Efficient Method for Solving TM-Polarized Plane Wave Scattering from Two-Dimensional Perfect Conductor Surfaces Using Fourier Series Approximation of the Green’s Function

The method of moments generates a matrix which is usually solved using iterative methods due to the high computational complexity of a direct inversion. The cost of matrix-vector multiplications and memory requirement at each iteration step is proportional to O(N), where N is the number of unknowns in the problem. To reduce the computational complexity, the Green’s function is approximated using Fourier series. This will allow to separate the source points from the observation points. Hence, aggregate all source points and then multiply it with each observation point with a small adjustment in the aggregation term. The proposed method is tested by solving electromagnetic wave scattering from perfect conductor twodimensional basic canonical shape, i.e., circular cylinder. The results showed that the proposed method is accurate and for large N it has a computational complexity less than the direct matrix-vector multiplication.


Introduction
Method of moments (MoM) is a numerical method that is used to solve boundary-integral equations [1] and produces a matrix. The direct inversion for the matrix is limited to small scale problems due to the high numerical cost. Hence, iterative methods are found to be numerically more effective [2] where matrix-vector multiplication (MVM) count and memory requirement (MR) are proportional to O(N 2 ) at each iteration step where N is the number of unknowns in the problem.
For the last several decades, two major groups of algorithms have been developed to decrease the numerical demands of MVM and the associated MR. The first group is the algebra-based methods such as the adaptive crossapproximation algorithm [3], [4], the multilevel matrix decomposition method [5], and the IE-QR algorithm [6]. They are kernel independent and improve the computational complexity through linear-algebra manipulations on the MoM matrix. The second group is the kernel-based methods. Their implementations, performances, and constructions depend on the specific integral kernels. This group uses two ideas. The first idea depends on the grid representation to enable the use of the fast Fourier transform (FFT) to solve the MoM matrix. One old and simple method is a conjugate gradient fast Fourier transform [7], [8]. It reduces the MVM operation count and MR proportional to O(NlogN) and O(N), respectively. However, this method does not work with all kinds of basis functions and hence its application is restricted [9]. To solve this issue, an adaptive integral method was developed which reduces the MVM operation count proportional to O(N 3/2 logN) operations and the MR proportional to O(N 3/2 ) [10], [11]. This method uses arbitrary basis functions that are projected on a uniform grid to enable the use of FFT. A similar idea is also employed in the precorrected-FFT [12], [13], sparsematrix/canonical grid [14], and integral equation-FFT methods [15]. The other idea is to replace the Green's function with an equivalent mathematical representation that separates the observation point from the source point. The most well-known method to adapt this idea is the fast multipole method (FMM) which reduces both MVM and MR to O(N 3/2 ) [16], [17]. The multilevel version of FMM reduces numerical complexity to O(NlogN) [18]. Several enhancements, adjustments, and approaches have been done or based on the conventional FMM over the years [19][20][21][22][23][24]. However, all [16][17][18][19][20][21][22][23][24] methods require intensive derivation to separate the observation point from the source point and sometimes it is impossible which make these methods useless. In addition, even though the methods [3][4][5][6] and [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24] introduce additional parameters on the MoM which will lead to increase the difficulty of the algorithm, their efficiency improves as N increases. Note MoM is usually used to solve problems that require millions of unknowns or more.
In this paper, the Green's function is replaced with the Fourier series (FS) approximation. This method will be called the Fourier series method of moments (FS-MoM). The resultant equation separates the observation point from the source point without any additional effort and regardless of the Green's function. Therefore, all source points will be aggregated and then multiplied by each observation point with a small adjustment to the aggregation term. The advantages of this approach are efficiency, accuracy, generality, and simplicity.
The outline of this paper is as follows. An FS representation analysis for two-dimensional (2D) function and an FS representation of the 2D Green's function for electromagnetic wave scattering (EWS) is described in Sec. 2.1. Section 2.2 discusses how the FS-MoM is incorporated with MoM to reduce the computational complexity. Finally, Section 3 presents the numerical results to validate the proposed method, comparison, and discussion when the FS-MoM and MoM are applied to solve the EWS from perfect electric conductor (PEC) basic canonical shape, i.e., circular cylinder.

Fourier Series Representation Analyses
in the x-direction and on [-L y /2  y  L y /2] in the y-direction where L x and L y are constants. The FS representation of f(x,y) is given by [25]   For 2D EWS, the Green's function is the zeroth-order Hankel function of the second kind f(x,y) = H 0 (2) (k 0 |ρ|) [9] where k 0 = 2π/λ is the wavenumber, λ is the wavelength and = xa x + ya y . The FS representation for it using (1) is where N x and N y are the upper limits of the summations, TR is the truncation error, and f N (x,y) is the approximation of f(x,y) when the summations are truncated by N x and N y .
Equation (3) is valid in the domain of the solution except when = 0. Truncation error arises in (3) due to replace the ± limits in the summations by N x and N y . Limits of the summations can be determined for the required mean square error (MSE) using [25]   y x Note is that the Fourier transform ( ) for 2D Green's function is given by [26] where k x and k y are the Fourier variables. Therefore, equation (4) can be used to evaluate the Fourier coefficient C n x n y as Using (4) instead of (2) will significantly reduce the load of calculate the Fourier coefficient.

Incorporating FS-MoM in MoM
Consider a problem of a scalar wave produced by a source E z inc in the presence of a 2D arbitrarily PEC shaped object centered at the origin and having maximum length of L x in the x-direction and maximum length of L y in the y-direction immersed in free space as shown in Fig. 1.
Assume that both the source and the object have no variation along the z-axis. The solution of the electric field integral equation (EFIE) for this case with TM z polarizing using MoM is given by [9]  E ZI (6) where Z is the impedance matrix given by [9]  • PEC Free space where Γ denotes the boundary enclosing the scatterer, = xa x + ya y and ' = x'a x + y'a y denote the observation and the source point respectively, ρ and ρ' , t m (ρ) denotes the testing functions, f n (ρ') denotes the basis functions, and  0 is the free space impedance, E 0 exp(jk 0 (x cosΦ i + y sinΦ i )) is the known incident electric field, E 0 is a constant, and Φ i is the incident angle, I = [I 1 …I n …I N ] is a column vector to be determined and the relation between it and the unknown current is Approximate the Green's function in (7) using (1) where T 0 x = 2L x and T 0 y = 2L y to cover all the possible values of |ρ -ρ'|. Equation (8) This approach, i.e., (9), can be used. However, note that the Green's function, i.e., H 0 (2) (k 0 |ρ|) goes to infinity when (k 0 |ρ|) goes to zero. As the result, high values for the upper limits in (9) are required. Therefore, to reduce the values of the upper limits which will lead to reduce the computational complexity for the proposed method, rather than use the FS-MoM for the whole interval, it will be used for (k 0 |ρ|) > (k 0 |ρ̄ |) where | | is a none zero constant where the FS-MoM starts. Hence, this method will use a combination of direct multiplication for (k 0 |ρ|) < (k 0 |ρ̄ |) and fast multiplication, i.e., FS-MoM for (k 0 |ρ|) > (k 0 |ρ̄ |). Therefore, (6) where | | is the value where fast multiplication starts.
The implementation of (10) is done as follows. First, determine the first observation point (m = 1) and then divide the source points into two groups, one corresponding to direct multiplication using n  |ρ -ρ'| < |ρ̄ | and the other to fast multiplication using n  |ρ -ρ'| > |ρ̄ |. Second, multiply the unknown currents with its source points using

Results and Discussion
In this section, the FS-MoM method is validated first by comparing the Green's function, i.e., H 0 (2) (k 0 |ρ|), with its FS representation, i.e., (3), and calculating the MSE between them. Then, the proposed method, i.e., (10), is used to solve EWS from canonical 2D shape, i.e., circular cylinder, with different sizes and MSE with MoM, i.e., (6), as a reference is calculated as a function of the upper limits of the summations and number of points on the scatterer. A comparison of the time and memory required to perform MVM are shown for the two methods. The results demonstrate the numerical complexity and accuracy of the proposed method.
: number of total basis functions.

FS-MoM
N x and N y : upper summation limits for the 2D FS-MoM. N direct : number of points that are going to multiply directly in the FS-MoM method. N fast = N -N direct : number of points that are going to multiply efficiency using the FS-MoM method.

Term
Computational Complexity Near-field interactions mn n n Z I Computational complexity for FS-MoM. For the next results, the following assumptions are used. First, the iterative method that is used in this section is the conjugate gradient method [2]. Second, (k 0 |ρ|) > (0.55) = (k 0 |ρ̄ |) is used. Third, E 0 = 1 and Φ i = 0 are assumed for the incident field. Forth, the wavelength λ = 1 is used. Finally, by divide Γ into small segments and using pulse basis functions and the Dirac delta testing function in (7), (7) becomes [9] To show the stability and accuracy of the proposed method, EWS from PEC circular cylinders as an example is used. The analytical solution for it is given by [27]      (13) where J i is the induced current at contour angle θ i , E 0 is the magnitude of the incident field, μ 0 = 4π10 -7 H/m is permeability of vacuum, a is the circular cylinder radius, and ω = 2π/f where f is the frequency used in the problem.
EWS from PEC circular cylinders with radii 1λ m and 2λ m are solved using FS-MoM as a function of N x and N y where N x = N y is used. For each radius, different number of points N = 2 8 , 2 9 , 2 10 , 2 11 , and 2 12 are used to test the behavior of the method with an increased number of points. Figure 3(a), and 3(b) show the results. It is clear that as the upper limits increase, the MSE reduces. Also, it can be noted that increasing N does not change much in MSE. This makes sense because the approximation has enough Fourier coefficients to represent the original function. This result is the essence of explaining why FS-MoM reduces the computational complexity. In MoM, as N in the problem increases, the computational complexity will increase in O(N 2 ) relation which is a square relation. On the other hand, for FS-MoM, after determining the values of the summation limits for the required error, the increases of N will affect the computational complexity in a linear relation as shown in Tab. 1, i.e., N  N direct + (2N y +1)(2N x + 1)  [N fast + 1 +N] where (2N y +1)(2N x + 1) is a constant. Figure 4 shows the unknown currents versus θ, where is the angle along contour of the scatterer for a PEC circular cylinder of radius 2λ m, solved using the analytical solution (13), MoM (11) and FS-MoM (12). The time and memory required to perform MVM between the two methods are shown in Fig. 5   that as N increased, the proposed method becomes faster and uses less memory than the MoM.

Conclusions
FS can be used to represent the Green's function, i.e., the zeroth-order Hankel function of the second kind. This representation can be used to reduce the computational complexity of the MoM. The results showed that the proposed method is accurate, efficient, and the most important, it can be used without any mathematical derivation on the Green's function that other methods required. Future work will focus on reducing the computational complexity more by using the FS properties and on studying different approaches to execute the method.