when increasing the grid spacing size. Besides, the preprocessing time of the VIE-FG-FFT is obviously less than that of the P-FFT for inhomogeneous nonmagnetic dielectric objects.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported in part by the National Basic Research Program of China (no. 2013CB329002) and in part by the National Nonprofit Industry Specific Research Program of China (no. 201110046-2). The authors would like to thank C. Cai and X. Li at Shenzhen Academy of Metrology and Quality Inspection, China, for their support and assistance.

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