## 1. Introduction

[2] For the solution of electromagnetics problems involving three-dimensional dielectric objects, the electric and magnetic current combined-field integral equation (JMCFIE) [*Ylä-Oijala and Taskinen*, 2005a, 2005b] is a preferable formulation in terms of accuracy and efficiency. In numerical solutions employing Rao-Wilton-Glisson (RWG) functions [*Rao et al.*, 1982] on triangles, JMCFIE is more accurate than the normal (N) formulations, such as the combined normal formulation (CNF) [*Ylä-Oijala et al.*, 2005b] and the modified normal Müller formulation (MNMF) [*Ylä-Oijala and Taskinen*, 2005b]. In addition, iterative solutions of problems involving large and complicated objects are more efficient with JMCFIE, which requires fewer iterations than MNMF and CNF [*Ergül and Gürel*, 2007, 2009]. For a given discretization with the RWG functions, the tangential (T) formulations, such as the combined tangential formulation (CTF) [*Ylä-Oijala et al.*, 2005b] and the Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) formulation [*Poggio and Miller*, 1973; *Chang and Harrington*, 1977; *Wu and Tsai*, 1977], may provide more accurate results than JMCFIE. On the other hand, matrix equations obtained with the T formulations are difficult to solve iteratively [*Ylä-Oijala et al.*, 2005b, 2008; *Ergül and Gürel*, 2007, 2009]. In fact, improving the accuracy of JMCFIE solutions to the levels of the T formulations by refining the discretization can be more efficient than using the T formulations with coarse discretizations. Moreover, JMCFIE becomes essential for large problems, which might not easily be solved with the T formulations.

[3] JMCFIE can easily be applied to electromagnetics problems involving multiple dielectric regions or composite structures with coexisting metallic and dielectric parts [*Ylä-Oijala and Taskinen*, 2005a, 2005b; *Ylä-Oijala*, 2008]. In general, equivalent problems, which are defined for all nonmetallic regions, are discretized with oriented basis and testing functions. Then, the related unknowns on the boundaries and the corresponding equations are combined to form a single matrix equation to solve. This procedure is detailed by *Ylä-Oijala et al.* [2005a, 2005b] in the context of a PMCHWT formulation, and is extended to JMCFIE in the work of *Ylä-Oijala and Taskinen* [2005a, 2005b]. As discussed by *Ylä-Oijala and Taskinen* [2005a, 2005b], JMCFIE is appropriate for complicated structures involving multiple dielectric and metallic regions.

[4] Electromagnetics problems involving large metallic, dielectric, and composite objects can be solved iteratively, where the required matrix-vector multiplications are performed efficiently with the multilevel fast multipole algorithm (MLFMA) [*Song et al.*, 1997; *Sheng et al.*, 1998; *Chew et al.*, 2001; *Donepudi et al.*, 2003]. Recently, MLFMA is used to solve electromagnetics problems involving homogeneous dielectric objects formulated with JMCFIE [*Ergül and Gürel*, 2007, 2009]. In this study, we extend the MLFMA solution of JMCFIE to those problems involving multiple dielectric and composite dielectric-metallic structures. We mainly focus on the efficiency of the solutions and investigate the number of iterations for increasingly large objects. We show that iterative solutions of JMCFIE become difficult as the contrast increases, i.e., when electromagnetic parameters change significantly across dielectric interfaces. For efficient solutions of JMCFIE, we present a four-partition block-diagonal preconditioner (4PBDP), which reduces the iteration counts significantly. This preconditioner, which was originally developed by *Ergül and Gürel* [2009] for homogeneous dielectric objects, is particularly useful when a standard two-partition block-diagonal preconditioner (2PBDP) fails to provide a rapid convergence. In this paper, we present 4PBDP to accelerate the solution of more complicated problems involving multiple dielectric and metallic regions.

[5] The rest of the paper is organized as follows. Section 2 presents the matrix equations obtained with the JMCFIE formulation of electromagnetics problems involving multiple dielectric and metallic regions. MLFMA solutions are considered in section 3, where we provide the specific details of our implementation. Block-diagonal preconditioning is discussed in section 4, followed by numerical examples in section 5, and our concluding remarks in section 6. Time-harmonic electromagnetic fields with *e*^{−iωt} time dependence are assumed throughout the paper.