Analyzing Three-Dimensional Laplace Equations Using the Dimension Coupling Method

Abstract

Due to the low computational efficiency of the Improved Element-Free Galerkin (IEFG) method, efficiently solving three-dimensional (3D) Laplace problems using meshless methods has been a longstanding research direction. In this study, we propose the Dimension Coupling Method (DCM) as a promising alternative approach to address this challenge. Based on the Dimensional Splitting Method (DSM), the DCM divides the 3D problem domain into a coupling of multiple two-dimensional (2D) problems which are handled via the IEFG method. We use the Finite Element Method (FEM) in the third direction to combine the 2D discretized equations, which has advantages over the Finite Difference Method (FDM) used in traditional methods. Our numerical verification demonstrates the DCM’s convergence and enhancement of computational speed without losing computational accuracy compared to the IEFG method. Therefore, this proposed method significantly reduces computational time and costs when solving 3D Laplace equations with natural or mixed boundary conditions in a dimensional splitting direction, and expands the applicability of the dimension splitting EFG method.

1. Introduction

In science and engineering fields, the meshless method has become an important tool in numerical methods for solving partial differential equations. Compared with the traditional Finite Element Method (FEM), the meshless method [1] is only based on node distribution, eliminating mesh constraints and allowing for higher-accuracy solutions to large deformation problems by establishing appropriate shape functions without requiring grid reconstruction.

The Element-Free Galerkin Method (EFGM) [2,3,4,5] is a noteworthy meshless method proposed by Belytschko et al. that utilizes the Moving Least-Squares (MLS) approximation to construct shape functions. The MLS approximate functions have been improved to accelerate computational speed, including the Improved Moving Least-Squares (IMLS) approximation [6], interpolating MLS approximation [7,8], and complex variable MLS approximation [9,10]. Using these methods to construct shape functions resulted in the presentation of the IEFG method [11,12,13,14], interpolating EFG method [15,16,17,18,19,20], and complex variable EFG method [10,21,22], respectively.

Due to the complexity of establishing trial functions, the meshless method has been known to be time-consuming when solving complex 3D mechanics problems. Therefore, finding ways to analyze complex 3D problems more efficiently has become a significant research direction in the field of numerical methods. Recently, researchers have made strides in this area, leading to the development of numerous dimensional split meshless methods, including the dimensional split complex variable EFG method [23,24,25,26], dimensional split EFG method [27,28,29,30], dimensional split reproducing kernel particle method [31,32,33,34], and interpolating dimensional split EFG method [35,36]. These hybrid meshless methods have demonstrated the capability to efficiently solve a wide range of 3D partial differential equations with smaller relative errors.

However, when natural or mixed boundary conditions occur in a dimensional splitting direction, traditional hybrid meshless methods require the 3D problems to be split into more layers to achieve higher accuracy. In [28], when a natural boundary condition existed in a split direction, Meng et al. had to choose 50 planes to obtain greater computational accuracy. In [33], the wave propagation problem with mixed boundary conditions was investigated. The splitting direction of the second numerical example had natural boundary conditions, so Peng and Cheng had to select 100 planes to achieve a smaller relative error of 0.62% with a computation time of 15.2 s. In contrast, if the splitting direction is selected as an essential boundary condition, only 10 layers are required, resulting in an error of 0.24% and a computation time of 1.1 s. Hence, it is worth noting that time consumption remains a concern when using the FDM in the dimensional split direction to address natural boundary conditions.

To address the above-mentioned issue of time consumption and explore a novel approach to enhance the convenience and efficiency of applying natural or mixed boundary conditions, we propose a modification that replaces the FDM with the FEM in the dimensional splitting direction. The FEM [37,38] is widely recognized for its advantages over the FDM in numerical simulations. Therefore, compared with the Dimensional Splitting EFG Method presented by Meng et al. [28], our proposed method can reduce CPU time when solving partial differential equations with mixed or natural boundary conditions in the splitting direction.

Due to the low computational efficiency of the EFG for solving 3D problems, researchers have been exploring alternative efficient meshless methods. The Dimension Coupling Method (DCM) [39] as a potential approach for solving the 3D Laplace equation is presented in this study. By introducing the Dimensional Split Method (DSM), the problem domain of the 3D Laplace equation is divided into a coupling of multiple 2D problems, and the IEFG method is employed to handle 2D forms. Two-dimensional discrete equations are combined by the FEM in the third direction and the final algebraic equation of the original 3D Laplace’s equation is derived.

The error formula of the DCM is given in numerical examples, and we discuss the influence of meshes of the FEM in the third direction, weight function, other parameters, and node distribution in 2D domains on precision. Furthermore, the convergence of the proposed method in this study is numerically verified. Through three numerical examples, it is demonstrated that the DCM for 3D Laplace equations can significantly improve computational efficiency without reducing computational accuracy.

2. IMLS Approximation

For a point $\mathit{x}$, the approximation of the corresponding function $\mathit{u}(\mathit{x})$ can be written as

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

(1)

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

(2)

$\tilde{\mathit{\Phi }}$ is the shape function and its form is

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

(3)

${\mathit{p}}^{\mathrm{T}}(\mathit{x})$ is the vector of the basis function,

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

(4)

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

(5)

$$\mathit{P}=\left[\begin{array}{cccc}{p}_1({\mathit{x}}_1)& p_2({\mathit{x}}_1)& \cdots & p_m({\mathit{x}}_1)\\ p_1({\mathit{x}}_2)& p_2({\mathit{x}}_2)& \cdots & p_m({\mathit{x}}_2)\\ ⋮& ⋮& \ddots & ⋮\\ p_1({\mathit{x}}_n)& p_2({\mathit{x}}_n)& \cdots & p_m({\mathit{x}}_n)\end{array}\right],$$

(6)

$$\mathit{W}=\left[\begin{array}{cccc}w(\mathit{x}-{\mathit{x}}_1)& 0& \cdots & 0\\ 0& w(\mathit{x}-{\mathit{x}}_2)& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & w(\mathit{x}-{\mathit{x}}_n)\end{array}\right],$$

(7)

and $w(\mathit{x}-{\mathit{x}}_I)$ is the weighting function.

Equations (1)–(7) are the IMLS approximation [6].

3. Dimension Coupling Method for 3D Laplace Equations

When solving 3D Laplace equations using the IEFG method, the numerical solution obtained has low computational speed due to the complexity of shape functions compared to 2D problems.

In this paper, we propose using the DCM to solve this issue. The 3D problem can be transformed into multiple 2D problems as shown in Figure 1, which are then discretized using the IEFG method. In the splitting direction, 2D discretized equations are coupled via the FEM.

The formula of the DCM for 3D Laplace equations is derived in this section. Considering the governing equation [23]

$$\Delta u=0, (x_1,x_2,x_3)\in \Omega ;$$

(8)

with boundary conditions

$$u=\overline{u}(x_1,x_2,x_3), (x_1,x_2,x_3)\in {\Gamma }_u,$$

(9)

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

(10)

$\overline{q}$ and $\overline{u}$ are given values, and $n_i$ is the unit outward normal to the boundary $\Gamma$ in direction $x_i$, $\Gamma ={\Gamma }_u\cup {\Gamma }_q$, ${\Gamma }_u\cap {\Gamma }_q=\varnothing$.

The governing equation of 2D form in the k-th layer based on the DCM is

$$\frac{{\partial }^2{u}^{(k)}}{\partial x_1^2}+\frac{{\partial }^2{u}^{(k)}}{\partial x_2^2}=-\frac{{\partial }^2{u}^{(k)}}{\partial x_3^2}, (x_1,x_2)\in {\Omega }^{(k)}, x_3=x_3^{(k)},$$

(11)

where ${\Omega }^{(k)}$ is the 2D domain of the k-th layer of $\Omega$, and

$$\Omega ={\displaystyle \underset{k=1}{\overset{L}{\cup }}\left\{{\Omega }^{(k)}\times [x_3^{(k)},x_3^{(k+1)}]\right\}}\cup {\Omega }^{(L+1)},$$

(12)

$$u^{(k)}=u(x_1,x_2,x_3^{(k)}),$$

(13)

with boundary conditions

$$u^{(k)}={\overline{u}}^{(k)}(x_1,x_2)=\overline{u}(x_1,x_2,x_3^{\left(k\right)}), (x_1,x_2)\in {\Gamma }_u^{(k)},$$

(14)

$$q^{(k)}={\overline{q}}^{(k)}(x_1,x_2)=\overline{q}(x_1,x_2,x_3^{\left(k\right)}), (x_1,x_2)\in {\Gamma }_q^{(k)},$$

(15)

where ${\Gamma }_u^{(k)}$ and ${\Gamma }_q^{(k)}$ are essential and natural boundaries. ${\Gamma }^{(k)}={\Gamma }_u^{(k)}\cup {\Gamma }_q^{(k)}$, and ${\Gamma }_u^{(k)}\cap {\Gamma }_q^{(k)}=\varnothing$.

Equations (11), (14) and (15) are then analyzed using the IEFG method. The discretization of the second-order partial derivative in the splitting direction is performed using the FEM, then we obtain the discretized equations.

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

(16)

Equation (16) is the equivalent functional, and the penalty method is selected for exerting boundary conditions, hence we can obtain that the modified functional of each 2D form is

$${\mathsf{\Pi }}^{\ast }=\mathsf{\Pi }+\frac{\alpha }{2}{\displaystyle {\int }_{{\Gamma }_u^{(k)}}(u-\overline{u})(u-\overline{u})\mathrm{d}{\Gamma }^{(k)}}.$$

(17)

Let

$$\delta {\mathsf{\Pi }}^{*}=0,$$

(18)

hence we have

$$\begin{gathered} {\displaystyle {\int }_{{\Omega }^{(k)}}\delta u\cdot \frac{{\partial }^{2}u}{\partial x_3^2}\mathrm{d}{\Omega }^{(k)}}-{\displaystyle {\int }_{{\Omega }^{(k)}}\delta {(\mathit{L}u)}^{\mathrm{T}}\cdot (\mathit{L}u)\mathrm{d}{\Omega }^{(k)}}-{\displaystyle {\int }_{{\Gamma }_q^{(k)}}\delta u\cdot \overline{q}\mathrm{d}{\Gamma }^{(k)}} \\ +\alpha {\displaystyle {\int }_{{\Gamma }_u^{(k)}}\delta u\cdot u\mathrm{d}{\Gamma }^{(k)}}-\alpha {\displaystyle {\int }_{{\Gamma }_u^{(k)}}\delta u\cdot \overline{u}\mathrm{d}{\Gamma }^{(k)}}=0, \end{gathered}$$

(19)

where

$$\mathit{L}(\cdot )=\left[\begin{array}{c}\frac{\partial }{\partial {\mathit{x}}_1}\\ \frac{\partial }{\partial {\mathit{x}}_2}\end{array}\right](\cdot ).$$

(20)

In the 2D domain ${\Omega }^{(k)}$, we select $M$ nodes ${\mathit{x}}_I$; thus, we can obtain the following form:

$$u({\mathit{x}}_I^{\left(k\right)})=u({\mathit{x}}_I^{\left(k\right)},x_3^{\left(k\right)})=u_I.$$

(21)

From Section 2, the expression of the approximate function of Equation (21) is

$$u(\mathit{x},x_3^{\left(k\right)})=\tilde{\mathit{\Phi }}\mathit{u}={\displaystyle \sum _{I=1}^n{\tilde{\mathrm{\Phi }}}_I(\mathit{x})u_I},$$

(22)

with

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

(23)

Thus,

$$\frac{{\partial }^{2}u(\mathit{x},x_3^{\left(k\right)})}{\partial x_3^2}=\frac{{\partial }^2}{\partial x_3^2}{\displaystyle \sum _{I=1}^n{\tilde{\mathrm{\Phi }}}_{I}u({\mathit{x}}_I,x_3^{\left(k\right)})}={\displaystyle \sum _{I=1}^n{\tilde{\mathrm{\Phi }}}_I\frac{{\partial }^2{u}_I}{\partial x_3^2}}=\tilde{\mathit{\Phi }}(\mathit{x})\mathit{u}",$$

(24)

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

(25)

where

$$\mathit{u}"={\left(\frac{{\partial }^{2}u({\mathit{x}}_1,x_3^{\left(k\right)})}{\partial x_3^2},\frac{{\partial }^{2}u({\mathit{x}}_2,x_3^{\left(k\right)})}{\partial x_3^2},\cdots ,\frac{{\partial }^{2}u({\mathit{x}}_n,x_3^{\left(k\right)})}{\partial x_3^2}\right)}^{\mathrm{T}},$$

(26)

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

(27)

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

(28)

Substituting Equations (22), (24) and (25) into Equation (19),

$$\begin{gathered} {\displaystyle {\int }_{{\Omega }^{(k)}}\delta {[\tilde{\mathit{\Phi }}\mathit{u}]}^{\mathrm{T}}\cdot [\tilde{\mathit{\Phi }}{\mathit{u}}^{\mathbf{″}}]\mathrm{d}{\Omega }^{\left(k\right)}}-{\displaystyle {\int }_{{\Gamma }_q^{(k)}}\delta {[\tilde{\mathit{\Phi }}\mathit{u}]}^{\mathrm{T}}\cdot \overline{q}\mathrm{d}{\Gamma }^{\left(k\right)}}-{\displaystyle {\int }_{{\Omega }^{\left(k\right)}}\delta {[\mathit{B}\mathit{u}]}^{\mathrm{T}}[\mathit{B}\mathit{u}]\mathrm{d}{\Omega }^{\left(k\right)}} \\ +\alpha {\displaystyle {\int }_{{\Gamma }_{\mathit{u}}^{(k)}}\delta {[\tilde{\mathit{\Phi }}\mathit{u}]}^{\mathrm{T}}\cdot [\tilde{\mathit{\Phi }}\mathit{u}]\mathrm{d}{\Gamma }^{(k)}}-\alpha {\displaystyle {\int }_{{\Gamma }_{\mathit{u}}^{(k)}}\delta {[\tilde{\mathit{\Phi }}\mathit{u}]}^{\mathrm{T}}\cdot \overline{\mathit{u}}\mathrm{d}{\Gamma }^{(k)}}=0. \end{gathered}$$

(29)

Next, we write Equation (29) in its matrix form

$${\displaystyle {\int }_{{\Omega }^{(k)}}\delta {[\tilde{\mathit{\Phi }}\mathit{u}]}^{\mathrm{T}}\cdot [\tilde{\mathit{\Phi }}{\mathit{u}}^{″}]\mathrm{d}{\Omega }^{(k)}=\delta {\mathit{u}}^{\mathrm{T}}\cdot [{\displaystyle {\int }_{{\Omega }^{(k)}}{\tilde{\mathit{\Phi }}}^{\mathrm{T}}\tilde{\mathit{\Phi }}\mathrm{d}{\Omega }^{(k)}]\cdot {\mathit{u}}^{″}}}=\delta {\mathit{u}}^{\mathrm{T}}\cdot \mathit{C}\cdot {\mathit{u}}^{\mathbf{″}},$$

(30)

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

(31)

$${\displaystyle {\int }_{{\Gamma }_q^{(k)}}\delta {[\tilde{\mathit{\Phi }}\mathit{u}]}^{\mathrm{T}}\cdot \overline{q}\mathrm{d}{\Gamma }^{\left(k\right)}}=\delta {\mathit{u}}^{\mathrm{T}}\cdot {\displaystyle {\int }_{{\Gamma }_q^{(k)}}{\tilde{\mathit{\Phi }}}^{\mathrm{T}}\overline{q}\mathrm{d}{\Gamma }^{\left(k\right)}}=\delta {\mathit{u}}^{\mathrm{T}}\cdot {\mathit{f}}_q,$$

(32)

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

(33)

$$\alpha {\displaystyle {\int }_{{\Gamma }_{\mathit{u}}^{(k)}}\delta {[\tilde{\mathit{\Phi }}\mathit{u}]}^{\mathrm{T}}\cdot \overline{u}\mathrm{d}{\Gamma }^{(k)}}=\delta u^{\mathrm{T}}\cdot \alpha {\displaystyle {\int }_{{\Gamma }_{\mathit{u}}^{(k)}}{\tilde{\mathit{\Phi }}}^{\mathrm{T}}}\overline{\mathit{u}}\mathrm{d}{\Gamma }^{(k)}=\delta {\mathit{u}}^{\mathrm{T}}\cdot {\mathit{f}}_{\alpha },$$

(34)

where

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

(35)

$$\mathit{C}={\displaystyle {\int }_{{\Omega }^{\left(k\right)}}{\tilde{\mathit{\Phi }}}^{\mathrm{T}}\tilde{\mathit{\Phi }}\mathrm{d}{\Omega }^{\left(k\right)}},$$

(36)

$${\mathit{K}}_{\alpha }=\alpha {\displaystyle {\int }_{{\Gamma }_{\mathit{u}}^{(k)}}{\tilde{\mathit{\Phi }}}^{\mathrm{T}}\tilde{\mathit{\Phi }}\mathrm{d}{\Gamma }^{(k)}},$$

(37)

$${\mathit{f}}_q={\displaystyle {\int }_{{\Gamma }_q^{(k)}}{\tilde{\mathit{\Phi }}}^{\mathrm{T}}\overline{q}\mathrm{d}{\Gamma }^{\left(k\right)}},$$

(38)

$${\mathit{f}}_{\alpha }=\alpha {\displaystyle {\int }_{{\Gamma }_{\mathit{u}}^{(k)}}{\tilde{\mathit{\Phi }}}^{\mathrm{T}}\overline{u}\mathrm{d}{\Gamma }^{(k)}}.$$

(39)

Substituting Equations (30)–(34) into Equation (29),

$$\delta {\mathit{u}}^{\mathrm{T}}\cdot (\mathit{C}{\mathit{u}}^{\mathbf{″}}+{\mathit{K}}_{\alpha }\mathit{u}-\mathit{K}\mathit{u}-{\mathit{f}}_{\alpha }-{\mathit{f}}_q)=0.$$

(40)

Let

$$\mathit{F}={\mathit{f}}_q+{\mathit{f}}_{\alpha },$$

(41)

$$\widehat{\mathit{K}}={\mathit{K}}_{\alpha }-\mathit{K}.$$

(42)

Hence, Equation (40) can be transformed as

$$\mathit{C}{\mathit{u}}^{\mathbf{″}}+\widehat{\mathbf{K}}\mathit{u}=\mathit{F}.$$

(43)

Let

$$\mathit{u}(x_3^{(2)})={\mathit{u}}^{(2)},$$

(44)

$$\mathit{u}(x_3^{(3)})={\mathit{u}}^{(3)},$$

(45)

$$⋮$$

$$\mathit{u}(x_3^{(L)})={\mathit{u}}^{(L)},$$

(46)

suppose $x_3\in [a,c]$, thus we have

$$\mathit{C}{\mathit{u}}^{\prime }(c)v(x_3^{(L+1)})-\mathit{C}{\mathit{u}}^{\prime }(a)v(x_3^{(1)})-\mathit{C}{\displaystyle {\int }_a^c{\mathit{u}}^{\prime }{v}^{\prime }\mathrm{d}{x}_3}+\widehat{\mathit{K}}{\displaystyle {\int }_a^c\mathit{u}v\mathrm{d}{x}_3}={\displaystyle {\int }_a^c\mathit{F}v\mathrm{d}{x}_3},$$

(47)

the test function $v={\phi }_i$ ($i=1,2,\cdots ,N+1$) is selected with the shape function based on piecewise linear interpolation functions, thus $v(x_3^{(1)})=0$ and $v(x_3^{(L+1)})=0$ when $x_3=x_3^{(2)},x_3^{(3)},\cdots ,x_3^{(L-1)},x_3^{(L)}$. Hence, Equation (47) is changed to the following form

$$\widehat{\mathit{K}}{\displaystyle \sum _{k=1}^{L+1}{\mathit{u}}^{(k)}}{\displaystyle {\int }_a^c{\phi }_k{\phi }_i\mathrm{d}{x}_3}-\mathit{C}{\displaystyle \sum _{k=1}^{L+1}{\mathit{u}}^{(k)}}{\displaystyle {\int }_a^c{{\phi }^{\prime }}_k{{\phi }^{\prime }}_i\mathrm{d}{x}_3}={\displaystyle {\int }_a^c{\mathit{F}}^{(k)}{\phi }_i\mathrm{d}{x}_3}, i=2,3,\cdots ,N.$$

(48)

Let

$${\mathit{H}}_{ik}=\left({\displaystyle \sum _{k=1}^{L+1}{\displaystyle {\int }_a^c{\phi }_k{\phi }_i\mathrm{d}{x}_3}}\right)\widehat{\mathit{K}}-\left({\displaystyle \sum _{k=1}^{L+1}{\displaystyle {\int }_a^c{{\phi }^{\prime }}_k{{\phi }^{\prime }}_i\mathrm{d}{x}_3}}\right)\mathit{C}, i=2,3,\cdots ,N,$$

(49)

$${\mathit{W}}_i={\displaystyle {\int }_a^c{\mathit{F}}^{(k)}{\phi }_i\mathrm{d}{x}_3}, i=1,2,\cdots ,N+1.$$

(50)

Therefore, Equation (48) is transformed as

$${\mathit{H}}_{ik}{\mathit{u}}^{(k)}={\mathit{W}}_i, i=2,3,\cdots ,N.$$

(51)

If the Dirichlet boundary conditions are known,

$${\mathit{u}}^{(L+1)}=\mathit{u}(c),$$

(52)

$${\mathit{u}}^{(1)}=\mathit{u}(a).$$

(53)

Hence, Equation (47) is written in the following form:

$$\widehat{\mathit{E}}{\mathit{u}}^{(1)}=\mathit{u}(a), x_3=x_3^{(1)};$$

(54)

$${\mathit{H}}_{21}{\mathit{u}}^{(1)}+{\mathit{H}}_{22}{\mathit{u}}^{(2)}+{\mathit{H}}_{22}{\mathit{u}}^{(3)}={\mathit{W}}_2, x_3=x_3^{(2)};$$

(55)

$${\mathit{H}}_{32}{\mathit{u}}^{(2)}+{\mathit{H}}_{33}{\mathit{u}}^{(3)}+{\mathit{H}}_{34}{\mathit{u}}^{(4)}={\mathit{W}}_3, x_3=x_3^{(3)};$$

(56)

$$⋮$$

$${\mathit{H}}_{L,L-1}{\mathit{u}}^{(L-1)}+{\mathit{H}}_{L,L}{\mathit{u}}^{(L)}+{\mathit{H}}_{L,L+1}{\mathit{u}}^{(L+1)}={\mathit{W}}_L, x_3=x_3^{(L)};$$

(57)

$$\widehat{\mathit{E}}{\mathit{u}}^{(L+1)}=\mathit{u}(c), x_3=x_3^{(L+1)};$$

(58)

Let

$$\mathit{H}=\left[\begin{array}{cccccc}\widehat{\mathit{E}}& & & & & \\ {\mathit{H}}_{21}& {\mathit{H}}_{22}& {\mathit{H}}_{23}& & & \\ & {\mathit{H}}_{32}& {\mathit{H}}_{33}& {\mathit{H}}_{34}& & \\ & & & \ddots & & \\ & & & {\mathit{H}}_{L,L-1}& {\mathit{H}}_{L,L}& {\mathit{H}}_{L,L+1}\\ & & & & & \widehat{\mathit{E}}\end{array}\right],$$

(59)

$$\mathit{W}={\left({\left(\mathit{u}(a)\right)}^{\mathrm{T}},{\mathit{W}}_2{}^{\mathrm{T}},\cdots ,{\left(\mathit{u}(c)\right)}^{\mathrm{T}}\right)}^{\mathrm{T}},$$

(60)

$$\mathit{U}={\left({\mathit{u}}^{(1)}{}^{\mathrm{T}},{\mathit{u}}^{(2)}{}^{\mathrm{T}},\cdots ,{\mathit{u}}^{(L+1)}{}^{\mathrm{T}}\right)}^{\mathrm{T}},$$

(61)

thus the final linear equation of the 3D Laplace equation is

$$\mathit{H}U=\mathit{W}.$$

(62)

If mixed boundary conditions are known,

$${\mathit{u}}^{\prime }(x_3^{(L+1)})={\mathit{u}}^{\prime }(c),$$

(63)

$$\mathit{u}(x_3^{(1)})=\mathit{u}(a),$$

(64)

Equations (58), (59), and (60) can be changed as follows:

$${\mathit{H}}_{L+1,L}{\mathit{u}}^{(L)}+{\mathit{H}}_{L+1,L+1}{\mathit{u}}^{(L+1)}={\mathit{W}}_{L+1}-\mathit{C}{\mathit{u}}^{\prime }(c), x_3=x_3^{(L+1)},$$

(65)

$$\mathit{H}=\left[\begin{array}{cccccc}\widehat{\mathit{E}}& & & & & \\ {\mathit{H}}_{21}& {\mathit{H}}_{22}& {\mathit{H}}_{23}& & & \\ & {\mathit{H}}_{32}& {\mathit{H}}_{33}& {\mathit{H}}_{34}& & \\ & & & \ddots & & \\ & & & {\mathit{H}}_{L,L-1}& {\mathit{H}}_{L,L}& {\mathit{H}}_{L,L+1}\\ & & & & {\mathit{H}}_{L+1,L}& {\mathit{H}}_{L+1,L+1}\end{array}\right],$$

(66)

and

$$\mathit{W}={\left({\left(\mathit{u}(a)\right)}^{\mathrm{T}},{\mathit{W}}_2{}^{\mathrm{T}},{\mathit{W}}_3{}^{\mathrm{T}},\cdots ,{\mathit{W}}_L{}^{\mathrm{T}},{\left({\mathit{W}}_{L+1}-\mathit{C}{\mathit{u}}^{\prime }(c)\right)}^{\mathrm{T}}\right)}^{\mathrm{T}},$$

(67)

respectively.

Equations (8)–(67) are the DCM for 3D Laplace equations.

4. Numerical Examples

In this section, four numerical examples are calculated using the DCM.

The formula of error is

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

(68)

We employ the IEFG method based on some distributed nodes and use linear basis functions to construct trial functions in the 2D computational domain. Additionally, 4 × 4 Gaussian points are used in each cell with two Gaussian points used in the dimensional split direction in each mesh.

The first example [40,41] is

$${\nabla }^{2}u=0,$$

(69)

and the problem domain is Ω = [0, 1]³,

$$u=\sin (\mathrm{\pi }{x}_2)\sin (\mathrm{\pi }{x}_3), (x_1=0),$$

(70)

$$u=2\sin (\mathrm{\pi }{x}_2)\sin (\mathrm{\pi }{x}_3), (x_1=1),$$

(71)

$$u=0, (x_2=x_3=0,x_2=x_3=1),$$

(72)

Equations (70)–(72) are boundary conditions.

$$u=[2\sin \mathrm{h}(\mathrm{\pi }\sqrt{2}{x}_1)+\sin \mathrm{h}(\mathrm{\pi }\sqrt{2}(1-x_1)]\frac{\sin (\mathrm{\pi }{x}_2)\sin (\mathrm{\pi }{x}_3)}{\sin \mathrm{h}(\mathrm{\pi }\sqrt{2})}.$$

(73)

Equation (73) is the analytical solution.

In this example, we investigate the convergence of the DCM before obtaining the greater computational accuracy of the numerical solution.

(1) Weight function.

The impact of weight functions on the relative error of the numerical solutions is discussed. In the case of using the cubic spline function, we choose 19 × 19 regular nodes and 18 × 18 integral cells. In the x₃ split direction, we employ the FEM with the mesh number of 10, with d_max = 1.48 and α = 2.1 × 10⁶. The resulting relative error of our proposed DCM is 0.1435%. For the case of using the quartic spline function, we maintain the same background integral grid and node distributions. In the x₃ split direction, the mesh number of the FEM is set to 18, with d_max = 1.35 and α = 3.9 × 10⁶. As a result, the relative error is 0.1516%. These results indicate that the relative error of the quartic spline function is slightly larger than that of the cubic spline function. Consequently, the cubic spline function is used in the following analysis.

(2) Scale parameter.

We choose 19 × 19 regular nodes and 18 × 18 integral cells. The mesh number of the FEM in the x₃ split direction is 18 and we set α = 2.1 × 10⁶. Figure 2 shows the relationship between the scale parameter d_max and the error. From Figure 2, we can conclude that when d_max = 1.4~1.5, the computational accuracy of the numerical solution is higher.

(3) Penalty factor.

We select 19 × 19 regular nodes and 18 × 18 integral cells. The mesh number of the FEM in the x₃ split direction is 18 with d_max = 1.48. The computational accuracy of the numerical solution is higher when α = 2.1 × 10⁶ in Figure 3.

Different node distributions in the k-th plane and meshes in the x₃ direction are analyzed in the following, respectively.

(4) Node distribution.

We select d_max = 1.48, α = 2.1 × 10⁶, and 18 meshes in the x₃ split direction. The relationship between the relative error and the number of nodes is shown in Figure 4. We can see that as the number of nodes increases, the relative error tends to decrease.

(5) Mesh number.

We use 19 × 19 regular nodes and 18 × 18 integral cells in each 2D domain, setting d_max = 1.48 and α = 2.1 × 10⁶. The relationship between the relative error and the mesh number is shown in Figure 5. We can see that as the number of meshes increases in the x₃ splitting direction, the computational accuracy steadily improves.

As discussed above, the DCM for the 3D Laplace equation is convergent.

Using the proposed DCM in this paper to solve Example 1, we select 19 × 19 regular nodes and 18 × 18 integral cells. In the x₃ splitting direction, we employ the FEM with a mesh number of 18, d_max = 1.48, and α = 2.1 × 10⁶. As a result, the relative error is 0.1435% and the computational time is 1.5 s.

In contrast, when applying the IEFG method to solve Example 1, we choose 19 × 19 × 19 regular nodes, 18 × 18 × 18 integral cells, and the cubic spline weight function. By selecting d_max = 1.32 and α = 1.5 × 10³, the resulting relative error is 0.2134% with a computational time of 63.9 s.

The comparison of the computational accuracy and time of the DCM and the IEFG method is shown in

Table 1. The comparison of computational accuracy and time of the DCM and the IEFG.

Method	Regular Nodes	Relative Error	CPU Time (s)
DCM	19 × 19 × 19	0.1435%	1.5
IEFG	19 × 19 × 19	0.2134%	63.9

.

A comparison is performed between the numerical solutions obtained using the DCM and the IEFG method, along with the exact solutions depicted in Figure 6, Figure 7 and Figure 8. It can be observed that the results of these two methods accord well with the analytical solutions.

We can see that the DCM not only exhibits superior calculation accuracy but also significantly enhances the computational efficiency of the IEFG method in the numerical results of Figure 6, Figure 7 and Figure 8.

The second example [1] is

$${\nabla }^{2}u=0,$$

(74)

this problem domain is Ω = [0, 1]³, and

$$u_{,1}{|}_{x_1=0}=u_{,1}{|}_{x_1=1}=u_{,2}{|}_{x_2=0}=u_{,2}{|}_{x_2=1}=0,$$

(75)

$$\mathit{u}{|}_{x_3=0}=-\frac{\cos (\pi x_1)\cos (\pi x_2)}{\sqrt{2}\pi \tanh (\sqrt{2}\pi )},$$

(76)

$$u_{,3}{|}_{x_2=1}=0.$$

(77)

Equations (75)–(77) are boundary conditions, and

$$u=\cos (\pi x_1)\cos (\pi x_2)\left[\frac{\sinh (\sqrt{2}\pi x_3)}{\sqrt{2}\pi }-\frac{\cosh (\sqrt{2}\pi x_3)}{\sqrt{2}\pi \tanh (\sqrt{2}\pi )}\right]$$

(78)

is the analytical solution.

In Example 2, 15 × 15 regular nodes and 14 × 14 integral cells are selected in each 2D domain. We employ the FEM in the x₃ split direction with a mesh number of 14 and d_max = 1.34. The resulting relative error is 0.3796% and the computational time is 0.6 s.

In contrast, when analyzing Example 2 using the IEFG method, we choose the cubic spline weight function along with 15 × 15 × 15 regular nodes and 14 × 14 × 14 integral cells. Moreover, we set d_max = 1.0 and α = 5.8 × 10². The corresponding relative error is 0.3745% and the computational time is 12.9 s.

The comparison of the computational accuracy and time of the DCM and the IEFG method is shown in

Table 2. The comparison of computational accuracy and time of the DCM and the IEFG.

Method	Regular Nodes	Relative Error	CPU Time (s)
DCM	15 × 15 × 15	0.3796%	0.6
IEFG	15 × 15 × 15	0.3745%	12.9

.

The numerical solution of the DCM is compared with that of the IEFG method and the exact solutions in Figure 9, Figure 10 and Figure 11. It can be observed that the results of these two numerical methods accord well with the exact ones.

The results demonstrate that the DCM exhibits higher efficiency compared to the IEFG method, despite both methods yielding similar errors. Furthermore, the DCM has the advantage of efficiently handling mixed boundary conditions without requiring excessive layers in the dimensional splitting direction.

The third example is

$${\nabla }^{2}u=0,$$

(79)

this problem domain is Ω = [0, 1]³, and

$$u=x_2^2+x_3^2, (x_1=0),$$

(80)

$$u=-2+x_2^2+x_3^2, (x_1=1)$$

(81)

$$u=-2x_1^2+x_3^2, (x_2=0),$$

(82)

$$u=-2x_1^2+1+x_3^2, (x_2=1),$$

(83)

$$u=-2x_1^2+x_2^2, (x_3=0),$$

(84)

$$u=-2x_1^2+x_2^2+1, (x_3=1).$$

(85)

Equations (80)–(85) are boundary conditions.

$$u=-2x_1^2+x_2^2+x_3^2$$

(86)

is the analytical solution.

Then, we apply the DCM to solve Example 3 and analyze three situations in which the FEM is applied in different directions.

(1) The FEM applied in the x₁ direction.

The mesh number is 18 with d_max = 1.51 and α = 2.5 × 10⁴. We select 11 × 11 regular nodes and 10 × 10 integral cells in each 2D domain. The resulting relative error is 0.1025% and the computational time is 0.3 s.

(2) The FEM applied in the x₂ or x₃ direction.

The mesh number is 10 with d_max = 1.32 and α = 4.2 × 10⁴. We select 11 × 11 regular nodes and 10 × 10 integral cells in each 2D domain. The resulting relative error is 0.1934% and the computational time is 0.2 s.

Furthermore, we employ the IEFG method for analysis. In this case, we choose 11 × 11 × 11 regular nodes, 10 × 10 × 10 integral cells, and the cubic spline weight function. The parameters used are d_max = 1.33 and α = 1.4 × 10³. The resulting error is 0.1544% with a calculation time of 7.1 s.

The comparison of the computational accuracy and time of the DCM and the IEFG method is shown in

Table 3. The comparison of computational accuracy and time of the DCM and the IEFG.

Method	Regular Nodes	Relative Error	CPU Time (s)
DCM (split in x1 direction)	11 × 11 × 11	0.1025%	0.3
DCM (split in x2 or x3 direction)	11 × 11 × 11	0.1934%	0.2
IEFG	11 × 11 × 11	0.1544%	7.1

.

When the FEM is used in the x₁-axis splitting direction, it can be observed that the DCM can obtain a smaller error. Comparing the numerical solution of the DCM with those of the IEFG method and the exact ones in Figure 12, Figure 13 and Figure 14, the results of these two methods agree well with the exact ones.

The fourth example [27,31] is in an irregular-shaped domain selected as

$${\nabla }^{2}u=0, (r\in [1,2],\hspace{0.33em}\theta \in [0,\pi ],\hspace{0.33em}{\mathit{x}}_3\in [0,1]),$$

(87)

The boundary conditions are

$$u(1,\theta ,x_3)=\sin \theta +x_3,$$

(88)

$$u(2,\theta ,x_3)=x_3,$$

(89)

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

(90)

$$u(r,\pi ,x_3)=x_3,$$

(91)

$$u(r,\theta ,0)=\frac{4}{3}\left(\frac{1}{r}-\frac{r}{4}\right)\sin \theta ,$$

(92)

$$u(r,\theta ,1)=\frac{4}{3}\left(\frac{1}{r}-\frac{r}{4}\right)\sin \theta +1.$$

(93)

The analytical solution of this problem is

$$u(r,\theta ,x_3)=\frac{4}{3}\left(\frac{1}{r}-\frac{r}{4}\right)\sin \theta +x_3.$$

(94)

For this example, the FEM is applied in the x₃ split direction with a mesh number of 10. A total of 9 × 31 nodes are distributed in a half-torus domain for the 2D problem. Among these nodes, 9 nodes are positioned along the radial direction r, and 31 nodes are uniformly distributed along the angle axis θ as shown in Figure 15 given in [27]. This means that the integral node distribution is 9 × 31 × 11 with d_max = 1.0 and α = 0.12. The resulting relative error is 0.1510% and the computational time is 1.3 s.

For this example, in order to verify the effectiveness and correctness of the DCM, the results of the DCM are compared with those of the IEFG method given in [27] using 9 × 31 × 11 nodes, 8 × 30 × 10 integral cells, and the cubic spline weight function. When d_max = 1.2 and α = 1.0 × 10⁴ are selected, the relative error and the computational time of the IEFG method are 0.2377% and 132.1 s, respectively.

The comparison of the computational accuracy and time of the present DCM and the IEFG method given in [27] is shown in

Table 4. The comparison of computational accuracy and time of the present DCM and the IEFG method given in [27].

Method	Nodes	Relative Error	CPU Time (s)
DCM	9 × 31 × 11	0.1510%	1.3
IEFG [27]	9 × 31 × 11	0.2377%	132.1

.

Comparing the numerical solutions of the present DCM with those of the IEFG method given in [27] and the exact ones in Figure 16, Figure 17 and Figure 18, the results of the DCM agree well with the exact ones.

5. Discussions

In this study, a new dimensional split EFG method called the DCM is studied. The proposed method is used to solve 3D Laplace equations.

The influence of meshes of the FEM in the third direction, the weight function, other parameters, and the node distribution in 2D domains on precision is discussed. Additionally, the convergence of the proposed method in this study is numerically verified. Four numerical examples are provided to illustrate that the new method significantly enhances computational efficiency without reducing computational accuracy compared to the IEFG method.

6. Conclusions

Due to the lower computational efficiency of the EFG and IEFG methods in solving 3D Laplace equations, we propose a new dimension splitting EFG method named the DCM, which efficiently solves the 3D Laplace equation. Moreover, it conveniently and efficiently handles mixed boundary conditions by using the FEM instead of the FDM in the splitting direction. Consequently, this study provides an efficient and potential numerical method that can reduce computational time and costs without sacrificing accuracy to solve 3D complex engineering problems, including but not limited to crack propagation, high-speed collisions and explosions, metal press forming, and large deformation problems in gradient materials. Because of the large-scale calculation of such problems, research into the DCM combined with parallel algorithms needs to be investigated in the future, which has not been addressed before.

However, the DCM still has some limitations. First, the MATLAB program for the DCM is more complex than that for the IEFG method. Moreover, challenges may arise when using the DCM presented in this study combined with parallel algorithms. Therefore, this new method still needs further improvement in future research.