## 1. Introduction

[2] Method of moments (MOM) techniques have been widely used in electromagnetics to solve radiation and scattering problems. For perfect electric conductor (PEC) geometries a surface integral equation formulation is typically used, and a suitable geometry modeling scheme along with appropriate basis functions has been demonstrated to have significant effect on the accuracy of the solution [*Antilla and Alexopoulos*, 1994; *Song and Chew*, 1995].

[3] Several methods have been used to formulate scattering and radiation problems involving dielectric materials. The finite element method (FEM), along with various mesh truncation schemes [*Volakis et al.*, 1998], is one of the most commonly used approaches. Among these FEMs the finite element boundary integral (FE-BI) method [*Volakis et al.*, 1998] provides an exact means of truncating the FEM mesh, hence keeping the FEM domain small. Keeping the FEM domain small is crucial in numerical simulations since the FEM is prone to error propagation. The necessity of using suitable geometry modeling schemes and basis functions in the FE-BI formulation has also been demonstrated to be important, both in terms of solution accuracy and in terms of convergence. When dealing with homogeneous regions, a surface integral equation formulation can be used [*Poggio and Miller*, 1973]. However, for regions with varying material properties a volume integral equation must be employed [*Livesay and Chen*, 1974; *Schaubert et al.*, 1984]. *Livesay and Chen* [1974] used cubic elements in this context, and tetrahedral elements were used by *Schaubert et al.* [1984]. For curved geometries, tetrahedral elements are suitable because of their flexibility in modeling. However, when dealing with thin layers, hexahedral elements are more suitable since they avoid elongated tetrahedra, which lead to ill-conditioned matrix systems.

[4] Also, so far, direct volume integral equations (VIEs) have not been exploited because of their excessive CPU and memory requirements. Nevertheless, the recent introduction of fast methods [*Coifman et al.*, 1993] is beginning to make VIE solutions more practical. In this paper, we consider direct VIE solutions of electromagnetics problems using curvilinear hexahedra, which are particularly suited for thin layers and curvilinear volumes.