## 1. Introduction

[2] The surface integral equation method is one of the most popular numerical methods in the electromagnetic scattering analysis of metallic, homogeneous dielectric and composite metallic and dielectric objects [see, e.g., *Kolundzija and Djordjevic*, 2002]. Traditional surface integral equation methods are, however, restricted to rather small-scale problems, because discretization of an integral equation results in a system matrix which is fully populated and expensive to store and solve. In order to consider complex large-scale problems including a high number of unknowns, advanced methods like the fast multipole method [*Rokhlin*, 1990; *Chew et al.*, 2001] and the adaptive integral method [*Bleszynski et al.*, 1996], are required. These fast integral equation methods are based on the iterative solution of the matrix equation. Consequently, for the fast methods to be “fast” it is important to have sufficiently rapidly converging iterative solutions. However, many of the traditional surface integral equation formulations lead to ill-conditioned matrix equations and slowly converging iterative solutions [*Zhao and Chew*, 2000]. Conditioning of the system matrix can be improved with a better choice of an integral equation formulation [*Wilton and Wheeler*, 1991; *Lloyd et al.*, 2004] and in some cases the number of iterations may be further decreased, for example, with preconditioning [*Kas and Yip*, 1987] or with a better choice of an iterative solver [*Kolundzija and Sarkar*, 2001].

[3] In the case of metallic objects the fundamental integral equations are the electric field integral equation (EFIE), the magnetic field integral equation (MFIE) and the combined field integral equation (CFIE) [*Mautz and Harrington*, 1978]. Iterative solution of these formulations have been studied [e.g., *Wilton and Wheeler*, 1991; *Song and Chew*, 1995; *Pocock and Walker*, 1997; *Makarov and Vedantham*, 2002].

[4] For homogeneous dielectric objects there are infinitely many possibilities to derive a set of coupled integral equation formulations [*Harrington*, 1989]. The most popular formulations are the PMCHWT (Poggio-Miller-Chang-Harrington-Wu-Tsai) formulation [*Mautz and Harrington*, 1979] and the Müller formulation [*Müller*, 1969]. In recent years iterative solution of these formulations have attained a lot attention in the literature. *Yeung* [1999] reported that the PMCHWT formulation leads to slowly converging iterative solutions. *Li et al.* [2002] and *Lloyd et al.* [2004] showed that convergence of the PMCHWT formulation can be improved with an appropriate choice of the coupling coefficients. *Lloyd et al.* [2004] and *Zhu and Gedney* [2004] observed that the Müller formulation leads to faster converging iterative solutions than the PMCHWT formulation.

[5] These integral equation formulations can be rather straightforwardly extended for piecewise dielectric and composite metallic and dielectric objects [see, e.g., *Kishk and Shafai*, 1986]. However, in the case where three or more subregions meet at a common edge, and form a junction, a special care is required [*Kolundzija*, 1999; *Ylä-Oijala et al.*, 2005]. *Lu and Luo* [2003] apply coupled EFIE-PMCHWT formulation for scattering by metallic objects with dielectric coating and observed that iterative solution converges very slowly. *Zhu and Gedney* [2004] apply CFIE-Müller and CFIE-PMCHWT formulations for a similar problem and found that the CFIE-Müller formulation gives a better conditioned matrix equation.

[6] In addition to the conditioning of the system matrix and the convergence of an iterative solver an important issue associated with the formulation of integral equations is the accuracy of the solution. For example, it is well known that in the case of nonsmooth metallic objects the MFIE formulation with the low-order basis functions, like the Rao-Wilton-Glisson (RWG) [*Rao et al.*, 1982] functions, leads to poorer accuracy than the EFIE formulation. Recently, *Ergül and Gürel* [2004] observed that the accuracy of the MFIE formulation can be significantly improved by applying the Trintinalia-Ling (TL) [*Trintinalia and Ling*, 2001] first-order basis functions. In addition, *Zhu and Gedney* [2004] reported that the accuracy of the Müller formulation decreases compared to the PMCHWT formulation when the permittivity of the object is increased.

[7] In this paper we study formulation of the surface integral equations and their properties in solving electromagnetic scattering by homogeneous dielectric and piecewise dielectric objects as well as by closed metallic and composite objects with iterative methods. Four types of formulations are considered: *T* formulations, *N* formulations, combination of the *T* and *N* formulations, that is, the CFIE formulation, and the Müller formulation. After having studied the properties of the integral equations and their testing in the Galerkin method, we derive “optimal” form for each formulation type. The developed new formulations lead to clear improvements in the conditioning of the system matrices and in the convergence rates when the matrix equations are solved iteratively with the generalized minimal residual (GMRES) method. Both the RWG and TL basis functions are used in expanding the unknown electric and magnetic surface current densities. The TL basis functions are required in the *N* formulations to maintain the solution accuracy in the case of metallic objects with sharp edges and nonsmooth dielectric objects with a high electric permittivity.