5328 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 61, NO. 10, OCTOBER 2013 Efficient Nyström Solutions of Electromagnetic Scattering by Composite Objects With Inhomogeneous Anisotropic Media Kuo Yang, Jia Cheng Zhou, Wei Tian Sheng, Zhen Ying Zhu, and Mei Song Tong Abstract—Analysis of electromagnetic (EM) scattering by composite ob- jects with inhomogeneous or anisotropic media requires an efficient solu- tion of volume integral equations (VIEs) or volume-surface integral equa- tions (VSIEs) in the integral equation approach. The traditional method of moments (MoM) with the Rao-Wilton-Glisson (RWG) basis function and the Schaubert-Wilton-Glisson (SWG) basis function may not be convenient because the basis functions are defined over a triangular or tetrahedral el- ement pairs and the SWG basis function could cause a surface charge den- sity on the common faces separating dissimilar media. In this work, we develop an efficient Nyström scheme as an alternative to solve the VIEs or VSIEs for those objects. The advantages of Nyström method include the simple mechanism of implementation, allowance of nonconforming meshes, and removal of basis or testing function, making the method more suitable for solving such problems. Numerical examples for EM scattering by typ- ical objects are presented to demonstrate the method. Index Terms—Electromagnetic scattering, inhomogeneous anisotropic media, Nyström method, volume-surface integral equation. I. INTRODUCTION Integral equation approach has been widely employed to solve electromagnetic (EM) problems [1] due to its unique merits compared with other numerical approaches like finite-difference time-domain (FDTD) [2] and finite element method (FEM) [3]. For composite ob- jects with inhomogeneous or anisotropic penetrable media which could be encountered in many applications, the problems are described with volume integral equations (VIEs) or volume-surface integral equations (VSIEs) if conductors exist simultaneously. Although the surface in- tegral equations (SIEs) can also be applied to penetrable materials in addition to the conducting surfaces and may be preferred when avail- able, they require a piecewise homogeneity in the materials and may not be suitable for arbitrarily inhomogeneous media. The SIEs include electric field integral equation (EFIE), magnetic field integral equation (MFIE), or combined field integral equation (CFIE), for the conducting surfaces. For the interfaces of piecewise homogeneous penetrable materials, there are also Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) formulation [4] and Müller formulation [5] in addition to those three SIEs for the conducting surfaces. If the penetrable materials are arbitrarily inhomogeneous or include many material interfaces, we have to resort to the VIEs which include electric field VIE (EFVIE) and magnetic field VIE (MFVIE) to describe the problems [1]. Cou- pled with the SIEs for conducting surfaces, we can form the VSIEs for the objects including both conductors and penetrable materials. If the penetrable materials are also anisotropic, the resultant VSIEs become complicated and seeking their accurate solutions efficiently could be challenging for some objects [6]–[8]. Manuscript received February 02, 2012; revised April 08, 2013; accepted July 01, 2013. Date of publication July 10, 2013; date of current version October 02, 2013. This work was supported by the National Natural Science Foundation of China with the Project No. 61271097. The authors are with the School of Electronics and Information Engineering, Tongji University, Shanghai, China (e-mail: mstong@tongji.edu.cn). Digital Object Identifier 10.1109/TAP.2013.2272671 Traditionally, the above VIEs or VSIEs are solved with the method of moments (MoM) in which the Rao-Wilton-Glisson (RWG) basis func- tion is used to expand the surface current on the conducting surface with a triangular tessellation [9] while the Schaubert-Wilton-Glisson (SWG) basis function is applied to represent the volume currents in the penetrable materials with a tetrahedral discretization [10]. Though the MoM with those well-designed basis functions is robust, it is found that its implementation could be inconvenient for some complex ob- jects [11]. This is because the RWG basis function is defined over a triangle pair and the SWG basis function is defined over a tetrahedron pair, and both basis functions require a higher mesh quality or con- formal meshes in geometric discretization. The MoM is sensitive to the defective (non-conformal) meshes which cannot form an effective triangle pair or tetrahedron pair, and they cannot be tolerated [12]. The defective meshes can be encountered frequently near the geometric junctions in the complex objects including many different parts and a tedious remeshing process should be performed when this happens. Also, the SWG basis function does not allow the inclusion of material interfaces in tetrahedral elements because of the violation of boundary conditions, prohibiting it from being used in the arbitrarily inhomoge- neous problems. If we have to clearly locate the material interfaces in geometric discretization, the discretization is tedious and solving the VIEs could be meaningless (the SIEs are preferred). In addition, we need to take care of the surface charge density residing on a common face separating two dissimilar media in the SWG basis function for the VIEs [10]. Treating the surface charge density requires an extra care and it could be very inconvenient in composite objects with many ma- terial interfaces [13]. In this work, we develop an efficient Nyström scheme as an alterna- tive to the MoM for solving those VIEs or VSIEs. Although the Nys- tröm method has been widely used to solve EM problems in recent years [14]–[17], it is mostly used to solve SIEs for simple homogeneous isotropic objects and it has not been applied to the EM analysis in- volving both conductors and inhomogeneous or anisotropic penetrable media. The Nyström method directly replaces the integral operator with an appropriate quadrature rule when the operator is smooth and the corresponding matrix entries can be generated by simply sampling in- tegral kernels at quadrature points without involving numerical inte- gration. The primary merits of the method include simple mechanism of implementation, easy discretization or lower requirement on mesh quality which allows the inhomogeneity of material in tetrahedrons, and no involvement of basis and testing function. These merits could be especially desirable in fast algorithms like multilevel fast multipole algorithm (MLFMA) [18], because they can greatly simplify the im- plementation. However, the Nyström method requires an efficient local correction scheme for evaluating self-interaction (singular) or near-in- teraction (near-singular) matrix elements since the numerical quadra- ture rules cannot be applied directly. Unlike in the MoM in which the well-designed basis and testing functions can help reduce the degree of singularity in the evaluation of singular or near-singular elements by using integration by parts, we have to handle the hypersingular in- tegrals which come from the double gradient operator of the dyadic Green’s function in the Nyström method. Fortunately, we have devel- oped a robust local correction scheme for both SIEs and VIEs in our previous work [19], [20] and it can be used for more complicated cases in this work by a revision. Though the Nyström method may not be preferred when the MoM can be easily implemented, they could be a good alternative to the MoM for complicated objects such as those in- cluding both conducting and inhomogeneous or anisotropic penetrable materials. Numerical examples for EM scattering by such objects are presented to illustrate the scheme and its merits can be observed. 0018-926X © 2013 IEEE