rely on the discretization of volumetric domains which is usually tedious. Also, the method of moments (MoM) with the the. Schaubert-Wilton-Glisson (SWG) basis ...
A Meshless Scheme for Solving Volume Integral Equations with Inhomogeneous Media K. Yang and M. S. Tong Department of Electronic Science and Technology Tongji University, Shanghai, China
Abstract—The solutions of Volume integral equations (VIEs) rely on the discretization of volumetric domains which is usually tedious. Also, the method of moments (MoM) with the the Schaubert-Wilton-Glisson (SWG) basis function does not allow a discontinuous inhomogeneity of material in tetrahedral elements and has to assume a homogeneous material in each tetrahedron for continuously inhomogeneous media. We propose a novel meshless scheme which can avoid the problems in the MoM based on the Green-Gauss theorem for solving the VIEs with inhomogeneous media. Numerical examples for EM scattering by inhomogeneous objects are presented to illustrate the scheme.
written as [1] E(r) =
H(r) =
G(r, r ) · JV (r )dr Einc (r) + iωμ0 V −∇ × G(r, r ) · MV (r )dr , r ∈ V (1) V G(r, r ) · MV (r )dr Hinc (r) + iω0 V +∇ × G(r, r ) · JV (r )dr , r ∈ V (2) V
inc
I. I NTRODUCTION The volume integral equations (VIEs) are indispensable for solving inhomogeneous electromagnetic (EM) problems and their solutions require an efficient discretization for integral domains [1]. However, this could be very tedious, especially for multiscale structures which require perfectly conformal meshes in different scales. Also, the VIEs are traditionally solved by the method of moments (MoM) with the SchaubertWilton-Glisson (SWG) basis function [2] and it does not allow a discontinuous inhomogeneity of materials in each tetrahedral element. Moreover, the basis function has to assume a homogeneity of material in each tetrahedron for continuously inhomogeneous media, yielding an extra numerical error. In this work, we develop a meshless scheme which can avoid the above problems. The scheme employs discrete nodes to replace meshes in the geometric discretization and uses the moving least square (MLS) approximation to represent the unknown currents. The VIEs are transformed into matrix equations by a point-matching procedure and the resultant volume integrals in matrix elements are evaluated by changing them into boundary (surface) integrals based on the GreenGauss theorem [3] with regularized integral kernels. Numerical examples for EM scattering by inhomogeneous dielectric objects are presented to illustrate the scheme. II. VOLUME I NTEGRAL E QUATIONS Consider the EM scattering by a three-dimensional (3D) inhomogeneous object with a permittivity (r ) and a permeability μ(r ). The object is embedded in the free space with a permittivity 0 and a permeability μ0 and the VIEs can be
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inc
where E (r) and H (r) are the incident electric and magnetic field while E(r) and H(r) are the total electric and magnetic field inside the object, respectively. Also, G(r, r ) is the dyadic Green’s function and JV (r ) = iω[0 − (r )]E(r )
(3)
MV (r ) = iω[μ0 − μ(r )]H(r )
(4)
are the volumetric electric and magnetic currents inside the object, respectively. If the object has the same permeability as the background, which is usually true, then MV (r ) = 0 and the above two equations can be reduced to inc E(r) = E (r) + iωμ0 G(r, r ) · JV (r )dr , r ∈ V (5) V G(r, r ) · JV (r )dr , r ∈ V. (6) H(r) = Hinc (r) + ∇ × V
We only need to solve the first equation to obtain the unknown current JV (r ) or total electric field. III. T HE M ESHLESS S CHEME FOR S OLVING THE VIE S To solve the above VIEs with a meshless scheme, we first enclose the object using a cylinder which circumscribes the object, see Fig. 1(a), and then choose some discrete nodes inside the object. The VIEs are transformed into a matrix equation by performing a point-matching procedure based on those chosen nodes. The value of unknown current at an arbitrary point is represented with some neighboring nodes within a compact support through the moving least square (MLS) approximation. The matrix elements are then obtained by evaluating the volume integrals over the object domain. To evaluate the volume integrals without using volumetric elements, we consider the following general volume integral I= ψ(x, y, z) dV = h(x, y, z) dV (7) V −V0
Ve −V0
where ψ(x, y, z) represents an integral kernel in the object domain of the VIEs and it is redefined as ψ(x, y, z), (x, y, z) ∈ V h(x, y, z) = (8) 0, (x, y, z) ∈ V
where Γ is the boundary or surface of a volume domain Ω and np with p = x, y, z is the p component of the unit normal vector n ˆ on the boundary. If we choose z u(x, y, z) = h(x, y, t) dt (10) c
where c is an arbitrary constant, then we have z h(x, y, z) dV = h(x, y, t) dt nz dΓ I= Ve −V0
Γe +Γ0
c
(11) where Γe is the boundary of the circumscribing cylinder Ve and Γ0 is the boundary of the small cylinder V0 . The above boundary integral can be reduced to u(x, y, z)nz dΓ = u(x, y, z)nz dΓ (12) I= S0
S0 −A0
where S0 = A0 + A1 + B0 + B1 is the total area of the top and bottom of the two cylinders. This is because nz = 0 on the walls of the two cylinders and also the integral over A0 vanishes if we choose c = 0. The evaluation of the above three integrals is convenient because we have nz = 1 on A1 and B1 , nz = −1 on B0 , and z is a constant on those surfaces. We can find I by discretizing those surfaces and evaluating the corresponding u(x, y, z) with numerical integration. IV. N UMERICAL E XAMPLES We present two numerical examples for EM scattering to illustrate the novel meshless scheme. The scatterer is a threelayered dielectric sphere as shown in Fig. 1(b). It is assumed that the incident wave is a plane wave with a frequency f = 300 MHz and is propagating along −z direction in free space. We calculate the bistatic radar cross section (RCS) observed along the principal cut (φ = 0◦ and θ = 0◦ to 180◦ ) for the scatterer in both vertical polarization (VV) and horizontal polarization (HH). The radii of three interfaces in the sphere are a1 = 0.2λ, a2 = 0.3λ, and a3 = 0.4λ, respectively. In the first example, we assume a constant relative permittivity in each layer and it is r1 = 4.0, r2 = 6.0, r3 = 3.0, respectively (the relative permeability μr = 1.0 is assumed). We select 5846 discrete nodes inside the object and Fig. 1(c)
dS
Ve
P B1 V0 Γ0 B0
Observation Node
Γe
a2
a3
V
oε
A0
o
y
y
r1
ε r3 , μr3
x
x (a)
(b)
10
10
Bistatic Radar Cross Section (dB)
VV, Exact HH, Exact VV, Meshless HH, Meshless
5
0
−5
−10
−15
a1
, μr1 ε r 2 , μr 2
Q
Bistatic Radar Cross Section (dB)
by expanding the domain to the circumscribing cylinder Ve . Since the integral kernel in the VIEs is singular, we choose a small cylinder V0 enclosing the observation node and exclude this small part in the volume integral so that the integrand is regular. For the integral over the small cylinder, we can specially treat it by using the singularity subtraction technique as described in [4]. The above volume integral can be changed to a boundary integral by applying the Green-Gauss theorem [3] ∂u(x, y, z) dV = u(x, y, z)np dΓ (9) ∂p Ω Γ
z
z A1
0
20
40
60
80 100 θ (Degrees)
120
140
160
180
VV, MoM HH, MoM VV, Meshless HH, Meshless
5
0
−5
−10
−15
0
20
(c)
40
60
80 100 θ (Degrees)
120
140
160
180
(d)
Fig. 1: (a) A descriptive geometry for the meshless scheme. (b) A three-layered dielectric sphere. (c) Solutions for the first example. (d) Solutions for the second example. shows the bistatic RCS solutions of the scatterer which are close to the exact analytical solutions. In the second example, we assume that the material of each layer is continuously inhomogeneous in the radial direction and the relative permittivity in each layer can be described by r1 (a) = 2.0a + 4.0, r2 (a) = −2.0a+6.4, r3 (a) = 2.0a+2.4, respectively, where a is the radial distance from the center of the sphere. Fig. 1(d) sketches the bistatic RCS solutions of the scatterer from both the scheme and traditional MoM and they are in good agreement. V. C ONCLUSION We propose a novel meshless scheme to solve the VIEs with inhomogeneous media. The scheme changes the volume integrals in matrix elements into the boundary integrals based on the Green-Gauss theorem after regularizing the integral kernels. The merits of the scheme include the removal of volumetric discretization and allowance of arbitrary inhomogeneity in materials. Numerical examples for EM scattering by inhomogeneous dielectric objects have demonstrated the effectiveness of the scheme. R EFERENCES [1] W. C. Chew, M. S. Tong, and B. Hu, Integral Equation Methods for Electromagnetic and Elastic Waves. San Rafael, CA: Morgan & Claypool, 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] C. R. Wylie and L. C. Barrett, Advanced Engineering Mathematics, 6th Ed. New York: McGraw-Hill, 1995. [4] M. S. Tong and W. C. Chew, “Super-hyper singularity treatment for solving 3D electric field integral equations”, Microw. Opt. Technol. Lett., vol. 49, no. 6, pp. 1383-1388, June 2007.
