Analysis of Homogeneous Waveguides via the Meshless Radial Basis Function-Generated-Finite-Difference Method

The radial basis function-generated-finite-difference (RBF-FD) method is applied to the analysis of homogenous waveguides. To this end, the Helmholtz equation and the boundary conditions are collocated on the waveguide cross section. At each collocation node, derivatives are locally approximated by the RBF-FD formulas based on the polyharmonic splines supplemented with high-degree polynomials. As a result, a sparse matrix eigenvalue problem is obtained, which allows cutoff wavenumbers and axial fields to be calculated. To illustrate the accuracy of the method, we consider a semicircular and an eccentric circular waveguide.


I. INTRODUCTION
T HE numerical solution of partial differential equations by classical finite-difference (FD) methods is based on using a mesh to approximate the unknowns and their derivatives. Alternatively, in the meshless methods, the approximation is based on scattered nodes without any mesh connectivity, which provides more flexibility and simplicity in complex geometries [1].
During the last decades, meshless methods have been successfully applied in many branches of science and engineering. However, they still remain relatively less developed within the computational electromagnetics field [2]- [4].
Among the meshless methods, the radial basis function generated-FD (RBF-FD) method can be viewed as a generalization of the classical FD method from structured meshes to scattered node sets. It has been recently shown that the RBF-FD method allows high-order local FD formulas to be obtained by using interpolants based on polyharmonic spline (PHS) RBFs supplemented with high-degree polynomials [5], [6]. Manuscript  In this letter, the RBF-FD method is applied to the analysis of homogenous waveguides. To this end, the collocation approach is applied to the Helmholtz equation with appropriate boundary conditions (BCs). At each collocation node, derivatives are locally approximated by suitable RBF-FD expressions. As a result, a sparse matrix eigenvalue problem is obtained, which allows cutoff wavenumbers and axial fields to be calculated. To illustrate the accuracy of the method, we consider a semicircular and an eccentric circular waveguide.
Other meshless methods based on RBFs have recently been applied to the analysis of homogeneous waveguides (see [7] and the references within). Comparing with them, the RBF-FD method adopted here does not require optimal shape parameter calculation. In addition, polynomials up to a high degree are appended, which largely control the rate of convergence of the solution.

II. THEORY
Electromagnetic waves in a homogeneous waveguide with cross section Ω and perfect electric conductor boundary ∂Ω are determined by solving the Helmholtz eigenvalue problem where L is the Laplacian in the transverse (xy) plane, k c is the cutoff wavenumber, and u = E z , H z for the TM and TE modes, respectively. The boundary operator L B is related to the type of mode. TM modes are subjected to Dirichlet BCs, E z = 0, while TE modes to Neumann BCs, ∇H z ·n = 0. With the goal of approximating (1) by a matrix eigenvalue problem, (1a) is collocated at N I distinct node points in Ω and (1b) at N B distinct node points on ∂Ω, with N I +N B = N. As shown in Fig. 1, at each collocation node r c , Lu (or L B u) is approximated by an FD formula with a stencil formed by the N s closest node points, including the collocation node itself where u i ≡ u( r i ) are the values of u at the stencil points, w i are the weights, and N s is the stencil size (N s N ). In (2), The calculation of w begins by considering an interpolant of the form [5], [6] where a j and b j are the coefficients. The functions p j ( r) are all the N p = d+2 d bivariate monomials up to degree d. For instance, for d = 3, there are N p = 10 monomials: 1, x, y, xy, x 2 , y 2 , x 2 y, xy 2 , x 3 , and y 3 . The RBFs φ j ( r − r j ) are N s PHS centered at each node of the stencil r j , and · denotes the Euclidean norm. The PHS radial functions are defined as φ(r) = r m if m is odd and φ(r) = r m ln r if m is even. For simplicity, only the odd case is considered here.
To obtain a j and b j , the interpolant (3) is enforced to be exact at the N s nodes of the stencil with the additional constrains for the polynomials Ns j=1 a j p i ( r j ) = 0 (i = 1, . . . , N p ).
This results in the following block linear system: Applying the differential operator L to the interpolant in (3) and evaluating at r c , we get Solving (5) for [a b] T , substituting the result into (6), and comparing with (2), the following linear system is obtained: where w contains the sought weights and the coefficients in v are discarded after solving. By solving the linear system (7) N I times, one for each collocation node in Ω, a sparse differentiation matrix A is constructed, which permits (1a) to be replaced by Repeating the same procedure for the N B nodes on ∂Ω, the Neumann BC is approximated by the homogeneous system Incorporating (9) into (8) leads to the following ordinary eigenvalue problem for the TE modes: For the TM case, the Dirichlet BC can be directly imposed by simply letting u B = 0 in (8).
The application of the RBF-FD method involves several parameters, which should be carefully selected [5], [6]. The error of the method is strongly dictated by the highest degree of the appended polynomials, d. This is because the error associated with (6) is the internodal distance and k is the order of the differential operator L. As mentioned above, for a given d, the total number of appended polynomials is N p = d+2 d . The stencil size is then taken as N s 2N p . Finally, the PHS exponent is commonly chosen as m = 2d + 1.

III. NUMERICAL RESULTS
To illustrate the accuracy of the RBF-FD method, we consider the calculation of the cutoff wavenumbers of the first modes of an empty semicircular waveguide of radius a, shown in Fig. 2. The exact solutions of this problem can be found in [8, Tables 1, 2]. Fig. 3 depicts the relative error in the cutoff wavenumber for the first six TM pn and six TE pn modes versus the square root of the number of interior nodes, √ N I . The used RBF-FD parameters have been d = 5 (N p = 21), m = 11, and N s = 45. It can be seen that the TM pn modes exhibit a fourthorder rate of convergence. In the TE case, the convergence rate reaches fifth-order accuracy. Fig. 4(a) and (b) shows the relative error in the cutoff wavenumber as a function of √ N I for the TM 11 mode. In Fig. 4(a), this error is plotted for several  values of the PHS exponent, m. It can be seen that the rate of convergence does not change with m. In Fig. 4(b), the error is plotted for several values of the stencil size N s . Analogously to Fig. 4(a), a vertical shifting in the error curves is only observed by varying N s .
Second, we consider an eccentric circular waveguide with the outer and inner conductors of radii a and b, respectively. The offset between the centers of both conductors is , as shown in Fig. 5(a). The cross section of this waveguide is symmetric with respect to the x-axis. Consequently, even and odd modes can be calculated separately. For the sake of brevity, we will focus on the even TM modes. To this end, the upper half waveguide is solved with Dirichlet BCs on the conductors and Neumann BCs on the x-axis, as illustrated in Fig. 5(b). Fig. 6 depicts the longitudinal field, E z , and the normalized cutoff wavenumber, k c a, for the first six even TM modes of an   Table I, these results agree up to the digits compared with those given in [9, Table III]. Moreover, the RBF-FD results outperform those calculated, also in [9], by the finite-element method (FEM) with 3028 degrees of freedom.

IV. CONCLUSION
An RBF-FD method based on PHSs with supplementary polynomials has been applied to the analysis of homogeneous waveguides. It has been shown that this technique is able to provide high-order rates of convergence in problems with arbitrary shape domains and smooth solutions. Moreover, the RBF-FD method does not require the choice of an optimal shape parameter and its local nature makes it suitable to be applied to large-scale problems. The efficiency of the RBF-FD method can be improved by using it in subdomains with complex shapes while using classical FD stencils in the remaining part of the waveguide cross section.