^{1,2}

^{1,2}

^{2}

^{2}

^{2}

^{1}

^{2}

It is well-known that if an

Electric field integral equation (EFIE) and magnetic field integral equation (MFIE) have been widely employed to analyze electromagnetic scattering of conducting bodies [

Several ways of dealing with this numerical problem have been proposed. Nowadays, the popular combined field integral equation (CFIE) technique to overcome this problem is a proper combination of the electric field integral equation and the magnetic field equation [

Here, we present an effective method to solve the scattering problem of conductor bodies at or in the neighborhood of the resonant frequencies in both 2D and 3D EM scattering. At the resonant frequencies, Inagaki modes, firstly applied to analyze the antenna array [

The principle is elaborated as follows. The solution of anill-conditioned system of the equation will consist of (1) the correct physical solution to the problem and (2) the resonant solution, which is in conjunction with Green’s Theorem which produces the nonzero complementary resonant modes. When the orthogonal modes are used to solve the electric field integral equation, the solutions will be divided into the induced modes and the resonant modes corresponding to eigenvalues. We can easily obtain the current density and exterior field from the induced modes.

Consider a PEC scatterer excited by an incident wave and the scatterer defined by the surface

PEC scatterer impressed by electromagnetic field.

According to the boundary condition, we get

As a result of interior resonance, the current density

If the incident

In other words, the solution to the inhomogeneous EFIE is not unique since the non-zero- solution to its corresponding homogeneous equation exits at some discrete frequencies. Namely, in addition to the induced current

Similar to the behavior of a conducting cavity, the resonant current density

For the sake of simplicity, only the electric field integral equation is involved here. To solve the EFIE by method of moments, first of all, we should choose a set of expansion functions and a set of weighting functions. Here, we choose basis functions aiming at the diagonalization of the moment matrix which was proposed by Inagaki and Garbacz [

At the interior resonances, the surface current is composed of the resonant current and the induced current. The resonant current is the solution to the corresponding homogeneous

Actually, the moment matrix will not be exactly singular, but rather, it will be highly ill-conditioned since the round-off and the truncated error exist during the computation of the matrix elements. According to the orthogonal property of Inagaki modes [

At the interior resonances, when eigenvalue ^{−1} by zero. The incident current

Theoretically, the scattered field and the radar cross section external to

In fact, the resonance current does not really exist on the surface of the conductors. The interior resonance problem is just caused by the deficiency of the selected mathematical model. After getting rid of the virtual resonance current, we can obtain the stable and reliable current density and exterior field of conductors at interior resonances.

Here we confine our attention to not only two-dimensional structures but also three-dimensional structures. The presented method is applied here to get the surface current density and exterior field of conductors when they are at (or near) the interior resonant frequencies.

As the first case for testing the approach described above, scattering from a circular conducting cylinder was examined here. It was found that when the TM plane wave normally illuminates an infinite circular cylinder, the interior resonance takesplace if _{11} mode of the same surface circular cylinder cavity.

From Figures

Resonant current on a circular cylinder,

Induced current on a circular cylinder,

Bistatic RCS of a circular cylinder,

Also, an infinitely long square conducting cylinder was considered here with a TM wave incident along the axis of the cylinder. The first and the second resonant frequency of the square cylinder are, respectively, at

Minimal eigenvalue of square cylinder.

Resonant current on a square cylinder,

Induced current on a square cylinder,

In Figure

Bistatic RCS on square cylinder,

In the end, two conducting spheres, respectively, at the

Bistatic RCS of a resonant sphere,

Bistatic RCS of a resonant sphere,

A new scheme for eliminating interior resonance problems associated with surface integral equation is presented. The orthogonal property of Inagaki modes is used here to isolate the resonant mode, which is then omitted in the computation to obtain the right property of the conductors at the interior resonances. This simple technique has been proven to be effective in attenuating the resonant modes and getting the unique and stable solution to EFIE when conductors are at (or near) the frequencies associated with interior resonances. Excellent numerical results, away from resonance problems, have been obtained for some shapes not only in two dimensions but also in three dimensions. Compared with other techniques, the advantages of this approach are only involved EFIE equation and easy to get the right results, but, due to the determination of eigenvalues and eigenvectors, there is a bit time-consuming for property computation at or in the interior resonance.

The authors declare that there is no conflict of interests regarding the publication of this paper.