AbstractâVolume integral equations (VIEs) are usually solved by the method of moments (MoM) with the Schaubert-Wilton-. Glisson (SWG) basis function.
A Point-Matching Method for Solving Volume Integral Equations with Inhomogeneous Media J. Zhang, Z. S. Wang, G. Sun, J. H. Zhou, and M. S. Tong Department of Electronic Science and Technology Tongji University, Shanghai, China
Abstract—Volume integral equations (VIEs) are usually solved by the method of moments (MoM) with the Schaubert-WiltonGlisson (SWG) basis function. The SWG basis function requires conformal meshes in geometric discretization and may be inconvenient for inhomogeneous problems. In this work, we propose a point-matching method for solving the VIEs without using any basis function. The method can allow an inhomogeneity of materials in each tetrahedron and may greatly facilitate the meshing of geometries. A numerical example for electromagnetic scattering by a multilayered dielectric object is presented to demonstrate the effectiveness of the method.
as [1]
G(r, r ) · JV (r )dr E(r) = Einc (r) + iωμ0 V −∇ × G(r, r ) · MV (r )dr , r ∈ V (1) V G(r, r ) · MV (r )dr H(r) = Hinc (r) + iω0 V +∇ × G(r, r ) · JV (r )dr , r ∈ V (2) V
inc
I. I NTRODUCTION Volume integral equations (VIEs) are necessary for solving electromagnetic (EM) problems with inhomogeneous or anisotropic materials by integral equation approach [1]. The VIEs are usually solved by the method of moments (MoM) with the Schaubert-Wilton-Glisson (SWG) basis function [2]. The SWG basis function is defined over a pair of tetrahedrons with a common face and requires conformal meshes which cannot stride across material boundaries in geometric discretization. However, why do we need to bother the VIEs if we must locate the material interfaces and discretize the materials without striding across the interfaces? The surface integral equations (SIEs) which only require to discretize the material boundaries are more convenient in this case. In this work, we propose a point-matching method to solve the VIEs whose value can be highlighted. The method does not use any basis function and only works with individual tetrahedrons instead of tetrahedron pairs. Also, the method can allow each tetrahedron to include inhomogeneous materials or stride across material boundaries without enforcing a conformity in geometric discretization. Due to this distinctive feature, the method can more conveniently handle arbitrarily inhomogeneous or anisotropic structures though requiring an efficient treatment of hypersingularity in the integral kernels [1]. A numerical example for electromagnetic scattering by a multilayered dielectric sphere is presented to demonstrate the method and its robustness can be observed. II. VOLUME I NTEGRAL E QUATIONS (VIE S ) Consider the EM scattering by a three-dimensional (3D) inhomogeneous object embedded in the free space with a permittivity 0 and a permeability μ0 , the VIEs can be written
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inc
where E (r) and H (r) are the incident electric and magnetic fields, respectively, while E(r) and H(r) are the total electric and magnetic fields inside the object, respectively. Also, the integral kernel ∇∇ G(r, r ) = I + 2 g(r, r ) (3) k0 is the 3D dyadic Green’s function in which I is the identity √ dyad, k0 = ω μ0 0 is the free-space wavenumber, and ik0 R /(4πR) is the 3D scalar Green’s function. In g(r, r ) = e addition, R = |r − r | is the distance between an observation point r and a source point r in the scalar Green’s function and JV (r ) = iω[0 − (r )]E(r )
(4)
MV (r ) = iω[μ0 − μ(r )]H(r )
(5)
are the induced volumetric electric and magnetic current densities inside the object with a permittivity (r ) and a permeability μ(r ), respectively. From the above equations, we can solve the unknown functions which could be the current densities or total fields or flux densities. III. P OINT-M ATCHING M ETHOD FOR S OLVING THE VIE S We propose a point-matching method to solve the VIEs. In the method, we first select some representative discrete nodes in each tetrahedron and then transform the VIEs into matrix equations by performing a point-matching procedure over those nodes. These nodes are chosen as equally as possible in each tetrahedron without the constraint of a quadrature rule, so the method is different from the Nystr¨om method in which the nodes are determined by a quadrature rule. The current densities instead of the total fields or flux densities are chosen as unknown functions to be solved in the method. From the preceding VIEs, we can see that there are not explicit material
AP-S 2013
f = a1 + a2 u + a3 v + a4 w = f (u, v, w)
(6)
where a1 , a2 , a3 , and a4 are the unknown coefficients to be determined. We can then obtain a set of equations by matching the unknown function on those chosen nodes within the same region, i.e. fl = a1 + a2 ul + a3 vl + a4 wl ,
l = 1, 2, · · · , L
(7)
where fl (l = 1, 2, · · · , L) are the values of unknown function on those chosen nodes and L is the total number of chosen nodes in the region. The set of equations can only be solved with the least square method (LSM) by defining a functional F =
L
{ξl [fl − (a1 + a2 ul + a3 vl + a4 wl )]2 }
(8)
l=1
which is the sum of weighted residue errors for the equations (ξl is the lth weight). Minimizing the functional as done in the moving least square (MLS) approximation for an unknown function in meshfree methods [3], we can determine the unknown coefficients and express the unknown function as f (u, v, w) = φT · f
(9)
where f = [f1 , · · · , fL ]T and φ = [φ1 , · · · , φL ]T is named shape function. With the shape function, we can represent the unknown current density within the same region of a tetrahedron as L φl (r )Jl (10) J(r ) = l=1
Jlu u ˆ
Jlv vˆ
+ + Jlw w ˆ is the value of unknown where Jl = current density at the lth chosen node in the region. After the unknown current density is represented, the matrix elements can be evaluated numerically when the observation node is outside a source tetrahedron. If the observation node is inside a source tetrahedron, we have to evaluate the hypersingular integrals resulting from the dyadic Green’s function and we have developed the relevant technique in recent years [1]. IV. N UMERICAL E XAMPLE To demonstrate the effectiveness of the proposed method for solving the VIEs, we present one numerical example for EM scattering by a multilayered concentric dielectric sphere as shown in Figure 1(a). It is assumed that the incident wave is a plane wave with a frequency f = 300 MHz and is propagating along the −z direction in free space. We calculate the bistatic
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15
z
VV, Exact HH, Exact VV, VIEs HH, VIEs
10
a4 a3
o εra1 1 a2
εr2
x
εr3
y εr 4
Bistatic Radar Cross Section (dB)
parameters of the object in the integrals and this characteristic allows the geometric discretization to stride across material boundaries or each tetrahedron to include different materials. To evaluate the integrals of the above equations, we have to represent the unknown current densities at an arbitrary point within a tetrahedron. This can be done by an interpolation scheme based on those chosen nodes within a local region. For example, we can approximate an unknown function f with a first-order polynomial function (linear approximation) in a local coordinate system (u, v, w), i.e.
5
0
−5
−10
−15
0
(a)
20
40
60
80 100 θ (Degrees)
120
140
160
180
(b)
Fig. 1: EM scattering by a four-layer concentric dielectric sphere. (a) Geometry. (b) Bistatic RCS solutions. radar cross section (RCS) observed along the principal cut (φ = 0◦ and θ = 0◦ − 180◦) for the scatterer with both vertical polarization (VV) and horizontal polarization (HH). We consider a four-layer concentric dielectric sphere which is piecewise-homogeneous but treated as an inhomogeneous object. The radii of four material interfaces are a1 = 0.2λ, a2 = 0.25λ, a3 = 0.3λ, and a4 = 0.35λ, respectively, where λ is the wavelength in free space. The relative permittivities of dielectric materials in the four layers are r1 = 3.0, r2 = 5.0, r3 = 2.0, and r4 = 4.0, respectively (the relative permeability μr = 1.0 is assumed for dielectric materials). The scatterer is discretized into 2735 tetrahedral elements and some elements could stride across the material interfaces. Fig. 1(b) plots the bistatic RCS solutions of the scatterer obtained from the method and we can see that they are close to the corresponding exact solutions. V. C ONCLUSION In the conventional MoM for solving the VIEs, the SWG basis function requires conformal meshes without striding across material boundaries in geometric discretization and this constraint could dramatically weaken the importance of VIEs. In this work, we develop a point-matching method to solve the VIEs. The method does not rely on any basis and testing functions and can allow an inhomogeneity of materials in each tetrahedron, resulting in much facilitation in meshing geometries. A numerical example for EM scattering by a multilayered dielectric sphere is presented to illustrate the method and good results have been observed. ACKNOWLEDGEMENT This work was supported by the Program of Pujiang Talents, Shanghai, China, with the Project No. 12PJ1408600. R EFERENCES [1] W. C. Chew, M. S. Tong, and B. Hu, Integral Equation Methods for Electromagnetic and Elastic Waves, Morgan & Claypool: San Rafael, 2008. [2] D. H. Schaubert, D. R. Wilton, and A. W. Glisson, “A tetrahedral modeling method for electromagnetic scattering by arbitrary shaped inhomogeneous dielectric bodies,” IEEE Trans. Antennas Propagat., vol. AP-32, no. 1, pp. 77-85, 1984. [3] M. S. Tong, “Meshfree solutions of volume integral equations for electromagnetic scattering by anisotropic objects,” IEEE Trans. Antennas Propagat., vol. 60, no. 9, pp. 4249-4258, 2012.
