Analyzing three-dimensional wave propagation with the hybrid reproducing kernel particle method based on the dimension splitting method

Abstract

By introducing the dimension splitting method into the reproducing kernel particle method (RKPM), a hybrid reproducing kernel particle method (HRKPM) for solving three-dimensional (3D) wave propagation problems is presented in this paper. Compared with the RKPM of 3D problems, the HRKPM needs only solving a set of two-dimensional (2D) problems in some subdomains, rather than solving a 3D problem in the 3D problem domain. The shape functions of 2D problems are much simpler than those of 3D problems, which results in that the HRKPM can save the CPU time greatly. Four numerical examples are selected to verify the validity and advantages of the proposed method. In addition, the error analysis and convergence of the proposed method are investigated. From the numerical results we can know that the HRKPM has higher computational efficiency than the RKPM and the element-free Galerkin method.

1 Introduction

Wave propagation exists in engineering fields, thus it is necessary to find efficient techniques for solving wave equations. Until now, the most commonly used approaches for these problems are the finite difference method [1,2,3,4] and the finite element method [5,6,7].

Meshless technique is a promising method, and its approximation function is based on a set of nodes [8,9,10,11]. Compared with the conventional numerical methods that need meshing the problem domain or boundary, the meshless methods do not require constant mesh reconstruction, so they will not cause the disadvantages of time consuming or sometimes difficult to use when dealing with some complicated problems [12,13,14,15]. Thus, the meshless methods are available to solve wave propagation problems. As one of the most widely used meshless methods, the element-free Galerkin (EFG) method which based on the moving least-squares (MLS) approximation has been used to solve the hyperbolic equation [16,17,18]. By orthogonalizing the basis functions, Zhang et al. employed the improved element-free Galerkin (IEFG) method for the solution of 3D wave equations [19]. Shivanian presented the meshless local Petrov–Galerkin (MLPG) method for solving 3D nonlinear wave equations [20]. Shivanian and Shaban studied the pseudospectral meshless radial point interpolation (PSMRPI) method which combined meshless methods with spectral collocation techniques to solve 3D wave equations [21]. Liew and Cheng presented the mesh-free kp-Ritz method for solving 3D wave equations [22]. Dehghan and Salehi also studied some special cases of the wave equations using the meshless technique [23, 24].

The reproducing kernel particle method (RKPM) used in this paper is one of the most important meshless methods which was developed from the smoothed particle hydrodynamics (SPH) method [25]. Cheng et al. studied diverse heat conduction problems using the RKPM [26, 27]. Ma et al. introduced the Hermit-type radial basis function into the RKPM to solve wave equations [28]. Dehghan and Abbaszadeh have done much research on the RKPM and applied this method to solve many problems [29, 30]. Like the MLS approximation, the RKPM is the approximation based on scalar functions [31, 32]. Then the shape functions must be computed at every point, which will lead to large computational cost. Therefore, Chen et al. presented the complex variable reproducing kernel particle method (CVRKPM) which is the approximation based on vector function and applied it to some problems [33,34,35,36]. Weng et al. also applied the CVRKPM for inverse heat conduction [37] and variable coefficient advection–diffusion problems [38]. However, the shape functions used in the RKPM and the CVRKPM are more complicated than the ones in finite element method, so they have lower computational efficiency, especially for 3D problems. Therefore, it is urgent to improve the computational efficiency of meshless methods to solve 3D problems.

By transforming the procedure of solving a 3D problem into solving a group of 2D problems, Li et al. presented the dimension splitting method (DSM) for many kinds of 3D problems [39,40,41]. Bragin and Rogov dealt with multidimensional scalar quasilinear hyperbolic conservation law using the DSM [42]. Recently, some meshless methods, such as the improved complex variable element-free Galerkin (ICVEFG) method [43,44,45,46,47], the IEFG method [48,49,50] and the RKPM [51], are combined with the DSM to form some approaches that can overcome the inefficiency of the meshless methods in solving 3D problems. And all these papers show a great improvement in computational efficiency.

By introducing the DSM into the RKPM, a hybrid reproducing kernel particle method (HRKPM) for solving 3D wave propagation problems is presented in this paper. Compared with the RKPM of 3D problems, the HRKPM needs only solving a set of 2D problems in some subdomains, rather than solving a 3D problem in the 3D problem domain. The shape functions of 2D problems are much simpler than those of 3D problems, which results in that the HRKPM can save the CPU time greatly. Four numerical examples are selected to verify the validity and advantages of the proposed method. In addition, the error analysis and convergence of the proposed method are investigated. From the numerical results we can know that the HRKPM has higher computational efficiency than the RKPM and the EFG method.

2 Basic equations in HRKPM scheme

In a 3D domain $\Omega$ with the boundary $\Gamma$, the hyperbolic equation governing the wave propagation $u$ is

$$\frac{{\partial^{2} u}}{{\partial t^{2} }} + r\frac{\partial u}{{\partial t}} + \overline{k}u - c^{2} \left( {\frac{{\partial^{2} u}}{{\partial x_{1}^{2} }} + \frac{{\partial^{2} u}}{{\partial x_{2}^{2} }} + \frac{{\partial^{2} u}}{{\partial x_{3}^{2} }}} \right) - f(x_{1} ,x_{2} ,x_{3} ,t) = 0,$$

$$((x_{1} ,x_{2} ,x_{3} ) \in \Omega ,\quad t \in [0,T]),$$

(1)

the boundary conditions are

$$u(x_{1} ,x_{2} ,x_{3} ,t) = \overline{u}(x_{1} ,x_{2} ,x_{3} ,t),\quad ((x_{1} ,x_{2} ,x_{3} ) \in \Gamma_{u} ),$$

(2)

$$q(x_{1} ,x_{2} ,x_{3} ,t) = c^{2} \left( {\frac{\partial u}{{\partial x_{{1}} }}n_{1} + \frac{\partial u}{{\partial x_{2} }}n_{2} + \frac{\partial u}{{\partial x_{3} }}n_{3} } \right) = {\mathbf{n}} \cdot c^{2} \nabla u(x_{1} ,x_{2} ,x_{3} ,t) = \overline{q}(x_{1} ,x_{2} ,x_{3} ,t),\quad ((x_{1} ,x_{2} ,x_{3} ) \in \Gamma_{q} ),$$

(3)

and the initial conditions are

$$u(x_{1} ,x_{2} ,x_{3} ,0) = \phi_{1} (x_{1} ,x_{2} ,x_{3} ),$$

(4)

$$\frac{{\partial u(x_{1} ,x_{2} ,x_{3} ,0)}}{\partial t} = \phi_{2} (x_{1} ,x_{2} ,x_{3} ),$$

(5)

where $c^{2}$ denotes wave speed, $r$ represents resistance, $\overline{k}$ is the coefficient of source term; $t$ is time, $T$ is the total time; ${\varvec{n}} = (n_{1} ,n_{2} ,n_{3} )$ is the outward unit normal vector on the boundary $\Gamma$, $\Gamma_{u}$ and $\Gamma_{q}$ are essential boundary and natural boundary, respectively, $\Gamma = \Gamma_{u} \cup \Gamma_{q}$; $f(x_{1} ,x_{2} ,x_{3} ,t)$ represents external applied force; and $\phi_{1} (x_{1} ,x_{2} ,x_{3} )$ and $\phi_{2} (x_{1} ,x_{2} ,x_{3} )$ are given functions.

This problem is a kind of classical mathematical and physical equations, and the corresponding existence and uniqueness of the solutions were analyzed [52].

Without loss of generality, we choose $x_{3}$ as the splitting direction. By introducing the DSM, the problem domain $\Omega$ can be divided into $L + 1$ layers in this direction as shown in Fig. 1. Then, there are $L + 1$ subdomains $\Omega^{(k)}$, $k = 0,{1}, \ldots ,L$ existed at the layer $x_{3} = x_{{3}}^{{{(}k{)}}}$,

$$a = x_{3}^{(0)} < x_{3}^{(1)} < \cdots < x_{{3}}^{(L - 1)} < x_{3}^{(L)} = b,$$

(6)

in which $x_{{3}}^{{{(}k{)}}}$ is the value of $x_{3}$ at the layer $x_{3} = x_{{3}}^{{{(}k{)}}}$, $a$ and $b$ represent the values of $x_{3}$ at the layer $x_{3} = x_{{3}}^{{{(}{0}{)}}}$ and $x_{3} = x_{{3}}^{{{(}L{)}}}$, respectively; $L$ is the step number.

Fig. 1

The idea of dimension splitting method

At each layer $x_{3} = x_{{3}}^{{{(}k{)}}}$, the plane rectangular coordinate system can be expressed as $O^{(k)} x_{1} x_{2}$, and the origin is $O^{(k)} (0,0,x_{3}^{(k)} )$.

The relationship between $\Omega$ and $\Omega^{(k)}$ can be expressed as

$$\Omega = \bigcup\limits_{{k = {0}}}^{{L{ - 1}}} {\left\{ {\Omega ^{{{(}k{)}}} \times {[}x_{{3}}^{{{(}k{)}}} {,}x_{{3}}^{{{(}k + {1)}}} {)}} \right\}} \cup\Omega ^{{{(}L{)}}} ,$$

(7)

where ${[}x_{{3}}^{{{(}k{)}}} {,}x_{{3}}^{{{(}k + {1)}}} {)}$ $(k = 0,1, \ldots ,L - 1)$ means the space between $\Omega ^{{{(}k{)}}}$ and $\Omega ^{{{(}k + 1{)}}}$.

In the subdomain $\Omega^{(k)}$, the 2D wave propagation problem can be expressed as

$$\frac{{\partial^{2} u^{(k)} }}{{\partial t^{2} }} + r\frac{{\partial u^{(k)} }}{\partial t} + \overline{k}u^{(k)} - c^{2} \left( {\frac{{\partial^{{2}} u^{(k)} }}{{\partial x_{{1}}^{{2}} }} + \frac{{\partial^{{2}} u^{(k)} }}{{\partial x_{{2}}^{{2}} }}} \right) = c^{2} \frac{{\partial^{{2}} u^{(k)} }}{{\partial x_{{3}}^{{2}} }} + f^{(k)} ,$$

$$((x_{1} ,x_{2} ) \in \Omega^{(k)} ,\quad x_{3} = x_{3}^{(k)} ),$$

(8)

where

$$u^{(k)} = u(x_{1} ,x_{2} ,x_{3}^{(k)} ,t),$$

(9)

$$f^{(k)} = f(x_{1} ,x_{2} ,x_{3}^{(k)} ,t).$$

(10)

The corresponding boundary conditions are

$$u^{(k)} (x_{1} ,x_{2} ,t) = \overline{u}^{(k)} (x_{1} ,x_{2} ,t) = \overline{u}(x_{1} ,x_{2} ,x_{3}^{(k)} ,t),\quad ((x_{1} ,x_{2} ) \in \Gamma_{u}^{(k)} ),$$

(11)

$$q^{(k)} (x_{1} ,x_{2} ,t) = \overline{q}^{(k)} (x_{1} ,x_{2} ,t) = \overline{q}(x_{1} ,x_{2} ,x_{3}^{(k)} ,t),\quad ((x_{1} ,x_{2} ) \in \Gamma_{q}^{(k)} ),$$

(12)

and the initial conditions are

$$u(x_{1} ,x_{2} ,x_{3}^{(k)} ,0) = \phi_{1} (x_{1} ,x_{2} ,x_{3}^{(k)} ),$$

(13)

$$\frac{{\partial u(x_{1} ,x_{2} ,x_{3}^{(k)} ,0)}}{\partial t} = \phi_{2} (x_{1} ,x_{2} ,x_{3}^{(k)} ),$$

(14)

where $\Gamma^{(k)}$ is the boundary of subdomain $\Omega^{(k)}$, $\Gamma^{(k)} = \Gamma_{u}^{(k)} \cup \Gamma_{q}^{(k)}$.

Then, the foregoing 2D wave propagation problem can be solved by the RKPM of 2D problems.

The equivalent functional of Eqs. (8)–(14) can be written as

$$\begin{aligned} \Pi & = \int_{{\Omega^{(k)} }} {u^{(k)} \left[ {\frac{{\partial^{2} u^{(k)} }}{{\partial t^{2} }} + r\frac{{\partial u^{(k)} }}{\partial t} + \frac{1}{2}\overline{k}u^{(k)} - c^{2} \frac{{\partial^{{2}} u^{(k)} }}{{\partial x_{3}^{{2}} }} - f^{(k)} } \right]{\text{d}}} \Omega^{(k)} \\ & \quad + \frac{1}{2}\int_{{\Omega^{(k)} }} {\left[ {c^{2} \left( {\frac{{\partial u^{(k)} }}{{\partial x_{{1}} }}} \right)^{{2}} + c^{2} \left( {\frac{{\partial u^{(k)} }}{{\partial x_{{2}} }}} \right)^{{2}} } \right]{\text{d}}} \Omega^{(k)} - \int_{{\Gamma_{q}^{(k)} }} {u^{(k)} \cdot \overline{q}^{(k)} {\text{d}}\Gamma^{{{(}k{)}}} } . \end{aligned}$$

(15)

Imposing essential boundary conditions by penalty function method

$$\Pi^{ * } = \Pi + \frac{\alpha }{2}\int_{{\Gamma_{u}^{(k)} }} {(u^{(k)} - \overline{u}^{(k)} )(u^{(k)} - \overline{u}^{(k)} ){\text{d}}\Gamma^{{{(}k{)}}} } ,$$

(16)

where $\alpha$ is the penalty factors.

The modified equivalent integral weak form is

$$\begin{aligned} \Pi^{ * } & = \int_{{\Omega^{(k)} }} {u^{(k)} \left[ {\frac{{\partial^{2} u^{(k)} }}{{\partial t^{2} }} + r\frac{{\partial u^{(k)} }}{\partial t} + \frac{1}{2}\overline{k} \cdot u{}^{(k)} - c^{2} \frac{{\partial^{{2}} u^{(k)} }}{{\partial x_{3}^{{2}} }} - f^{(k)} } \right]{\text{d}}} \Omega^{(k)} \\ &\quad + c^{2} \int_{{\Omega^{(k)} }} {({\mathbf{L}}u^{(k)} {)}^{{\text{T}}} \cdot ({\mathbf{L}}u^{(k)} {\text{)d}}} \Omega^{(k)} - \int_{{\Gamma_{q}^{(k)} }} {u^{(k)} \cdot \overline{q}^{(k)} {\text{d}}\Gamma^{{{(}k{)}}} }\\ &\quad + \int_{{\Gamma_{u}^{(k)} }} {u^{(k)} \cdot \alpha (u^{(k)} - \overline{u}^{(k)} ){\text{d}}\Gamma^{{{(}k{)}}} } ,\end{aligned}$$

(17)

where

$${\mathbf{L}}{(} \cdot {)} = \left[ {\begin{array}{*{20}c} {\tfrac{\partial }{{\partial x_{{1}} }}} \\ {\tfrac{\partial }{{\partial x_{{2}} }}} \\ \end{array} } \right]( \cdot ).$$

(18)

3 The HRKPM for 3D wave propagation

3.1 The approximation function of the RKPM

Using the RKPM [26, 30], the approximation function of $u^{(k)}$ at any points ${\mathbf{x}} = (x_{1} ,x_{2} ) \in \Omega^{(k)}$ can be expressed as

$$u^{h} ({\mathbf{x}},x_{3}^{(k)} ,t) = \int_{{\Omega^{(k)} }} {u({\mathbf{x^{\prime}}})\overline{w}({\mathbf{x}} - {\mathbf{x^{\prime}}}){\text{d}}{\mathbf{x^{\prime}}}} ,$$

(19)

where $\overline{w}({\mathbf{x}} - {\mathbf{x^{\prime}}})$ is the correction kernel function,

$$\overline{w}({\mathbf{x}} - {\mathbf{x^{\prime}}}) = C({\mathbf{x}};{\mathbf{x}} - {\mathbf{x^{\prime}}})w({\mathbf{x}} - {\mathbf{x^{\prime}}}),$$

(20)

in which $w({\mathbf{x}} - {\mathbf{x^{\prime}}})$ is weight function which has a compact support domain, and $C({\mathbf{x}}; \,{\mathbf{x}} - {\mathbf{x}}^{\prime})$ is the correction function,

$$C({\mathbf{x}};{\mathbf{x}} - {\mathbf{x^{\prime}}}) = \sum\limits_{{i = {1}}}^{m} {p_{i} ({\mathbf{x}} - {\mathbf{x^{\prime}}})b_{i} ({\mathbf{x}}) = {\mathbf{p}}^{{\text{T}}} ({\mathbf{x}} - {\mathbf{x^{\prime}}}){\mathbf{b}}} ({\mathbf{x}}),\quad ({\mathbf{x}} \in \Omega^{(k)} ),$$

(21)

where $m$ is the number of basis functions, ${\mathbf{p}}({\mathbf{x}} - {\mathbf{x^{\prime}}})$ is the vector of the basis functions $p_{i} ({\mathbf{x}} - {\mathbf{x^{\prime}}})$, ($i = 1,2, \ldots ,m$); and $b_{i} ({\mathbf{x}})$ are the coefficients of basis functions, ${\mathbf{b}}({\mathbf{x}})$ is the vector of the coefficients,

$${\mathbf{b}}({\mathbf{x}}) = (b_{1} ({\mathbf{x}}),b_{2} ({\mathbf{x}}), \ldots ,b_{m} ({\mathbf{x}}))^{{\text{T}}} .$$

(22)

In general, the basis function can be chosen as linear basis,

$${\mathbf{p}}^{{\text{T}}} = (1,x_{1} - x^{\prime}_{1} ,x_{2} - x^{\prime}_{2} ),$$

(23)

or quadratic basis,

$${\mathbf{p}}^{{\text{T}}} = (1,x_{1} - x^{\prime}_{1} ,x_{2} - x^{\prime}_{2} ,(x_{1} - x^{\prime}_{1} )^{2} ,(x_{1} - x^{\prime}_{1} )(x_{2} - x^{\prime}_{2} ),(x_{2} - x^{\prime}_{2} )^{2} ).$$

(24)

The trapezoidal integral method is used to obtain the discretization approximation, and then Eq. (19) can be written as

$$u^{h} ({\mathbf{x}},x_{3}^{(k)} ,t) = \sum\limits_{I = 1}^{n} {C({\mathbf{x}};{\mathbf{x}} - {\mathbf{x}}_{I} )w({\mathbf{x}} - {\mathbf{x}}_{I} )u_{I}^{(k)} \Delta V_{I} } ,$$

(25)

where ${\mathbf{x}}_{I}$ ($I = 1,2, \ldots ,n$) are nodes in the neighborhood of the point ${\mathbf{x}}$, $u_{I}^{(k)}$ is the value of $u^{(k)}$ at the node ${\mathbf{x}}_{I}$,

$$u_{I}^{(k)} = u({\mathbf{x}}_{I} ,x_{3}^{(k)} ,t),\quad I = 1,2, \ldots ,n;$$

(26)

and $\Delta V$ is a regional measure related to ${\mathbf{x}}_{I}$,

$$\sum\limits_{I = 1}^{n} {\Delta V_{I} = {\mathbf{V}}} .$$

(27)

The matrix form of Eq. (25) can be expressed as

$$u^{h} ({\mathbf{x}},x_{3}^{(k)} ,t) = {\mathbf{C}}({\mathbf{x}}){\mathbf{W}}({\mathbf{x}}){\mathbf{Vu}},$$

(28)

where ${\mathbf{u}}$ is the vector of the variable at the nodes,

$${\mathbf{u}} = (u_{{1}}^{(k)} ,u_{2}^{(k)} , \ldots ,u_{n}^{(k)} )^{{\text{T}}} ,$$

(29)

${\mathbf{W}}({\mathbf{x}})$ is the matrix of weight functions,

$${\mathbf{W}}({\mathbf{x}}) = \left[ {\begin{array}{*{20}c} {w({\mathbf{x}} - {\mathbf{x}}_{1} )} & 0 & \cdots & 0 \\ 0 & {w({\mathbf{x}} - {\mathbf{x}}_{2} )} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & {w({\mathbf{x}} - {\mathbf{x}}_{n} )} \\ \end{array} } \right],$$

(30)

${\mathbf{V}}$ is the matrix of regional measures related to ${\mathbf{x}}_{I}$,

$${\mathbf{V}} = \left[ {\begin{array}{*{20}c} {\Delta V_{1} } & 0 & \cdots & 0 \\ 0 & {\Delta V_{2} } & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & {\Delta V_{n} } \\ \end{array} } \right],$$

(31)

${\mathbf{C}}({\mathbf{x}})$ is the vector of correction functions,

$${\mathbf{C}}({\mathbf{x}}) = (C_{1} ({\mathbf{x}}),C_{2} ({\mathbf{x}}), \ldots ,C_{n} ({\mathbf{x}})) = {\mathbf{b}}^{{\text{T}}} ({\mathbf{x}}){\mathbf{P}},$$

(32)

$$C_{I} ({\mathbf{x}}) = C({\mathbf{x}};{\mathbf{x}} - {\mathbf{x}}_{I} ),$$

(33)

and ${\mathbf{P}}$ is the matrix of basis functions,

$${\mathbf{P}} = \left[ {\begin{array}{*{20}c} {p_{1} ({\mathbf{x}} - {\mathbf{x}}_{1} )} & {p_{1} ({\mathbf{x}} - {\mathbf{x}}_{2} )} & \cdots & {p_{1} ({\mathbf{x}} - {\mathbf{x}}_{n} )} \\ {p_{2} ({\mathbf{x}} - {\mathbf{x}}_{1} )} & {p_{2} ({\mathbf{x}} - {\mathbf{x}}_{2} )} & \cdots & {p_{2} ({\mathbf{x}} - {\mathbf{x}}_{n} )} \\ \vdots & \vdots & \ddots & \vdots \\ {p_{m} ({\mathbf{x}} - {\mathbf{x}}_{1} )} & {p_{m} ({\mathbf{x}} - {\mathbf{x}}_{2} )} & \cdots & {p_{m} ({\mathbf{x}} - {\mathbf{x}}_{n} )} \\ \end{array} } \right].$$

(34)

The coefficients b_i(x) are determined by the reproducing conditions of the approximation function. Let

$${\overline{\mathbf{M}}}({\mathbf{x}}) = {\mathbf{M}}({\mathbf{x}}){\mathbf{b}}({\mathbf{x}}) = {\mathbf{H}},$$

(35)

where

$${\mathbf{M}}({\mathbf{x}}) = \sum\limits_{I = 1}^{n} {{\mathbf{p}}({\mathbf{x}} - {\mathbf{x}}_{I} ){\mathbf{p}}^{{\text{T}}} ({\mathbf{x}} - {\mathbf{x}}_{I} )w({\mathbf{x}} - {\mathbf{x}}_{I} )\Delta V_{I} } ,$$

(36)

$${\mathbf{H}} = (1,0, \ldots ,0)^{{\text{T}}} ,$$

(37)

then we have

$${\mathbf{b}}({\mathbf{x}}) = {\mathbf{M}}^{ - 1} ({\mathbf{x}}){\mathbf{H}}.$$

(38)

Finally, the approximation function $u^{h} ({\mathbf{x}},x_{3}^{(k)} ,t)$ is obtained as

$$u^{h} ({\mathbf{x}},x_{3}^{(k)} ,t) = \sum\limits_{I = 1}^{n} {\Phi_{I}^{(k)} ({\mathbf{x}})} u_{I}^{(k)} = {{\varvec{\Phi}}}({\mathbf{x}}){\mathbf{u}},\quad k = 0,{1}, \ldots ,L,$$

(39)

where ${{\varvec{\Phi}}}({\mathbf{x}})$ is the shape function,

$${{\varvec{\Phi}}}({\mathbf{x}}) = (\Phi_{1}^{(k)} ({\mathbf{x}}),\Phi_{2}^{(k)} ({\mathbf{x}}), \ldots ,\Phi_{n}^{(k)} ({\mathbf{x}})) = {\mathbf{C}}({\mathbf{x}}){\mathbf{W}}({\mathbf{x}}){\mathbf{V}}.$$

(40)

3.2 The discretized equation of the HRKPM for 3D wave propagation

According to Eq. (39), we can obtain the following approximations related to $u^{(k)}$,

$$\frac{{\partial^{{2}} u^{(k)} }}{{\partial x_{3}^{{2}} }} = \frac{\partial }{{\partial x_{3}^{{2}} }}\sum\limits_{I = 1}^{n} {\Phi_{I}^{(k)} ({\mathbf{x}})} \cdot u_{I}^{(k)} = \sum\limits_{I = 1}^{n} {\Phi_{I}^{(k)} ({\mathbf{x}})} \frac{{\partial^{{2}} u_{I}^{(k)} }}{{\partial x_{3}^{{2}} }} = {{\varvec{\Phi}}}({\mathbf{x}})D_{{x_{3} x_{3} }} {\mathbf{u}},$$

(41)

$$\frac{{\partial u^{(k)} }}{\partial t} = \frac{\partial }{\partial t}\sum\limits_{I = 1}^{n} {\Phi_{I}^{(k)} ({\mathbf{x}})} \cdot u_{I}^{(k)} = \sum\limits_{I = 1}^{n} {\Phi_{I}^{(k)} ({\mathbf{x}})} \frac{{\partial u_{I}^{(k)} }}{\partial t} = {{\varvec{\Phi}}}({\mathbf{x}}){\dot{\mathbf{u}}},$$

(42)

$$\frac{{\partial^{2} u^{(k)} }}{{\partial t^{2} }} = \frac{{\partial^{2} }}{{\partial t^{2} }}\sum\limits_{I = 1}^{n} {\Phi_{I}^{(k)} ({\mathbf{x}})} \cdot u_{I}^{(k)} = \sum\limits_{I = 1}^{n} {\Phi_{I}^{(k)} ({\mathbf{x}})} \frac{{\partial^{2} u_{I}^{(k)} }}{{\partial t^{2} }} = {{\varvec{\Phi}}}({\mathbf{x}}){\mathbf{\ddot{u}}},$$

(43)

$${\mathbf{L}}u^{(k)} = \sum\limits_{I = 1}^{n} {\left[ {\begin{array}{*{20}c} {\frac{\partial }{{\partial x_{1} }}} \\ {\frac{\partial }{{\partial x_{2} }}} \\ \end{array} } \right]} \Phi_{I}^{(k)} ({\mathbf{x}}) \cdot u_{I}^{(k)} = \sum\limits_{I = 1}^{n} {{\mathbf{B}}_{I}^{(k)} ({\mathbf{x}})u_{I}^{(k)} } = {\mathbf{B}}({\mathbf{x}}){\mathbf{u}},$$

(44)

where

$$D_{{x_{3} x_{3} }} {\mathbf{u}} = \left( {\frac{{\partial^{{2}} u_{1}^{(k)} }}{{\partial x_{3}^{{2}} }},\frac{{\partial^{{2}} u_{{2}}^{(k)} }}{{\partial x_{3}^{{2}} }}, \ldots ,\frac{{\partial^{{2}} u_{n}^{(k)} }}{{\partial x_{3}^{{2}} }}} \right)^{{\text{T}}} ,$$

(45)

$${\dot{\mathbf{u}}} = \left( {\frac{{\partial u_{1}^{(k)} }}{\partial t},\frac{{\partial u_{2}^{(k)} }}{\partial t}, \ldots ,\frac{{\partial u_{n}^{(k)} }}{\partial t}} \right)^{{\text{T}}} ,$$

(46)

$${\mathbf{\ddot{u}}} = \left( {\frac{{\partial^{2} u_{1}^{(k)} }}{{\partial t^{2} }},\frac{{\partial^{2} u_{2}^{(k)} }}{{\partial t^{2} }}, \ldots ,\frac{{\partial^{2} u_{n}^{(k)} }}{{\partial t^{2} }}} \right)^{{\text{T}}} ,$$

(47)

$${\mathbf{B}}({\mathbf{x}}) = ({\mathbf{B}}_{1}^{(k)} ({\mathbf{x}}),{\mathbf{B}}_{2}^{(k)} ({\mathbf{x}}), \ldots ,{\mathbf{B}}_{n}^{(k)} ({\mathbf{x}})),$$

(48)

$${\mathbf{B}}_{I}^{(k)} ({\mathbf{x}}) = \left[ {\begin{array}{*{20}c} {\Phi_{I,1}^{(k)} ({\mathbf{x}})} \\ {\Phi_{I,2}^{(k)} ({\mathbf{x}})} \\ \end{array} } \right].$$

(49)

Substitute Eqs. (41)–(44) into Eq. (17) yields

$$\begin{aligned} \Pi^{ * } &= \int_{{\Omega^{(k)} }} {\left[ {{\mathbf{B}}({\mathbf{x}}){\mathbf{u}}} \right]^{{\text{T}}} \cdot c^{{2}} \cdot \left[ {{\mathbf{B}}({\mathbf{x}}){\mathbf{u}}} \right]} {\text{d}}\Omega^{{{(}k{)}}} + \int_{{\Omega^{(k)} }} {{{\varvec{\Phi}}}({\mathbf{x}}){\mathbf{u}} \cdot {{\varvec{\Phi}}}({\mathbf{x}}){\mathbf{\ddot{u}}}} {\text{d}}\Omega^{{{(}k{)}}} \\ &\quad + \int_{{\Omega^{(k)} }} {{{\varvec{\Phi}}}({\mathbf{x}}){\mathbf{u}} \cdot r \cdot {{\varvec{\Phi}}}({\mathbf{x}}){\dot{\mathbf{u}}}} {\text{d}}\Omega^{{{(}k{)}}} + \frac{1}{2}\int_{{\Omega^{(k)} }} {{{\varvec{\Phi}}}({\mathbf{x}}){\mathbf{u}} \cdot \overline{k} \cdot {{\varvec{\Phi}}}({\mathbf{x}}){\mathbf{u}}} {\text{d}}\Omega^{{{(}k{)}}} \\ &\quad - \int_{{\Omega^{(k)} }} {{{\varvec{\Phi}}}({\mathbf{x}}){\mathbf{u}} \cdot c^{2} \cdot {{\varvec{\Phi}}}({\mathbf{x}})D_{{x_{3} x_{3} }} {\mathbf{u}}} {\text{d}}\Omega^{{{(}k{)}}} - \int_{{\Omega^{(k)} }} {{{\varvec{\Phi}}}({\mathbf{x}}){\mathbf{u}} \cdot f^{(k)} } {\text{d}}\Omega^{{{(}k{)}}} \\ &\quad - \int_{{\Gamma_{q}^{(k)} }} {{{\varvec{\Phi}}}({\mathbf{x}}){\mathbf{u}} \cdot \overline{q}}^{(k)} {\text{d}}\Gamma^{{{(}k{)}}} + \int_{{\Gamma_{u}^{(k)} }} {{{\varvec{\Phi}}}({\mathbf{x}}){\mathbf{u}} \cdot \alpha \cdot {{\varvec{\Phi}}}({\mathbf{x}}){\mathbf{u}}} {\text{d}}\Gamma^{{{(}k{)}}} \\ &\quad - \int_{{\Gamma_{u}^{(k)} }} {{{\varvec{\Phi}}}({\mathbf{x}}){\mathbf{u}} \cdot \alpha \cdot {\overline{u}}^{(k)} } {\text{d}}\Gamma^{{{(}k{)}}} . \end{aligned}$$

(50)

The variation formulation of Eq. (50) is

$$\begin{aligned} \delta \Pi^{ * } &= \int_{{\Omega^{(k)} }} {\delta {\mathbf{u}}^{{\text{T}}} \left[ {{\mathbf{B}}^{{\text{T}}} ({\mathbf{x}}) \cdot c^{{2}} \cdot {\mathbf{B}}({\mathbf{x}})} \right]} {\mathbf{u}}{\text{d}}\Omega^{{{(}k{)}}} + \int_{{\Omega^{(k)} }} {\delta {\mathbf{u}}^{{\text{T}}} \left[ {{{\varvec{\Phi}}}^{{\text{T}}} ({\mathbf{x}}) \cdot {{\varvec{\Phi}}}({\mathbf{x}})} \right]{\mathbf{\ddot{u}}}} {\text{d}}\Omega^{{{(}k{)}}} \\ &\quad + \int_{{\Omega^{(k)} }} {\delta {\mathbf{u}}^{{\text{T}}} \left[ {{{\varvec{\Phi}}}^{{\text{T}}} ({\mathbf{x}}) \cdot r \cdot {{\varvec{\Phi}}}({\mathbf{x}})} \right]{\dot{\mathbf{u}}}} {\text{d}}\Omega^{{{(}k{)}}} + \overline{k}\int_{{\Omega^{(k)} }} {\delta {\mathbf{u}}^{{\text{T}}} \left[ {{{\varvec{\Phi}}}^{{\text{T}}} ({\mathbf{x}}) \cdot {{\varvec{\Phi}}}({\mathbf{x}})} \right]{\mathbf{u}}} {\text{d}}\Omega^{{{(}k{)}}} \\ &\quad - \int_{{\Omega^{(k)} }} {\delta {\mathbf{u}}^{{\text{T}}} \left[ {{{\varvec{\Phi}}}^{{\text{T}}} ({\mathbf{x}}) \cdot c^{2} \cdot {{\varvec{\Phi}}}({\mathbf{x}})} \right]D_{{x_{3} x_{3} }} {\mathbf{u}}} {\text{d}}\Omega^{{{(}k{)}}} - \int_{{\Omega^{(k)} }} {\delta {\mathbf{u}}^{{\text{T}}} \left[ {{{\varvec{\Phi}}}^{{\text{T}}} ({\mathbf{x}}) \cdot f^{(k)} } \right]} {\text{d}}\Omega^{{{(}k{)}}} &\quad - \int_{{\Gamma_{q}^{(k)} }} {\delta {\mathbf{u}}^{{\text{T}}} \left[ {{{\varvec{\Phi}}}^{{\text{T}}} ({\mathbf{x}}) \cdot \overline{q}^{(k)} } \right]} {\text{d}}\Gamma^{{{(}k{)}}} + \int_{{\Gamma_{u}^{(k)} }} {\delta {\mathbf{u}}^{{\text{T}}} \left[ {{{\varvec{\Phi}}}^{{\text{T}}} ({\mathbf{x}}) \cdot \alpha \cdot {{\varvec{\Phi}}}({\mathbf{x}})} \right]} {\mathbf{u}}{\text{d}}\Gamma^{{{(}k{)}}} \\ &\quad - \int_{{\Gamma_{u}^{(k)} }} {\delta {\mathbf{u}}^{{\text{T}}} \left[ {{{\varvec{\Phi}}}^{{\text{T}}} ({\mathbf{x}}) \cdot \alpha \cdot {\overline{{u}}}^{(k)} } \right]} {\text{d}}\Gamma^{{{(}k{)}}} . \end{aligned}$$

(51)

Since $\delta {\mathbf{u}}$ is arbitrary, from $\delta \Pi^{ * } = {0}$, the discrete system equation can be obtained as

$${\mathbf{C\ddot{u}}} + r{\mathbf{C\dot{u}}} - c^{2} {\mathbf{C}}D_{{x_{3} x_{3} }} {\mathbf{u}} + {\hat{\mathbf{K}}\mathbf{u}} = {\hat{\mathbf{F}}},$$

(52)

where

$${\hat{\mathbf{K}}} = {\mathbf{K}} + \overline{k}{\mathbf{C}} + {\mathbf{K}}^{\alpha } ,$$

(53)

$${\hat{\mathbf{F}}} = {\mathbf{F}}^{(1)} + {\mathbf{F}}^{(2)} + {\mathbf{F}}^{\alpha } ,$$

(54)

$${\mathbf{C}} = \int_{{\Omega^{(k)} }} {{{\varvec{\Phi}}}{}^{{\text{T}}}{(}{\mathbf{x}}{)}{{\varvec{\Phi}}}{(}{\mathbf{x}}{)}} {\text{d}}\Omega^{{{(}k{)}}} ,$$

(55)

$${\mathbf{K}} = c^{2} \int_{{\Omega^{(k)} }} {{\mathbf{B}}{}^{{\text{T}}}{(}{\mathbf{x}}{)}{\mathbf{B}}{(}{\mathbf{x}}{)}} {\text{d}}\Omega^{{{(}k{)}}} ,$$

(56)

$${\mathbf{K}}^{\alpha } = \alpha \int_{{\Gamma_{u}^{(k)} }} {{{\varvec{\Phi}}}{}^{{\text{T}}}{(}{\mathbf{x}}{)}{{\varvec{\Phi}}}{(}{\mathbf{x}}{)}} {\text{d}}\Gamma^{{{(}k{)}}} ,$$

(57)

$${\mathbf{F}}^{(1)} = \int_{{\Omega^{(k)} }} {{{\varvec{\Phi}}}^{{\text{T}}} ({\mathbf{x}})f^{(k)} } {\text{d}}\Omega^{{{(}k{)}}} ,$$

(58)

$${\mathbf{F}}^{(2)} = \int_{{\Gamma_{q}^{(k)} }} {{{\varvec{\Phi}}}^{{\text{T}}} ({\mathbf{x}})\overline{q}^{(k)} } {\text{d}}\Gamma^{{{(}k{)}}} ,$$

(59)

$${\mathbf{F}}^{\alpha } = \alpha \int_{{\Gamma_{u}^{(k)} }} {{{\varvec{\Phi}}}^{{\text{T}}} ({\mathbf{x}}){\overline{u}}^{(k)} } {\text{d}}\Gamma^{{{(}k{)}}} .$$

(60)

According to finite difference method, we have

$$D_{{x_{3} x_{3} }} {\mathbf{u}} \approx \frac{{{\mathbf{u}}^{(k - 1)} - 2{\mathbf{u}}^{(k)} + {\mathbf{u}}^{(k + 1)} }}{{(\Delta x_{3} )^{2} }},\quad k = 1,2, \ldots ,L - 1.$$

(61)

Then Eq. (52) can be written as

$${\mathbf{C\ddot{u}}}^{(k)} + r{\mathbf{C\dot{u}}}^{(k)} - c^{2} {\mathbf{C}}\frac{{{\mathbf{u}}^{(k - 1)} - 2{\mathbf{u}}^{(k)} + {\mathbf{u}}^{(k + 1)} }}{{(\Delta x_{3} )^{2} }} + {\hat{\mathbf{K}}\mathbf{u}}^{(k)} = {\hat{\mathbf{F}}}.$$

(62)

And the discrete system formulas in each subdomain can be expressed as

$${\mathbf{C\ddot{u}}}^{(1)} + r{\mathbf{C\dot{u}}}^{(1)} - c^{2} {\mathbf{C}}\frac{{{\mathbf{u}}^{(0)} - 2{\mathbf{u}}^{(1)} + {\mathbf{u}}^{(2)} }}{{(\Delta x_{3} )^{2} }} + {\hat{\mathbf{K}}\mathbf{u}}^{(1)} = {\hat{\mathbf{F}}}^{(1)} ,$$

(63)

$${\mathbf{C\ddot{u}}}^{(2)} + r{\mathbf{C\dot{u}}}^{(2)} - c^{2} {\mathbf{C}}\frac{{{\mathbf{u}}^{(1)} - 2{\mathbf{u}}^{(2)} + {\mathbf{u}}^{(3)} }}{{(\Delta x_{3} )^{2} }} + {\hat{\mathbf{K}}\mathbf{u}}^{(2)} = {\hat{\mathbf{F}}}^{(2)} ,$$

(64)

$$\begin{gathered} {\mathbf{C\ddot{u}}}^{{({3})}} + r{\mathbf{C\dot{u}}}^{{({3})}} - c^{2} {\mathbf{C}}\frac{{{\mathbf{u}}^{{({2})}} - 2{\mathbf{u}}^{{({3})}} + {\mathbf{u}}^{{({4})}} }}{{(\Delta x_{3} )^{2} }} + {\hat{\mathbf{K}}\mathbf{u}}^{{({3})}} = {\hat{\mathbf{F}}}^{(3)} , \hfill \\ \vdots \hfill \\ \end{gathered}$$

(65)

$${\mathbf{C\ddot{u}}}^{(L - 2)} + r{\mathbf{C\dot{u}}}^{(L - 2)} - c^{2} {\mathbf{C}}\frac{{{\mathbf{u}}^{(L - 3)} - 2{\mathbf{u}}^{(L - 2)} + {\mathbf{u}}^{(L - 1)} }}{{(\Delta x_{3} )^{2} }} + {\hat{\mathbf{K}}\mathbf{u}}^{(L - 2)} = {\hat{\mathbf{F}}}^{(L - 2)} ,$$

(66)

$${\mathbf{C\ddot{u}}}^{(L - 1)} + r{\mathbf{C\dot{u}}}^{(L - 1)} - c^{2} {\mathbf{C}}\frac{{{\mathbf{u}}^{(L - 2)} - 2{\mathbf{u}}^{(L - 1)} + {\mathbf{u}}^{(L)} }}{{(\Delta x_{3} )^{2} }} + {\hat{\mathbf{K}}\mathbf{u}}^{(L - 1)} = {\hat{\mathbf{F}}}^{(L - 1)} ,$$

(67)

where

$${\mathbf{u}}^{(k)} = {\mathbf{u}}{(}{\mathbf{x}},x_{3}^{(k)} ,t{),}$$

(68)

$${\mathbf{u}}^{{({0})}} = {\mathbf{u}}{(}{\mathbf{x}}{,}a{,}t{),}$$

(69)

$${\mathbf{u}}^{(L)} = {\mathbf{u}}{(}{\mathbf{x}}{,}b{,}t{),}$$

(70)

$$\Delta x_{{3}} = x_{{3}}^{(k)} - x_{3}^{{(k - {1})}} = {{{(}b - a{)}} \mathord{\left/ {\vphantom {{{(}b - a{)}} L}} \right. \kern-\nulldelimiterspace} L}.$$

(71)

The matrix form of Eqs. (63)-(67) is

$${\mathbf{R}}\left[ {\begin{array}{*{20}c} {\mathbf{\ddot{u}}^{{({1})}} } \\ {\mathbf{\ddot{u}}^{{({2})}} } \\ {\mathbf{\ddot{u}}^{{({3})}} } \\ \vdots \\ {\mathbf{\ddot{u}}^{{(L - {2})}} } \\ {\mathbf{\ddot{u}}^{{(L - {1})}} } \\ \end{array} } \right] + r{\mathbf{R}}\left[ {\begin{array}{*{20}c} {\dot{\mathbf{u}}^{{({1})}} } \\ {\dot{\mathbf{u}}^{{({2})}} } \\ {\dot{\mathbf{u}}^{{({3})}} } \\ \vdots \\ {\dot{\mathbf{u}}^{{(L - {2})}} } \\ {\dot{\mathbf{u}}^{{(L - {1})}} } \\ \end{array} } \right] - \tilde{k}\left[ {\begin{array}{*{20}c} {\mathbf{H}} & {\mathbf{C}} & {} & {} & {} & {} \\ {\mathbf{C}} & {\mathbf{H}} & {\mathbf{C}} & {} & {} & {} \\ {} & {\mathbf{C}} & {\mathbf{H}} & {\mathbf{C}} & {} & {} \\ {} & {} & \ddots & \ddots & \ddots & {} \\ {} & {} & {} & {\mathbf{C}} & {\mathbf{H}} & {\mathbf{C}} \\ {} & {} & {} & {} & {\mathbf{C}} & {\mathbf{H}} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {{\mathbf{u}}^{{({1})}} } \\ {{\mathbf{u}}^{{({2})}} } \\ {{\mathbf{u}}^{{({3})}} } \\ \vdots \\ {{\mathbf{u}}^{{(L - {2})}} } \\ {{\mathbf{u}}^{{(L - {1})}} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\hat{\mathbf{F}}^{{({1})}} + {\mathbf{M}}} \\ {\hat{\mathbf{F}}^{{({2})}} } \\ {\hat{\mathbf{F}}^{{({3})}} } \\ \vdots \\ {\hat{\mathbf{F}}^{{(L - {2})}} } \\ {\hat{\mathbf{F}}^{{(L - {1})}} + {\mathbf{W}}} \\ \end{array} } \right],$$

(72)

where

$$\tilde{k} = \frac{{c^{2} }}{{{(\Delta }x_{{3}} )^{2} }},$$

(73)

$${\mathbf{M}} = \frac{{c^{2} {\mathbf{Cu}}^{{({0})}} }}{{{(\Delta }x_{{3}} )^{2} }},$$

(74)

$${\mathbf{W}} = \frac{{c^{2} {\mathbf{Cu}}^{(L)} }}{{{(\Delta }x_{{3}} )^{2} }},$$

(75)

$${\mathbf{R}} = \left[ {\begin{array}{*{20}c} {\mathbf{C}} & {} & {} & {} & {} & {} \\ {} & {\mathbf{C}} & {} & {} & {} & {} \\ {} & {} & {\mathbf{C}} & {} & {} & {} \\ {} & {} & {} & \ddots & {} & {} \\ {} & {} & {} & {} & {\mathbf{C}} & {} \\ {} & {} & {} & {} & {} & {\mathbf{C}} \\ \end{array} } \right],$$

(76)

$${\mathbf{H}} = - 2{\mathbf{C}} - \frac{{\left( {\Delta x_{3} } \right)^{2} }}{{c^{2} }}{\hat{\mathbf{K}}}.$$

(77)

Let

$${\tilde{\mathbf{u}}} = \left( {{\mathbf{u}}^{{(1)^{{\text{T}}} }} ,{\mathbf{u}}^{{(2)^{{\text{T}}} }} ,{\mathbf{u}}^{{(3)^{{\text{T}}} }} , \ldots ,{\mathbf{u}}^{{(L - 2)^{{\text{T}}} }} ,{\mathbf{u}}^{{(L - 1)^{{\text{T}}} }} } \right)^{{\text{T}}} ,$$

(78)

$${\mathbf{\ddot{\tilde{u}}}} = \left( {{\mathbf{\ddot{u}}}^{{(1)^{{\text{T}}} }} ,{\mathbf{\ddot{u}}}^{{(2)^{{\text{T}}} }} ,{\mathbf{\ddot{u}}}^{{(3)^{{\text{T}}} }} , \ldots ,{\mathbf{\ddot{u}}}^{{(L - 2)^{{\text{T}}} }} ,{\mathbf{\ddot{u}}}^{{(L - 1)^{{\text{T}}} }} } \right)^{{\text{T}}} ,$$

(79)

$${\mathbf{\dot{\tilde{u}}}} = \left( {{\dot{\mathbf{u}}}^{{(1)^{{\text{T}}} }} ,{\dot{\mathbf{u}}}^{{(2)^{{\text{T}}} }} ,{\dot{\mathbf{u}}}^{{(3)^{{\text{T}}} }} , \ldots ,{\dot{\mathbf{u}}}^{{(L - 2)^{{\text{T}}} }} ,{\dot{\mathbf{u}}}^{{(L - 1)^{{\text{T}}} }} } \right)^{{\text{T}}} ,$$

(80)

$${\mathbf{E}} = \tilde{k}\left[ {\begin{array}{*{20}c} {\mathbf{H}} & {\mathbf{C}} & {} & {} & {} & {} \\ {\mathbf{C}} & {\mathbf{H}} & {\mathbf{C}} & {} & {} & {} \\ {} & {\mathbf{C}} & {\mathbf{H}} & {\mathbf{C}} & {} & {} \\ {} & {} & \ddots & \ddots & \ddots & {} \\ {} & {} & {} & {\mathbf{C}} & {\mathbf{H}} & {\mathbf{C}} \\ {} & {} & {} & {} & {\mathbf{C}} & {\mathbf{H}} \\ \end{array} } \right],$$

(81)

$${\mathbf{G}} = ({({\hat{{\mathbf{F}}}^{(1)}} + {\mathbf{M}})^{{\text{T}}}}, {\hat{{\mathbf{F}}}^{(2){{\text{T}}}}},{\hat{{\mathbf{F}}}^{(3){{\text{T}}}}}, \ldots , {\hat{{\mathbf{F}}}^{(L - 2){{\text{T}}}}},{({\hat{{\mathbf{F}}}^{(L - 1)}} + {\mathbf{W}})^{{\text{T}}})^{{\text{T}}}}.$$

(82)

Equation (72) can be derived as following equation

$${\mathbf{R}}\mathbf{\ddot{\tilde{u}}} + r{\mathbf{R}}\mathbf{\dot{\tilde{u}}} - {\mathbf{E}}\tilde{\mathbf{u}} = {\mathbf{G}}.$$

(83)

We obtain the time discretization of Eq. (83) by the center difference method

$${\mathbf{R}}\frac{{\tilde{\mathbf{u}}^{(i + 2)} - 2\tilde{\mathbf{u}}^{(i + 1)} + \tilde{\mathbf{u}}^{(i)} }}{{\left( {\Delta t} \right)^{2} }} + r{\mathbf{R}}\frac{{\tilde{\mathbf{u}}^{(i + 1)} - \tilde{\mathbf{u}}^{(i)} }}{\Delta t} - {\mathbf{E}}\frac{{\tilde{\mathbf{u}}^{(i + 1)} + \tilde{\mathbf{u}}^{(i)} }}{2} = \mathbf{G,}$$

(84)

that is

$$2{\mathbf{R}}\tilde{\mathbf{u}}^{(i + 2)} = (4{\mathbf{R}} - 2r\Delta t{\mathbf{R}} + (\Delta t)^{2} {\mathbf{E}})\tilde{\mathbf{u}}^{(i + 1)} + (2r\Delta t{\mathbf{R}} + (\Delta t)^{2} {\mathbf{E}} - 2{\mathbf{R}})\tilde{\mathbf{u}}^{(i)} + 2(\Delta t)^{2} \mathbf{G,}$$

(85)

where $\Delta t$ represents time step length.

4 Numerical examples

To test whether the proposed method can improve the computational efficiency, the numerical results of the following examples obtained by the HRKPM are compared with analytical ones and the ones of the RKPM and the EFG method.

4.1 Wave propagation problem with Dirichlet boundary conditions

In the 3D domain $\Omega = [0,{\uppi} ] \times [0,{\uppi }] \times [0,{\uppi ]}$, the hyperbolic equation governing the wave propagation $u$ is [19]

$$\frac{{\partial^{{2}} u}}{{\partial t^{{2}} }} = \frac{{\partial^{2} u}}{{\partial x_{{1}}^{{2}} }} + \frac{{\partial^{2} u}}{{\partial x_{{2}}^{{2}} }} + \frac{{\partial^{2} u}}{{\partial x_{{3}}^{{2}} }} - u,\quad ({(}x_{{1}} {,}x_{{2}} {,}x_{{3}} {)} \in {\Omega ,}\;t \in [0,T]),$$

(86)

with the boundary conditions

$$u(0,x_{2} ,x_{3} ,t) = - u(\uppi ,x_{2} ,x_{3} ,t) = \sin x_{2} \sin (x_{3} + 2t),$$

(87)

$$u(x_{1} ,0,x_{3} ,t) = - u(x_{1} ,\uppi ,x_{3} ,t) = \sin x_{1} \sin (x_{3} + 2t),$$

(88)

$$u(x_{1} ,x_{{2}} ,{0},t) = - u(x_{1} ,x_{{2}} ,{\uppi },t) = \sin (x_{1} + x_{2} )\sin (2t),$$

(89)

and the initial conditions

$$u(x_{1} ,x_{2} ,x_{3} ,0) = \sin (x_{1} + x_{2} )\sin (x_{3} ),$$

(90)

$$\frac{{\partial u(x_{1} ,x_{2} ,x_{3} ,0)}}{\partial t} = 2\sin (x_{1} + x_{2} )\cos (x_{3} ).$$

(91)

The analytical solution used to compare with the approximate solution is

$$u(x_{1} ,x_{2} ,x_{3} ,t) = \sin (x_{1} + x_{2} )\sin (x_{3} + 2t).$$

(92)

In the HRKPM scheme for this example, we discuss factors that affect relative errors when $x_{{3}}$ is the selected splitting direction, such as the scale parameter of the influence domain $d_{\max }$, penalty factor $\alpha$, the time step length $\Delta t$ and node distribution. Figure 2 shows the relative errors of the HRKPM with different $d_{\max }$ when $\alpha = 5.22 \times 10^{3}$, $\Delta t = 0.001$, $T = 0.1$ and the node distribution is ${11} \times {11} \times {11}$. Figure 3 shows the relative errors of the HRKPM with different $\alpha$ when $d_{\max } = {1}{\text{.1}}$, $\Delta t = 0.001$, $T = 0.1$ and the node distribution is ${11} \times {11} \times {11}$. Figure 4 shows the relative errors of the HRKPM with different $\Delta t$ when $d_{\max } = {1}{\text{.1}}$, $\alpha = {4}{\text{.36}} \times 10^{{2}}$, $T = 0.1$ and the node distribution is ${11} \times {11} \times {11}$. Figure 5 shows the relative errors of the HRKPM with different number of nodes when $d_{\max } = {1}{\text{.1}}$, $\Delta t = 0.001$, $T = 0.1$, and $l$ is the is the distance between adjacent nodes.

Fig. 2

Relative errors versus scale parameter $d_{{\text{max}}}$

Fig. 3

Relative errors versus penalty factor $\alpha$

Fig. 4

Relative errors versus time step length $\Delta t$

Fig. 5

Relative error versus the distance between adjacent nodes $l$

From Figs. 2 and 3, we can know that the HRKPM can obtain higher computational accuracy when $d_{\max }$ is in the interval from ${1}{\text{.07}}$ to $1.{5}$, $\alpha$ is in the interval from ${2} \times {10}^{{2}}$ to ${1} \times {10}^{{3}}$. It can be seen from Fig. 4 that when the time step length $\Delta t$ is less than 0.001, the relative error tends to be stable. It can be seen from Fig. 5 that when there are more nodes in the domain, the distance between nodes is smaller, and the computational accuracy is higher. Therefore, the numerical result is convergent about the distance between nodes, and the order of convergence is 2.

In this example, we also use following polynomial basis [53]

$${\mathbf{p}}^{{\text{T}}} (x_{1} ,x_{2} ) = (1,(x_{1} - x^{\prime}_{1} )/h,(x_{2} - x^{\prime}_{2} )/h),$$

(93)

where the fill distance $h$ for a set of nodes $\left\{ {{\mathbf{x}}_{i} } \right\}_{i = 1}^{M}$ in the subdomain $\Omega ^{{{(}k{)}}}$ can be defined as [10]

$$h = \mathop {\sup }\limits_{{{\mathbf{x}} \in \Omega^{(k)} }} \mathop {\min }\limits_{1 \le j \le M} \left\| {{\mathbf{x}} - {\mathbf{x}}_{j} } \right\|_{2} .$$

(94)

When the basis function in Eq. (93) is employed to the HRKPM, the factors affecting the relative errors, such as the scale parameter of the influence domain $d_{\max }$ and penalty factor $\alpha$, are discussed. Figure 6 shows the relative errors of the HRKPM with different $d_{\max }$ when $\alpha = {4}{\text{.4}} \times 10^{{2}}$, $\Delta t = 0.001$, $T = 0.1$ and the node distribution is ${11} \times {11} \times {11}$. Figure 7 shows the relative errors of the HRKPM with different $\alpha$ when $d_{\max } = {1}{\text{.18}}$, $\Delta t = 0.001$, $T = 0.1$ and the node distribution is ${11} \times {11} \times {11}$. From Figs. 6 and 7, the HRKPM has higher computational accuracy when $d_{\max }$ is in the interval from ${1}{\text{.01}}$ to $1.{25}$, $\alpha$ is in the interval from ${2} \times {10}^{{2}}$ to ${6} \times {10}^{{2}}$.

Fig. 6

Relative errors using the basis function in Eq. (93) versus scale parameter $d_{{\text{max}}}$

Fig. 7

Relative errors using the basis function in Eq. (93) versus penalty factor $\alpha$

Using the original basis function in Eq. (23) for this example, when $L = {10}$, node distribution in each subdomain $\Omega^{(k)}$ is ${11} \times {11}$, $d_{\max } = 1.{1}$, $\alpha = {4}{\text{.36}} \times 10^{{2}}$, $\Delta t = 0.0{0}1$ and $T = 0.1$, the relative error is ${0}{\text{.0016}}$, the CPU time is ${19}{\text{.65}}$. Using the basis function in Eq. (93) for this example, when $L = {10}$, node distribution in each subdomain $\Omega^{(k)}$ is ${11} \times {11}$, $d_{\max } = 1.{18}$, $\alpha = {4}{\text{.74}} \times 10^{{2}}$, $\Delta t = 0.0{0}1$ and $T = 0.1$, the relative error is ${0}{\text{.0082}}$, the CPU time is ${19}{\text{.89}}$. The comparison of the results shows that the original basis function can obtain higher computational accuracy than the basis function in Eq. (93). Therefore, the original basis function is used in this paper.

Table 1 lists the numerical results of the HRKPM, the RKPM and the EFG method at $T = 0.1$, $T = 0.{3}$ and $T = 0.{5}$. It is seen that the HRKPM can obtain more accurate and efficient numerical results. Especially when the time value is larger, the superiority of the HRKPM in computing speed is more obvious.

Table 1 The relative errors and CPU time of the HRKPM, the RKPM and the EFG method at different times

Table 2 shows the relative errors and CPU time of the HRKPM with different Gaussian points. From this table, we can see that when we choose more Gaussian points, the computational accuracy is improved, but the CPU time also increases. To balance the computational efficiency and the computational accuracy, we need to select the appropriate Gaussian points to compute the appeared integrals in Eqs. (55)–(60).

Table 2 The relative errors and CPU time of the HRKPM with different Gaussian points

Figure 8 shows that the numerical solution obtained by the HRKPM, the RKPM and the EFG method are in good agreement with the analytical solution in the direction $x_{{1}}$, $x_{{2}}$ and $x_{{3}}$ at $T = 0.1$, $T = 0.{3}$ and $T = 0.{5}$. Compared with the RKPM and the EFG method, the HRKPM is implemented in a more time-saving way.

Fig. 8

Wave propagation obtained by the HRKPM, the RKPM and the EFG method a in the direction $x_{{1}}$; b in the direction $x_{{2}}$ and c in the direction $x_{{3}}$

4.2 Wave propagation problem with mixed boundary conditions

In the 3D domain $\Omega = [0,{\uppi }] \times [0,{\uppi }] \times [0,{\uppi ]}$, the hyperbolic equation governing the wave propagation $u$ is [19]

$$\frac{{\partial^{{2}} u}}{{\partial t^{{2}} }} = {6}\left( {\frac{{\partial^{2} u}}{{\partial x_{{1}}^{{2}} }} + \frac{{\partial^{2} u}}{{\partial x_{{2}}^{{2}} }} + \frac{{\partial^{2} u}}{{\partial x_{{3}}^{{2}} }}} \right),\quad ({(}x_{{1}} {,}x_{{2}} {,}x_{{3}} {)} \in {\Omega ,}\;t \in [0,T]),$$

(95)

with the boundary conditions

$$u(0,x_{2} ,x_{3} ,t) = u(\uppi ,x_{2} ,x_{3} ,t) = u(x_{1} ,0,x_{3} ,t) = u(x_{1} ,\uppi ,x_{3} ,t) = 0,$$

(96)

$$\frac{{\partial u(x_{1} ,x_{2} ,0,t)}}{{\partial x_{{3}} }} = \frac{{\partial u(x_{1} ,x_{2} ,\uppi ,t)}}{{\partial x_{{3}} }} = 0,$$

(97)

and the initial conditions

$$u(x_{1} ,x_{2} ,x_{3} ,0) = \sin x_{1} \sin x_{2} \cos (2x_{3} ),$$

(98)

$$\frac{{\partial u(x_{1} ,x_{2} ,x_{3} ,0)}}{\partial t} = 0.$$

(99)

The analytical solution used to compare with the approximate solution is

$$u(x_{1} ,x_{2} ,x_{3} ,t) = \sin x_{1} \sin x_{2} \cos (2x_{3} )\cos (6t).$$

(100)

In the HRKPM scheme for this example, the discussion about different types of boundary conditions is applied.

• 1. When Neumann boundary condition exists in the splitting direction

  Set $x_{{3}}$ as splitting direction, the boundary conditions in this direction are $\frac{{\partial u(x_{{1}} ,x_{2} ,{0})}}{{\partial x_{{3}} }} = {0}$ and $\frac{{\partial u(x_{{1}} ,x_{2} ,{\uppi })}}{{\partial x_{{3}} }} = {0}$. It means that the smaller $\Delta x_{{3}}$ is, the closer the value of $u$ on plane $x_{{3}} = {0}$ is to that on plane $x_{{3}} = \Delta x_{{3}}$, and the closer the value of $u$ on plane $x_{{3}} = {\uppi }$ is to that on plane $x_{{3}} = {{\uppi - }}\Delta x_{{3}}$.

  When $L = {100}$, node distribution in each subdomain is ${11} \times {11}$, $d_{\max } = 1.{05}$, $\alpha = {1}{\text{.01}} \times 10^{{4}}$, $\Delta t = 0.001$ and $T = 0.1$, the relative error is ${0}{\text{.0062}}$, the computing time is ${15}{\text{.82}}$.

• 2. When Dirichlet boundary condition exists in the splitting direction

Set $x_{{2}}$ as splitting direction, the boundary conditions in this direction are $u(x_{1} ,0,x_{3} ,t) = 0$ and $u(x_{1} ,\uppi ,x_{3} ,t) = 0$, which are exactly the values of $u$ on plane $x_{{2}} = {0}$ and $x_{{2}} = {\uppi }$, respectively.

When $L = {10}$, node distribution in each sudomain is ${11} \times {11}$, $d_{\max } = 1.{23}$, $\alpha = {1}{\text{.7}} \times 10^{{3}}$, $\Delta t = 0.001$ and $T = 0.1$, the relative error is ${0}{\text{.0024}}$, the computing time is ${1}{\text{.1}}$.

According to the above description and discussion, better results are obtained when Dirichlet boundary condition is set as the splitting direction. Therefore, to obtain higher accuracy, it is necessary to choose more suitable splitting direction according to boundary conditions.

We employ the HRKPM, the RKPM and the EFG method to solve this 3D example. A good agreement between the numerical solutions obtained by these methods and the analytical solution are shown in Fig. 9 when $T = 0.1$ and $T = 0.{5}$. Moreover, the HRKPM has higher numerical efficiency, which is exactly what we want to achieve with the proposed method.

Fig. 9

Wave propagation obtained by the HRKPM, the RKPM and the EFG method a in the direction $x_{{1}}$; b in the direction $x_{{2}}$ and c in the direction $x_{{3}}$

4.3 Wave propagation problem in cylindrical coordinates

Consider the following equation in cylindrical coordinates [50]

$$\frac{{\partial^{2} u}}{{\partial t^{2} }} = \frac{{\partial^{2} u}}{{\partial r^{2} }} + \frac{{\partial^{2} u}}{{\partial \theta^{2} }} + \frac{{\partial^{2} u}}{{\partial x_{{3}}^{{2}} }} + u,\quad (r \in [1,2],\;\theta \in [0,{\uppi }],\;x_{3} \in [0,1]),$$

(101)

with the boundary conditions

$$u({1},\theta ,x_{3} ,t) = (\sin \theta + x_{3} ) \cdot e^{t} ,$$

(102)

$$T({2},\theta ,x_{3} ,t) = x_{3} \cdot e^{t} ,$$

(103)

$$T(r,0,x_{3} ,t) = x_{3} \cdot e^{t} ,$$

(104)

$$T(r,{\uppi },x_{3} ,t) = x_{3} \cdot e^{t} ,$$

(105)

$$T(r,\theta ,0,t) = \frac{4}{3}\left( {\frac{1}{r} - \frac{r}{4}} \right)\sin \theta \cdot e^{t} ,$$

(106)

$$T(r,\theta ,1,t) = \left[ {\frac{4}{3}\left( {\frac{1}{r} - \frac{r}{4}} \right)\sin \theta + 1} \right] \cdot e^{t} ,$$

(107)

and the initial conditions

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

(108)

The analytical solution used to compare with the approximate solution is

$$u(r,\theta ,x_{3} ,t) = \left[ {\frac{4}{3}\left( {\frac{1}{r} - \frac{r}{4}} \right)\sin \theta + x_{3} } \right] \cdot e^{t} .$$

(109)

In the HRKPM scheme for this example, the step number in the selected splitting direction $x_{{3}}$ is 20, the node distribution in each 2D subdomain $\Omega^{(k)}$ is $9 \times 31$. The arrangement of nodes can be shown in Fig. 10. Using the RKPM and the EFG method for this example, the node distribution in the 3D domain $\Omega$ is ${9} \times {31} \times {21}$.

Fig. 10

Node distribution in 2D subdomain of a half torus

Table 3 lists the numerical results obtained by the HRKPM, the RKPM and the EFG method at $T = 0.01$, $T = 0.0{3}$ and $T = 0.0{5}$. We can observe that the HRKPM has higher computational accuracy and efficiency. Especially when the time value is larger, the superiority of the HRKPM in computing speed is more obvious.

Table 3 The relative errors and CPU time of the HRKPM, the RKPM and the EFG method at different times

We employ the HRKPM, the RKPM and the EFG method to solve this 3D example. A good agreement between the numerical solutions obtained by these methods and the analytical solution are shown in Fig. 11. Again, the HRKPM has higher computational efficiency than the RKPM and the EFG method for solving 3D wave propagation problems.

Fig. 11

Wave propagation obtained by the HRKPM, the RKPM and the EFG method a in the direction $r$; b in the direction $\theta$ and c in the direction $x_{{3}}$

For randomly node distribution in each subdomain, as shown in Fig. 12, when the node distribution is ${9} \times {31} \times {21}$, the relative errors of the HRKPM, the RKPM and the EFG method are ${4}{\text{.6}} \times {10}^{{ - 4}}$, ${0}{\text{.0011}}$ and ${0}{\text{.001}}$, respectively. The CPU times are ${28}{\text{.5}}$, ${780}{\text{.53}}$ and ${515}{\text{.67}}$, respectively. Based on the results, we can know that the HRKPM can also obtain higher computational efficiency than the RKPM and the EFG method with randomly node distribution.

Fig. 12

Randomly distribution in 2D subdomain of a half torus

The numerical solutions in the directions $r$, $\theta$ and $x_{3}$ obtained by the HRKPM, the RKPM and the EFG method with randomly node distribution are presented in Fig. 13. And these numerical solutions are in agreement with the analytical one.

Fig. 13

Wave propagation obtained by the HRKPM, the RKPM and the EFG method a in the direction $r$; b in the direction $\theta$ and c in the direction $x_{{3}}$

4.4 Wave propagation problem with non-smooth solution

In the 3D domain $\Omega = [0,{\uppi }] \times [0,{\uppi }] \times [0,{\uppi ]}$, the hyperbolic equation governing the wave propagation $u$ is

$$\frac{{\partial^{{2}} u}}{{\partial t^{{2}} }} = \frac{{\partial^{2} u}}{{\partial x_{{1}}^{{2}} }} + \frac{{\partial^{2} u}}{{\partial x_{{2}}^{{2}} }} + \frac{{\partial^{2} u}}{{\partial x_{{3}}^{{2}} }} + \cos x_{1} + \cos x_{2} ,\quad ({(}x_{{1}} {,}x_{{2}} {,}x_{{3}} {)} \in {\Omega ,}\;t \in [0,T]),$$

(110)

with the boundary conditions

$$\frac{{\partial u(0,x_{2} ,x_{3} ,t)}}{{\partial x_{{1}} }} = \frac{{\partial u({\uppi },x_{2} ,x_{3} ,t)}}{{\partial x_{{1}} }} = \frac{{\partial u(x_{1} ,{0},x_{3} ,t)}}{{\partial x_{{2}} }} = \frac{{\partial u(x_{1} ,\uppi ,x_{3} ,t)}}{{\partial x_{{2}} }} = {0,}$$

(111)

$$u(x_{1} ,x_{{2}} ,{0},t) = u(x_{1} ,x_{{2}} ,{\uppi },t) = \cos x_{1} + \cos x_{2} ,$$

(112)

and the initial conditions

$$u(x_{1} ,x_{2} ,x_{3} ,0) = \cos x_{1} + \cos x_{2} ,$$

(113)

$$\frac{{\partial u(x_{1} ,x_{2} ,x_{3} ,0)}}{\partial t} = \left\{ {\begin{array}{*{20}c} {\sin x_{3} } & {x_{1} < \frac{8}{10}{\uppi }} \\ {5\sin 5x_{3} } & {x_{1} \ge \frac{8}{10}{\uppi }} \\ \end{array} } \right..$$

(114)

The analytical solution used to compare with the approximate solution is

$$u(x_{1} ,x_{2} ,x_{3} ,t) = \left\{ {\begin{array}{*{20}l} {\cos x_{1} + \cos x_{2} + \sin x_{3} \cdot \sin t} & {x_{1} < \frac{8}{10}{\uppi }} \\ {\cos x_{1} + \cos x_{2} + \sin 5x_{3} \cdot \sin 5t} & {x_{1} \ge \frac{8}{10}{\uppi }} \\ \end{array} } \right..$$

(115)

Using the HRKPM scheme for this example, when $L = {10}$, the node distribution in each subdomain $\Omega^{(k)}$ is ${11} \times {11}$, $d_{\max } = 1.{5}$, $\Delta t = 0.01$ and $T = 0.1$, the relative error is ${0}{\text{.0036}}$, the CPU time is ${11}{\text{.12}}$. Using the RKPM scheme for this example, when the node distribution in 3D domain $\Omega$ is ${11} \times {11} \times {11}$, $d_{\max } = 1.{21}$, $\alpha = {2}{\text{.7}} \times {10}^{{2}}$, $\Delta t = 0.01$ and $T = 0.1$, the relative error is ${0}{\text{.0037}}$, the CPU time ${118}{\text{.1}}$. Using the EFG method for this example, when the node distribution in 3D domain $\Omega$ is ${11} \times {11} \times {11}$, $d_{\max } = 1.{01}$, $\alpha = {2}{\text{.6}} \times {10}^{{2}}$, $\Delta t = 0.01$ and $T = 0.1$, the relative error is ${0}{\text{.0052}}$, the CPU time is ${36}{\text{.04}}$. From these results, it can be seen that the HRKPM is more efficient than the RKPM and the EFG method.

We employ the HRKPM, the RKPM and the EFG method to solve this 3D example. A good agreement between the numerical solutions obtained by these methods and the analytical solution are shown in Fig. 14. Again, the HRKPM has higher computational efficiency than the RKPM and the EFG method for solving 3D wave propagation problems.

Fig. 14

Wave propagation obtained by the HRKPM, the RKPM and the EFG a in the direction $x_{{1}}$; b in the direction $x_{{2}}$ and c in the direction $x_{{3}}$

5 Conclusions

This paper studies the HRKPM for solving 3D wave propagation problems. Four selected numerical examples are used to evaluate the effectiveness and superiority of the proposed method. Compared with the RKPM and the EFG method, the following conclusions can be obtained:

• 1. The HRKPM has greater computational precision when $d_{\max }$ is in the interval from ${1}{\text{.07}}$ to $1.{5}$, $\alpha$ is in the interval from ${2} \times {10}^{{2}}$ to ${1} \times {10}^{{3}}$, the number of nodes increase or the time step length decrease. Moreover, the proper splitting direction is also one of the factors affecting the calculation accuracy.

• 2. The HRKPM greatly improves the computational efficiency. Especially when the time value is larger, the superiority of HRKPM in computing speed is more obvious.