Radio Science

Evaluation of singular Fourier coefficients in solving electromagnetic scattering by body of revolution

Authors


Abstract

[1] We develop an efficient evaluation scheme for the singular Fourier coefficients in solving electromagnetic scattering by body of revolution (BOR). For the singular modal Green's function (MGF), the scheme first evaluates the integrals over generating arc segments analytically and then distinguishes the singular part from the regular part for the integral over the angle in the revolution direction. This allows us to avoid the approximate calculation for the elliptical integral and the separated singular part is exactly evaluated in a closed form. For the singular MGF's derivatives generated from the gradient of the Green's function, we subtract the kernels with more similar singular integrands in the vicinity of singularity so that the kernels are better regularized. The procedure differs from the existent Glisson's method in several aspects and numerical experiments show that the scheme is simpler in implementation and more efficient for calculation.

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/R2 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.

2. 2D Integral Equation

[6] Electromagnetic scattering by an arbitrarily shaped objects can be described by 3D integral equations. Depending on the property of the object, the integral equation can be electric field integral equation (EFIE), magnetic field integral equation (MFIE), combined field integral equation (CFIE), or volume integral equation (VIE). Since the treatments for the singular kernels in those equations are quite similar, we exemplarily consider the EFIE for scattering by a conducting BOR. The EFIE is given by

equation image

where g(r,r′) = equation image is the 3D Green's function and R = ∣r − r′∣ is the distance between a field point and a source point. Also, κ is the wave number, η is the intrinsic impedance of the medium, J(r′) is the unknown surface current on the object surface S, and Einc is the incident electric field. The subscript tan denotes the tangential component. If the object is a BOR as shown in Figure 1, we can expand the integral kernel and unknown function into Fourier series over the revolution direction. Since the current is a surface vector, we can decompose it into two tangential components along equation image and equation image which are the unit tangential vectors at the generating arc and in the revolution direction of the BOR, respectively, i.e.

equation image

and its divergence is given by Glisson [1978]

equation image

where ρ′ is the radial coordinate of a source point (denoted by a prime) in a cylindrical coordinate system. Multiplying (1) by equation image and equation image with a dot product, we can form the following two scalar equations

equation image

We expand all quantities in (4) into Fourier series, i.e.

equation image

where

equation image

and Jnt(ρ′, z′) and Jnϕ(ρ′, z′) (only n′ = n terms exist) are the unknown Fourier expansion coefficients for the currents which will be solved. In the above, we have defined β = ϕ − ϕ′, R = equation image, v1 = (iκR − 1)eiκR/(4πR3) and v2 = (ρρ′cosβ)sinα + (zz′)cosα. Also, gn is known as the modal Green's function (MGF) with the modal number n, and Gnt and Gnϕ are two tangential components of the MGF's gradient. Here, we use (ρ,ϕ,z) to denote a field point and (ρ′, ϕ′, z′) to denote a source point in a cylindrical coordinate system.

Figure 1.

Geometry of a body of revolution (BOR). equation image and equation image are the two unit tangential vectors at the surface along the generating arc and revolution direction, respectively.

[7] Substituting those Fourier expansions to (4) and making use of the orthogonality of Fourier basis, we obtain

equation image

where C is the generating arc of BOR as shown in Figure 2 and

equation image

Here α is the angle between the tangential direction of the generating arc and the +z axis, as defined in Figure 1. The above equations are 2D integral equations which hold for each Fourier modal index n. They represent an infinite set of the coupled integral equations from which the unknown Jnt and Jnϕ can be solved independently for each n using a numerical method like MoM.

Figure 2.

Discretization of BOR generating arc. (ρ1, z1) and (ρ2, z2) are the two end points of a segment and (ρ, z) is an observation point on the segment.

3. Evaluation of Singular Fourier Coefficients

[8] Since the Fourier coefficients with regular kernels can be evaluated by numerical integration conveniently, we only consider the evaluation for singular Fourier coefficients. They are MGF and its gradients as shown in (6). We treat MGF and its gradients separately as follows.

3.1. For MGF

[9] We assume that the generating arc of BOR is discretized into straight line segments as shown in Figure 2. Since the singularity appears when ρ = ρ′, z = z′ and ϕ = ϕ′ or β = 0, we incorporate the evaluation of MGF and its gradient with the outer integral over the self segment. For MGF, the integral will be, with the consideration of constant term in the basis of current expansion,

equation image

In the above, we have used the even property of the integrand and dℓ′ = Ja′. Here Ja is the Jacobi which is a constant for the straight line approximation of an arc segment and we have suppressed it in all expressions. The first term in the above integral is regular now because the singular kernel has been subtracted with a similar singular integrand and the numerical integration can be applied. The second term is the subtracted similar singular integrand and we need to handle it specially. Figure 3 illustrates the regularization of a singular kernel by the subtracting a similar singular integrand. Generally, the construction of a similar singular integrand should satisfy three conditions, i.e., (1) the integrand f0(R) should be in the same degree of singularity as the kernel f1(R) so that the subtracted kernel Δ f(R) will be bounded at the singularity, (2) f0(R) should be as similar as possible to f1(R) in the vicinity of singularity so that Δ f(R) will be regular near the singularity, and (3) f0(R) should be integrable either analytically or numerically. In constructing the similar singular integrand f0(R) for the kernel in (9), we may expand eiκR into a series form and extract the singular term which is cos()/R as the similar singular integrand. Such a f0(R) is more similar to f1(R) in the neighbor of the singularity than f0(R) = 1/R which was chosen by Glisson [1978] and Abdelmageed [2000] by letting β = 0. The reason of selecting f0(R) = 1/R instead of f0(R) = cos()/R by Glisson [1978] and Abdelmageed [2000] is probably due to the inability of integrating f0(R) = cos()/R, but Gedney and Mittra [1990] provided a solution by using a recursive relation in a complete elliptic integral and we have also derived different solution for the integration of f0(R) = cos()/R. However, since the singularity is weak in this case, the two choices for f0(R) above do not show an obvious difference in the results, and we also choose the latter one to facilitate the treatment. Switching the order of integrations in the second term, we have

equation image

where Ik0s (k = 1, 2, 3) is the kth integral in the preceding equation and

equation image

The integral in (10) includes both regular and singular parts. If ρ2 < ρ < ρ1, then I20s is singular and I30s is regular, but conversely if ρ2 > ρ > ρ1. I10s is always regular. The regular parts are evaluated numerically and the singular part is performed analytically. The singular part in I20s or I30s can be extracted further

equation image

The second term above is regular and evaluated numerically, but the first term is singular and can be evaluated analytically with the aid of Gradshteyn and Ryzhik [1965]

equation image

If MGF incorporates the first-order term in the basis of current expansion, the subtracted singular integral is

equation image

The second integral I21s above is very similar to Ig0s in (10) and is handled in a similar way. The first integral I11s can be written as

equation image

Similar to the last three integrals in (10), Ib1s is singular and Ic1s is regular if ρ2 < ρ < ρ1, but conversely if ρ2 > ρ > ρ1, and Ia1s is always regular. The singular part in Ib1s or Ic1s can be separated as

equation image

and the singular core is again evaluated analytically with the aid of Gradshteyn and Ryzhik [1965]

equation image
Figure 3.

Illustration of regularization for a singular kernel. f1(R) is the singular kernel, f0(R) is the subtracted similar singular integrand, and Δf(R) is the resulting regular kernel.

3.2. For MGF's Derivatives

[10] When incorporating the outer integral over an arc length segment with the constant term in the basis for current expansion, the t component of MGF's derivatives can be written as

equation image

where R0equation image (see (11) for h) and d0 = sinα + bcosα. The first integral is regular now and can be evaluated numerically. The second integral is singular and will be handled specially. The above singularity subtraction is based on the fact that 1 − cosβ = 2sin2(β/2) ≈ β2/2 when β → 0. The subtracted similar singular integrand f0(R) in this case is also different from the one in Glisson's method Glisson and Wilton [1980] in which ρρ′ is used in R0 to enable the derivation of closed-form solutions. Since our subtracted singular integrand f0(R) is more similar to the kernel f1(R) nearby the singularity, the regularized kernel is numerically integrable and the required quadrature rule can be of lower order in numerical integration. Since the singularity is strong in this case, the difference in the subtracted similar singular integrand is important. The singular integral above can be found in a closed form as Gradshteyn and Ryzhik [1965]

equation image

where Rjequation image, j = 1,2. Similarly, we can derive the subtracted singular integral when Gnt incorporates the first-order term in the basis Gradshteyn and Ryzhik [1965], i.e.

equation image

For the ϕ component of MGF's derivatives incorporating the constant term in the basis, we have

equation image

The extracted singular integral is

equation image

where Ra = equation image (see (11) for w). The first integral is numerically integrable and evaluated numerically. The second integral is singular and evaluated analytically as

equation image

where

equation image

If the ϕ component of MGF's derivatives incorporates the first-order term in the basis, the extracted singular integral becomes

equation image

The first integral is regular again and evaluated numerically. The second integral is singular and evaluated analytically as

equation image

where

equation image

4. Numerical Examples

[11] The new evaluation scheme is verified using the scattering by three different conducting objects as shown in Figure 4. The incident wave is a horizontally or vertically polarized plane wave propagating along the (θinc, ϕinc) direction. The sphere has a radius of a = 0.5λ and the cone has a base radius of r = 0.5λ and a height of h = 2λ. The step-radius cylinder consists of four segments with different radii and thicknesses. The radii are r1 = 0.6λ, r2 = 0.4λ, r3 = 0.2λ, and r4 = 0.8λ, respectively, and the thicknesses are h1 = 0.3λ, h2 = 0.5λ, h3 = 0.8λ, and h4 = 0.4λ, respectively. Figure 5 shows the normalized magnitude of the tangential components Jt (vertical polarization) and Jϕ (horizontal polarization) of the surface current along the principal cut (ϕ = 0° and θ = 0° ≈ 180°) when (θinc, ϕinc) = (30°, 30°) and using 80 segments. We can see that the BOR solutions agree well with the exact solutions derived from the Mie series. Figures 6 and 7 plot the bistatic radar cross section (RCS) for the conducting cone and the step-radius cylinder. The propagating direction of the incident wave is (30°, 30°) for the cone and (60°, 60°) for the step-radius cylinder, and the observation for the scattered wave is also taken along the principal cut. The BOR solutions with 100 segments for the cone and 200 segments for the step-radius cylinder are compared to the MoM solutions obtained by treating the objects as full 3D scatterers. The cone and step-radius cylinder are discretized with 3080 and 4146 triangle patches, respectively. It can be seen that the solutions based on the two approaches are in good agreement with each other, except for the case of the cone in the vertical polarization. The reason for this is that the geometry has a sharp circular corner which is recognized as a challenging geometry sometimes. Because of the difficulty of generating good meshes around the corner, the solution from the full 3D formulations may not be accurate enough. We believe that the BOR solution is more accurate. Since the geometrical singularity or discontinuity is in the θ direction, it has a large influence on the solution of the vertical polarization case.

Figure 4.

Geometries of three conducting BOR scatterers: (a) Sphere, (b) Cone, and (c) Step-radius cylinder.

Figure 5.

Current distribution along the principal cut on a conducting sphere surface.

Figure 6.

Bistatic RCS for a conducting cone in horizontal and vertical polarizations.

Figure 7.

Bistatic RCS for a conducting step-radius cylinder in horizontal and vertical polarizations.

5. Conclusion

[12] We propose a different evaluation scheme for the singular Fourier expansion coefficients in solving electromagnetic scattering by BOR. The scheme incorporates the Fourier coefficients with the integration over generating arc segments and performs the segment integration first with closed-form expressions. This allows us to avoid the approximate calculation of the elliptical integrals in Glisson's method. Also, we subtract the kernels of MGF's derivative by using more similar singular integrands, resulting in numerically integrable kernels. Finally, we consider the first-order basis representation for the unknown current density in the evaluation and thus the applications of the technique can be more extensive.

Ancillary