Numerical Investigation of Electromagnetic Scattering Problems Based on the Compactly Supported Radial Basis Functions

1 Introduction

In recent years, the interest in electromagnetic (EM) scattering problems has increased due to their many practical applications in numerous fields of science and engineering such as optics, acoustics, geoscience, and remote sensing. These practical applications led to theoretical investigation of the EM scattering problems. Many of the wave scattering phenomena can be modelled as surface integral equations [1], [2], [3]. The most commonly used surface integral equations are electric field integral equations (EFIEs), magnetic field integral equations (MFIEs), combined field integral equation (CFIE), PMCHWT formulation, and Müller’s integral equation. The EFIE and MFIE formulations are commonly adopted to model scattering from perfectly conducting surfaces [4], [5]. The CFIE, which is a linear combination of EFIE and MFIE, is introduced and formulated to investigate the scattering problem from finite perfectly conducting surfaces [6]. Moreover, the PMCHWT and Müller’s integral formulations are linear combinations of EFIE and MFIE, which are used for analysing the scattering problem from dielectric and composite surfaces [7], [8]. So, to deal with an EM scattering problem, it needs to compute the solution of a proper surface integral equation. In many cases, these surface integral equations are ill-posed and finding the analytic solutions for them is often impossible or very difficult. Consequently, a proper numerical technique should be employed for dealing with these problems. One of the most efficient and powerful numerical methods for solving the surface integral equations is the method of moments (MoM). This method is widely used for solving several types of the EM scattering problems. The efficiency and accuracy of this method have been demonstrated in many reports [2], [3], [5], [9], [10]. In recent years, many numerical techniques such as the mixed-domain Galerkin expansion method [11], the wavelets collocation method [12], [13], the characteristic basis functions method [14], the singular integral equation approach [15], spectral methods based on the block pulse functions [16], [17], etc. have been introduced and improved for solving different types of wave scattering problems.

Nowadays, a new class of developed and advanced computational methods based on the radial basis functions (RBFs) [18], [19], [20], has been introduced and successfully employed for solving several types of practical mathematical models arising from engineering and applied sciences [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35]. The RBF’s meshless techniques are very powerful and flexible computational tools for dealing with complicated practical models such as models with complex geometric domains or high-dimensional problems. The standard RBFs based methods can be categorised into two main classes, the numerical methods based on the globally supported RBFs and the compactly supported RBFs (CSRBFs) (see [21] for more details).

In this paper, a numerical technique based on the CSRBFs will be introduced and formulated to solve the EFIE and MFIE arising from the EM scattering problem from infinite perfectly conducting cylinders with arbitrary cross sections. The organisation of this paper is as follows. Section 2 presents some mathematical modelling of EM scattering from perfectly conducting objects. A brief review of compactly supported RBFs and their applications on evaluation surface integral equations will be explained in Section 3. In Section 4, three test problems and some practical examples of scattering from some infinite perfectly conducting cylinders with arbitrary cross sections will be presented to show the accuracy and applicability of the method.

2 Mathematical Modelling of EM Scattering from Perfectly Conducting Objects

The solution of any scattering problem depends on a knowledge of the current density distribution on the surface of the scatterer. Once it is known, then the scattered field can be found using the standard radiation integrals. In scattering problems, the current density is an unknown quantity. Therefore, solving the scattering problems involves two steps:

1. Solving an integral equation for unknown current J created by a known incident field E^inc or H^inc.

2. Integrating the induced currents J to obtain the scattered field E^scat and H^scat.

In this section, we will formulate the well-known electric and MFIEs of scattering from infinite perfectly conducting objects. There is not any reflected wave from the ends of an infinitely long object, so the current does not change along the length and the problem, therefore, becomes two dimensional. For more details on this section, the interested reader can see [1], [3], [5].

Figure 1 shows a two-dimensional scatterer of arbitrary cross section with contour C in both transverse magnetic (TM) and transverse electric (TE) modes.

Figure 1:

Geometry for two-dimensional scatterer of arbitrary cross section. (a) TM polarisation. (b) TE polarisation.

3 Surface Integral Equations Evaluation

As observed in the previous section, the wave scattering problems from the perfectly conducting surfaces have been modelled by linear Fredholm integral equations. So, to analyse the scattering from the infinite conducting cylindrical objects in TM or TE modes, a linear Fredholm integral equations of the first kind (EFIE) (7) or a linear Fredholm integral equations of the second kind (MFIE) (14) should be solved, respectively. Note that the kernels of these surface integral equations are singular, and moreover, the first kind of Fredholm integral equations are ill-posed. Thus, most standard and calssical numerical techniques cannot achieve a good accuracy in solving these singular and ill-posed problems. Now, our interest is to formulate and apply an advanced computational method based on the compactly supported radial functions for solving these surface integral equations.

3.2 EFIE Evaluation

Here, we recall the following EFIE (7)

Ezinc(ρ) = kη04∫CJz(ρ′)H0(2)  (k∥ρ − ρ′∥)dc′.

Now, the unknown current function J_z(ρ′) is approximated using a linear combination of CSRBFs as

(21)Jz(ρ′) = ∑j = 1Nλjϕ(rj), (21)

where ϕ(.) is a proper compactly supported radial function and r_j=∥ρ′–ρ_j∥ denotes the Euclidean distance between ρ′ and ρ_j. ρ_j, j=1, …, N are centre points on the perimeter of cross-section C. Moreover, {λj}j = 1N is a set of N unknown coefficients to be determined. Substituting (21) in EFIE (7) gives

kη04∫CH0(2)  (k∥ρ − ρ′∥)(∑j = 1Nλjϕ(∥ρ′ − ρj∥))dc′ = Ezinc(ρ),

and therefore,

(22)∑j = 1Nλj(∫CH0(2)  (k∥ρ − ρ′∥)ϕ(∥ρ′ − ρj∥)dc′) = 4kη0Ezinc(ρ). (22)

In the following, the perimeter of cross section, C, is discretised into N straight boundary elements C_i, i=1, …, N, and the collocation points ρ˜i are chosen as the centre of C_i, see Figure 2. Now, a point matching scheme can be implemented by satisfying (22) at collocation points ρ˜i, i = 1, … , N, as

Figure 2:

Descretisation the perimeter of cross section C.

(23)∑j = 1Nλj(∫CH0(2)  (k∥ρ˜i − ρ′∥)ϕ(∥ρ′ − ρj∥)dc′) = 4kη0Ezinc(ρ˜i),  i = 1, 2, …, N. (23)

Using the compact support property of the selected trial functions, the above equations can be reduced to the following:

(24)∑j = 1Nλj(∫SjH0(2)  (k∥ρ˜i − ρ′∥)ϕ(∥ρ′ − ρj∥)dc′) = 4kη0Ezinc(ρ˜i),  i = 1, 2, …, N, (24)

where according to the definition of Wendland’s CSRBFs, Sj = {ρ′ ∈ C:∥ρ′ − ρj∥ ≤ 1ϵ} ⊂ C, denotes the support of trial function ϕ(r_j). The accuracy and the stability of the process is highly dependent on the scaling parameter ε. The optimal value for the scaling parameter can be found by the Contour-Pade algorithm [21].

Clearly, (24) is an N×N linear system of equations, which can be presented as follows:

(25)Ξλ = 4kη0Ei, (25)

where

λ = [λ1, λ2, … , λN]T,   Ei = [Ezinc(ρ˜1), Ezinc(ρ˜2), …, Ezinc(ρ˜N)]T,

also the elements of the square matrix Ξ are given by

(26)Ξij = ∫SjH0(2)  (k∥ρ˜i − ρ′∥)ϕ(∥ρ′ − ρj∥)dc′,   i, j = 1, …, N. (26)

Now, the unknown coefficient vector λ can be obtained by solving the linear system of equation (25).

In the practical implementation of the method the boundary integral, (26) should be evaluated numerically. For this purpose, the boundary integral (26) is presented as follows:

(27)∫SjH0(2)  (k∥ρ˜i − ρ′∥)ϕ(∥ρ′ − ρj∥)dc′ = ∑p = 1Nj(∫CpH0(2)  (k∥ρ˜i − ρ′∥) ϕ(∥ρ′ − ρj∥)dc′), (27)

where Sj = ∪p = 1NjCp. Now, for computing the above boundary integral, when the collocation point ρ˜i is not on C_p, a simple centroid formulation is employed to evaluate the boundary integrals, which is

(28)∫CpH0(2)  (k∥ρ˜i − ρ′∥)ϕ(∥ρ′ − ρj∥)dc′ = |Cp|H0(2)  (k∥ρ˜i − ρ˜p∥) ϕ(∥ρ˜p − ρj∥), (28)

where ρ˜p is the centre of C_p. When the collocation point ρ˜i belongs to the segment C_p, then the Hankel function in the boundary integral is a singular kernel and so, it requires a special treatment (note that the boundary integral is convergent). Several numerical and analytical procedures have been introduced in the literature to deal with this singular boundary integral. In particular, an efficient numerical approach based on the sinh transformation has been introduced and formulated by Elliot and Johnson for solving nearly weakly singular integrals with Hankel kernels [39]. This technique is employed for dealing with some scattering problems [40], [41]. As an alternative procedure, it is clear that for small argument ρ, the expansion of the Hankel function is as follows:

(29)H0(2)  (ρ) = 1 − j2πlnγρ2 + O(ρ2, ρ2lnρ),   ρ → 0, (29)

where γ=1.781 is Euler’s constant. So, by substituting the relation (29) in the singular boundary integral, we have

(30)∫CpH0(2)  (k∥ρ˜i − ρ′∥)ϕ(∥ρ′ − ρj∥)dc′ ≈ ∫Cp(1 − j2πlnkγ∥ρ˜i − ρ′∥2) ϕ(∥ρ′ − ρj∥)dc′, =∫Cpϕ(∥ρ′ − ρj∥)dc′ − j2π∫Cplnkγ∥ρ˜i − ρ′∥2 ϕ(∥ρ′ − ρj∥)dc′. (30)

In our implementation, the first above boundary integral is calculated exactly and the singular integral with logarithmic kernel is evaluated according to [42].

4 Results and Discussion

In the current section, some numerical experiments are presented to illustrate the accuracy and efficiency of the presented method. In the first three examples, some test problems that their exact solutions are available have been investigated. The solutions of these problems are approximated with the presented method using CSRBFs and the MOM using the block pulse trial functions, and then, the absolute errors are computed to analyse the convergence and accuracy of the method. Next, some practical scattering problems will be numerically investigated to show efficiency of the method. In fact, the presented method is used for computing the current density and RCS for some infinite perfectly conducting cylinders with square, triangular, and elliptical cross sections.

In all cases, the Wendland’s CSRBF are chosen as trial function and also a proper and optimal value for the scaling factor ε is computed using the Contour-Pade algorithm for each case.

Example 1: Test problem 1

As the first example consider the following linear Fredholm integral equation of the first kind

The exact solution of the integral equation is [16], [17]

u(t) = t2e2t + jln|t + b − a|.

The source function f(s) is extracted from the exact solution and computed numerically using the composite trapezoidal rule. In this case, we let: C=[–0.5, 0.5] and k=2π. We discretise C into N=200 equal segments such that h = b − aN. Two common Wendland’s CSRBFs in one-dimensional ϕ1,0(r) = (1 − ϵr)+1 and ϕ1,1(r) = (1 − ϵr)+3(1 + 3ϵr) are used as the trial functions. As mentioned before, the Contour-Pade algorithm is applied for determining the optimal shape parameter ε. These optimal values are ϵ = 16h for ϕ_1,0(r) and ϵ = 17.55h for ϕ_1,1(r). If we define root mean square error (RMS) as

RMS − error = 1N∑i = 1N[uexact(xi) − uapproximate(xi)]2,

in order to confirm Contour-Pade algorithm results, we plot the variations in the RMS-error in respect to ε in Figure 3a and b for ϕ_1,0(r) and ϕ_1,1(r), respectively, and the optimal values of ε are shown. In Figure 4a and b, the RMS-error as a function of N is plotted. This figure confirms that when N increases, the approximate solution will be converge to exact solution. In Table 2, the absolute error for nine interior points is calculated for both Wendland’s CSRBFs and compared to MOM with N=200 block pulse trial functions. Clearly, all the reported results show the accuracy and efficiency of the presented method.

Figure 3:

Optimal ε in the first test problem for N=200 uniformly spaced points based on (a) ϕ_1,0, (b) and ϕ_1,1.

Figure 4:

RMS-error for the first test problem with (a) ϕ_1,0 and (b) ϕ_1,1 based on N uniformly spaced points in [–0.5, 0.5] and optimal ε for each N.

Example 2: Test problem 2

Consider the following second-kind Fredholm integral equation

with exact solution [16], [17]

u(t) = t2e2t + jln|t + b − a|.

The source function f(s) is obtained by inserting the exact solution in (37) using the composite trapezoidal rule. The centre and collocation points, trial functions, and optimal value of ε used in this example are the same as previous example. Table 3 shows the absolute error of the approximate solution obtained by current method for k=2π, a=–b=–0.5, and compare them with the MOM. These results confirm that this method works well and is more accurate than MOM with N=200 block pulse trial functions.

Example 3: Test problem 3 (Circular cylinder)

As a practical test problem, suppose that an infinitely perfect conducting circular cylinder of radius a is encountered by a plane wave. Because the circular cylinder has a regular shape, it has been analysed using the method of separation of variables. First assume that the incident wave is a TM plane wave of unit amplitude (E₀=1) and zero incident angle. We express it in terms of cylindrical waves as

The analytical solution for surface current density on the perimeter of circular cylinder can be expressed as [1]

We solve EFIE (7) and MFIE (14) for TM and TE modes, respectively, by the presented method to calculate J. In this case, we consider the radius equal to 2 wavelength, or a=2λ and a two-dimensional Wendland’s CSRBF ϕ2,1 = (1 − ϵr)+4(1 + 4ϵr) is used as the basis function. The presented technique is employed to approximate the current density by spreading N=300 identical centre and collocation nodes uniformly along the perimeter of cross section C. The shape parameter ε varies in [1], [10], and its optimal value for N=300 is shown in Figure 5a. RMSE is computed for various value of N and plotted in Figure 5b. It confirms the convergence of the approximate solution to the exact solution for large value of N. It should be mentioned that for each N, its optimal shape parameter is considered to calculate the RMSE. To show the validity of this method, EFIE is also solved using MOM ([3]) and absolute error of solution as a function of angle along the perimeter of cylinder is compared to the current method in Figure 6a and b).

Figure 5:

Result for EFIE in TM mode; (a) RMS-error vs. shape parameter ε for N=300. (b) RMS-error vs. N.

Figure 6:

Absolute error of the current density as a function of angle along the perimeter of the circular cylinder for N=300, k=2π for solution of EFIE in TM mode: (a) by MOM, (b) by presented method.

Now, let us assume that a TE plane wave is normally incident upon a perfectly conducting circular cylinder of radius a (θ_i=0). The incident magnetic field can be written as

The current density on the surface of the cylinder can be expressed analytically as [1]:

Similar to TM mode, MFIE is solved numerically by the method presented in the previous section and the corresponding result of optimal shape parameter and convergency analysis is shown in Figure 7a and b, respectively. A comparison between MOM ([3]) and our method is presented in Figure 8a and b, respectively, in respect to absolute error of current density along the perimeter of cylinder.

Figure 7:

Result for MFIE in TE mode: (a) RMS-error vs. shape parameter ε for N=300. (b) RMS-error vs. N.

Figure 8:

Absolute error of the current density as a function of angle along the perimeter of the circular cylinder for N=300, k=2π for solution of MFIE in TE mode, (a) by MOM, (b) by presented method.

This was an interesting test problem to analyse the performance of the method in actual scattering problem. A brief look at the graphs and a comparison between two methods justifies validity, accuracy, and efficiency of the presented method.

Example 4: Square cylinder

In this example, we obtain the current density along a perfectly conducting square cylinder of half-length side a which encountered by a plane wave as shown in Figure 9a. The numerical solution of EFIE (7) and MFIE (14) for a=λ is calculated by the presented method using ϕ_2,1 as the basic function and N=296 centre and collocation points. The magnitude of the current density has been shown in Figure 9b. The bistatic RCS is given for TM polarisation in Figure 9c and for TE polarisation in Figure 9d.

Figure 9:

(a) Cross section of square cylinder scatterer. (b) Current density magnitude along a perfectly conducting square cylinder for k=2π, a=λ, θ_i=0, N=296, and ε=5.5 obtained by the presented method for TM and TE polarisations. Its RCS (c) for TM, (d) for TE polarisation.

Example 5: Triangular cylinder

The magnitude of current density along the perimeter of a triangular cylinder of length side a which encountered by a plane wave is numerically calculated by the presented method for a=2λ using ϕ_2,1 as the basic function and N=296 centre and collocation points. Figure 10a shows its cross section. The results of solving EFIE (7) and MFIE (14) is plotted in Figure 10b and Figure 10c, respectively. The bistatic RCS pattern is plotted in Figure 10d.

Figure 10:

(a) Cross section of triangular cylinder scatterer. (b) Current density magnitude along a perfectly conducting triangular cylinder for k=2π, a=2λ, θ_i=0, N=296, and ε=5.5 obtained by the presented method for TM polarisations. (c) The result for TE polarisation. (d) Their bistatic RCS for TM and TE polarisation.

Example 6: Elliptical cylinder

In the last example, we obtain the RCS and the magnitude of current density along the perimeter of two perfectly conducting elliptical cylinder which encountered by a plane wave. Figure 11a shows the cross section of an elliptical cylinder. This problem is solved by the presented method using the CSRBF ϕ_2,1 and N=300 centre and collocation points. Figure 11b shows the result for a = λ4, b = 4a, for both TM and TE polarisations. The bistatic RCS is given for TM in Figure 11c and for TE in Figure 11d.

Figure 11:

(a) Cross-section of elliptical cylinder scatterer. (b) Current density magnitude along a perfectly conducting elliptical cylinder for k=2π, a = λ4, b = 4a,θ_i=0, N=300, and ε=5.5 obtained by the presented method for TM and TE polarisations. Its bistatic RCS (c) for TM, (d) for TE polarisation.

Figure 12a shows the current density along the perimeter of another elliptical cylinder for a = λ, b = a4 for both TM and TE polarisations. The bistatic RCS is given for TM and TE polarisations in Figure 12b. These figures can be compared with figures in [5].

Figure 12:

(a) Current density magnitude along a perfectly conducting elliptical cylinder for k=2π, a = λ,b = a4,θ_i=0, N= 300, and ε=5.5 obtained by presented method for TM and TE polarisations. (b) Their RCS for both TM and TE polarisations.

5 Conclusion

The EM scattering from infinite perfectly conducting cylinders with arbitrary cross sections has been modelled mathematically. The EFIE and MFIE for TM and TE modes, respectively, have been derived. The efficient meshless technique based on CSRBFs has been applied to approximate the solution of these surface integrals. The compactly supported and radial property of CSRBFs reduce the computational cost and improve the efficiency of the method. The accuracy, robustness, and efficiency of the method have been tested and validated through two test problems that their exact solutions are available. This method is also well used to compute the RCS and the electric current density on the perimeter of infinite perfectly conducting cylinders with different cross sections. In the case of circular cylinder, the analytical solution exists and the comparison between the analytical and numerical solutions shows a good agreement and confirms the accuracy and validity of the method.