## 1. Introduction

[2] Electromagnetic scattering by a body of revolution (BOR) was first studied by *Andreasen* [1965], and followed by many other researchers based on the method of moments (MoM) [*Mautz and Harrington*, 1969; *Wu and Tsai*, 1977; *Mautz and Harrington*, 1978, 1979; *Morgan and Mei*, 1979; *Mautz and Harrington*, 1982; *Glisson*, 1978; *Glisson and Wilton*, 1980; *Govind et al.*, 1984; *Abdelmageed*, 2000; *Datthanasombat and Prata*, 1999; *Kucharski*, 2000]. Because of the geometrical symmetry, one can make Fourier expansions in the direction of revolution for the quantities in the governing equations of the problem and reduce the three-dimensional (3D) equations to two-dimensional (2D) equations. This reduction will dramatically lower computational costs and allow one to solve very large BOR problems with moderate expense. The BOR problems can also be solved using different methods, such as finite element method (FEM), finite difference time domain (FDTD) or some hybrid methods, and many papers have addressed the solution techniques for various BOR structures [*Khebir et al.*, 1993; *Greenwood and Jin*, 1999; *Dunn et al.*, 2006; *Mittra and Gordon*, 1989; *Sun and Rusch*, 1994; *Yang and Hesthaven*, 1999; *van der Vorst and de Maagt*, 2002; *Medgyesi-Mitschang and Wang*, 1983; *Gedney and Mittra*, 1990; *Altman and Mittra*, 1996].

[3] However, in the MoM approach for solving the 2D BOR integral equations, one will encounter the problem on how to evaluate the singular Fourier expansion coefficients. Since the integral kernels related to the 3D Green's function are singular for the self interactions between source points and field points, the Fourier coefficients involved are also singular and should be treated specially. The efficient evaluation for the singular Fourier coefficients plays a critical role in the solution process because it is related to the generation of diagonal entries in the system matrix, which have very significant influence on the solution. Unfortunately, there had not been explicit description for the indispensable treatment technique in those early published literatures [*Andreasen*, 1965; *Mautz and Harrington*, 1969; *Wu and Tsai*, 1977; *Mautz and Harrington*, 1978, 1979; *Morgan and Mei*, 1979; *Mautz and Harrington*, 1982], until Glisson's dissertation appeared [*Glisson*, 1978]. Glisson developed a systematic treatment scheme for various kernels in his dissertation and the scheme has been demonstrated to be effective in applications [*Glisson and Wilton*, 1980; *Govind et al.*, 1984]. Some other schemes were developed after Glisson [*Gedney and Mittra*, 1990; *Mohsen and Kucharski*, 2000], but they only dealt with the weakly singular coefficient related to the Green's function. Abdelmageed also presented an evaluation method for those Fourier coefficients [*Abdelmageed*, 2000], but the method was only for the regular Fourier coefficients for which the numerical quadrature rules can be used conveniently.

[4] The electromagnetic integral equations usually include two singular kernels. One is the Green's function and the other is its gradient. Correspondingly, there are two kinds of singular Fourier coefficients in the 2D BOR equations. The coefficient related to the Green's function is called modal Green's function (MGF) [*Abdelmageed*, 2000] and the coefficient related to the gradient of the Green's function is named MGF's derivatives by us. MGF has a weak 1/*R* singularity and MGF's derivatives include a strong 1/*R*^{2} singularity. To handle these singular integrals, we need to regularize the kernels by subtracting the singularities. This can be done by constructing a singular integrand which coincides with the kernel at the singularity and similar to it in the vicinity of the singularity. The constructed singular integrand which is subtracted from the kernel should be tractable either analytically or numerically and the regularized kernel should be as smooth as possible so that ordinary numerical quadrature rules can be applied.

[5] We proposed a new evaluation scheme for the singular Fourier expansion coefficients in the MoM solution. The scheme differs from Glisson's scheme on several aspects. First, the scheme incorporates the Fourier coefficients with the line integral in the 2D equation and performs the integrals over generating arc segments first with analytical expressions. Second, the scheme gets rid of the elliptical integrals which are handled approximately in Glisson's scheme. Third, when performing the outer integral over revolution angle, the singular parts are further distinguished from the regular parts in the kernels and evaluated exactly in a closed form. Fourth, we construct a different singular integrand in subtracting the kernel of MGF's derivatives so that the kernel is better regularized. Fifth, we use the triangle basis and testing function in the MoM, so both the constant (zeroth-order) term and the first-order term in the basis function are incorporated with the kernels in the evaluation. Because of the above improvements, our scheme is simpler in implementation and more efficient in calculation. Numerical examples have demonstrated the effectiveness of the scheme.