The Improved Element-Free Galerkin Method for 3D Helmholtz Equations

Abstract

The improved element-free Galerkin (IEFG) method is proposed in this paper for solving 3D Helmholtz equations. The improved moving least-squares (IMLS) approximation is used to establish the trial function, and the penalty technique is used to enforce the essential boundary conditions. Thus, the final discretized equations of the IEFG method for 3D Helmholtz equations can be derived by using the corresponding Galerkin weak form. The influences of the node distribution, the weight functions, the scale parameters of the influence domain, and the penalty factors on the computational accuracy of the solutions are analyzed, and the numerical results of three examples show that the proposed method in this paper can not only enhance the computational speed of the element-free Galerkin (EFG) method but also eliminate the phenomenon of the singular matrix.

1. Introduction

As an important elliptic differential equation, the Helmholtz equation has been widely applied in many different fields, such as mechanics, acoustics, physics, electromagnetics, engineering, and so on. It is well known that how to achieve the numerical solutions of Helmholtz equations effectively and accurately is one of the important directions in the scientific research.

Currently, many meshless methods have been used for researching Helmholtz equations, such as the element-free Galerkin (EFG) method [1], meshless Galerkin least-square method [2], meshless hybrid boundary-node method [3], boundary element-free method [4], and complex variable boundary element-free method [5,6]. Compared to the traditional finite difference method [7,8,9,10] and the finite element method, meshless methods [11,12,13,14,15] are based on scattered point approximation, which can avoid the mesh reconstruction, and thus a higher accuracy of the numerical solutions can be obtained.

As an important meshless method, the EFG method [16] was studied by Belytschko et al. In this method, a trial function is established by using the moving least-squares (MLS) approximation. Cheng et al. analyzed the error estimates of EFG method for potential problems [17]. Because the MLS approximation is based on the least-squares method [18,19,20,21,22], the disadvantages of the least-squares method also exist in the MLS approximation, in which sometimes ill-conditional or singular matrices occur.

In order to eliminate the singular matrices, the improved moving least-squares (IMLS) approximation [23] was proposed by Cheng et al., in which the orthogonal function system with a weight function is used as basis function, and thus can make up for the deficiency of the MLS approximation and has greater computational efficiency, using the IMLS approximation to establish the trial function. Thus, the improved element-free Galerkin (IEFG) method was applied for potential [24], transient heat conduction problems [25], the wave equation [26], the Schrödinger equation [27], advection–diffusion [28], elastodynamics [29], elastoplasticity [30], viscoelasticity [31], and diffusional drug release [32] problems. From these studies, we can see that under similar computational accuracy, the IEFG method has higher computational speed than the EFG method. As we know, meshless methods are based on node approximation without mesh reconstruction. When solving large deformation problems and dynamic propagation of cracks, the meshless method can obtain greater precision than the finite element method. In order to take advantage of the IEFG method further, Zhang et al. [33] developed the enriched IEFG method to solve 2D fracture problems. In this method, the enriched basis function is used at the tip of the crack. As a result, the singularity of the stresses at the tip of the crack can be shown better than in the IEFG method. Cai et al. [34] used the IEFG method for solving large elastoplasticity deformation problems. Three numerical examples are given to show that the numerical solutions are in good agreement with the solutions of finite element method software ANASYS and can enhance the computational efficiency of the EFG method.

By introducing the singular weight function into the MLS approximation, Lancaster et al. presented an interpolating MLS method [35]. The boundary conditions could be enforced directly in the corresponding meshless method. Based on the concept of an inner production, Ren et al. improved the interpolating MLS method [36] by using the singular weight function in interpolating points and orthogonalizing some of the base functions. Thus, the corresponding interpolating EFG method was presented for potential [37], transient heat conduction [38], and some mechanics [39,40,41] problems. Compared with the traditional EFG method, the interpolating EFG method has higher computational efficiency. Additionally, the interpolating smoothed particle method was developed by Qin et al. [42].

Using the nonsingular weight function, Wang et al. developed the improved interpolating MLS method [43], which can overcome the difficulties caused by the singular weight function in the interpolating MLS method, and used this method to construct the trial function. The improved interpolating EFG method was presented for potential [43] and several large deformation problems [44,45,46].

Based on the approximation of the vector function, the complex variable moving least-squares (CVMLS) approximation was presented by Cheng et al. [47]. Based on the CVMLS approximation and Galerkin weak form, the complex variable element-free Galerkin (CVEFG) method [48] was presented. Moreover, based on the conjugate basis function, Bai et al. proposed the improved CVMLS approximation to construct the shape function, and the improved CVEFG method was presented for elasticity problems [49]. The improved CVEFG method has higher computational accuracy and efficiency than the EFG method, but it cannot be applied to 3D problems directly because the complex theory is used. Chen et al. [50,51] proposed the complex variable reproducing kernel particle method.

By combining meshless methods and the finite difference method, the hybrid CVEFG method [52,53,54,55,56], dimension-splitting EFG method [57,58,59,60], dimension-splitting reproducing kernel particle method [61,62,63,64], interpolating dimension-splitting EFG method [65] and hybrid generalized interpolated EFG method [66] were proposed. These methods can greatly improve the computational efficiency of the traditional meshless method for solving multi-dimensional problems.

The IEFG method has some advantages over the traditional EFG method, such as higher computational efficiency, avoiding matrix inversion, and eliminating singular matrix. Therefore, it has been applied to many science and engineering problems. However, 3D Helmholtz equations have not been studied by the IEFG method yet, and the corresponding parameters cannot be discussed; thus, the computational accuracy and efficiency of the IEFG method for 3D Helmholtz equations are also uncertain. In order to overcome the disadvantage of the lower efficiency of the EFG method, this paper presents the IEFG method for solving 3D Helmholtz equations. The trial function was established by using the IMLS approximation, using the penalty technique to enforce the essential boundary conditions. The final discretized equations could be derived by using the corresponding weak form. Thus, we obtained the final formulate of the IEFG method for 3D Helmholtz equations.

In Section 4, the influences of the node distribution, the weight functions, the scale parameters, and the penalty factors on the computational accuracy of the solutions are analyzed by giving examples. It is shown that the IEFG method for Helmholtz equations is convergent. Compared with the EFG method, the IEFG method has greater computational speed. Moreover, the singular matrix can be eliminated.

2. The IMLS Approximation

The approximation of a function $u(\mathit{x})$ is

$$u^h(\mathit{x})={\displaystyle \sum _{i=1}^m{p}_i(\mathit{x})\cdot a_i(\mathit{x})}={\mathit{p}}^{\mathrm{T}}(\mathit{x})\cdot \mathit{a}(\mathit{x}), (\mathit{x}\in \Omega ),$$

(1)

where ${\mathit{p}}^{\mathrm{T}}(\mathit{x})$ is the basis function vector, $m$ is the basis function number, and

$${\mathit{a}}^{\mathrm{T}}(\mathit{x})=(a_1(\mathit{x}),a_2(\mathit{x}),\cdots ,a_m(\mathit{x}))$$

(2)

is the coefficient vector of ${\mathit{p}}^{\mathrm{T}}(\mathit{x})$.

In general,

$${\mathit{p}}^{\mathrm{T}}(\mathit{x})=(1,x_1,x_2,x_3),$$

(3)

$${\mathit{p}}^{\mathrm{T}}(\mathit{x})=(1,x_1,x_2,x_3,x_1^2,x_2^2,x_3^2,x_1{x}_2,x_2{x}_3,x_1{x}_3).$$

(4)

The local approximation is

$$u^h(\mathit{x},\widehat{\mathit{x}})={\displaystyle \sum _{i=1}^m{p}_i(\widehat{\mathit{x}})\cdot a_i(\mathit{x})}={\mathit{p}}^{\mathrm{T}}(\widehat{\mathit{x}})\cdot \mathit{a}(\mathit{x}).$$

(5)

Define

$$J={\displaystyle \sum _{I=1}^{n}w(\mathit{x}-{\mathit{x}}_I)}{[u^h(\mathit{x},{\mathit{x}}_I)-u_I]}^2={\displaystyle \sum _{I=1}^{n}w(\mathit{x}-{\mathit{x}}_I)}{\left[{\displaystyle \sum _{i=1}^m{p}_i({\mathit{x}}_I)\cdot a_i(\mathit{x})-u_I}\right]}^2,$$

(6)

where $w(\mathit{x}-{\mathit{x}}_I)$ is a weighting function, and ${\mathit{x}}_I$ ($I=1,2,\cdots ,n$) are the nodes with influence domains covering point $\mathit{x}$.

Equation (6) can be written as

$$J={(\mathit{P}\mathit{a}-\mathit{u})}^{\mathrm{T}}\mathit{W}(\mathit{x})(\mathit{P}\mathit{a}-\mathit{u}),$$

(7)

where

$${\mathit{u}}^{\mathrm{T}}=(u_1,u_2,\cdots ,u_n),$$

(8)

$$\mathit{P}=\left[\begin{array}{cccc}{p}_1(x_1)& p_2(x_1)& \cdots & p_m(x_1)\\ p_1(x_2)& p_2(x_2)& \cdots & p_m(x_2)\\ ⋮& ⋮& \ddots & ⋮\\ p_1(x_n)& p_2(x_n)& \cdots & p_m(x_n)\end{array}\right],$$

(9)

and

$$\mathit{W}(\mathit{x})=\left[\begin{array}{cccc}w(x-x_1)& 0& \cdots & 0\\ 0& w(x-x_2)& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & w(x-x_n)\end{array}\right].$$

(10)

From

$$\frac{\partial J}{\partial \mathit{a}}=\mathit{A}(\mathit{x})\mathit{a}(\mathit{x})-\mathit{B}(\mathit{x})\mathit{u}=0,$$

(11)

we have

$$\mathit{A}(\mathit{x})\mathit{a}(\mathit{x})=\mathit{B}(\mathit{x})\mathit{u},$$

(12)

where

$$\mathit{A}(\mathit{x})={\mathit{P}}^{\mathrm{T}}\mathit{W}(\mathit{x})\mathit{P},$$

(13)

$$\mathit{B}(\mathit{x})={\mathit{P}}^{\mathrm{T}}\mathit{W}(\mathit{x}).$$

(14)

Equation (12) sometimes forms a singular or ill-conditional matrix. In order to make up for this deficiency, for basis functions

$$\mathit{q}=(q_i)=(1,x_1,x_2,x_3,x_1^2,x_2^2,x_3^2,x_1{x}_2,x_2{x}_3,x_3{x}_1,\cdots ),$$

(15)

using the Gram–Schmidt process, we can obtain

$$p_i=q_i-{\displaystyle \sum _{k=1}^{i-1}\frac{(q_i,p_k)}{(p_k,p_k)}}{p}_k, (i=1,2,3,\cdots ),$$

(16)

and

$$(p_i,p_j)=0, (i\ne j).$$

(17)

Then, from Equation (12), $a(\mathit{x})$ can be obtained as

$$\mathit{a}(\mathit{x})={\mathit{A}}^{*}(\mathit{x})\mathit{B}(\mathit{x})\mathit{u},$$

(18)

where

$${\mathit{A}}^{*}(\mathit{x})=\left[\begin{array}{cccc}\frac{1}{(p_1,p_1)}& 0& \cdots & 0\\ 0& \frac{1}{(p_2,p_2)}& 0& 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & \frac{1}{(p_n,p_n)}\end{array}\right].$$

(19)

Substituting Equation (18) into Equation (5), we have

$$u^h(\mathit{x})={\mathit{\Phi }}^{*}(\mathit{x})u={\displaystyle \sum _{I=1}^n{\mathit{\Phi }}_I^{*}(\mathit{x})u_I},$$

(20)

where

$${\mathit{\Phi }}^{*}(\mathit{x})=({\mathit{\Phi }}_1^{*}(\mathit{x}),{\mathit{\Phi }}_2^{*}(\mathit{x}),\cdots ,{\mathit{\Phi }}_n^{*}(\mathit{x}))={\mathit{p}}^{\mathrm{T}}(\mathit{x}){\mathit{A}}^{*}(\mathit{x})\mathit{B}(\mathit{x})$$

(21)

is the shape function.

This is the IMLS approximation [23], in which the shape function can be obtained more easily than the MLS approximation. Moreover, the IMLS approximation can also avoid the singular matrix. Thus, it can enhance the computational efficiency of the MLS approximation.

3. The IEFG Method for 3D Helmholtz Equations

The governing equation is

$$\Delta u+k^{2}u=f(\mathit{x}), (\mathit{x}=(x_1,x_2,x_3)\in \Omega )$$

(22)

and the boundary conditions are

$$u(\mathit{x})=\overline{u}, (\mathit{x}\in {\Gamma }_u),$$

(23)

$$q(\mathit{x})=\frac{\partial u(\mathit{x})}{\partial x_1}{n}_1+\frac{\partial u(\mathit{x})}{\partial x_2}{n}_2+\frac{\partial u(\mathit{x})}{\partial x_3}{n}_3=\overline{q}, (\mathit{x}\in {\Gamma }_q),$$

(24)

where $k^2$ is the wave number, $f(\mathit{x})$ is the given function, $\overline{u}$ and $\overline{q}$ are the given values, and $\Gamma ={\Gamma }_u\cup {\Gamma }_q$, ${\Gamma }_u\cap {\Gamma }_q=\varnothing$, $n_i$ ($i=1,2,3$) is the unit outward normal to the boundary $\Gamma$ in direction $x_i$.

For 3D Helmholtz equations, the equivalent functional is

$$\Pi ={\displaystyle {\int }_{\Omega }u(\frac{1}{2}{k}^{2}u-f)\mathrm{d}\Omega }-{\displaystyle {\int }_{\Omega }\frac{1}{2}\left[{\left(\frac{\partial u}{\partial x_1}\right)}^2+{\left(\frac{\partial u}{\partial x_2}\right)}^2+{\left(\frac{\partial u}{\partial x_3}\right)}^2\right]}\mathrm{d}\Omega -{\displaystyle {\int }_{{\Gamma }_q}u\overline{q}\mathrm{d}\Gamma }.$$

(25)

By introducing the penalty technique to apply the boundary conditions, we can obtain the modified functional

$${\Pi }^{*}=\Pi +\frac{\alpha }{2}{\displaystyle {\int }_{{\Gamma }_u}(u-\overline{u})(u-\overline{u})}\mathrm{d}\Gamma ,$$

(26)

where $\alpha$ is the penalty factor.

Let

$$\delta {\Pi }^{*}=0.$$

(27)

We can obtain the following equivalent integral weak form

$$\begin{array}{c}{\displaystyle {\int }_{\Omega }\delta u\cdot k^{2}u}\mathrm{d}\Omega -{\displaystyle {\int }_{\Omega }\delta {(\mathit{L}u)}^{\mathrm{T}}\cdot (\mathit{L}u)\mathrm{d}\Omega }-{\displaystyle {\int }_{\Omega }\delta u\cdot f}\mathrm{d}\Omega -{\displaystyle {\int }_{{\Gamma }_q}\delta u\cdot \overline{q}\mathrm{d}\Gamma }\\ +\alpha {\displaystyle {\int }_{{\Gamma }_u}\delta u\cdot u\mathrm{d}\Gamma }-\alpha {\displaystyle {\int }_{{\Gamma }_u}\delta u\cdot \overline{u}\mathrm{d}\Gamma }=0,\end{array}$$

(28)

where

$$\mathit{L}(\cdot )=\left[\begin{array}{c}\frac{\partial }{\partial x_1}\\ \frac{\partial }{\partial x_2}\\ \frac{\partial }{\partial x_3}\end{array}\right](\cdot ).$$

(29)

In the cubic domain $\Omega$, we employ $M$ nodes ${\mathit{x}}_I$ ($I=1,2,\cdots ,M$). Thus, we have

$$u_I=u({\mathit{x}}_I).$$

(30)

From the IMLS approximation, we can obtain

$$u(\mathit{x})={\mathit{\Phi }}^{\ast }(\mathit{x})\mathit{u}={\displaystyle \sum _{I=1}^n{\mathit{\Phi }}_I^{\ast }(\mathit{x})u_I},$$

(31)

where

$$\mathit{u}={(u_1,u_2,\cdots ,u_n)}^{\mathrm{T}}.$$

(32)

From Equations (29) and (31), we have

$$\mathit{L}u(\mathit{x})={\displaystyle \sum _{I=1}^n\left[\begin{array}{c}\frac{\partial }{\partial x_1}\\ \frac{\partial }{\partial x_2}\\ \frac{\partial }{\partial x_3}\end{array}\right]}{\mathit{\Phi }}_I^{\ast }(\mathit{x})u_I={\displaystyle \sum _{I=1}^n{\mathit{B}}_I{u}_I=\mathit{B}(\mathit{x})}u,$$

(33)

where

$$\mathit{B}(\mathit{x})=({\mathit{B}}_1,{\mathit{B}}_2,\cdots ,{\mathit{B}}_n),$$

(34)

$${\mathit{B}}_I=\left[\begin{array}{c}{\mathit{\Phi }}_{I,1}^{\ast }(\mathit{x})\\ {\mathit{\Phi }}_{I,2}^{\ast }(\mathit{x})\\ {\mathit{\Phi }}_{I,3}^{\ast }(\mathit{x})\end{array}\right].$$

(35)

Substituting Equations (31) and (33) into Equation (28), we have

$$\begin{array}{c}{\displaystyle {\int }_{\Omega }\delta {[{\mathit{\Phi }}^{*}(\mathit{x})\mathit{u}]}^{\mathrm{T}}\cdot k^2\cdot [{\mathit{\Phi }}^{*}(\mathit{x})u]\mathrm{d}\Omega -}{\displaystyle {\int }_{\Omega }\delta {[\mathit{B}(\mathit{x})\mathit{u}]}^{\mathrm{T}}[\mathit{B}(\mathit{x})u]\mathrm{d}\Omega }\\ -{\displaystyle {\int }_{{\Gamma }_q}\delta {[{\mathit{\Phi }}^{*}(\mathit{x})\mathit{u}]}^{\mathrm{T}}\cdot \overline{q}\mathrm{d}\Gamma }-{\displaystyle {\int }_{\Omega }\delta {[{\mathit{\Phi }}^{*}(\mathit{x})\mathit{u}]}^{\mathrm{T}}\cdot f\mathrm{d}\Omega }\\ \alpha {\displaystyle {\int }_{{\Gamma }_u}\delta {[{\mathit{\Phi }}^{*}(\mathit{x})\mathit{u}]}^{\mathrm{T}}\cdot [{\mathit{\Phi }}^{*}(\mathit{x})u]\mathrm{d}\Gamma }-\alpha {\displaystyle {\int }_{{\Gamma }_u}\delta {[{\mathit{\Phi }}^{*}(\mathit{x})\mathit{u}]}^{\mathrm{T}}\cdot \overline{u}\mathrm{d}\Gamma }=0.\end{array}$$

(36)

In Equation (36), the form of $\mathit{u}$ is the same as Equation (32), and $n=M$.

All integral terms in Equation (36) are analyzed as follows:

$${\displaystyle {\int }_{\Omega }\delta {[{\mathit{\Phi }}^{*}(\mathit{x})\mathit{u}]}^{\mathrm{T}}\cdot k^2\cdot [{\mathit{\Phi }}^{*}(\mathit{x})\mathit{u}]\mathrm{d}\Omega }=\delta {\mathit{u}}^{\mathrm{T}}\cdot k^2[{\displaystyle {\int }_{\Omega }{\mathit{\Phi }}^{*}{}^{\mathrm{T}}(\mathit{x}){\mathit{\Phi }}^{*}(\mathit{x})\mathrm{d}\Omega }]\cdot u,$$

(37)

$${\displaystyle {\int }_{\Omega }\delta {[\mathit{B}(\mathit{x})\mathit{u}]}^{\mathrm{T}}\cdot [\mathit{B}(\mathit{x})\mathit{u}]\mathrm{d}\Omega }=\delta {\mathit{u}}^{\mathrm{T}}\cdot [{\displaystyle {\int }_{\Omega }{\mathit{B}}^{\mathrm{T}}(\mathit{x})\mathit{B}(\mathit{x})\mathrm{d}\Omega }]\cdot \mathit{u},$$

(38)

$${\displaystyle {\int }_{\Omega }\delta {[{\mathit{\Phi }}^{*}(\mathit{x})\mathit{u}]}^{\mathrm{T}}\cdot f\mathrm{d}\Omega =}\delta {\mathit{u}}^{\mathrm{T}}\cdot [{\displaystyle {\int }_{\Omega }{\mathit{\Phi }}^{*}{}^{\mathrm{T}}(\mathit{x})f\mathrm{d}\Omega }],$$

(39)

$${\displaystyle {\int }_{{\Gamma }_q}\delta {[{\mathit{\Phi }}^{*}(\mathit{x})\mathit{u}]}^{\mathrm{T}}\cdot \overline{q}\mathrm{d}\Gamma =}\delta {\mathit{u}}^{\mathrm{T}}\cdot [{\displaystyle {\int }_{{\Gamma }_q}{\mathit{\Phi }}^{*}{}^{\mathrm{T}}(\mathit{x})\overline{q}\mathrm{d}\Gamma }],$$

(40)

$$\alpha {\displaystyle {\int }_{{\Gamma }_u}\delta {[{\mathit{\Phi }}^{*}(\mathit{x})\mathit{u}]}^{\mathrm{T}}\cdot [{\mathit{\Phi }}^{*}(\mathit{x})\mathit{u}]\mathrm{d}\Gamma }=\delta {\mathit{u}}^{\mathrm{T}}\cdot [\alpha {\displaystyle {\int }_{{\Gamma }_u}{\mathit{\Phi }}^{*}{}^{\mathrm{T}}(\mathit{x}){\mathit{\Phi }}^{*}(\mathit{x})\mathrm{d}\Gamma }]\cdot \mathit{u},$$

(41)

$$\alpha {\displaystyle {\int }_{{\Gamma }_u}\delta {[{\mathit{\Phi }}^{*}(\mathit{x})\mathit{u}]}^{\mathrm{T}}\cdot \overline{u}\mathrm{d}\Gamma }=\delta {\mathit{u}}^{\mathrm{T}}\cdot [\alpha {\displaystyle {\int }_{{\Gamma }_u}{\mathit{\Phi }}^{*}{}^{\mathrm{T}}(\mathit{x})\overline{u}\mathrm{d}\Gamma }].$$

(42)

Let

$$\mathit{C}=k^2{\displaystyle {\int }_{\Omega }{\mathit{\Phi }}^{*}{}^{\mathrm{T}}(\mathit{x}){\mathit{\Phi }}^{*}(\mathit{x})\mathrm{d}\Omega },$$

(43)

$$\mathit{K}={\displaystyle {\int }_{\Omega }{\mathit{B}}^{\mathrm{T}}(\mathit{x})\mathit{B}(\mathit{x})\mathrm{d}\Omega },$$

(44)

$${\mathit{F}}_1={\displaystyle {\int }_{\Omega }{\mathit{\Phi }}^{*}{}^{\mathrm{T}}(\mathit{x})f\mathrm{d}\Omega },$$

(45)

$${\mathit{F}}_2={\displaystyle {\int }_{{\Gamma }_q}{\mathit{\Phi }}^{*}{}^{\mathrm{T}}(\mathit{x})\overline{q}\mathrm{d}\Gamma },$$

(46)

$${\mathit{K}}^{\alpha }=\alpha {\displaystyle {\int }_{{\Gamma }_u}{\mathit{\Phi }}^{*}{}^{\mathrm{T}}(\mathit{x}){\mathit{\Phi }}^{*}(\mathit{x})\mathrm{d}\Gamma },$$

(47)

$${\mathit{F}}^{\alpha }=\alpha {\displaystyle {\int }_{{\Gamma }_u}{\mathit{\Phi }}^{*}{}^{\mathrm{T}}(\mathit{x})\overline{u}\mathrm{d}\Gamma }.$$

(48)

Substituting Equations (37)–(42) into Equation (36), we can obtain

$$\delta {\mathit{u}}^{\mathrm{T}}(\mathit{C}\mathit{u}-\mathit{K}\mathit{u}-{\mathit{F}}_1-{\mathit{F}}_2-{\mathit{F}}^{\alpha }+{\mathit{K}}^{\alpha }\mathit{u})=0.$$

(49)

The $\delta {\mathit{u}}^{\mathrm{T}}$ is arbitrary; thus we can obtain

$$\tilde{\mathit{K}}u=\tilde{\mathit{F}},$$

(50)

where

$$\tilde{\mathit{K}}=C-\mathit{K}+{\mathit{K}}^{\alpha },$$

(51)

$$\tilde{\mathit{F}}={\mathit{F}}_1+{\mathit{F}}_2+{\mathit{F}}^{\alpha }.$$

(52)

This is the IEFG method for 3D Helmholtz equations.

4. Numerical Examples

The formula of the relative error is

$${\Vert u-u^h\Vert }_{L^2(\Omega )}^{rel}=\frac{{\Vert u-u^h\Vert }_{L^2(\Omega )}}{{\Vert u\Vert }_{L^2(\Omega )}},$$

(53)

where

$${\Vert u-u^h\Vert }_{L^2(\Omega )}={\left({\displaystyle {\int }_{\Omega }{(u-u^h)}^2\mathrm{d}\Omega }\right)}^{1/2}.$$

(54)

In order to illustrate the advantages of the IFFG method, we chose three examples from other literature. The nodes distributed in the problem domains of these numerical examples were regular, the linear basis function was selected, and 3 × 3 × 3 Gaussian points were selected in each integral cell. The IEFG and the EFG methods are used to solve these examples.

The following equation is considered in the first example:

$$\Delta u+u=(12x_1^2-x_1^4)\sin x_2\cos x_3.$$

(55)

The boundary conditions are

$$u(0,x_2,x_3)=0,$$

(56)

$$u(\pi ,x_2,x_3)={\pi }^4\sin x_2\cos x_3,$$

(57)

$$u(x_1,0,x_3)=u(x_1,\pi ,x_3)=0,$$

(58)

$$u(x_1,x_2,0)=-u(x_1,x_2,\pi )=x_1^4\sin x_2.$$

(59)

The problem domain is $\Omega =[0,\pi ]\times [0,\pi ]\times [0,\pi ]$, and

$$u=x_1^4\sin x_2\cos x_3$$

(60)

is the analytical solution.

In order to study the convergence of the EFG and the IEFG methods for Helmholtz equations, all parameters of both methods were kept the same. The cubic spline weight functions were used, d_max = 1.35, $\mathsf{\alpha }=2.0\times {10}^4$.

Table 1. Relative errors and CPU times of the improved element-free Galerkin (IEFG) and element-free Galerkin (EFG) methods with the increase in node distribution.

Nodes	Relative Error		Time (s)
	IEFG	EFG	IEFG	EFG
7 × 7 × 7	4.3092%	4.3092%	5.5	5.8
11 × 11 × 11	1.3234%	1.3234%	26.7	28.5
13 × 13 × 13	0.8832%	0.8832%	49.9	53.1
15 × 15 × 15	0.6296%	0.6296%	92.1	98.0
17 × 17 × 17	0.4706%	0.4706%	152.0	161.9
21 × 21 × 21	0.2905%	0.2905%	384.5	398.6
25 × 25 × 25	0.1967%	0.1967%	903.3	940.2
29 × 29 × 29	0.1520%	0.1520%	1907.2	1937.1
33 × 33 × 33	0.1075%	0.1075%	4287.2	4408.3

shows the relationship between relative errors and node distribution. It is shown that, with the increase in nodes, the precision of numerical solutions improves as well, but the computational efficiency is reduced gradually. Therefore, the two methods in this paper are convergent. Both the computational accuracy and efficiency are considered, and 15 × 15 × 15 regularly distributed nodes are selected.

The effects of the weight function, the scale parameter of the influence domain, and the penalty factor on solution of the IEFG method will be discussed.

(1)

Weight function

When the cubic spline function is used, 15 × 15 × 15 regularly distributed nodes and 14 × 14 × 14 background integral cells are selected, $\mathsf{\alpha }=2.0\times {10}^4$, d_max = 1.35. Thus, the smaller relative error is 0.6296%. When the quartic spline function is used, and the same regularly distributed nodes and background integral grids are used, $\mathsf{\alpha }=2.2\times {10}^4$, d_max = 1.28, the smaller relative error is 0.6274%. It is shown that the similar relative errors can be obtained when using two weight functions.

In addition, the singular matrix can be avoided in the IEFG method when using the cubic spline function. If d_max = 1.0, the quartic spline function is selected. Unfortunately, the singular matrix occurs and the final result cannot be obtained. When the cubic spline function is used, the relative error is 0.6451%.

Thus, the cubic spline function is selected.

(2)

Scale parameter

The same node distribution and background integral grids are selected, $\mathsf{\alpha }=2.0\times {10}^4$, and the cubic spline function is used. Figure 1 shows the relationship between d_max and relative errors. Because of the error of computer itself, the relative error become larger when d_max = 1.2. It is shown that when d_max = 1.35, the relative error is smaller.

(3)

Penalty factor

The same node distribution, background integral grids, and weight function are selected, d_max = 1.35. Figure 2 shows the relationship between $\mathsf{\alpha }$ and relative errors. It is shown that when $\mathsf{\alpha }=2.0\times {10}^4$, the relative error is smaller.

The IEFG method is selected to solve it, 15 × 15 × 15 regularly distributed nodes and 14 × 14 × 14 background integral cells are selected, and the cubic spline function is used, $\mathsf{\alpha }=2.0\times {10}^4$, d_max = 1.35. When using the EFG method to solve it, the same parameters are selected, and thus the relative errors of two methods are equal to 0.6296%.

Figure 3, Figure 4 and Figure 5 show the comparison between numerical solutions and analytical ones, and the CPU times of the IEFG method and the EFG method are 92.1 s and 98.0 s, respectively. Obviously, higher computational efficiency can be obtained when using the IEFG method.

Additionally, the singular matrix can be avoided when constructing the shape functions when the IEFG is used. If d_max = 1.0 and other parameters are the same, two methods are used to solve it, and two different results are obtained. When the EFG method is used, the singular matrix occurs and the final result cannot be obtained. However, using the IEFG method to solve it, the relative error of the numerical solutions is 0.6451%. The numerical and analytical results are compared in Figure 6; it is shown that the numerical results are in good agreement with the analytical ones.

The second example [67] is

$$\Delta u-k^{2}u=0.$$

(61)

The boundary conditions are

$$u(0,x_2,x_3)=e^{({\xi }_2{x}_2+{\xi }_3{x}_3)},$$

(62)

$$u(1,x_2,x_3)={}^{({\xi }_1+{\xi }_2{x}_2+{\xi }_3{x}_3)},$$

(63)

$$u(x_1,0,x_3)=e^{({\xi }_1{x}_1+{\xi }_3{x}_3)},$$

(64)

$$u(x_1,1,x_3)=e^{({\xi }_1{x}_1+{\xi }_2+{\xi }_3{x}_3)},$$

(65)

$$u(x_1,x_2,0)=e^{({\xi }_1{x}_1+{\xi }_2{x}_2)},$$

(66)

$$u(x_1,x_2,1)=e^{({\xi }_1{x}_1+{\xi }_2{x}_2+{\xi }_3)}.$$

(67)

The problem domain is $\Omega =[0,1]\times [0,1]\times [0,1]$, and

$$u=e^{({\xi }_1{x}_1+{\xi }_2{x}_2+{\xi }_3{x}_3)}$$

(68)

is the analytical solution.

We set k = 2, ${\xi }_1$ = 1, and ${\xi }_2$ = 0.5. The IEFG method is used to solve it, $\mathsf{\alpha }=1.7\times {10}^3$, d_max = 1.21. The 15 × 15 × 15 regularly distributed nodes and 14 × 14 × 14 background integral grids are used. When using the EFG method to solve it, the same parameters are selected, and thus the same computational accuracy can be obtained. The relative errors of both methods are equal to 0.0844%. Figure 7, Figure 8 and Figure 9 show the comparison of the numerical solutions of the two methods and the analytical ones. The CPU times of the IEFG method and the EFG method are 92.1 s and 98.0 s, respectively. We can see that the computational results of both methods are in very good agreement with the analytical ones.

When different parameters are selected, k = 5, ${\xi }_1$ = 3, and ${\xi }_2$ = 2.7. Using two methods to solve it, the same parameters are used. Thus, the relative errors of both methods are equal to 0.5295%. Figure 10, Figure 11 and Figure 12 show the comparison of the numerical solutions of the two methods and the analytical ones, and the CPU times of the IEFG method and the EFG method are 92.1 s and 98.0 s, respectively. We can see that the computational results of both methods are in good agreement with the analytical ones.

We can select k = 10, ${\xi }_1$ = 5.8, and ${\xi }_2$ = 6.2. Using the two methods to solve it, the same parameters are used, and the relative errors of both methods are equal to 2.3884%. Figure 13, Figure 14 and Figure 15 show the comparison of the numerical solutions of the two methods and the analytical ones. The computational results of both methods are in good agreement with the analytical ones.

From this example, we can draw two conclusions: On the one hand, the IEFG method has greater computational efficiency; on the other hand, the bigger the wave numbers are, the lower the computational accuracy.

Similarly, if d_max = 1.0, we select k = 10, ${\xi }_1$ = 5.8, and ${\xi }_2$ = 6.2. When the EFG method is used, unfortunately, the singular matrix occurs. When the IEFG method is used, the relative error is 2.4229%. The numerical solutions and analytical ones are compared in Figure 16. It is shown that the numerical results are in good agreement with the analytical ones.

The third example [68] is

$$\Delta u+k^{2}u=(k^2-3{\pi }^2)\cos (\pi x_1)\sin (\pi x_2)\sin (\pi x_3).$$

(69)

The boundary conditions are

$$\frac{\partial u(0,x_2,x_3)}{\partial x_1}=\frac{\partial u(1,x_2,x_3)}{\partial x_1}=0,$$

(70)

$$u(x_1,0,x_3)=u(x_1,1,x_3)=0,$$

(71)

$$u(x_1,x_2,0)=u(x_1,x_2,1)=0.$$

(72)

The problem domain is $\Omega =[0,1]\times [0,1]\times [0,1]$, and

$$u=\cos (\pi x_1)\sin (\pi x_2)\sin (\pi x_3)$$

(73)

is the analytical solution.

The IEFG method is used to solved it. The wave number is selected as 100, and 19 × 19 × 19 regularly distributed nodes and 18 × 18 × 18 background integral cells are used, $\mathsf{\alpha }=1.9\times {10}^7$, d_max = 1.1. When using the EFG method to solve it, the same parameters are selected, and the relative errors of both methods are equal to 0.8646%. Figure 17, Figure 18 and Figure 19 show the comparison of the numerical solutions and the analytical ones. We can see that numerical solutions are in good agreement with the analytical ones. The CPU times of the IEFG method and the EFG method are 200.6 s and 208.1 s, respectively.

A similar computational accuracy can be obtained when using the two methods, but the higher computational speed can be obtained when using the IEFG method.

Similarly, if d_max = 1.0, when the EFG method is used, the singular matrix occurs and the final result cannot be obtained. However, when the IEFG method is selected, the relative error is 0.8648%. The numerical solutions and the analytical one are compared in Figure 20, where it is shown that the numerical results are in good agreement with the analytical ones.

5. Conclusions

In order to solve 3D Helmholtz equations efficiently, the IEFG method is proposed in this paper.

Some numerical examples are given in Section 4, and the convergence of the IEFG method is proven numerically. From these examples, we can see that the IEFG method in this paper can not only enhance the computational speed of the traditional EFG method, but also eliminate the phenomenon of the singular matrix.