Solving the Eigenfrequencies Problem of Waveguides by Localized Method of Fundamental Solutions with External Source

Abstract

The localized method of fundamental solutions (LMFS) is a domain-type, meshless numerical method. Compared with numerical methods that have a high grid dependence, it does not require grid generation and numerical integration, so it can effectively improve computational efficiency and avoid complex integration processes. Moreover, it is formed using the traditional method of fundamental solutions (MFS) and the localization approach. Previous studies have shown that the MFS may produce a dense and ill-conditioned matrix. However, the proposed LMFS can yield a sparse system of linear algebraic equations, so it is more suitable and effective in solving complicated engineering problems. In this article, LMFS was used to solve eigenfrequency problems in electromagnetic waves, which were controlled using two-dimensional Helmholtz equations. Additionally, the resonant frequencies of the eigenproblem were determined by the response amplitudes. In order to determine the eigenfrequencies, LMFS was applied for solving a sequence of inhomogeneous problems by introducing an external source. Waveguides with different shapes were analyzed to prove the stability of the present LMFS in this paper.

1. Introduction

A waveguide is a hollow metal device with a very clean inner wall, which can trap energy in itself instead of radiation. Waveguides are used to transfer energy from one place to another. Waveguides can have many shapes, such as rectangular, circular, and elliptical. Due to the effect of transferring and guiding electromagnetic waves, waveguides are widely used as optical fiber and microwave elements and components, etc., in optic and electronic applications. Waveguide devices are also very common in our daily life; examples include mobile phone charging cables, network cables, etc. Studies have shown that the eigenfrequencies of a waveguide have a significant effect on its electromagnetic wave propagation performance. Hence, it is important to determine the eigenfrequencies of a waveguide when electromagnetic waves with specific frequencies have to propagate in a designed direction. The purpose of our study was to analyze the characteristic frequencies of waveguides with different shapes and find the specific wavelength at which different waveguides resonate.

In the past, many researchers have carried out a lot of research to determine the eigenfrequencies of waveguides. There are many numerical methods proposed for solving eigenproblems, which can be roughly classified as the mesh generation method and meshfree ones. The traditional grid method requires a grid to be set up and produces complicated calculations, while meshless methods do not depend on meshing and do not require numerical integration. Hence, an increasing number of researchers prefer meshless methods. With the development of computer technology, many meshless numerical methods have been developed to solve complicated engineering science problems. For example, Young et al. [1] adopted the method of fundamental solutions (MFS) [2,3,4,5,6] with the technique of singular value decomposition (SVD) as the solution to the eigenproblems of waveguides. Cai et al. [7] used the meshless Galerkin method to solve Dirichlet problems. Fan et al. [8] researched the numerical solutions of boundary detection problems using the modified collocation Trefftz method [9,10,11,12,13]. Xiong et al. [14] combined the BKM and the localization concept to create the localized BKM to solve two-dimensional Laplace and bi-harmonic equations. Jiang et al. [15] adopted the radial basis functions collocation method (RBFCM) [16,17,18] to research eigenproblems of elliptic waveguides. In the application of RBFCM, not only boundary nodes but also interior nodes are needed to participate in the calculation. Therefore, this method has some limitations. Following this, the local RBFCM (LRBFCM) [19,20,21], which can produce a sparse matrix of a linear system instead of a full matrix, especially an ill-conditioned one, was proposed to solve complicated engineering science problems [22,23]. Reutskiy [24,25,26,27] proposed an innovative numerical scheme that utilized the method of external source (MES) to deal with eigenproblems. Then, MFS and MES were used by Fan et al. [28] to find the numerical solutions for eigenfrequencies of the waveguides. Fang, H. M. et al. [29] adopted the least squares Trefftz method (LSTM) and the method of external source (MES) to solve eigenfrequency problems with Helmholtz equation as the governing equation. In recent years, Fan et al. [30] proposed the localized method of fundamental solutions (LMFS). This is a localized version of MFS. Based on the concept of localization, in 2021, Liu et al. [31] proposed the localized Trefftz method (LTM), which improved the traditional Trefftz method and turned it into a localized numerical method.

The localized method of fundamental solution (LMFS) is a domain-type, meshless numerical method. Since LMFS is developed from the traditional MFS, it is a truly meshless method that does not require numerical quadrature. As a localized method, both interior and boundary nodes should be applied into the calculation. Similar to the localized principle in LRBFCM [32,33] and GFDM [34,35], a particular node and its neighbors are used to form the subdomain. Then, a linear system can be obtained entirely by taking consideration of the governing equation or boundary conditions for each specific node in its subdomain. The subdomain centered on each of the inner points is inside the computation domain, so there is no doubt that the Helmholtz equation can be satisfied in the subdomain by the solution of LMFS. Using MFS, the subdomains needs to be focused on.

Based on our understanding of previous research regarding the eigenfrequencies of waveguides and many meshless numerical methods, in this study, we used the LMFS and MES to solve eigenfrequency problems of electromagnetic waves governed by two-dimensional Helmholtz equations with computational domains. The LMFS with an external source transformed the eigenproblem from a homogeneous equation into several inhomogeneous equations, and the eigenfrequencies could be known through the solutions of direct problems. In addition, the proposed localized MFS could efficiently and accurately analyze problems with complicated domains, for example, multi-connected domains. This article represents the first attempt to use LMFS and MES to analyze eigenfrequency problems governed by Helmholtz equations.

This article is divided into the following chapters: in Section 1, the physical problems and the research purpose are briefly introduced. We describe the governing equations and boundary conditions in Section 2. In Section 3, the numerical method of the proposed LMFS and ESM are clearly shown. The numerical results of the proposed LMFS and ESM and comparisons of elliptic, concentric annular, eccentric annular, and multi-connected domain waveguides are analyzed in Section 4. To conclude, Section 5 presents the summaries and a discussion regarding the research in this article.

2. Governing Equations and Boundary Conditions

A homogeneous Helmholtz equation was used to govern the problems of waveguide, which is described as follows:

$$\left({\nabla }^2+k^2\right)\psi \left(x,y\right)=0, \begin{array}{c}\left(x,y\right)=\mathrm{\Omega }\end{array},$$

(1)

where ${\nabla }^2$ is the Laplace operator, $k=2\pi /\lambda$ is the wavenumber, and $\lambda$ is the cutoff wavelength. When the cutoff wavenumber is known, the cutoff wavelengths can be calculated according to the conversion formula. $\psi \left(x,y\right)$ is the unknown variable, and $\mathrm{\Omega }$ is the computational domain. The electromagnetic waves are classified as two basic waves, $\psi \left(x,y\right)=E_Z$ represents the transverse magnetic (TM) wave, while $\psi \left(x,y\right)=H_Z$ represents the transverse electric (TE) wave.

TM waves satisfy the following homogeneous Dirichlet boundary condition:

$$\psi \left(x,y\right)=E_Z\left(x,y\right)=0, x,y\in {\mathrm{\Gamma }}^D.$$

(2)

Additionally, TE waves satisfy the following Neumann boundary condition:

$$\frac{\partial \psi }{\partial n}\left(x,y\right)=\frac{\partial H_Z}{\partial n}\left(x,y\right)=0, \left(x,y\right)\in {\mathrm{\Gamma }}^N.$$

(3)

Here, ${\mathrm{\Gamma }}^D$ is the boundary portion with the Dirichlet boundary condition, while ${\mathrm{\Gamma }}^N$ is that with the Neumann boundary condition. The boundary points must satisfy the homogeneous boundary conditions, and the inner points must satisfy the homogeneous governing Equation (1). In this study, the homogeneous Helmholtz equation and the homogeneous boundary condition formed the eigenproblems for the waveguides. The resonance wavenumbers could be determined using the numerical solution of the governing equation and boundary conditions.

3. Numerical Methods

In this study, different from the solution procedure of the original homogeneous problem, the eigenfrequency problems of the waveguide were solved by LMFS with an external source. By adding the external source, the eigenfrequency problem is converted to a homogeneous Helmholtz equation with an inhomogeneous boundary condition. The external source with known strength can be located anywhere outside the computational domain. The external source is described as ${\overrightarrow{X}}_{ext}=\left(x_{ext},y_{ext}\right)$. Additionally, the inhomogeneous governing equation is expressed as follows:

$$\left({\nabla }^2+k^2\right)\psi \left(\overrightarrow{x}\right)=\delta \left(\left|\overrightarrow{x}-{\overrightarrow{X}}_{ext}\right|\right),$$

(4)

in which $\delta$ is the Dirac delta function. In addition, the numerical solution can be decomposed into the following format:

$$\psi \left(\overrightarrow{x}\right)={\psi }_h\left(\overrightarrow{x}\right)+{\psi }_p\left(\overrightarrow{x}\right),$$

(5)

in which ${\psi }_h\left(\overrightarrow{x}\right)$ represents a homogeneous solution and ${\psi }_p\left(\overrightarrow{x}\right)$ represents a particular solution. At the same time, the fundamental solution of the Helmholtz equation, which is obtained by the Fourier transform theory, is used as the particular solution. Thus, ${\psi }_p\left(\overrightarrow{x}\right)$ can be written as follows:

$${\psi }_p\left(\overrightarrow{x}\right)=\frac{i}{4}{H}_0^{\left(2\right)}\left(k\left|\overrightarrow{x}-{\overrightarrow{X}}_{ext}\right|\right),$$

(6)

in which $H_0^{\left(2\right)}\left(\right)$ is the Hankel function. Equation (6) is the fundamental solution of the Helmholtz equation. In this paper, LMFS was used to solve the homogeneous equation with an inhomogeneous boundary condition. The particular solution satisfies the inhomogeneous Equation (4). Finally, the eigenfrequency problems transfer into the homogeneous Equation (7) and the inhomogeneous boundary condition (8), like the following equations:

$${\nabla }^2{\psi }_h\left(\overrightarrow{x}\right)+k^2{\psi }_h\left(\overrightarrow{x}\right)=0,$$

(7)

$$T_{B.C.}\left[{\psi }_h\left(\overrightarrow{x}\right)\right]=-T_{B.C.}\left[{\psi }_p\left(\overrightarrow{x}\right)\right].$$

(8)

Here, ${\psi }_h\left(\overrightarrow{x}\right)$ is the homogeneous solution that satisfies the homogeneous equation and the modified inhomogeneous boundary conditions. $T_{B.C.}\left[\hspace{0.33em}\right]$ is the partial differential operator for the Dirichlet or Neumann boundary conditions.

In this study, LMFS was suitable for the series of eigenfrequency problems with different cutoff wavenumbers. The numerical procedures of the LMFS for two-dimensional Helmholtz equations are clearly described in the following section.

As depicted in Figure 1a, $n_i$ nodes are randomly distributed inside the computational domain $\mathrm{\Omega }$, and $n_b$ nodes are arbitrarily distributed along the whole boundary $\mathrm{\Gamma }$. $N=n_i+n_b$ denotes the total number of points. The schematic diagram of the computational nodes is displayed in Figure 1a. We chose the ith point as an example to explain the numerical process of the LMFS, as follows: Calculate the distance between the ith node and the other nodes inside the computational domain. Then, take the $m$ nearest nodes around the ith node and form a small area, which is occupied by the ith node and the $m$ nearest nodes. This small area is called the subdomain of the ith node, ${\mathrm{\Omega }}_i$, as shown in Figure 1b. The subdomain of the ith node is inside the computational domain and along the boundary, so the solution of the subdomain also satisfies the Helmholtz equations. We can use MFS to obtain the solution of the subdomain of the ith node, and the solution can be represented as follows:

$${\psi }^{\left(i\right)}={\displaystyle \sum _{j=1}^{m+1}{\alpha }_j^{\left(i\right)}G}\left(\overrightarrow{x},{\overrightarrow{s}}_j{}^{\left(i\right)}\right),$$

(9)

in which $G\left(\overrightarrow{x},{\overrightarrow{s}}_j{}^{\left(i\right)}\right)=-\frac{i}{4}{H}_0^{\left(2\right)}\left(k\left|\overrightarrow{x}-{\overrightarrow{s}}_j^{\left(i\right)}\right|\right)$ is the fundamental solution of two-dimensional Helmholtz equations and ${\left\{{\alpha }_j^{\left(i\right)}\right\}}_{j=1}^{m+1}$ is the unknown coefficients. $\left|\overrightarrow{x}-s_j^{\left(i\right)}\right|$ is the distance between the subdomain nodes and the jth source $s_j^{\left(i\right)}=\left(s_j^{x\left(i\right)},s_j^{y\left(i\right)}\right)$. The number of source points is equal to the number of subdomain nodes. The sources are evenly distributed along a fictitious circular boundary, ${\mathrm{\Gamma }}_s^{\left(i\right)}$, which is centered at the ith node. $R_s=R\times \underset{1\le i\le N}{\max }\left({\lambda }_i\right)$ is the radius of the fictitious circular boundary. $R>1$ is a parameter, and ${\lambda }_i$ is the farthest distance between the ith node and the $m$ nearest nodes. In this paper, $R$ was set to 6 for all of the numerical examples. We substituted all of the points in the subdomain of the ith node into Formula (9) and obtained the following system:

$${\psi }^{\left(i\right)}=\mathrm{C}{\alpha }^{\left(i\right)},$$

(10)

in which $\mathrm{C}$ is the coefficients matrix and its entry is $G\left({\overrightarrow{x}}_{ij},{\overrightarrow{s}}_j^{\left(i\right)}\right)$. ${\alpha }^{\left(i\right)}={\left[{\alpha }_1^{\left(i\right)}\hspace{0.33em}{\alpha }_2^{\left(i\right)}\hspace{0.33em}{\alpha }_3^{\left(i\right)}\hspace{0.33em}\dots \hspace{0.33em}{\alpha }_{m+1}^{\left(i\right)} \right]}^T$ is the vector of the unknown coefficients, and ${\mathsf{\psi }}^{\left(i\right)}={\left[{\psi }_1^{\left(i\right)}\hspace{0.33em}{\psi }_2^{\left(i\right)}\hspace{0.33em}{\psi }_3^{\left(i\right)}\hspace{0.33em}\dots \hspace{0.33em}{\psi }_{m+1}^{\left(i\right)}\right]}^T$ represents the vector of unknown variables at the $m$ node. The symbol $T$ denotes the transpose of the vector. By multiplying the inverse of $\mathrm{C}$,${\mathrm{C}}^{-1}$, to Equation (10), another form of Equation (10) concerning the unknown coefficients can be represented as follows:

$${\alpha }^{\left(i\right)}={\mathrm{C}}^{-1}{\psi }^{\left(i\right)}.$$

(11)

In this paper, we noticed that coefficient matrix $\mathrm{C}$ was ill-conditioned. So, we used the function pinv of MATLAB to calculate the ${\mathrm{C}}^{-1}$, and the precision was set to be 10⁻⁸. The numerical solution of the ith node can be written in the following form by substituting the spatial coordinate of the ith node.

$$\psi \left({\overrightarrow{x}}_i\right)={\psi }_i={\displaystyle \sum _{j=1}^{m+1}{\alpha }_j^{\left(i\right)}G\left({\overrightarrow{x}}_{ij},{\overrightarrow{s}}_j^{\left(i\right)}\right)}=c^{\left(i\right)T}{\alpha }^{\left(i\right)}=c^{\left(i\right)T}{\mathrm{C}}^{-1}{\mathsf{\psi }}^{\left(i\right)}={\displaystyle \sum _{j=1}^{m+1}{w}_j^{\left(i\right)}{\psi }_j^{\left(i\right)}}.$$

(12)

Equation (12) shows the relation of unknown variables between the $m$ nearest nodes and the ith node, and the unknown variables in the subdomain satisfy the Helmholtz equations. $c^{\left(i\right)}={\left[G\left({\overrightarrow{x}}_{i1},{\overrightarrow{s}}_1^{\left(i\right)}\right)\hspace{0.33em}G\left({\overrightarrow{x}}_{i2},{\overrightarrow{s}}_2^{\left(i\right)}\right)\hspace{0.33em}G\left({\overrightarrow{x}}_{i3},{\overrightarrow{s}}_3^{\left(i\right)}\right)\hspace{0.33em}\dots \hspace{0.33em}G\left({\overrightarrow{x}}_{im+1},{\overrightarrow{s}}_{m+1}^{\left(i\right)}\right)\right]}^T$ is the vector of fundamental solution at the ith node. ${\left\{w_j^{\left(i\right)}\right\}}_{j=1}^{m+1}$ is the weighting coefficients connecting with the sources, the fundamental solution of the Helmholtz equations, and the spatial coordinates of the $m+1$ nodes in the subdomain.

With regard to Equation (9), we can directly take the derivative to acquire the derivatives of unknown variables in the conventional MFS. Similar procedures can be applied to Equations (9) and (12) in the proposed LMFS:

$${\left.\frac{\partial \psi }{\partial x}\right|}_i={\left.{\displaystyle \sum _{j=1}^{m+1}{\alpha }_j^{\left(i\right)}\frac{\partial }{\partial x}}G\left(\overrightarrow{x},{\overrightarrow{s}}_j{}^{\left(i\right)}\right)\right|}_i=c_x^{\left(i\right)T}{\alpha }^{\left(i\right)}=c_x^{\left(i\right)T}{\mathrm{C}}^{-1}{\mathsf{\psi }}^{\left(i\right)}={\displaystyle \sum _{j=1}^{m+1}{w}_j^{x\left(i\right)}{\psi }_j^{\left(i\right)}},$$

(13)

$${\left.\frac{\partial \psi }{\partial y}\right|}_i={\left.{\displaystyle \sum _{j=1}^{m+1}{\alpha }_j^{\left(i\right)}\frac{\partial }{\partial y}}G\left(\overrightarrow{x},{\overrightarrow{s}}_j{}^{\left(i\right)}\right)\right|}_i=c_y^{\left(i\right)T}{\alpha }^{\left(i\right)}=c_y^{\left(i\right)T}{\mathrm{C}}^{-1}{\mathsf{\psi }}^{\left(i\right)}={\displaystyle \sum _{j=1}^{m+1}{w}_j^{y\left(i\right)}{\psi }_j^{\left(i\right)}},$$

(14)

in which

$$c_x^{\left(i\right)}={\left[{\left.\frac{\partial G\left(\overrightarrow{x},{\overrightarrow{s}}_1{}^{\left(i\right)}\right)}{\partial x}\right|}_i\hspace{0.33em}{\left.\frac{\partial G\left(\overrightarrow{x},{\overrightarrow{s}}_2{}^{\left(i\right)}\right)}{\partial x}\right|}_i\hspace{0.33em}\dots \hspace{0.33em}{\left.\frac{\partial G\left(\overrightarrow{x},{\overrightarrow{s}}_{m+1}^{\left(i\right)}\right)}{\partial x}\right|}_i\right]}^T,c_y^{\left(i\right)}={\left[{\left.\frac{\partial G\left(\overrightarrow{x},{\overrightarrow{s}}_1{}^{\left(i\right)}\right)}{\partial y}\right|}_i\hspace{0.33em}{\left.\frac{\partial G\left(\overrightarrow{x},{\overrightarrow{s}}_2{}^{\left(i\right)}\right)}{\partial y}\right|}_i\hspace{0.33em}\dots \hspace{0.33em}{\left.\frac{\partial G\left(\overrightarrow{x},{\overrightarrow{s}}_{m+1}^{\left(i\right)}\right)}{\partial y}\right|}_i\right]}^T$$

are the vectors of the derivatives of fundamental solutions at the ith node.

All of the collocation nodes are regarded to be the ith node, and the numerical procedure of every node can be implemented from Equations (9)–(14). In order to satisfy Equation (12) in the interior domain, the following formula is obtained:

$${\psi }_i-{\displaystyle \sum _{j=1}^{m+1}{w}_j^{\left(i\right)}{\psi }_j^{\left(i\right)}}=0,\hspace{0.33em}i=1,2,3,\hspace{0.33em}\dots \hspace{0.33em},n_i.$$

(15)

conditions, we can acquire the resultant sparse system:

$$\mathrm{H}\mathsf{\psi }=\mathrm{f},$$

(16)

in which $\mathsf{\psi }={\left[{\psi }_1\hspace{0.33em}{\psi }_2\hspace{0.33em}{\psi }_3\hspace{0.33em}\dots \hspace{0.33em}{\psi }_N\right]}^T$ represents the vector of unknown variables for each node, ${\mathrm{H}}_{N\times N}$ is the coefficient matrix which represents the sparse linear system of linear algebraic equations, and ${\mathrm{f}}_{N\times 1}$ is the vector of the governing equation and the given boundary condition. The numerical solution at every node can be efficiently obtained by resolving the sparse linear system of Equation (16).

Finally, this work records the resonant responses of numerical solutions with different wavenumbers. Additionally, the resonant responses can be obtained using the formula below:

$$F\left(k\right)=\sqrt{\frac{1}{N_i}{{\displaystyle \sum _{j=1}^{N_t}\left|{\psi }_h\left({\overrightarrow{x}}_j\right)\right|}}^2},$$

(17)

$$F_d\left(k\right)=\frac{F\left(k\right)}{F\left(k_0\right)}.$$

(18)

Here, $k_0$ means a reference wavenumber that was defined as a unit. $F_d\left(k\right)$ is a dimensionless value. $N_i$ is the number of points randomly distributed in the computational domain. Different wavenumbers correspond to different eigenvalues.

A peak appears in the resonant curve when the input wavenumber hits the inherent resonance frequency of the system. Then, the eigenfrequency can be obtained.

Additionally, we can obtain a series of eigenfrequencies for a kind of waveguide using the same method. We calculate the eigenfrequencies of elliptic, concentric annular, eccentric annular, and multi-connected domain waveguides in the following section.

4. Numerical Results and Comparisons

In this paper, eigenfrequency problems governed by the two-dimensional Helmholtz equation were researched through the proposed method, using LMFS and MES. The elliptic, concentric annular, eccentric annular, and multi-connected domain waveguides shown in Figure 2 were investigated to prove the accuracy and convenience of LMFS. For ease of description, some abbreviations were used herein to express the different variables in those examples. $N_b$ represents the quantity of boundary nodes on ${\mathrm{\Gamma }}^D$ or ${\mathrm{\Gamma }}^N$. $N_i$ represents the quantity of interior nodes distributed inside the computational domain. $P$ is the number of nodes in a subdomain. $k$ is the wavenumber. Sometimes, a resonance curve was not smooth, so we had to add a smoothing process by changing the test spacing of the wavenumbers. $\mathrm{\Delta }k$ was set to 0.0001 in this study.

4.1. Case 1

In the first case, we tested the elliptic waveguide with the Dirichlet or Neumann boundary condition along the whole boundary. There are two kinds tests for TM mode and TE mode. No matter what the boundary conditions were, we always set the external source at (15, 15). A parametric equation was introduced to describe the elliptic waveguide:

$$\mathrm{\Gamma }=\left\{\left(x,y\right)\left|x=a\cos \theta ,y=b\sin \theta ,0\le \theta \le 2\pi \right.\right\},$$

where the major axis $a=1$, minor axis $b=\sqrt{1-e^2}$, and $e=0.9$. For the TM mode, $N_i=1064$, $N_b=120$, and $P=20$. For the TE mode, $N_i=1159$, $N_b=240$, and $P=25$.

In

Table 1. Comparison of the first six cutoff wavelengths for the elliptic waveguide in case 1.

$\left(\mathbf{a}\right) \mathbf{TM} \mathbf{Mode} \left({\mathit{N}}_{\mathit{i}}=1064,{\mathit{N}}_{\mathit{b}}=120\right)$
Mode	LMFS P = 15	LMFS P = 20	LMFS P = 30	MFS-SVD [1]	MFS-ES [28]	RBFCM [15]	Analytical Solution
1	1.4906	1.4906	1.4906	1.4908	1.4906	1.4906	1.4906
2	1.1607	1.1607	1.1607	1.1607	1.1607	1.1607	1.1607
3	0.9375	0.9375	0.9375	0.9375	0.9375	0.9376	0.9375
4	0.8093	0.8093	0.8093	0.8093	0.8093	0.8093	0.8093
5	0.7803	0.7803	0.7803	0.7799	0.7803	0.7804	0.7803
6	0.7083	0.7083	0.7083	0.7071	0.7083	0.7083	0.7083
$\left(\mathbf{b}\right) \mathrm{TE} \mathrm{Mode} \left({\mathit{N}}_{\mathit{i}}=1159,{\mathit{N}}_{\mathit{b}}=240\right)$
Mode	LMFS P= 20	LMFS P = 25	LMFS P= 35	MFS-SVD [1]	MFS-ES [28]	RBFCM [15]	Analytical Solution
1	3.3482	3.3482	3.3482	3.3481	3.3482	3.3479	3.3482
2	1.8287	1.8287	1.8287	1.8287	1.8287	1.8284	1.8287
3	1.5649	1.5649	1.5649	1.5649	1.5650	1.5650	1.5650
4	1.2654	1.2654	1.2654	1.2654	1.2654	1.2656	1.2654
5	1.2292	1.2292	1.2292	1.2292	1.2292	1.2293	1.2292
6	0.9986	0.9986	0.9986	0.9986	0.9986	0.9984	0.9986

, the numerical solutions obtained using the proposed LMFS with MES are compared with analytical solutions and other numerical results, which are in good agreement. Additionally, the corresponding resonance curves for the TM mode and TE mode are shown in Figure 3. Obviously, there are many peaks in the resonance curve, ranging from 0 to 10. Compared with three studies in the literature [1,15,28] and the analytical solutions, it can be seen that, when the wavenumber increases (mode 1 to mode 6), the cutoff wavelengths obtained using LMFS were still accurate and were more accurate than those obtained using the traditional MFS-SVD. The first four eigenmodes of the TM mode and TE mode are shown in Figure 4 and Figure 5, respectively, which are very close to the analytical solutions.

4.2. Case 2

In the second case, we resolved the eigenfrequency problem of a concentric annular waveguide, whose radii of the outer and inner circles were 2 and 0.5, respectively. The center of the concentric circle was set to be (0, 0). The external source was located at (15, 15). The other parameters were as follows: for the TM wave, $N_i=1360$, $N_b=250$, and $P=30$. For the TE wave, $N_i=1360$, $N_b=250$, and $P=25$.

Table 2. Comparison of the first five cutoff wavelengths for concentric annular waveguide in case 2.

$\left(\mathbf{a}\right) \mathbf{TM} \mathbf{Mode} \left({\mathit{N}}_{\mathit{i}}=1360,{\mathit{N}}_{\mathit{b}}=250\right)$
Mode	LMFS P = 20	LMFS P = 30	LMFS P = 50	LSTM [29]	MFS-DDSM [36]	Analytical Solution
1	3.0650	3.0650	3.0650	3.0650	3.0650	3.0650
2	2.8254	2.8253	2.8254	2.8302	2.8302	2.8176
3	2.3621	2.3621	2.3621	2.3621	2.3621	2.3621
4	1.9574	1.9574	1.9574	1.9574	1.9574	1.9574
5	1.6535	1.6535	1.6535	1.6535	1.6535	1.6535

shows the first five eigenfrequencies of the TM mode. The results of the calculation are in very good agreement with the values obtained with the analytical solution and the other numerical solutions. Figure 6 shows many peaks in the resonance curves of (a) the TM mode and (b) the TE mode for the concentric annular waveguide. The first three eigenmodes of the TM mode and TE mode are clearly shown in Figure 7 and Figure 8, respectively. The waveguide resonates at the characteristic frequencies, and the numerical distribution is different for every characteristic frequency. From the numerical distribution, we know that the numerical solution is a good and accurate solution.

4.3. Case 3

In the third case, a multi-connected domain, which is depicted in Figure 2c, was considered. This waveguide had an eccentric annular shape. Similarly, the radii of the outer and inner boundaries were 1 and 0.5, respectively. The center of the outer circle was at (0, 0), while the center of the inner circle was at (0, −0.2). The following parameters were used in case 3: $X_{ext}=\left(15,15\right)$. For the TM wave, $N_i=1270$, $N_b=150$ and $P=20$. For the TE wave, $N_i=1024$, $N_b=300$, and $P=25$.

Figure 9 shows the TM and TE resonance curves of the eccentric annular waveguide with k from 0 to 10. In

Table 3. Comparison of the first five cutoff wavelengths for the eccentric annular waveguide in case 3.

$\left(\mathbf{a}\right) \mathbf{TM} \mathbf{Mode} \left({\mathit{N}}_{\mathit{i}}=1270,{\mathit{N}}_{\mathit{b}}=150\right)$
Mode	LMFS P = 10	LMFS P = 20	LMFS P = 30	MFS-ES [28]	Lin et al. [37]	Kuttler [38]
1	1.3061	1.3061	1.3061	1.3061	1.3054	1.3057
2	1.14	1.14	1.14	1.14	1.1371	1.1398
3	1.0179	1.0179	1.0179	1.018	1.0118	1.0177
4	0.9241	0.9241	0.9241	0.9241	0.9189	0.9239
5	0.8497	0.8497	0.8497	0.8497	0.842	0.8495
$\left(\mathbf{b}\right) \mathbf{TE} \mathbf{Mode} \left({\mathit{N}}_{\mathit{i}}=1024,{\mathit{N}}_{\mathit{b}}=300\right)$
Mode	LMFS P= 15	LMFS P= 25	LMFS P= 35	MFS-ES [28]	Lin et al. [37]	Kuttler [38]
1	4.6466	4.6466	4.6466	4.6421	4.6152	4.6466
2	4.4628	4.4628	4.4628	4.4586	4.4492	4.4571
3	2.341	2.341	2.341	2.3399	2.3138	2.3409
4	2.3391	2.3391	2.3392	2.338	2.2989	2.339
5	1.599	1.5989	1.599	1.5984	1.5817	1.5988

a, it can be seen from the results in the first three columns that the results using different local points are stable and similar to the previous studies. However, in Table 3b, as the TE wave was solved including the Neumann boundary condition, under the same mode, there are slight differences compared with other methods. In Figure 10 and Figure 11, the details about the first four eigenmodes for the TM and TE modes are given, respectively. The results regarding the eccentric annular waveguide agreed with other numerical solutions, which was accurately demonstrated.

4.4. Case 4

To verify the applicability of LMFS with MES, a multi-connected domain was tested as the computational domain in the last case. The calculation domain is described in Figure 2d. The radii of the outer and inner boundaries were 2.5 and 0.5, respectively. The center of the outer circle was at (0, 0). (1, 1), (1, −1), (−1, −1), and (−1, 1) were the centers of the inner circles. For the other parameters, we used the following information: $X_{ext}=\left(15,15\right)$, $N_i=1132$, $N_b=430$, and $P=30$. The TM mode and TE mode had the same number of interior and boundary points.

In Figure 12, the resonance curves of the TM and TE modes displayed the first four peaks in the range of k from 0 to 5. This multi-connected computational domain was designed by us, so there was no literature to compare it to, and we can only draw the corresponding eigenmodes using the calculated k value. The first four eigenmodes in the TM and TE modes are demonstrated in Figure 13 and Figure 14, respectively. It can be seen that when the eigenfrequency increases, even if the computational domain is complex and multi-connected, the contour curves in the domain are still smooth and conform to the physical behavior of the TM waves and TE waves.

5. Conclusions

In this paper, the eigenfrequency problems, which were governed by the two-dimensional Helmholtz equation, were efficiently and accurately studied using the proposed LMFS with MES. LMFS combines the traditional MFS and the conception of localization. It is a meshless method that does not take a lot of time to generate meshes and numerical quadrature in computation. It only needs some arbitrarily distributed coordinate points to construct an interpolation function to discrete the governing equations, so flow fields with various complex shapes can be conveniently simulated. Within a small subdomain, the unknown variables of the source nodes in each subdomain can be obtained using the traditional MFS. Then, the resultant system of the algebraic equation is sparse, and it is easy and efficient to obtain the numerical solutions of LMFS. By adding an external source outside the computational domain, the resonant excitation is created. The cutoff wavenumber of the waveguide can be recorded using the responses system.

In our study, elliptic, concentric annular, eccentric annular, and multi-connected domain waveguides were examined to verify the applicability and efficiency of the meshless numerical method, the LMFS with the MES. The position of the external source was not fixed so long as it was outside the computational domain. The characteristic wavelength of a waveguide can be determined using this meshless numerical method, and its correctness can be verified by drawing the numerical distribution. By testing different cases, the proposed LMFS was shown to be able to handle a complex computational domain in realistic engineering applications. Furthermore, the computational efficiency and accuracy were shown to be high.