An Extrinsic Enriched Finite Element Method Based on RBFs for the Helmholtz Equation

Abstract

The traditional finite element method (FEM) usually exhibits significant numerical dispersion error for solving the Helmholtz equation in relatively high-frequency range, resulting in insufficiently accurate solutions. To address this problem, this paper proposes a novel enriched finite element method (EFEM) based on radial basis functions (RBFs) which are frequently used in meshless numerical techniques. In the proposed method, the partition of unity (PU) framework is retained, and nodal interpolation functions are formed using the RBFs. Furthermore, the linear dependence (LD) problem commonly encountered in many of the PU-based methods using polynomial basis functions (PBFs) is effectively avoided by using the present RBFs. To enrich the approximation space generated by the RBFs, the PBFs are introduced to construct the local enrichment functions. Several typical numerical experiments are conducted in this work. The results indicate that the proposed method can significantly reduce the dispersion error and yield accurate solutions even for relatively high-frequency Helmholtz problems. More importantly, the proposed method can be directly implemented with standard quadrilateral meshes as in FEM. Therefore, the proposed method represents a promising numerical scheme for solving relatively high-frequency Helmholtz problems.

1. Introduction

The traditional finite element method (FEM) has been widely adopted across various engineering fields during the past few decades because of its inherent advantages. For example, the global matrices generated by the traditional FEM are typically banded and sparse, and the FEM is well-suited for handling structural-acoustic coupling and heterogeneous problems [1,2]. However, there are still several challenges remaining unresolved for the traditional FEM when solving acoustic problems where the linear Helmholtz equation often needs to be addressed efficiently [3]. Due to the differential nature of the Helmholtz equation, the traditional FEM generally suffers from significant dispersion errors at relatively large wave numbers, which significantly degrade computational accuracy. Therefore, the satisfactory numerical solutions can usually only be provided at relatively small wave numbers [4,5,6].

Currently, researchers have developed various alternatives to the standard FEM for acoustic simulations, including finite volume method (FVM) [7], finite difference method (FDM) [8,9,10,11,12,13], boundary element method (BEM) [14,15,16], and various meshfree methods [17,18,19,20]. Among these, the BEM is a widely used and reliable approach for underwater acoustic scattering problems. The BEM is particularly suitable for problems defined over unbounded domains due to its inherent capability. Furthermore, BEM typically requires meshing only the domain boundary, resulting in reduced meshing overhead compared to FEM. However, the BEM tends to yield dense and non-symmetric global matrices, leading to high computational costs for large-scale problems [21,22,23]. Recently, several meshfree techniques, including the reproducing kernel particle method [24], the element-free Galerkin method (EFGM) [25,26,27,28], the point interpolation method (PIM), and the radial point interpolation method (RPIM) [29,30,31,32,33], have also been applied to acoustic problems. Compared to FEM, these methods can control dispersion errors better at relatively large wave numbers and often deliver higher computational accuracy. In the PIM, the nodal interpolation functions retain the Kronecker delta property and can be constructed directly from the scattered nodes. However, the method may suffer from a singular moment matrix and the nodal interpolation functions may oscillate significantly when too many nodes are used for interpolation. In contrast, the RPIM constructs the nodal shape functions with the radial basis functions (RBFs), avoiding singularities and offering improved stability and robustness. Moreover, the RPIM has been shown to outperform the traditional FEM in terms of convergence and accuracy in acoustic analysis [34,35,36].

Numerous RBF-based methods have been developed for solving partial differential equations (PDEs) [37,38,39,40]. For instance, Kien et al. proposed an RBF-FEM to produce the shape functions simply with increasing element node number, which has been applied to solid mechanics problems [41,42]. Ma et al. developed a novel space–time collocation method based on multiquadric-radial basis functions (MQ-RBF) to solve the Benjamin–Bona–Mahony–Burgers (BBMB) equation in shallow water waves [43]. Zhang et al. proposed a modified RPIM (M-RPIM) for PDEs governing steady point current source fields [44]. Liu et al. introduced a novel approach combining second-order cone programming (SOCP) with node-based smoothed RPIM (NSRPIM) to address the limitations of traditional Newton-type methods in solving elastoplastic boundary value problems [45]. Recently, a specialized RBF-FD method has been proposed and advanced for solving a series of PDEs [46,47,48,49,50,51,52], with numerical results confirming its effectiveness. Additionally, the local discontinuous Galerkin (LDG) method has also been developed and applied to a range of PDEs [53,54,55,56,57].

However, FEM remains a predominant and essential numerical method for solving acoustic problems due to its unique advantages. In general, the interpolation error and numerical pollution/dispersion error constitute the primary numerical error sources in the numerical solutions for Helmholtz equation [58]. Several studies have shown that the magnitude of these errors is jointly determined by the physical parameters and discretization scheme. At relatively small wave numbers, the total numerical error is usually dominated by interpolation error. Here, following the rule of thumb $kh=\mathrm{constant}$, in which $k$ represents the wave number and $h$ denotes the mesh size, the interpolation error can be efficiently controlled. However, at relatively large wave numbers, the dispersion error becomes the primary contributor to the total numerical error. Even if the used mesh pattern satisfies the rule of thumb, the accuracy may still be insufficient, indicating that the dispersion error grows faster than the interpolation error as the wave number increases. In essence, the dispersion error reflects phase lag in the numerical solution and is inherently difficult to eliminate completely.

To address the dispersion error, researchers have developed many FEM variants, including the Galerkin/least-squares FEM [59,60], the generalized FEM (GFEM) [61], the smoothed FEM (SFEM) [62,63,64,65], the overlapping FEM (OFEM) [66,67,68,69], and the enriched FEM (EFEM) [70]. These approaches are generally able to suppress the dispersion error more effectively than the traditional FEM. It should be noted that these approaches still suffer from limitations in completely eliminating the dispersion error, especially for 2D and 3D problems. Furthermore, the linear dependence (LD) problem is commonly encountered in many of the partition of unity (PU)-based methods using the polynomial basis functions (PBFs). A previous EFEM based on the PU concept, in which standard FEM shape functions are used, has been proposed [71]. Compared to the traditional FEM, the EFEM performs much better in reducing the dispersion error at relatively large wave numbers. However, the EFEM may suffer from the LD problem because the linearly dependent nodal shape functions are used to construct the local numerical approximation, particularly in quadrilateral elements [72,73,74].

Inspired by the prior research, this paper proposes a novel EFEM based on the RBFs to reduce the dispersion error, in which the PU framework is retained and the nodal interpolation functions are formed using the RBFs. In the proposed method, the LD problem is effectively avoided by using the present RBFs. Additionally, to enrich the approximation space generated by the RBFs, the PBFs are introduced to construct the local enrichment functions, which further improves the capability of the proposed method in solving relatively high-frequency Helmholtz problems. Actually, an arbitrary order of the PBFs can be applied to construct the local enrichment functions. Generally, the higher order of the PBFs can provide higher computational accuracy but incur higher computational costs. For brevity, only the linear PBFs are used to construct the local enrichment functions in this work, leading to three additional degrees of freedom (DOFs) at each node. Therefore, the proposed method is referred to as RBF-N3. Several typical numerical experiments are conducted in this work. The findings clearly show that the RBF-N3 delivers significantly higher computational accuracy and yields lower dispersion error than the traditional FEM, especially for relatively high-frequency problems. More importantly, in the RBF-N3 no special modifications for the mesh structure are required and the nodal shape functions retain the Kronecker delta property. Therefore, for relatively high-frequency Helmholtz problems, the RBF-N3 presents a promising and efficient alternative to the traditional FEM.

In Section 2, the matrix formulation of the Helmholtz equation is constructed by using the RBFs. In Section 3, the structure of the EFEM based on the RBFs for the Helmholtz equation is given. In Section 4, a dispersion error analysis of the RBF-N3 is provided and compared with the traditional FEM. Several typical numerical experiments are conducted in Section 5.

2. Formulation of the Helmholtz Equation

For acoustic problems in an ideal fluid medium, the problem domain is denoted as $\mathsf{\Omega }$ and the boundary as $\mathsf{\Gamma }$. Based on the assumptions of linearization and adiabatic processes, the linear governing equation can be derived as

$${\nabla }^{2}p={\displaystyle \frac{1}{c^2}}{\displaystyle \frac{{\partial }^{2}p}{\partial t^2}},$$

(1)

where $p$ denotes the acoustic pressure, ${\nabla }^2$ is the Laplace operator in the Cartesian coordinate system, and $c$ represents the acoustic propagation speed.

Assuming that $p$ exhibits a steady harmonic feature, i.e., $p=P{e}^{j\omega t}$, where $\omega$ represents the angular frequency and $P$ denotes the amplitude of acoustic pressure distribution, the Helmholtz equation is derived as

$${\nabla }^{2}P+k^{2}P=0,$$

(2)

in which $k=\omega /c_0$ denotes the wave number.

Based on the Helmholtz equation, the practical acoustic problems can be analyzed and solved by applying appropriate boundary conditions. In simple cases, the analytical solutions can be obtained directly. However, the real-world scenarios are often very complex and the actual boundary conditions are typically difficult to describe precisely in mathematical form. As a result, the exact analytical solutions are often unattainable. In such cases, suitable numerical methods are usually employed to obtain sufficiently accurate approximations for practical acoustic problems.

The boundary conditions usually fall into the following three categories:

$$\left\{\begin{array}{ll}P=P_D,\hfill & \mathrm{on} {\mathsf{\Gamma }}_D\hfill \\ {\displaystyle \frac{\partial P}{\partial n}}=-j\rho \omega v_n,\hfill & \mathrm{on} {\mathsf{\Gamma }}_N\hfill \\ {\displaystyle \frac{\partial P}{\partial n}}=-j\rho \omega A_{n}P,\hfill & \mathrm{on} {\mathsf{\Gamma }}_R\hfill \end{array}\right.,$$

(3)

in which ${\mathsf{\Gamma }}_D$ denotes the Dirichlet boundary prescribing the acoustic pressure amplitude, ${\mathsf{\Gamma }}_N$ denotes the Neumann boundary enforcing the normal particle velocity $v_n$, and ${\mathsf{\Gamma }}_R$ represents the Robin boundary defining the acoustic impedance condition. Here, $\rho$ is fluid density, $A_n$ is the acoustic admittance coefficient, and $n$ denotes the outward unit normal vector.

The equivalent integral form for Equation (2) is obtained as

$$-{\displaystyle {\int }_{\mathsf{\Omega }}w\left({\nabla }^{2}P+k^{2}P\right)\mathrm{d}\mathsf{\Omega }}+{\displaystyle {\int }_{{\mathsf{\Gamma }}_R}w\left(\frac{\partial P}{\partial n}+j\rho \omega A_{n}P\right)\mathrm{d}\mathsf{\Gamma }}={\displaystyle {\int }_{{\mathsf{\Gamma }}_N}w\left(\frac{\partial P}{\partial n}+j\rho \omega v_n\right)\mathrm{d}\mathsf{\Gamma }},$$

(4)

in which $w$ denotes an arbitrary weight function. By applying Green’s theorem and performing integration by parts, the equation can be rewritten as

$${\displaystyle {\int }_{\mathsf{\Omega }}\nabla w\cdot \nabla P\mathrm{d}\mathsf{\Omega }-}{k}^2{\displaystyle {\int }_{\mathsf{\Omega }}w\cdot P\mathrm{d}\mathsf{\Omega }}+j\rho \omega A_n{\displaystyle {\int }_{{\mathsf{\Gamma }}_R}w\cdot P\mathrm{d}\mathsf{\Gamma }}=-j\rho \omega {\displaystyle {\int }_{{\mathsf{\Gamma }}_N}w\cdot v_n\mathrm{d}\mathsf{\Gamma }},$$

(5)

In Equation (5), $P$ denotes the unknown scalar field variable to be solved. Clearly, the solution $P$ obtained from the equation is exact if the weight function $w$ is chosen arbitrarily. However, it is generally impossible to solve for the exact solution directly based on this equation. By applying the Galerkin method to Equation (5), an approximate form can be derived, in which the variation $\delta P$ of the acoustic pressure is chosen as a particular weight function:

$$w\left(x\right)=\delta P\left(x\right),$$

(6)

Substituting Equation (6) into Equation (5) yields the corresponding approximate equation:

$${\displaystyle {\int }_{\mathsf{\Omega }}\delta \left(\nabla P\right)\cdot \nabla P\mathrm{d}\mathsf{\Omega }-}{k}^2{\displaystyle {\int }_{\mathsf{\Omega }}\delta P\cdot P\mathrm{d}\mathsf{\Omega }}+j\rho \omega A_n{\displaystyle {\int }_{{\mathsf{\Gamma }}_R}\delta P\cdot P\mathrm{d}\mathsf{\Gamma }}=-j\rho \omega {\displaystyle {\int }_{{\mathsf{\Gamma }}_N}\delta P\cdot v_n\mathrm{d}\mathsf{\Gamma }},$$

(7)

The above equation represents the Galerkin weak form of the Helmholtz equation. In fact, obtaining the exact acoustic pressure solution directly from Equation (7) is not inherently easier than solving the original Helmholtz equation. However, this weak form facilitates the construction of corresponding discrete numerical models.

In this work, the RBFs are used for the construction of corresponding discrete numerical models. Consider an acoustic pressure field $P\left(x\right)$ defined over a d-dimensional problem domain discretized into $Ne$ elements and each element containing $n$ nodes. To construct the nodal shape functions, $P\left(x\right)$ is expressed as

$$\tilde{P}\left(x\right)=\sum _{i=1}^n{R}_i\left(x\right)a_i+\sum _{j=1}^m{q}_j\left(x\right)b_j=R\left(x\right)a+q\left(x\right)b,$$

(8)

In Equation (8), $\tilde{P}\left(x\right)$ represents the numerical approximation of $P\left(x\right)$. $R\left(x\right)$ denotes the RBFs used for interpolation. $n$ denotes the number of the employed RBFs, which is typically equal to the number of nodes used for interpolation. The standard quadrilateral mesh is employed to construct shape functions in this work, so $n=4$. $q\left(x\right)$ represents a sequence of the PBFs, which are employed to enhance the numerical stability and to enrich the approximation space. The number of PBFs is denoted by $m$. In this work, $m=3$, that is, $q\left(x\right)=\left[\begin{array}{ccc}1& x& y\end{array}\right]$. The coefficients $a_i$ and $b_i$ are the unknown parameters to be determined. There exist many different types of RBFs, here the Multi-Quadrics (MQ) type of RBF is chosen as

$$R_i\left(x\right)={\left[r_i^2+{\left({\alpha }_c{d}_c\right)}^2\right]}^q,$$

(9)

in which $d_c$ is a characteristic length that is related to the average nodal spacing. $a_c$ and $q$ are the shape parameters of the MQ-RBF to be determined. $R_i\left(x\right)$ represents the RBF corresponding to node $i$, whose variable depends solely on the distance $r_i$ between the field point $x$ and node $x_i$. For two-dimensional problems, this distance is given by

$$r_i=\sqrt{{\left(x-x_i\right)}^2+{\left(y-y_i\right)}^2},$$

(10)

In this work, the values $q=1.03$ and $a_c=2$ are chosen based on well-established practices from prior research on acoustic problems. Specifically, it is recommended by previous numerical investigations that $q=1.03$ and $a_c=2$ can be utilized to yield satisfactory numerical accuracy [34,75]. Furthermore, this parameter choice has been verified by You et al. [30] to provide very accurate simulations for acoustic problems. To determine the coefficients $a_i$ and $b_i$, the nodal acoustic pressures are substituted into Equation (8), leading to the following matrix equation:

$$P_s=R_{0}a+Qb,$$

(11)

Additionally, an extra constraint condition is introduced:

$$\sum _{i=1}^n{q}_j\left(x_i\right)a_i=Q^{T}a=0, j=1,2,\cdots ,m,$$

(12)

Hence, Equation (11) can be rewritten as

$${\tilde{P}}_s=\left[\begin{array}{c}{P}_s\\ 0\end{array}\right]=\underset{G}{\underbrace{\left[\begin{array}{cc}{R}_0& Q\\ Q^T& 0\end{array}\right]}}\left[\begin{array}{c}a\\ b\end{array}\right]=G{a}_0,$$

(13)

where

$$R_0={\left[\begin{array}{cccc}{R}_1\left(x_1\right)& R_2\left(x_1\right)& \cdots & R_n\left(x_1\right)\\ R_1\left(x_2\right)& R_2\left(x_2\right)& \cdots & R_n\left(x_2\right)\\ \cdots & \cdots & \ddots & \cdots \\ R_1\left(x_n\right)& R_2\left(x_n\right)& \cdots & R_n\left(x_n\right)\end{array}\right]}_{n\times n},$$

(14)

$$Q^T={\left[\begin{array}{cccc}1& 1& \cdots & 1\\ x_1& x_2& \cdots & x_n\\ y_1& y_2& \cdots & y_n\\ \cdots & \cdots & \ddots & \cdots \\ q_m\left(x_1\right)& q_m\left(x_2\right)& \cdots & q_m\left(x_n\right)\end{array}\right]}_{m\times n},$$

(15)

Here, $x_i$ denotes the coordinate vector for node $i$. $P_s$ represents the vector of the nodal acoustic pressures and can be expressed as $P_s={\left[\begin{array}{ccc}{P}_1& P_2& \begin{array}{cc}\cdots & P_n\end{array}\end{array}\right]}^T$, where $P_i=P\left(x_i\right)$ represents the acoustic pressure at node $i$. According to Equation (13), the unknown coefficients $a_0$ can be represented as $a_0=G^{-1}{\tilde{P}}_s$. Based on the employed quadrilateral mesh, the matrices $G$ and ${\tilde{P}}_s$ can be computed explicitly, and thus the unknown coefficients $a_0$ can be determined. Substituting $a_0$ into Equation (8) yields

$$\tilde{P}\left(x\right)=R\left(x\right)a+q\left(x\right)b=\left[\begin{array}{cc}R\left(x\right)& q\left(x\right)\end{array}\right]G^{-1}{\tilde{P}}_s=\tilde{\Phi }\left(x\right)\left[\begin{array}{c}{P}_s\\ 0\end{array}\right],$$

(16)

in which $\tilde{\Phi }\left(x\right)$ is a vector-valued function of length $n+m$, which can be expressed as follows:

$$\tilde{\Phi }\left(x\right)=\left[\begin{array}{cc}R\left(x\right)& q\left(x\right)\end{array}\right]G^{-1}=\left[{\varphi }_1\left(x\right) {\varphi }_2\left(x\right) \cdots {\varphi }_n\left(x\right) \cdots {\varphi }_{n+m}\left(x\right)\right],$$

(17)

Taking the first n terms as the nodal shape functions $\Phi \left(x\right)$, we have

$$\tilde{P}\left(x\right)=\Phi \left(x\right)P_s={\displaystyle \sum _{i=1}^n{\varphi }_i\left(x\right)P_i},$$

(18)

in which ${\varphi }_i\left(x\right)$ represents the nodal shape function which possesses the Kronecker delta property:

$${\varphi }_i\left(x_j\right)={\delta }_{ij}=\left\{\begin{array}{cc}1,& i=j\\ 0,& i\ne j\end{array}\right.,$$

(19)

Similarly, the variation in the acoustic pressure can also be defined as

$$\delta \tilde{P}\left(x\right)=\Phi \left(x\right)\delta P_{\mathrm{s}}={\displaystyle \sum _{i=1}^n{\varphi }_i\left(x\right)\delta P_i},$$

(20)

Substituting Equations (18) and (20) into Equation (7) yields the following expression:

$$\begin{array}{l}{\displaystyle {\int }_{\mathsf{\Omega }}{\left(\nabla \Phi \right)}^T}\left(\nabla \Phi \right)\mathrm{d}\mathsf{\Omega }P-k_w^2{\displaystyle {\int }_{\mathsf{\Omega }}{\Phi }^T}\Phi \mathrm{d}\mathsf{\Omega }P+j\rho \omega {\displaystyle {\int }_{{\mathsf{\Gamma }}_R}{A}_n}{\Phi }^T\Phi \mathrm{d}\mathsf{\Gamma }\\ =-j\rho \omega {\displaystyle {\int }_{{\mathsf{\Gamma }}_N}{\Phi }^T}{v}_n\mathrm{d}\mathsf{\Gamma }\end{array},$$

(21)

Equation (21) can also be expressed as

$$\left[\begin{array}{c}K-k_w^{2}M+j\rho \omega C\end{array}\right]P=-j\rho \omega F,$$

(22)

where

$$\left\{\begin{array}{l}K={\displaystyle {\int }_{\mathsf{\Omega }}{\left(\nabla \Phi \right)}^T}\left(\nabla \Phi \right)\mathrm{d}\mathsf{\Omega }\hfill \\ M={\displaystyle {\int }_{\mathsf{\Omega }}{\Phi }^T}\Phi \mathrm{d}\mathsf{\Omega }\hfill \\ C={\displaystyle {\int }_{{\mathsf{\Gamma }}_R}{A}_n}{\Phi }^T\Phi \mathrm{d}\mathsf{\Gamma }\hfill \\ F={\displaystyle {\int }_{{\mathsf{\Gamma }}_N}{\Phi }^T}{v}_n\mathrm{d}\mathsf{\Gamma }\hfill \end{array}\right.,$$

(23)

In Equation (23), the system matrices K, M, and C represent the global stiffness matrix, global mass matrix, and global damping matrix, respectively; F denotes the global load vector and P is the global nodal acoustic pressure vector. Based on Equation (23), the process of solving the original Helmholtz equation is thus converted into solving the corresponding matrix equation. This, in turn, transforms the problem into evaluating integrals of the shape functions over the problem domain, which can then be computed using various numerical integration techniques (e.g., Gaussian quadrature). The treatment of boundary conditions for the Helmholtz equation can similarly be transformed into appropriate modifications of the matrices in Equation (23). Finally, the nodal acoustic pressure vector P can be obtained by solving the resulting matrix equation.

3. The EFEM Based on RBFs for Helmholtz Equation

The EFEM improves computational accuracy by introducing additional DOFs at each node without altering the original finite element mesh topology. Consider an acoustic pressure field $P\left(x\right)$ defined over a d-dimensional problem domain which is discretized into $n$ nodes. In the EFEM based on RBFs, the interpolation of $P\left(x\right)$ can be defined as [76,77,78]

$$\tilde{P}\left(x\right)={\displaystyle \sum _{i=1}^n{\varphi }_i\left(x\right)P_i}+{\displaystyle \sum _{i=1}^n{N}_i^{*}\left(x\right)\psi \left(x\right)a_i},$$

(24)

In Equation (24), the interpolation of $P\left(x\right)$ is divided into two parts. The first part is the standard interpolation constructed based on the RBFs and the second part represents the enriched numerical approximation resulting from the introduction of additional DOFs. Here, ${\varphi }_i\left(x\right)$ represents the nodal shape function constructed using the RBFs as shown in Section 2, $P_i$ is the nodal acoustic pressure, $N_i^{*}\left(x\right)$ denotes a newly constructed nodal interpolation function, $\psi \left(x\right)$ is a specially designed enrichment function, $a_i$ is the corresponding unknown coefficient.

In fact, $N_i^{*}\left(x\right)$ should satisfy the PU property, i.e., ${\displaystyle \sum N_i^{*}\left(x\right)}=1$. In this work, $N_i^{*}\left(x\right)$ is directly chosen as the nodal shape function ${\varphi }_i\left(x\right)$ for simplicity, although this is not typically required. Hence, Equation (24) can be rewritten as:

$$\tilde{P}\left(x\right)={\displaystyle \sum _{i=1}^n{\varphi }_i\left(x\right)P_i}+{\displaystyle \sum _{i=1}^n{\varphi }_i\left(x\right)\psi \left(x\right)a_i},$$

(25)

In general, the enrichment function $\psi \left(x\right)$, which actually serves as a key factor in improving the accuracy of the EFEM, can be chosen as a sequence of polynomial functions or trigonometric functions, or as a combination of both. Furthermore, the enrichment function $\psi \left(x\right)$ can be specially designed for different types of problems, thereby yielding the EFEM adapted to various application scenarios. However, it is evident that the approximation of $P\left(x\right)$ at a node generally does not equal the actual nodal acoustic pressure when a general enrichment function is used. $\psi \left(x\right)$ is used. This discrepancy is shown in Equation (27) and can indeed affect the imposition of Dirichlet boundary conditions.

$$\tilde{P}\left(x_i\right)={\displaystyle \sum _{i=1}^n{\varphi }_i\left(x_i\right)P_i}+{\displaystyle \sum _{i=1}^n{\varphi }_i\left(x_i\right)\psi \left(x_i\right)a_i}=P_i+\psi \left(x_i\right)a_i\ne P_i,$$

(26)

Consider $\psi \left(x_i\right)=0$ at node $i$, we have

$${\displaystyle \sum _{i=1}^n{\varphi }_i\left(x_i\right)\psi \left(x_i\right)a_i}=\psi \left(x_i\right)a_i=0,$$

(27)

According to Equation (26), this leads to $\tilde{P}\left(x_i\right)=P_i$, thus allowing Dirichlet boundary conditions to be properly imposed. Therefore, Equation (25) needs to be modified as follows:

$$\tilde{P}\left(x\right)={\displaystyle \sum _{i=1}^n{\varphi }_i\left(x\right)P_i}+{\displaystyle \sum _{i=1}^n{\varphi }_i\left(x\right)\left[\frac{\psi \left(x\right)-\psi \left(x_i\right)}{h}\right]a_i},$$

(28)

in which h is the characteristic length of the element introduced to enhance numerical stability.

In two-dimensional problems, if a polynomial sequence is used, the enrichment function $\psi \left(x\right)$ can be expressed as

$$\psi \left(x\right)=\left\{\begin{array}{cccccccccc}{\overline{x}}_i,& {\overline{y}}_i,& {\overline{x}}_i^2,& {\overline{x}}_i{\overline{y}}_i,& {\overline{y}}_i^2,& \dots ,& {\overline{x}}_i^q,& \dots ,& {\overline{y}}_i^q& \dots ,\end{array}\right\}$$

(29)

If a trigonometric function sequence is used, $\psi \left(x\right)$ is given as [79]

$$\psi \left(x\right)=\left\{\begin{array}{l}\begin{array}{cccc}\cos \left({\displaystyle \frac{\mathsf{\pi }{\overline{x}}_i}{h}}\right),& \sin \left({\displaystyle \frac{\mathsf{\pi }{\overline{x}}_i}{h}}\right),& \cos \left({\displaystyle \frac{\mathsf{\pi }{\overline{y}}_i}{h}}\right),& \sin \left({\displaystyle \frac{\mathsf{\pi }{\overline{y}}_i}{h}}\right),\end{array}\\ \begin{array}{cccc}\cos \left({\displaystyle \frac{\mathsf{\pi }{\overline{x}}_i}{h}}+{\displaystyle \frac{\mathsf{\pi }{\overline{y}}_i}{h}}\right),& \sin \left({\displaystyle \frac{\mathsf{\pi }{\overline{x}}_i}{h}}+{\displaystyle \frac{\mathsf{\pi }{\overline{y}}_i}{h}}\right),& \cos \left({\displaystyle \frac{\mathsf{\pi }{\overline{x}}_i}{h}}-{\displaystyle \frac{\mathsf{\pi }{\overline{y}}_i}{h}}\right),& \sin \left({\displaystyle \frac{\mathsf{\pi }{\overline{x}}_i}{h}}-{\displaystyle \frac{\mathsf{\pi }{\overline{y}}_i}{h}}\right),\end{array}\\ \begin{array}{ccc}& \cdots ,& \end{array}\\ \begin{array}{cccc}\cos \left({\displaystyle \frac{\mathsf{\pi }q{\overline{x}}_i}{h}}\right),& \sin \left({\displaystyle \frac{\mathsf{\pi }q{\overline{x}}_i}{h}}\right),& \cos \left({\displaystyle \frac{\mathsf{\pi }q{\overline{y}}_i}{h}}\right),& \sin \left({\displaystyle \frac{\mathsf{\pi }q{\overline{y}}_i}{h}}\right),\end{array}\\ \begin{array}{cccc}\cos \left({\displaystyle \frac{\mathsf{\pi }q{\overline{x}}_i}{h}}+{\displaystyle \frac{\mathsf{\pi }q{\overline{y}}_i}{h}}\right),& \sin \left({\displaystyle \frac{\mathsf{\pi }q{\overline{x}}_i}{h}}+{\displaystyle \frac{\mathsf{\pi }q{\overline{y}}_i}{h}}\right),& \cos \left({\displaystyle \frac{\mathsf{\pi }q{\overline{x}}_i}{h}}-{\displaystyle \frac{\mathsf{\pi }q{\overline{y}}_i}{h}}\right),& \sin \left({\displaystyle \frac{\mathsf{\pi }q{\overline{x}}_i}{h}}-{\displaystyle \frac{\mathsf{\pi }q{\overline{y}}_i}{h}}\right)\end{array}\end{array}\right\},$$

(30)

in which ${\overline{x}}_i=x-x_i$ and ${\overline{y}}_i=y-y_i$ denote the relative coordinates within the element. The variable $q$ denotes the order of the polynomial or trigonometric function employed. If a combination of trigonometric and polynomial functions is used, then enrichment functions $\psi \left(x\right)$ can be selected from both Equations (29) and (30).

Generally, the higher order of the enrichment functions can theoretically provide higher computational accuracy but incur higher computational costs. For brevity, only the linear PBFs are used to construct the local enrichment functions in this work, i.e., $\psi \left(x\right)=\left\{\begin{array}{cc}{\overline{x}}_i,& {\overline{y}}_i\end{array}\right\}$, and then there exists three DOFs at each node. Therefore, the EFEM based on the RBFs is referred to as RBF-N3 in this work.

Many of the PU-based methods using the PBFs commonly suffer from the LD problem, which compromises the numerical stability of the method, particularly when quadrilateral meshes are employed. In the RBF-N3, the LD problem can be effectively avoided by using the RBFs. Furthermore, the approximation space generated by the RBFs is enriched by the local enrichment functions, which further improves the capability of the proposed method in solving the Helmholtz problems in relatively high-frequency range.

4. Numerical Dispersion Error Analysis

When the numerical methods are applied to the Helmholtz equation, a phase lag is generally introduced to the exact solution. Specifically, the underestimated numerical wave number $k_h<k$ causes waves to propagate faster numerically, resulting in delayed phase accumulation at observation points. At relatively small wave numbers, the traditional FEM can provide sufficiently accurate solutions due to negligible dispersion error. However, for problems involving relatively large wave numbers, the traditional FEM often exhibits significant dispersion error, which severely impacts computational accuracy. Therefore, a suitable numerical method should be capable of mitigating dispersion error at relatively large wave numbers. The dispersion error is analyzed for the proposed RBF-N3, traditional FEM, EFEM-N3 (using linear polynomial enrichment functions), and RPIM; in this work the used regular triangular and quadrilateral meshes are shown in Figure 1. For the RPIM, the same nodes distribution with h = 1 is employed and the support domain size is set to 3h, a choice proven effective in reducing dispersion errors [29,31].

In Figure 1, the acoustic domain is uniformly discretized into nine nodes for the purpose of dispersion error analysis. Here, $h$ denotes the mesh size, $\theta$ represents the wave propagation direction, and $n$ is the unit direction vector of wave propagation. For the Helmholtz equation without any boundary conditions, the discretized matrix equation is derived as

$$\left[\begin{array}{c}K-k^{2}M\end{array}\right]P=0,$$

(31)

Assume that

$$P_h=A_h{e}^{j{k}_{h}nx},$$

(32)

where $x$ represents the coordinate vector, $P_h$ and $k_h$ are the numerical acoustic pressure and numerical wave number, respectively. $A_h$ represents the vector of nodal acoustic pressures which can be expressed as

$$A_h={\left[\begin{array}{ccccccccc}{A}_{1,1}& A_{1,2}& \cdots & A_{1,ad},& A_{2,1}& A_{2,1}& \cdots & A_{2,ad},& \cdots \end{array}\right]}^T,$$

(33)

Here the vector ${\left[\begin{array}{cccc}{A}_{i,1}& A_{i,2}& \cdots & A_{i,ad}\end{array}\right]}^T$ denotes the nodal acoustic pressure amplitude vector, $ad$ represents the number of DOFs at each node. In the traditional FEM, $ad=1$, whereas in the RBF-N3, $ad=3$. Using Equations (31) and (32), we obtain

$$\left[\begin{array}{c}{D}_{stiff}-k^2{D}_{mass}\end{array}\right]A_h=0,$$

(34)

The coefficient matrices $D_{stiff}$ and $D_{mass}$ (with dimension $ad\times ad$) are given by [80,81]

$$\begin{array}{ll}{D}_{\mathrm{stiff}}=& K_{n,n}+K_{n,n-1}{e}^{-j{k}_{h}h\cos \theta }+K_{n,n+1}{e}^{j{k}_{h}h\cos \theta }+\\ & K_{n,n-2}{e}^{j{k}_{h}h\left(\cos \theta -\sin \theta \right)}+K_{n,n+2}{e}^{j{k}_{h}h\left(-\cos \theta +\sin \theta \right)}+\\ & K_{n,n-3}{e}^{-j{k}_{h}h\sin \theta }+K_{n,n+3}{e}^{j{k}_{h}h\sin \theta }+\\ & K_{n,n-4}{e}^{j{k}_{h}h\left(-\cos \theta -\sin \theta \right)}+K_{n,n+4}{e}^{j{k}_{h}h\left(\cos \theta +\sin \theta \right)}\end{array},$$

(35)

$$\begin{array}{ll}{D}_{\mathrm{mass}}=& M_{n,n}+M_{n,n-1}{e}^{-j{k}_{h}h\cos \theta }+M_{n,n+1}{e}^{j{k}_{h}h\cos \theta }+\\ & M_{n,n-2}{e}^{j{k}_{h}h\left(\cos \theta -\sin \theta \right)}+M_{n,n+2}{e}^{j{k}_{h}h\left(-\cos \theta +\sin \theta \right)}+\\ & M_{n,n-3}{e}^{-j{k}_{h}h\sin \theta }+M_{n,n+3}{e}^{j{k}_{h}h\sin \theta }+\\ & M_{n,n-4}{e}^{j{k}_{h}h\left(-\cos \theta -\sin \theta \right)}+M_{n,n+4}{e}^{j{k}_{h}h\left(\cos \theta +\sin \theta \right)}\end{array},$$

(36)

From $D_{stiff}$ and $D_{mass}$, the actual wave number $k$ can be calculated as

$$k=eig\left(\sqrt{{\displaystyle \frac{D_{stiff}}{D_{mass}}}}\right),$$

(37)

It is evident that both $D_{stiff}$ and $D_{mass}$ are actually the function of the numerical wave number $k_h$, mesh size $h$, and wave propagation direction $\theta$. Therefore, for a given numerical wave number $k_h$, the actual wave number $k$ can be determined from Equation (37) if $h$ and $\theta$ are known. In most cases, the dispersion error causes $k\ne k_h$. The ratio $\epsilon =k/k_h$ is used to quantify the dispersion error.

In this section, the meshes shown in Figure 1 are used for the dispersion error analysis. The numerical dispersion errors for different numerical methods under various wave propagation directions $\theta$ are computed using Equation (37). Figure 2 displays the dispersion errors versus the normalized wave number $kh/\pi$ for different numerical methods, namely, FEM-T3 (triangular elements) in Figure 2a, FEM-Q4 (quadrilateral elements) in Figure 2b, EFEM-N3 in Figure 2c, RPIM in Figure 2d, and the proposed RBF-N3 in Figure 2e.

As shown in Figure 2, the numerical dispersion errors obtained with the FEM-T3, FEM-Q4, EFEM-N3, and RPIM are larger than those of the RBF-N3 and vary significantly with different wave propagation angles, indicating that these methods suffer from pronounced numerical anisotropy. In contrast, the proposed RBF-N3 yields the smallest numerical dispersion errors over the entire range of computed normalized wave numbers and exhibits much weaker numerical anisotropy across wave propagation directions. It is worth noting that the same node discretization is employed. Therefore, these findings indicate that the approximation space generated by the RBFs, which is enriched by the linear PBFs, is particularly well-suited for suppressing the dispersion error. Moreover, the excellent performance of the RBF-N3 also suggests that the RBF-N3 could serve as a more accurate and reliable numerical method than the traditional FEM for solving practical Helmholtz problems.

5. Numerical Examples

Multiple practical numerical experiments are investigated using the traditional FEM and the RBF-N3. The numerical results obtained by different methods are compared to evaluate the performance of the RBF-N3 for practical Helmholtz problems.

5.1. The 2D Tube Acoustic Problem

This subsection investigates the 2D tube acoustic problem and the corresponding acoustic model is shown in Figure 3. Figure 3a gives the geometry of the tube which has a length $l=1 \mathrm{m}$ and width $b=0.01 \mathrm{m}$. The tube is filled with water with density $\rho =1000 {\mathrm{kg}/\mathrm{m}}^3$ and sound speed $c=1500 \mathrm{m}/\mathrm{s}$. The Neumann boundary conditions $v_n=1 \mathrm{m}/\mathrm{s}$ are applied at the left end of the tube, while all other boundaries are treated as rigid walls (zero normal velocity). The x-axis is aligned with the tube length. Figure 3b presents the structured triangular mesh used for the FEM-T3 and Figure 3c shows the structured quadrilateral mesh used for the FEM-Q4 and the RBF-N3. All the used meshes share an identical nodal distribution and exhibit uniform structuring.

The analytical solution for this 2D acoustic problem is given by

$$\left\{\begin{array}{l}P=-j\rho c{v}_n{\displaystyle \frac{\cos \left[k\left(1-x\right)\right]}{\sin k}}\\ v={\displaystyle \frac{v_n\sin \left[k\left(1-x\right)\right]}{\sin k}}\end{array}\right.,$$

(38)

in which $v$ represents the acoustic particle velocity.

Two different error norms are used to evaluate the computational accuracy in this work, defined as [58]

$$e_p=\left|{\displaystyle \frac{p_e-p_h}{p_e}}\right|\times 100\%,$$

(39)

$$\eta =\sqrt{{\displaystyle \frac{{\displaystyle {\int }_{\mathsf{\Omega }}{\left({\overline{v}}_e-{\overline{v}}_h\right)}^T\left(v_e-v_h\right)\mathrm{d}\mathsf{\Omega }}}{{\displaystyle {\int }_{\mathsf{\Omega }}{\overline{v}}^T{v}_e\mathrm{d}\mathsf{\Omega }}}}}\le C_{1}kh+C_2{k}^3{h}^2,$$

(40)

in which the symbol $e$ represents the analytical solutions, and the symbol $h$ represents the corresponding numerical solutions. The symbol $\overline{v}$ represents the complex conjugate of $v$. Equation (39) describes a local relative error, while Equation (40) evaluates the global error over the entire problem domain. In Equation (40), $C_{1}kh$ represents the component of interpolation error, and $C_2{k}^3{h}^2$ represents the component of dispersion error.

For different frequencies, the acoustic problem is solved using the FEM-T3, the FEM-Q4, and the RBF-N3. The numerical and analytical acoustic pressure distributions along the tube length are then shown in Figure 4.

In Figure 4, the analytical solutions are indicated by dotted lines, while the results obtained using the RBF-N3 are shown as solid black curves. At relatively low frequencies, it is evident that all three numerical methods can provide sufficiently accurate results that closely match the analytical solutions. However, as the frequency increases, the curves generated by the FEM-T3 and FEM-Q4 begin to deviate from the analytical solutions, indicating that their computational accuracy significantly deteriorates. As shown in Figure 4d, at 6600 Hz the results from the FEM-T3 and FEM-Q4 display noticeable phase shifts relative to the analytical solutions due to the dispersion error. In contrast, the RBF-N3 method maintains excellent agreement with the analytical solutions even at 6600 Hz, suggesting that the proposed RBF-N3 remains highly accurate in solving relatively high-frequency Helmholtz problems. Furthermore, the RBF-N3 effectively suppresses the numerical dispersion error. These findings demonstrate that, particularly at relatively high frequencies, the RBF-N3 offers considerably more accurate and reliable results than the traditional FEM.

A comparative analysis of the relative error $e_{\mathrm{p}}$ is conducted for two frequencies: 2200 Hz and 4400 Hz. Based on Equation (39), the distributions of the local relative error $e_{\mathrm{p}}$ along the tube length are calculated for each numerical method and are presented in Figure 5 and Figure 6. For reference, the corresponding analytical particle velocity at both frequencies is also plotted in the same figures.

As shown in Figure 5a and Figure 6a, it is evident that the relative errors from the FEM-T3 and FEM-Q4 are significantly larger than the RBF-N3 along the tube length at 2200 Hz and 4400 Hz. Moreover, the relative error curves of the FEM-T3 and FEM-Q4 display noticeable oscillatory behaviors along the tube length. By analyzing the corresponding analytical particle velocity distributions in Figure 5b and Figure 6b, it can be observed that these oscillations in relative error become particularly pronounced at locations where the velocity magnitude reaches its peak. Especially, at 4400 Hz, the relative errors from the FEM-T3 and FEM-Q4 can even exceed 200% near such region. In contrast, the relative errors obtained using the RBF-N3 consistently remain at much lower levels across the entire tube length. Furthermore, compared to the FEM-T3 and FEM-Q4, the oscillation amplitudes in the relative error distributions produced by the RBF-N3 are significantly reduced, which remains below 0.6% even at the frequency of 4400 Hz. These findings indicate that the FEM-T3 and FEM-Q4 generally exhibit significant numerical errors in solving the Helmholtz equation, while the proposed RBF-N3 consistently performs well and provides sufficiently accurate solutions. This comparative analysis of the three numerical methods further confirms that the RBF-N3 is more suitable for solving the Helmholtz equation and consistently yields significantly smaller dispersion errors than the traditional FEM, particularly at relatively high frequencies. These observations demonstrate that the RBF-N3 offers improved accuracy and better numerical stability compared to the traditional FEM in solving Helmholtz problems.

In addition to completing the analysis of the local relative error $e_p$ for all three numerical methods, the global error $\eta$ is subsequently evaluated based on Equation (40). In this process, the wave number $k$ is taken as the independent variable. For a given set of wave numbers, the global error $\eta$ is computed for all three numerical methods, and the resulting variation in global error with respect to $k$ is plotted in Figure 7 for comparative analysis.

Figure 7 shows the global error $\eta$ curves for all three numerical methods versus wave number $k$ ranging from 5 to 50. The curve corresponding to the RBF-N3 is marked with inverted triangle symbols. It can be observed that when the wave number is relatively small, all three methods exhibit low global error. However, as the wave number increases over 25, the global errors of the FEM-T3 and FEM-Q4 begin to grow significantly. With further increases in wave number, their global error curves display rapid growth accompanied by pronounced oscillatory behaviors. In contrast, the global error of the RBF-N3 remains consistently low for wave numbers less than 50. Therefore, even from the perspective of global error analysis, the same conclusion can be drawn: the proposed RBF-N3 exhibits significant advantages in solving the Helmholtz equation and can always yield significantly more accurate solutions than the traditional FEM. In particular, when dealing with relatively large wave numbers, the RBF-N3 is still capable of delivering results with high accuracy.

As shown in Equation (40), the numerical error mainly consists of two components: the interpolation error and dispersion error, which are generally controlled by setting $kh=\mathrm{constant}$ and $k^3{h}^2=\mathrm{constant}$, respectively. The global error curves for all three numerical methods are shown in Figure 8a ($kh=\mathrm{constant}$) and Figure 8b ($k^3{h}^2=\mathrm{constant}$). Additionally, the global error curves of the FEM-T3 and FEM-Q4 clearly confirm the effectiveness of these two control strategies.

As shown in Figure 8, it is evident that the RBF-N3 consistently yields significantly smaller global errors and dispersion errors compared to the traditional FEM when solving the Helmholtz equation. Specifically, Figure 8a indicates that even when the interpolation error is controlled, the traditional FEM still exhibits significant numerical errors at relatively higher wave numbers due to the influence of dispersion error. In contrast, the RBF-N3 maintains low error levels under the same conditions. These results further confirm that the RBF-N3 is more effective at suppressing dispersion error and can provide more accurate solutions than the traditional FEM at relatively large wave numbers for the Helmholtz equation.

5.2. The 2D Square Domain Acoustic Problem

This subsection investigates the 2D square domain acoustic problem, and the corresponding acoustic model is shown in Figure 9. In Figure 9a, both the length and width of the domain are denoted by $L$, and the interior is filled with an ideal fluid medium characterized by a density $\rho =1 {\mathrm{kg}/\mathrm{m}}^3$ and a sound speed $c=1 \mathrm{m}/\mathrm{s}$. As shown in Figure 9a, the Robin and Dirichlet boundary conditions are applied in the numerical model. Here, $\beta =\mathsf{\pi }/4$ denotes the direction angle of wave propagation. The defined path indicated in the figure is used to plot and compare the acoustic pressure distributions obtained from different methods. In this subsection, the traditional FEM, EFEM-N3, meshfree RPIM, and the proposed RBF-N3 are used to solve the 2D square domain acoustic problem. All methods share a common nodal arrangement. Specifically, Figure 9b shows the structured triangular mesh used by the FEM-T3 and EFEM-N3, Figure 9c shows the structured quadrilateral mesh used by the FEM-Q4 and RBF-N3, and Figure 9d shows the node distribution used by the RPIM.

The analytical solution for the model is given by

$$p=\cos \left[k\left(x\cos \beta +y\sin \beta \right)\right]+j\sin \left[k\left(x\cos \beta +y\sin \beta \right)\right],$$

(41)

The acoustic model is solved using all five numerical methods under different wave numbers $\left(k=2, 5, 10, 16\right)$. The numerical solutions and the analytical solutions are shown in Figure 10. Specifically, Figure 10a presents the results for $k=2$, Figure 10b for $k=5$, Figure 10c for $k=10$, and Figure 10d for $k=16$. In each case, the analytical solutions are denoted by dotted curves. At wave numbers $k=2$, corresponding to relatively low-frequency scenarios, sufficiently accurate solutions are provided by all methods. However, as the wave number increases to $k=5$ and $k=10$, significant discrepancies begin to appear in the results from the FEM-T3 and FEM-Q4, which include pronounced phase shifts relative to the analytical solutions. These phase shifts indicate severe dispersion error, rendering the traditional FEM results unreliable at relatively large wave numbers. Notably, at $k=16$, the results illustrate that the RBF-N3 provides the most accurate solution.

To further quantify the numerical error, the global error $\eta$ versus wave number $k$ is subsequently evaluated using Equation (40). As shown in Figure 11, all methods yield small global errors at relatively small wave numbers. As the wave number increases, the global errors of the FEM-T3, FEM-Q4, and RPIM grow significantly. With further increase, the global errors of the EFEM-N3 also rise rapidly. In contrast, the RBF-N3 induces the smallest global errors across the entire tested range. This numerical example confirms that the proposed RBF-N3 offers very accurate solutions and effectively reduces dispersion errors, especially at relatively large wave numbers.

In this work, the RBF-N3 is introduced to avoid the LD problem, thereby ensuring the system matrix is not singular. The condition number of the matrix is typically used to assess its singularity. Accordingly, a condition number analysis is conducted for different wave number k (2, 5, 10, 16) and mesh size h (0.2, 0.1, 0.05, 0.025) to discuss this.

For the analysis, the numerical procedure developed by Gui et al. [82] is employed to eliminate the LD issue inherent to the EFEM-N3. The computed results are presented in

Table 1. Condition numbers for different wave number k (mesh size h = 0.2).

k	FEM-T3	FEM-Q4	EFEM-N3	RPIM	RBF-N3
2	3.86 × 102	3.38 × 102	6.27 × 104	8.34 × 102	4.05 × 104
5	5.24 × 102	4.24 × 102	7.31 × 104	5.83 × 102	3.89 × 104
10	3.88 × 103	7.39 × 102	8.57 × 104	7.51 × 102	1.11 × 105
16	6.37 × 102	1.45 × 103	7.42 × 104	3.70 × 102	9.77 × 104

,

Table 2. Condition numbers for different wave number k (mesh size h = 0.1).

k	FEM-T3	FEM-Q4	EFEM-N3	RPIM	RBF-N3
2	1.73 × 103	1.38 × 103	9.21 × 105	3.55 × 103	4.45 × 105
5	1.47 × 103	1.07 × 103	9.54 × 105	2.76 × 103	4.48 × 105
10	2.41 × 103	2.06 × 103	1.08 × 106	1.63 × 104	4.65 × 105
16	3.96 × 103	5.63 × 103	1.13 × 106	1.56 × 104	4.56 × 105

,

Table 3. Condition numbers for different wave number k (mesh size h = 0.05).

k	FEM-T3	FEM-Q4	EFEM-N3	RPIM	RBF-N3
2	7.78 × 103	5.99 × 103	1.45 × 107	1.54 × 104	6.38 × 106
5	5.79 × 103	4.80 × 103	1.46 × 107	1.24 × 104	6.41 × 106
10	3.08 × 104	6.92 × 104	1.51 × 107	5.65 × 104	6.49 × 106
16	3.76 × 104	5.77 × 104	1.59 × 107	7.23 × 104	6.68 × 106

and

Table 4. Condition numbers for different wave number k (mesh size h = 0.025).

k	FEM-T3	FEM-Q4	EFEM-N3	RPIM	RBF-N3
2	3.53 × 104	2.66 × 104	2.32 × 108	6.83 × 104	1.02 × 108
5	2.81 × 104	2.19 × 104	2.32 × 108	5.67 × 104	1.02 × 108
10	1.75 × 105	1.27 × 105	2.34 × 108	2.21 × 105	1.02 × 108
16	7.34 × 104	6.72 × 104	2.37 × 108	3.40 × 105	1.03 × 108

. It is observed that the FEM-T3, FEM-Q4, and RPIM yield similar condition number, which consistently increase for larger k or smaller h. In contrast, the condition number for the proposed RBF-N3 is lower than that of the EFEM-N3 and is influenced only by h. The analysis confirms that the system matrix for the proposed RBF-N3 is not singular, demonstrating that the LD problem has been efficiently avoided.

In practical engineering applications using the FEM, distorted meshes are frequently encountered, adversely affecting computational accuracy. In this study, distorted meshes are generated from original regular meshes using the following equations [73]:

$$\left\{\begin{array}{c}{x}_{ir}=x_{re}+h\cdot r_c\cdot {\beta }_{ir}\\ y_{ir}=y_{re}+h\cdot r_c\cdot {\beta }_{ir}\end{array}\right.,$$

(42)

where $x_{re}$ and $y_{re}$ represent the nodal coordinates of the original regular mesh, while $x_{ir}$ and $y_{ir}$ denote the corresponding coordinates of the generated distorted mesh. The parameter h is defined as the characteristic element size. The random number $r_c\in \left[-1,1\right]$ is introduced to generate irregular nodal positions. The distortion coefficient ${\beta }_{ir}\in \left[0,0.5\right]$ controls the degree of mesh distortion. A larger value of ${\beta }_{ir}$ results in a more severely distorted mesh.

Two types of distorted meshes are generated from the regular mesh using distortion coefficients ${\beta }_{ir}=0.1$ and ${\beta }_{ir}=0.2$, as illustrated in Figure 12. Acoustic pressure distributions are computed for different wave number k using these distorted meshes. The results are presented in Figure 13 (${\beta }_{ir}=0.1$) and Figure 14 (${\beta }_{ir}=0.2$).

The results show that accurate solutions are provided by all method at $k=2$. At $k=5$, the solutions obtained using FEM-T3 exhibit obvious numerical errors, while the other methods still maintain satisfactory accuracy. At $k=10$, it is observed that the proposed RBF-N3 performs optimally under the same irregular node distributions. Notably, the proposed RNF-N3 continues to provide relatively accurate solutions even at relatively large wave number and on distorted meshes.

5.3. The 2D Car Acoustic Problem

This subsection investigates the 2D car acoustic problem, and the corresponding acoustic model is shown in Figure 15. Unlike the previous case, this numerical model cannot be given an analytical solution. Therefore, a highly refined numerical solution is employed as the reference solution for validation. The schematic of the model is presented in Figure 15a. The defined path indicated in the figure is used to plot and compare the acoustic pressure distributions obtained from different methods. As shown in Figure 15, the Robin and Neumann boundary conditions are applied in the numerical model, which are highlighted with bold blue lines. The corresponding normal particle velocity and acoustic admittance coefficient are specified as $v_n=0.01 \mathrm{m}/\mathrm{s}$ and $A_n=0.00144 \mathrm{m}/(\mathrm{Pa}·\mathrm{s})$, respectively. All other boundaries are assumed to be rigid walls. The interior of the model is filled with air, with a density $\rho =1.25 {\mathrm{kg}/\mathrm{m}}^3$ and a sound speed $c=340 \mathrm{m}/\mathrm{s}$. Figure 15b shows the structured triangular mesh used for the FEM-T3, while Figure 15c shows the structured quadrilateral mesh used for the FEM-Q4 and the RBF-N3. Both meshes have identical sizing.

The acoustic model described in Figure 15a is analyzed at two different frequencies: 320 Hz and 650 Hz. The computed results from all three numerical methods and the reference solutions are plotted in Figure 16. Each curve represents the acoustic pressure distributions along the defined path, and the reference solutions are marked with dots for clarity. Figure 16a,b show the results at 320 Hz and 650 Hz, respectively. As observed, at 320 Hz, all three methods produce results that agree well with the reference solutions. However, at 650 Hz, the results obtained using the FEM-T3 and FEM-Q4 deviate significantly from the reference solutions, while RBF-N3 continues to match the reference solutions closely. This observation is consistent with the conclusions drawn from the examples in Section 5.1 and Section 5.2: all three methods yield accurate results for relatively low-frequency acoustic problems, but as the frequency increases, the numerical errors of the traditional FEM grow rapidly. In contrast, RBF-N3 maintains high accuracy even at relatively high frequencies.

Additionally, the acoustic pressure amplitude contour plots computed by the three methods at the two frequencies are shown in Figure 17 and Figure 18. These contour plots clearly illustrate the differences in solution quality across the total problem domain. At 320 Hz (Figure 17), all three methods produce contour fields that closely match the reference solutions. However, at 650 Hz (Figure 18), noticeable discrepancies appear between the contour plots of the FEM-T3 and FEM-Q4 and that of the reference solutions, particularly in certain regions. In contrast, the contour plot obtained by the RBF-N3 remains more consistent with the reference solutions. These results further demonstrate that, from a global perspective, the RBF-N3 offers superior accuracy compared to the traditional FEM, especially for relatively high-frequency problems, providing more reliable and precise numerical solutions.

5.4. The 3D Cabin Acoustic Problem

To examine the performance of the proposed RBF-N3 in solving 3D acoustic problems, this subsection investigates a practical and complex case inside a 3D cabin model (length L = 4.36 m, width B = 2.37 m, height H = 1.89 m; see Figure 19). The main noise sources are the air-conditioning outlets at the cabin top, which are simplified as point sources with an acoustic particle velocity of $v_n={10}^{-4} \mathrm{m}/\mathrm{s}$. The domain is discretized using eight-node hexahedral elements (FEM-H8) with an average nodal spacing of h = 0.15 m. This mesh also serves as the background integration cells for the RPIM. At a frequency of 220 Hz, the computed acoustic pressure distributions from different methods are plotted in Figure 20. Results from the proposed RBF-N3 show high consistency with the reference solution obtained using a highly refined mesh (h = 0.03 m). In contrast, visible numerical errors appear in the RPIM and FEM-H8 solutions. This example demonstrates that the proposed RBF-N3 can effectively reduce the numerical errors and provide accurate solutions for complex 3D acoustic problems.

6. Concluding Remarks

In this work, a novel formulation of the Helmholtz equation is constructed based on the RBFs, and the theoretical formulation of the RBF-N3 is derived. In the proposed RBF-N3, the interpolation functions constructed using the RBFs are employed to avoid the LD problem and the polynomial enrichment functions are used to enrich the approximation space generated by the RBFs, which further improves the capability of the proposed method in solving relatively high-frequency Helmholtz problems. Compared to the standard EFEM, the proposed RBF-N3 yields a system matrix of the same size, resulting in no significant increase in computational cost. Dispersion error analysis shows that the numerical dispersion error is efficiently reduced by the RBF-N3 at relatively large wave numbers. Condition number analysis confirms that the RBF-N3 maintains a satisfactory condition number and successfully avoids the LD problem. Comparative analysis using various numerical examples demonstrates that the proposed RBF-N3 consistently achieves accurate solutions and small numerical errors for acoustic problems, even at relatively large wave numbers. Importantly, in the proposed RBF-N3, the nodal shape functions retain the Kronecker delta property, facilitating the imposition of essential boundary conditions. Furthermore, no special modifications to the mesh structure are required, making the RBF-N3 convenient for practical applications. Therefore, the proposed RBF-N3 represents a promising numerical scheme for solving relatively high-frequency Helmholtz problems.