(MoM) with the Schaubert-Wilton-Glisson (SWG) basis function, but we propose a point-matching scheme which does not relies on any basis and testing ...
Fast Solution of Volume Integral Equations with Complex Materials J. Zhang, S. C. Yan, C. X. Yang, and M. S. Tong Department of Electronic Science and Technology Tongji University, Shanghai, China
Abstract—Accurate solution of electromagnetic (EM) problems with complex materials requires the formulation of volume integral equations (VIEs) in the integral equation approach. The VIEs are traditionally solved by the method of moments (MoM) with the Schaubert-Wilton-Glisson (SWG) basis function, but we propose a point-matching scheme which does not relies on any basis and testing functions and allows the use of nonconforming meshes. Also, the scheme enables the VIEs to be free of material parameters in the integral kernel, facilitating the incorporation with the multilevel fast multipole algorithm (MLFMA). Numerical examples are presented to demonstrate the scheme.
I. I NTRODUCTION
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Solving electromagnetic (EM) problems with complex materials requires the formulation of volume integral equations (VIEs) in the integral equation approach [1]. The complex materials could be inhomogeneous, anisotropic, nonlinear, dispersive, magnetic, or lossy-conducting and they are characterized by constitutive parameters. Traditionally, the VIEs are solved by the method of moments (MoM) with the divergenceconforming Schaubert-Wilton-Glisson (SWG) basis function [2] or curl-conforming edge basis function [3], but the integrands are material-dependent and the implementation may be inconvenient for complex materials. Also, the MoM requires conforming meshes in geometric discretization, resulting in a higher cost for preprocessing. In this work, we propose a point-matching scheme to solve the VIEs for complex materials. The scheme does not use any basis and testing functions and allows a geometric discretization of nonconforming meshes, leading to a convenient implementation. Moreover, the scheme can choose current densities instead of flux densities as the unknowns to be solved so that the integral kernels of VIEs are free of material parameters (the material parameters are implied in the current densities). The integrals of matrix elements can thus be evaluated in an identical and regular way no matter how the material property varies. The scheme is very suitable for incorporating fast algorithms like multilevel fast multipole algorithm (MLFMA) [4] because the implementation will be almost unchanged for different materials. Typical numerical examples are presented to illustrate the scheme and its merits can be observed. II. VOLUME I NTEGRAL E QUATIONS Complex materials, including lossy conductors, are penetrable and we can use the VIEs to describe their EM feature.
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Consider EM scattering by such a material body embedded in the free space with a permittivity ǫ0 and a permeability µ0 , the VIEs can be written as [1] Z G(r, r′ ) · JV (r′ )dr′ E(r) = Einc (r) + iωµ0 V Z −∇× G(r, r′ ) · MV (r′ )dr′ , r ∈ V (1) V Z G(r, r′ ) · MV (r′ )dr′ H(r) = Hinc (r) + iωǫ0 V Z +∇× G(r, r′ ) · JV (r′ )dr′ , r ∈ V (2) inc
where E (r) and Hinc (r) are the incident electric field and magnetic field, respectively, while E(r) and H(r) are the total electric field and magnetic field inside the body, respectively. The integral kernel is the dyadic Green’s function given by ∇∇ G(r, r′ ) = I + 2 g(r, r′ ) (3) k0 √ where I is the identity dyad, k0 = ω ǫ0 µ0 is the free-space wavenumber (ω is the angular frequency), and g(r, r′ ) = eik0 R /(4πR) is the scalar Green’s function in which R = |r − r′ | is the distance between an observation point r and a source point r′ . The unknown functions to be solved are the volumetric electric current density and magnetic current density inside the body, which are respectively related to the total electric field and total magnetic field by JV (r′ ) = iω[ǫ0 −ǫ(r′ )]E(r′ ), MV (r′ ) = iω[µ0 −µ(r′ )]H(r′ ) (4) where ǫ(r′ ) and µ(r′ ) are the permittivity and permeability of the body (they will be tensors for anisotropic materials), respectively. The above VIEs can be applicable to any penetrable bodies including lossy conductors although the property of resulting matrix equations could be quite different due to the significant difference in material parameters. III. P OINT-M ATCHING S CHEME AND MLFMA ACCELERATION The above VIEs can be solved with a point-matching scheme in which the current densities are chosen as the unknown functions to be solved. Therefore, the integrands of the VIEs do not include the material parameters and the numerical implementation for different materials can be unified. The VIEs are first expressed into a scalar form since
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m = 1, 2, · · · , N (5) x ′c Z N M (r ) n X V c dV ′ iωǫ0 G(rcm , r′ ) · MVy (r′ n ) c z ′ n=1 ∆Vn MV (r n ) c c y gy (rcm , r′ )JVz (r′ n ) − gz (rcm , r′ )JV (r′ n ) c c c ′ x ′ c ′ z ′ + gz (rm , r )JV (r n ) − gx (rm , r )JV (r n ) c c gx (rcm , r′ )JVy (r′ n ) − gy (rcm , r′ )JVx (r′ n ) inc c x c Hx (rm ) MV (rm ) 1 MVy (rcm ) = − Hyinc (rcm ) , − iω [µ0 − µ(rcm )] MVz (rcm ) Hzinc (rcm ) m = 1, 2, · · · , N
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represents the center of the mth tetrahedron (obserwhere c vation element) and r′ n denotes the center of the nth tetrahedron (source element). Also, {gx (r, r′ ), gy (r, r′ ), gz (r, r′ )} are the three components of the gradient of scalar Green’s function. The matrix elements in the above can be evaluated by integrating the components of the dyadic Green’s function or the gradient of scalar Green’s function over each small tetrahedral element ∆Vn (n = 1, 2, · · · , N ) for different observation point rcm (m = 1, 2, · · · , N ). The dyadic Green’s function has nine components but only six are independent due to its symmetry. When m 6= n, the integral kernel is regular and we can use a numerical quadrature rule like the GaussLegendre quadrature rule to evaluate the matrix elements. When m = n, however, the integral kernel is hypersingular and we need to specially treat it. Since the integral kernels do not include the material parameters, we can use the singularity treatment technique developed for regular dielectric media to evaluate the relevant matrix elements [5]. The point-matching scheme can be incorporated with the MLFMA to accelerate the solution for electrically large problems. Because the integral kernels are free of the material property, the implementation can be unified for different materials except the calculation of self-interaction terms (with constitutive parameters) on which the MLFMA does not act. IV. N UMERICAL E XAMPLES We consider the EM scattering by a lossy spherical conductor to demonstrate the approach. The incident wave is a plane wave with a frequency 300 MHz and is propagating
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Fig. 1: Bistatic RCS solutions for a lossy spherical conductor which is also assumed to be magnetic. (a) VV. (b) HH.
along the +z direction. The spherical conductor has a radius 10.0 m and a conductivity σ = 1.0 S/m. The real part of relative permittivity is chosen as ǫ′r = 3.0. The scatterer is also assumed to be magnetic and its relative permeability is µr = 4.0. We calculate the bistatic radar cross section (RCS) for the scatterer and the solution is accelerated by the MLFMA since it is electrically large. Fig. 1 (a) shows the bistatic RCS solutions in vertical polarization (VV) while Fig. 1(b) depicts the solutions in horizontal polarization (HH). It can seen that they are in good agreement with the corresponding exact Mieseries solutions.
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we do not use any vector basis function to represent the unknown current densities. After discretizing the material body into N small tetrahedral elements, we can transform the VIEs into x ′c N Z JV (r n ) X c dV ′ iωµ0 G(rcm , r′ ) · JVy (r′ n ) c n=1 ∆Vn JVz (r′ n ) c c gy (rcm , r′ )MVz (r′ n ) − gz (rcm , r′ )MVy (r′ n ) c c − gz (rcm , r′ )MVx (r′ n ) − gx (rcm , r′ )MVz (r′ n ) c c gx (rcm , r′ )MVy (r′ n ) − gy (rcm , r′ )MVx (r′ n ) x c inc c Ex (rm ) JV (rm ) 1 JVy (rcm ) = − Eyinc (rcm ) , − iω [ǫ0 − ǫ(rcm )] JVz (rcm ) Ezinc (rcm )
V. C ONCLUSION We propose a point-matching scheme to solve the VIEs for complex materials. The scheme does not employ any basis and testing functions and allows the use of nonconforming meshes. Moreover, the scheme can choose the current densities as unknowns and enable the integral kernels to be free of the material parameters. We also incorporate the scheme with the MLFMA to accelerate the solution of large problems and the numerical implementation can be unified for different materials. Typical numerical examples are presented to illustrate the scheme and its advantages have been validated. ACKNOWLEDGMENT This work was supported by the International Collaboration Foundation for Science and Technology, Shanghai, China (Project No. 13520722600). R EFERENCES [1] W. C. Chew, M. S. Tong, and B. Hu, Integral Equation Methods for Electromagnetic and Elastic Waves, Morgan & Claypool, San Rafael, CA, 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, Jan. 1984. [3] L. E. Sun and W. C. Chew, “A novel formulation of the volume integral equation for electromagnetic scattering,” Waves Random Complex Media, vol. 19, no. 1, pp. 162-180, Feb. 2009. [4] W. C. Chew, J. M. Jin, E. Michielssen, and J. M. Song, Fast and Efficient Algorithms in Computational Electromagnetics, Artech House, Boston, 2001. [5] M. S. Tong and W. C. Chew, “A novel approach for evaluating hypersingular and strongly singular surface integrals in electromagnetics,” IEEE Trans. Antennas Propagat., vol. 58, no. 11, pp. 3593-3601, Nov. 2010.
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