“Mixed” Meshless Time-Domain Adaptive Algorithm for Solving Elasto-Dynamics Equations

Abstract

A time-domain adaptive algorithm was developed for solving elasto-dynamics problems through a mixed meshless local Petrov-Galerkin finite volume method (MLPG5). In this time-adaptive algorithm, each time-dependent variable is interpolated by a time series function of n-order, which is determined by a criterion in each step. The high-order series of expanded variables bring high accuracy in the time domain, especially for the elasto-dynamic equations, which are second-order PDE in the time domain. In the present mixed MLPG5 dynamic formulation, the strains are interpolated independently, as are displacements in the local weak form, which eliminates the expensive differential of the shape function. In the traditional MLPG5, both shape function and its derivative for each node need to be calculated. By taking the Heaviside function as the test function, the local domain integration of stiffness matrix is avoided. Several numerical examples, including the comparison of our method, the MLPG5–Newmark method and FEM (ANSYS) are given to demonstrate the advantages of the presented method: (1) a large time step can be used in solving a elasto-dynamics problem; (2) computational efficiency and accuracy are improved in both space and time; (3) smaller support sizes can be used in the mixed MLPG5.

1. Introduction

Structural vibration analysis is an important system dynamics problem in engineering. This dynamics problem is governed by partial differential equations of elasto-dynamics associated with a group of boundary conditions and initial conditions. The elasto-dynamics equation is a second-order PDE in both time and space domains. Exact analyses are usually very difficult, and only few analytical solutions are obtained [1]. Therefore, numerical methods have been developed to solve these complex problems, such as the finite difference method (FDM) [2], the stepwise integration method [3], the Newmark method [4], the Wilson-θ method and the Houbolt method [5] for the time domain; and the FDM [6], the finite element method (FEM) [7], the boundary element method (BEM) [8,9,10,11], the differential quadrature method (DQM) [12] and the meshless methods (MMs) [2] for the space domain.

Since the dynamics equation of vibration is a high-order partial differential equation in the time domain, many efforts have been made studying the time-domain methods [13]. Thus, the methods that are commonly used in solving vibration problems can be divided into two categories: mode superposition methods and direct integration methods. In a mode superposition method, the computational cost comes from solving the n-order generalized eigenvalue problem. If the structure vibrates in a short period, this method is less efficient than a direct integration method. In contrast, if the vibration lasts for some time, it is more effective to use the mode superposition method. In the direct integration method, the dynamics equation is directly solved by integration of, e.g., the FDM [2], the Newmark method [4], the Wilson-θ method or the Houbolt method [5]. In these methods, the computational cost in one time step is proportional to the product of the freedom degrees and the square of the average bandwidth of the matrix. All variables are usually constant or linear in every time step, which may lead to inefficient calculation, incomplete accuracy and even divergent results if the time step is selected improperly [14].

To improve the computing accuracy and reduce the error caused by an improper time step size, the time-domain adaptive algorithm was proposed by Haitian. Y. It is an unconditionally stable direct integration method [15]. Differently from the difference method [16], all the time-dependent variables are expanded into series in discrete time steps, and the expansion coefficients can be obtained by solving the recursive equation. Therefore, the variables in each discrete time step can be more accurately described without any assumptions made for nonlinearity. Since the computing accuracy can be controlled by the truncation error, the time step size can be freely selected in a large range. This means large time steps are allowed in the time domain. This method has been successfully used together with FEM in a large time step and has shown advantages in terms of calculation accuracy [17].

Besides the discretization in the time domain, the elasto-dynamics equations also need to be discretized in the space domain. Although the FEM is the most widely used method for structure vibration analysis in engineering, it still encounters many challenges due to the mesh distortion and remeshing when solving large deformation problems, such as high-speed impact, dynamic crack propagation and strain localization [18]. However, these disadvantages of the FEM can be avoided in MMs, as they do not need elements. Over the past two decades, some efforts were devoted to solving the elasto-dynamics equations with MMs [2].

Speaking of MMs, Atluri proposed the meshless local Petrov-Galerkin Method (MLPG) to avoid the background mesh [19]. In this method, the idea of eliminating residuals in the subdomain was firstly proposed. When combined with the moving least squares (MLS) approximation, a true meshless method is realized which does not need interpolation mesh or integral mesh [20]. Since the MLPG method establishes a residual equation on each subdomain separately, the equations from different subdomains are relatively independent, so different weighted residual methods can be easily mixed and used. Additionally, it provides a good platform for the coupling of various methods [21]. Nowadays, the MLPG is a general term for a series of methods (MLPG1–MLPG6) [22]. Nevertheless, MLPG has the problem of low efficiency. An effective way to overcome this shortcoming is to eliminate or simplify the domain integral in the stiffness matrix. In the above six methods, MLPG2, MLPG4 and MLPG5 have no domain integrals. However, MLPG2 relies too much on the configuration of nodes, and MLPG4 has singular integrals. Only MLPG5 includes neither domain integration in the stiffness matrix nor singularity integration, along with only local boundary integrals. Therefore, the MLPG5 is an attractive method that has high computational efficiency [23]. In this paper, MLPG5 is developed for solving elasto-dynamics equations through a “mixed” approach. Independent meshless approximations are used for both strains and displacements. The strain-displacement compatibility is enforced at nodal points by using the collocation method; thus, the independent nodal strains are expressed in terms of nodal displacements. The “mixed” approach eliminates the expensive process of differentiating the shape function, which greatly increases the computational efficiency.

In this paper, attention is devoted to the meshless time-domain adaptive method for structural vibration analyses of two-dimensional solids. Local weak forms are developed using the weighted residual method from the elasto-dynamics equations. In Section 2, the MLS approximation is introduced to establish shape functions for a set of regularly or randomly distributed nodes, and the Heaviside function is used as a test function. In Section 3, all the time related-variables are expanded in every time step, and then the spatiotemporally coupled dynamics equations are converted into a series of recursively solved spatial problems. In Section 4, the validity and accuracy of the proposed method are verified by several numerical examples.

2. Moving Least Square (MLS) Approximation

The moving least square (MLS) interpolation is generally considered to be one of the best schemes with which to interpolate random data with a reasonable accuracy, because of its completeness, robustness and continuity [24]. In this section, a briefing of MLS approximation is given. Consider a local sub-domain ${\mathsf{\Omega }}_s$, which is the neighborhood of a point $\mathit{X}=\left[x,y\right]$. The distribution of a function $u$ can be approximated, over a number of scattered local points $\left\{x_i\right\}, \left(i=1,2,\dots ,n\right)$, as

$$u\left(\mathit{X}\right)\approx u^h\left(\mathit{X}\right)={\displaystyle \sum }_{j=1}^m{p}_j\left(\mathit{X}\right)a_j\left(\mathit{X}\right)={\mathit{p}}^{\mathit{T}}\left(\mathit{X}\right)\mathit{a}\left(\mathit{X}\right)$$

(1)

where $\mathit{p}\left(\mathit{X}\right)$ is a monomial basis function of order $m$. In two dimensions, it is given by

$${\mathit{p}}^{\mathit{T}}\left(\mathit{X}\right)=\left[1,x,y,x^2,xy,y^2,\dots \right]$$

(2)

The vector $\mathit{a}\left(\mathit{X}\right)$ containing coefficients are functions of the global Cartesian coordinates, depending on the monomial basis. They are determined by minimizing a weighted discrete $L_2$ norm, defined as:

$$J={\displaystyle \sum }_{i=1}^n\stackrel{◠}{W}\left(\mathit{X}-{\mathit{X}}_{\mathit{i}}\right){\left[{\mathit{p}}^{\mathit{T}}\left({\mathit{X}}_{\mathit{i}}\right)\mathit{a}\left(\mathit{X}\right)-u_i\right]}^2$$

(3)

where $\stackrel{◠}{W}$ are the weight functions and $u_i$ are the fictitious nodal values.

The stationarity of $J$ in Equation (3) with respect to $\mathit{a}\left(\mathit{X}\right)$ leads to the following relationship:

$$\mathit{a}\left(\mathit{X}\right)={\mathit{A}}^{-1}\left(\mathit{X}\right)\mathit{B}\left(\mathit{X}\right){\mathit{U}}_{\mathit{s}}$$

(4)

Substituting $\mathit{a}\left(\mathit{X}\right)$ into Equation (1), we have

$$u^h\left(X\right)={\displaystyle \sum }_{i=1}^n{\varphi }_i\left(\mathit{X}\right)u_i$$

(5)

where the MLS shape function ${\varphi }_i\left(\mathit{X}\right)$ can be defined as:

$${\varphi }_i\left(\mathit{X}\right)={\displaystyle \sum }_{j=1}^m{p}_j\left(\mathit{X}\right){\left({\mathit{A}}^{-1}\left(\mathit{X}\right)\mathit{B}\left(\mathit{X}\right)\right)}_{ji}={\mathit{p}}^{\mathit{T}}\left(\mathit{X}\right){\left({\mathit{A}}^{-1}\mathit{B}\right)}_i$$

(6)

where $\mathit{A}\left(\mathit{x}\right)$ and $\mathit{B}\left(\mathit{x}\right)$ are defined by

$$\mathit{A}\left(\mathit{X}\right)={\displaystyle \sum }_{i=1}^n\stackrel{◠}{W}\left(\mathit{X}-{\mathit{X}}_{\mathit{i}}\right)p\left(X_i\right)\mathit{p}\left(\mathit{X}\right){\mathit{p}}^{\mathit{T}}\left({\mathit{X}}_{\mathit{i}}\right)={\displaystyle \sum }_{\mathit{i}=1}^{\mathit{n}}\stackrel{◠}{W}\left(\mathit{X}-{\mathit{X}}_{\mathit{i}}\right)\left[\begin{array}{ccc}1& x_i& y_i\\ x_i& x_i^2& x_i{y}_i\\ y_i& x_i{y}_i& y_i^2\end{array}\right]$$

(7)

$$\mathit{B}\left(\mathit{X}\right)=\left[W_1\left(\mathit{X}\right)\mathit{p}\left({\mathit{X}}_1\right),W_2\left(\mathit{X}\right)\mathit{p}\left({\mathit{X}}_2\right),\dots ,W_n\left(\mathit{X}\right)\mathit{p}\left({\mathit{X}}_{\mathit{n}}\right)\right]$$

(8)

The weight function is very important for the MLS interpolation, because the smoothness of the shape function and its derivatives depends on the order of the weight function. In two-dimensional problems, discontinuities in derivatives can be produced if the order of the spline is not sufficient, and unwanted oscillations in nodal shape functions are produced when a high order of spline function is used. It has been found that the best nodal shape function and its first derivative come from 4th order spline function [25]. Thus, in this paper, the following 4-order spline function is used:

$${\stackrel{◠}{W}}_i\left(\mathit{X}\right)=\{\begin{array}{c}1-6{\left(\frac{d_i}{r_w}\right)}^2+8{\left(\frac{d_i}{r_w}\right)}^3-3{\left(\frac{d_i}{r_w}\right)}^4 0\le d_i\le r_w\\ 0 d_i>r_w\end{array}$$

(9)

where $d_i=\left|{\mathit{x}}_Q-{\mathit{x}}_i\right|$ is the distance from node ${\mathit{x}}_i$ to the sampling point ${\mathit{x}}_Q$, and $r_w$ is the support size for the weight function.

3. Recursive Governing Equations

3.1. Recursive Elasto-Dynamics Equations

Consider a linear elastic body in a 2D domain $\mathsf{\Omega }$, with a boundary $\Gamma$, shown in Figure 1. The solid is assumed to undergo infinitesimal deformations. The governing differential equation for small displacement elasto-dynamics can be written as:

$${\sigma }_{ij,j}+b_i-\rho {\ddot{u}}_i-c{\dot{u}}_i=0; {\sigma }_{ij}={\sigma }_{ji}$$

(10)

where ${\sigma }_{ij}$ is the stress tensor, which corresponds to the displacement field $u_i$; $b_i$ is the body force; $\rho$ is the mass density; $c$ is the damping coefficient; ${\dot{u}}_i=\frac{\partial u_i}{\partial t}$ is the velocity; ${\ddot{u}}_i=\frac{{\partial }^2{u}_i}{\partial t^2}$ is the acceleration.

The boundary conditions are given as follows,

$$u_i={\tilde{u}}_i on {\Gamma }_u$$

(11)

$$t_i\equiv n_i{\sigma }_{ij}={\tilde{p}}_i on {\Gamma }_t$$

(12)

where ${\overline{u}}_i$ and ${\overline{t}}_i$ are the prescribed displacements and tractions, respectively, on the displacement boundary ${\Gamma }_u$ and on the traction boundary ${\Gamma }_t$; and $n_i$ is the unit outward normal to the boundary $\Gamma$. The initial conditions are defined by

$$\mathit{u}\left(\mathit{X},t_0\right)={\mathit{u}}_{\mathit{i}\mathit{n}}\left(\mathit{X}\right) \mathit{X}\in \Omega$$

(13)

$$\dot{\mathit{u}}\left(\mathit{X},t_0\right)={\mathit{v}}_{\mathit{i}\mathit{n}}\left(\mathit{X}\right) \mathit{X}\in \Omega$$

(14)

With ${\mathit{u}}_{\mathit{i}\mathit{n}}$ and ${\mathit{v}}_{\mathit{i}\mathit{n}}$ being the initial displacements and velocities at the initial time $t_0$, respectively.

To improve the computing accuracy, exploiting a discretized expanding technique is of interest. At each discretized time subdomain, all variables can be described as

$$\sigma ={\displaystyle \sum }_{m=0}{\sigma }_m{s}^m$$

(15)

$$\epsilon ={\displaystyle \sum }_{m=0}{\epsilon }_m{s}^m$$

(16)

$$c={\displaystyle \sum }_{m=0}{c}_m{s}^m$$

(17)

$$b={\displaystyle \sum }_{m=0}{b}_m{s}^m$$

(18)

$$u={\displaystyle \sum }_{m=0}{u}_m{s}^m$$

(19)

$$\tilde{u}={\displaystyle \sum }_{m=0}{\tilde{u}}_m{s}^m$$

(20)

$$\tilde{p}={\displaystyle \sum }_{m=0}{\tilde{p}}_m{s}^m$$

(21)

$$s=\frac{t-t_{k-1}}{{{\rm T}}_k}$$

(22)

where $t_{k-1}$ and ${{\rm T}}_k$ represent the beginning time and size of time step, respectively; $\tilde{u}$ and $\tilde{p}$ represent the prescribed displacement and traction on the boundary, respectively; and ${\sigma }_m$, ${\epsilon }_m$, $c_m$, $b_m$, $u_m$, ${\tilde{u}}_m$, $p_m$ and ${\tilde{p}}_m$ are the expanding coefficients of $\sigma$, $\epsilon$, c, $b$, $u$, $\tilde{u}$, $p$ and $\tilde{p}$, respectively.

The derivative with respect to t can be written in the form

$$\frac{d}{dt}=\frac{1}{{{\rm T}}_k}\frac{d}{ds}$$

(23)

Substitute Equations (15), (18) and (19) into Equation (10), and obtain

$${\displaystyle \sum }_{m=0}{\sigma }_m{}_{ij,j}{s}^m+{\displaystyle \sum }_{m=0}{b}_m{}_i{s}^m-\rho {\displaystyle \sum }_{m=0}\frac{\left(m+2\right)\left(m+1\right)}{{{\rm T}}_k^2}{u}_{m+2}{}_i{s}^m-c{\displaystyle \sum }_{m=0}\frac{m+1}{{{\rm T}}_k}{u}_{m+1}{}_i{s}^m=0$$

(24)

Equate every power of $s^m, m=1,2,3\dots$, and obtain

$${\sigma }_m{}_{ij,j}+b_m{}_i-\rho \frac{\left(m+2\right)\left(m+1\right)}{{{\rm T}}_k^2}{u}_{m+2}{}_i-c\frac{m+1}{{{\rm T}}_k}{u}_{m+1}{}_i=0$$

(25)

Equation (25) is the recursive governing equation by the time-domain adaptive method.

In the first time step, $u_0$ and $u_1$ are the initial conditions of displacement $u$ and velocity $\dot{u}$, which can be described as:

$$u_0=u\left(X,0\right)$$

(26)

$$u_1=\dot{u}\left(X,0\right)T_k$$

(27)

Then, $u_m$ can be obtained by solving Equation (37) iteratively. In the $\left(m+1\right)\mathrm{th}$ time step, the displacement and velocity can be obtained by:

$$u_{n+1}={\displaystyle \sum }_{m=0}^n{u}_{m_k}$$

(28)

$${\dot{u}}_{n+1}=\frac{1}{T_k}{\displaystyle \sum }_{m=0}^{n}m{u}_{m_k}$$

(29)

In each time step, the expanded order $m$ could be obtained adaptively from the following criteria:

$$\frac{||u_m{s}^m{||}_2}{||{{\displaystyle \sum }}_0^{m-1}{u}_m{s}^m{||}_2}\le \beta$$

(30)

where $\beta$ is an error bound (for example $\beta ={10}^{-6}$)-; $||{||}_2$ represents the 2-norm of the matrix.

3.2. Numerical Implementation of MLPG5

In the MLPG approaches, one may write a weak form over a local sub-domain ${\mathsf{\Omega }}_{\mathrm{s}}$, which can be: (1) a circle, (2) an ellipse, (3) a rectangle or any other regular or irregular shape. A generalized local weak form of the differential Equation (25) over a local sub-domain ${\mathsf{\Omega }}_{\mathrm{s}}$ can be written as:

$${\displaystyle {{\displaystyle \int }}_{{\Omega }_q}}{W}_I({\sigma }_m{}_{ij,j}+b_m{}_i-\rho \frac{\left(m+2\right)\left(m+1\right)}{{{\rm T}}_k^2}{u}_{m+2}{}_i-c\frac{m+1}{{{\rm T}}_k}{u}_{m+1}{}_i)d\Omega =0$$

(31)

where $W_I$ is the test function. In mixed MLPG5, the Heaviside function is used as the test function in the local weak form. It is defined as:

$$W_I=\{\begin{array}{c}1 inside {\Omega }_q\\ 0 outside {\Omega }_q\end{array}$$

(32)

By substituting Equation (32) into Equation (31) and applying the divergence theorem, Equation (31) may be rewritten in a symmetric weak form as:

$${{\displaystyle \int }}_{{\Omega }_q}\left(\rho \frac{\left(m+2\right)\left(m+1\right)}{{{\rm T}}_k^2}{u}_{m+2}{}_i+c\frac{m+1}{{{\rm T}}_k}{u}_{m+1}{}_i\right)d\Omega -{{\displaystyle \int }}_{{\Gamma }_{qi}}{\sigma }_m{}_{ij}{n}_{j}d\Gamma -{{\displaystyle \int }}_{{\Gamma }_{qu}}{\sigma }_m{}_{ij}{n}_{j}d\Gamma ={{\displaystyle \int }}_{{\Gamma }_{qt}}{\sigma }_m{}_{ij}{n}_{j}d\Gamma +{{\displaystyle \int }}_{{\Omega }_q}{b}_m{}_{i}d\Omega$$

(33)

where ${\Gamma }_{qi}$ is a part of the local boundary, which is inside the solution domain; ${\Gamma }_{qu}$ is the intersection between the local boundary and the global displacement boundary; and ${\Gamma }_{qt}$ is a part of the boundary over which the natural boundary conditions are specified, as shown in Figure 1.

In this paper, it is assumed that the body force is zero. By substituting the Equation (12) into Equation (33), we have:

$${{\displaystyle \int }}_{{\Omega }_q}\left(\rho \frac{\left(m+2\right)\left(m+1\right)}{{{\rm T}}_k^2}{u}_{m+2}{}_i+c\frac{m+1}{{{\rm T}}_k}{u}_{m+1}{}_i\right)d\Omega -{{\displaystyle \int }}_{{\Gamma }_{qi}}{t}_{m_i}d\Gamma -{{\displaystyle \int }}_{{\Gamma }_{qu}}{t}_{m_i}d\Gamma ={{\displaystyle \int }}_{{\Gamma }_{qt}}{\tilde{p}}_{m_i}d\Gamma$$

(34)

with the constitutive relations of an isotropic linear elastic homogeneous solid, the tractions in Equation (34) can be written in terms of the strains:

$$t_{m_i}=\mathit{n}{\mathit{\sigma }}_{\mathit{m}}=\mathit{n}\mathit{D}{\mathit{\epsilon }}_{\mathit{m}}$$

(35)

$${\mathit{\epsilon }}_{\mathit{m}}=\mathit{L}{\mathit{u}}_{\mathit{m}}$$

(36)

where

$$\mathit{n}=\left[\begin{array}{ccc}{n}_x& 0& n_y\\ 0& n_y& n_x\end{array}\right]$$

(37)

$$\mathit{D}=\frac{E}{1-{\nu }^2}\left[\begin{array}{ccc}1& \nu & 0\\ \nu & 1& 0\\ 0& 0& \frac{1-\nu }{2}\end{array}\right] for plane stress$$

(38)

In the mixed MLPG5 method, the interpolation of nodal displacements ${\mathit{u}}_{{\mathit{m}}_{\mathit{i}}}$ and strains ${\mathit{\epsilon }}_{{\mathit{m}}_{\mathit{i}}}$ can be accomplished by using the shape function mentioned in Section 2, as

$$u_m^h\left(X\right)={\displaystyle \sum }_{k=1}^n{\varphi }_k\left(X\right)u_{m_k}=\mathit{\Phi }{\mathit{U}}_{\mathit{m}}$$

(39)

$${\epsilon }_m^h\left(X\right)={\displaystyle \sum }_{k=1}^n{\varphi }_k\left(X\right){\epsilon }_{m_k}=\mathit{\Phi }{\mathit{\epsilon }}_{\mathit{m}}$$

(40)

where $\mathit{\Phi }$ is the shape function matrix; and ${\mathit{U}}_{\mathit{m}}$ and ${\mathit{\epsilon }}_{\mathit{m}}$ are the vector of virtual nodal displacement and strain, respectively.

With Equations (35)–(40), one may discretize the local symmetric weak-form of Equation (34), as:

$$\frac{\left(m+2\right)\left(m+1\right)}{{{\rm T}}_k^2}{{\displaystyle \int }}_{{\displaystyle {\Omega }_q}}\rho \mathit{\Phi }d\Omega {\mathit{U}}_{\mathit{m}+2}+\frac{m+1}{{{\rm T}}_k}{{\displaystyle \int }}_{{\displaystyle {\Omega }_q}}c\mathit{\Phi }d\Omega {\mathit{U}}_{\mathit{m}+1}-{{\displaystyle \int }}_{{\displaystyle {\Gamma }_{qi}}}\mathit{n}\mathit{D}\mathit{\Phi }d\Gamma \left(\mathit{L}\mathit{\Phi }{\mathit{U}}_{\mathit{m}}\right)-{{\displaystyle \int }}_{{\displaystyle {\Gamma }_{qu}}}\mathit{n}\mathit{D}\mathit{\Phi }d\Gamma \left(\mathit{L}\mathit{\Phi }{\mathit{U}}_{\mathit{m}}\right)={{\displaystyle \int }}_{{\displaystyle {\Gamma }_{qt}}}{\tilde{\mathit{p}}}_{\mathit{m}}d\Gamma$$

(41)

Obviously, it can be found that no derivatives of the shape functions are involved in the local integrals. While both shape function and its derivative at each point need to be calculated in the traditional MLPG, which greatly increases the computing cost. The final system equation can be rewritten in a matrix form:

$$\frac{\left(m+2\right)\left(m+1\right)}{{{\rm T}}_k^2}\mathit{M}{\mathit{U}}_{\mathit{m}+2}+\frac{m+1}{{{\rm T}}_k}\mathit{C}{\mathit{U}}_{\mathit{m}+1}-\mathit{K}{\mathit{U}}_{\mathit{m}}=\mathit{F}$$

(42)

where $\mathit{M}$, $\mathit{C}$, $\mathit{K}$ and $\mathit{F}$ are the matrixes of mass, damping and stiffness, and the vector of force, respectively. They are defined as follows.

$$\mathit{M}=\mathit{\rho }{{\displaystyle \int }}_{{\displaystyle {\Omega }_q}}\mathit{\Phi }d\Omega =\mathit{\rho }\mathit{S}$$

(43)

$$\mathit{C}=\mathit{c}{{\displaystyle \int }}_{{\displaystyle {\Omega }_q}}\mathit{\Phi }d\Omega =\mathit{c}\mathit{S}$$

(44)

$$\mathit{F}={{\displaystyle \int }}_{{\displaystyle {\Gamma }_{qt}}}{\tilde{\mathit{p}}}_{\mathit{m}}d\Gamma$$

(45)

$$\mathit{K}={{\displaystyle \int }}_{{\displaystyle {\Gamma }_{qi}}}\mathit{n}\mathit{D}\mathit{\Phi }d\Gamma \left(\mathit{L}\mathit{\Phi }\right)+{{\displaystyle \int }}_{{\displaystyle {\Gamma }_{qu}}}\mathit{n}\mathit{D}\mathit{\Phi }d\Gamma \left(\mathit{L}\mathit{\Phi }\right)$$

(46)

Once nonlinear damping is introduced, the system equation can be rewritten in another form:

$$\frac{\left(m+2\right)\left(m+1\right)}{{{\rm T}}_k^2}\mathit{M}{\mathit{U}}_{\mathit{m}+2}+\frac{c_0\left(m+1\right)}{{{\rm T}}_k}\mathit{S}{\mathit{U}}_{\mathit{m}+1}+\left(\frac{c_{1}m}{{{\rm T}}_k}\mathit{S}-\mathit{K}\right){\mathit{U}}_{\mathit{m}}=\mathit{F}$$

(47)

where $c_0$ and $c_1$ represent the expanding coefficients of damping coefficient $c$.

In the present study, the Gauss quadrature is used for the subdomain integration in Equations (43)–(46).

3.3. Natural Frequency Solved by Mixed MLPG5

The natural frequencies and corresponding mode shapes are often referred to as the dynamic characteristics of the structure. While the mass matrix $\mathit{M}$ and the stiffness matrix $\mathit{K}$ in the vibration system are obtained from Equation (42), the elasto-dynamics equation of the undamped system can be written as a typical eigenvalue equation as follows:

$$\lambda =eig\left({\mathit{M}}^{-1}\mathit{K}\right)$$

(48)

where $\lambda$ is the vector of eigenvalues. Finally, the natural frequency can be solved by:

$$f_i=\frac{\sqrt{{\lambda }_i}}{2\pi }$$

(49)

4. Numerical Examples

4.1. Free Vibration Analysis

Example 1:

A Variable-Cross-Section Beam

In this example, the presented method is used in free vibration analysis of a cantilever beam with a variable cross-section, as shown in Figure 2. The problem is solved for the plane stress case with the following parameters: the length $L=10 \mathrm{m}$, the height $H1=5 \mathrm{m}$, $H2=3 \mathrm{m}$, the density $\rho =1\mathrm{kg}/{\mathrm{m}}^3$, the Young’s modulus $E=3\times {10}^7\mathrm{Pa}$ and the Poisson ratio $\nu =0.3$.

Regular and irregular nodal configurations were used, as shown in Figure 3. For comparison, the problem was also analyzed by FEM software ANSYS (Mechanical). Additionally, the number of nodes used in ANSYS was 3978, which is 13 times more that used in the presented method.

The natural frequencies of the first five modes were calculated by the presented mixed MLPG method and the FEM software, as listed in

Table 1. Comparison of the natural frequencies of the variable-cross-section beam obtained by the meshless algorithm and ANSYS (FEM).

Mode	Mixed MLPG5 Regular		Mixed MLPG5 Random		ANSYS (FEM) (Hz)
	Frequency (Hz)	Relative Error (%)	Frequency (Hz)	Relative Error (%)
1	42.65	2.45	42.37	1.78	41.63
2	145.57	0.36	147.43	0.92	146.09
3	153.44	1.27	153.88	1.56	151.51
4	293.04	0.64	293.52	0.48	294.94
5	412.59	0.30	414.02	0.64	411.37

. It can be seen that the natural frequencies obtained by the presented method are in good agreement with that of ANSYS, whether a regular or random node distribution is used.

The first four eigenmodes obtained with the present mixed MLPG5 method are plotted in Figure 4. Compared with the FEM results obtained by ANSYS, the results are identical. As fewer nodes are used, the presented method has higher computational efficiency.

4.2. Forced Vibration Analysis

Example 2:

A cantilever beam with horizontal traction.

In the second example, the forced vibration analysis of a 2D cantilever beam is considered, as shown in Figure 5. The parameters are taken as, length $L=20 \mathrm{m}$, width $D=4 \mathrm{m}$, Young’s modulus $E=1\times {10}^5 \mathrm{Pa}$, Poisson ratio $\nu =0.3$, density $\rho =1 \mathrm{kg}/{\mathrm{m}}^3$ and damping coefficient $c=1 \mathrm{Ns}/\mathrm{m}$. The initial conditions are defined as:

$${\mathit{u}}_0={\mathit{u}}_{\mathit{i}\mathit{n}}\left(\mathit{X},0\right)=0$$

(50)

$${\mathit{u}}_1={\dot{\mathit{u}}}_{\mathit{i}\mathit{n}}\left(\mathit{X},0\right)=0$$

(51)

The transient response of the beam subjected to a suddenly loaded traction P = 300 Pa is considered. A regular uniform nodal configuration is used with nodal distances $d=1 \mathrm{m}$, as shown in Figure 6.

The presented method was used to obtain the transient response. The Newmark method and the present time-adaptive method were utilized in this analysis. The results of time steps $\Delta t=0.001,0.002,0.005,0.01,0.02\mathrm{s}$ were obtained. For comparison, solutions for this problem were also obtained using the FEM software ANSYS (Mechanical).

The horizontal displacement $u_x$ at point A of different time steps by the Newmark method is plotted in Figure 7. Additionally, the parameters $\beta =0.3 \mathrm{and} \gamma =0.6$ were used. One can observe that for $\Delta t=0.001 \mathrm{s}$, the results are in good agreement with FEM. However, it should be noted that the computational error would increase with the increase in time step in the Newmark method due to the dissipation and dispersion errors. When the time step is too large (e.g., $\Delta t=0.02 \mathrm{s}$), the accuracy would become unacceptable. Thus, a straightforward way of reducing the dispersion and dissipation error in Newmark method is to use a smaller time step size.

The horizontal displacement $u_x$ at point A for different time steps via the time-adaptive method is plotted in Figure 8. As the results are quite close to the reference solution when the time steps are 0.001, 0.002 and 0.005 s, it is hard to distinguish them on the figure. Only the results of the time steps $\Delta t=0.01,0.02,0.04\mathrm{s}$ are plotted in Figure 8. It can be found that all the results obtained by time-adaptive methods are in good agreement with FEM, even if the time step is very large and the peak of displacement cannot be accurately obtained. Since high-order expansion in each time step is adopted in a time-domain adaptive method, a high precision result can be obtained by this time-adaptive method.

The calculation time and relative error at point A obtained by the two methods at different time steps are shown in

Table 2. The computing time, the order of expanding and the mean relative error at point A for two time-discretization methods.

Method	DeltaT (s)	Computing Time (s)	Order of Expanding	Relative Error at Point A (%)
Newmark	0.001	0.283	/	3.30
	0.002	0.164		3.99
	0.005	0.072		7.45
	0.01	0.042		14.40
	0.02	0.026		27.87
	0.04	0.018		52.91
Adaptive	0.001	1.183	7	3.47
	0.002	0.693	9	3.42
	0.005	0.419	15	3.72
	0.01	0.314	24	3.17
	0.02	0.259	42	3.29
	0.04	0.227	79	3.13

. One can observe that the computing time of the Newmark method is less than that of our method, which shows that efficiency of Newmark method is higher at any sized time step. Although not as fast as the Newmark method, our method has higher accuracy when a large time step is used. Only when Δt = 0.001, were the efficiency and accuracy of Newmark method higher than those of our method. However, in engineering, it is difficult to predict the “best time step size.” As shown in Table 2, the expansion order would increase with the increase in time step while the relative error would remain unchanged. This is very useful for the forced vibration analysis in engineering applications, especially in long-time response analysis, as a large time step is preferred.

The test-domain size $S_t$ and the support size $S_s$ (or the size of the influence domain) are the key components for the mixed MLPG5 method. Both of them affect the computational efficiency and the accuracy. In the present study, the test domain size and the support size were chosen to be proportional to the nodal distance d.

In practice, the test-domain size is chosen to be less than 1.0 d to ensure that the local sub-domains of the internal nodes are entirely within the solution domain, without being intersected by the global boundary. In the present study, five test-domain sizes were used, 0.4, 0.5, 0.6, 0.7 and 0.8 d. Additionally, the support size was fixed as 1.5 d and the nodal distance was fixed as 1 m. The displacements $u_x$ at point A (shown in Figure 5) were used to examine the effects of the different test-domain size, as shown Figure 9. It can be seen that accuracy is excellent when the test-domain size is less than 0.7 d. It is noticeable that the accuracy would become unacceptable when the test-domain size is larger than 0.8 d, as the sub-domains are obviously over-lapping. Our study has found that the test size $S_t=0.4-0.7$ works for most of forced vibration problems. Additionally, $S_t=0.6$ was used in the following calculations.

For a small support size, the meshless approximation algorithms may be singular, and the shape function cannot be constructed because of too few nodes. In the present study, four support sizes (1.2, 1.5, 1.8 and 2.1 d) were used for the MLPG5 mixed approach. The nodal distance was fixed as 1 m. The displacements $u_x$ at point A (shown in Figure 5) were also used to examine the effects of the support-domain size, as shown in Figure 10.

It can be seen that good accuracy is obtained when the support sizes are 1.5 d and 1.8 d. However, the result becomes more unstable when the support size is 2.1 d, as the continuous shape function leads to smoother results but lower accuracy. Our study has found that support sizes of $S_s=1.2-1.8$ work for most of forced vibration problems. Additionally, $S_s=1.5$ was used in the following calculations.

Example 3:

A cantilever beam with vertical traction.

In this example, our method is used to analyze the forced vibration of a cantilever beam subjected to a vertical traction with different damping coefficient, as shown in Figure 11. The problem is solved for the plane stress case with: length $L=48 \mathrm{m}$, width $D=12 \mathrm{m}$, Young’s modulus $E=3\times {10}^7 \mathrm{Pa}$, Poisson ratio $\nu =0.3$ and density $\rho =1 \mathrm{kg}/{\mathrm{m}}^3$. A regular set of 85 scattered nodes with nodal distances d = 3 m is employed here, as shown in Figure 12. The same initial conditions are defined as:

$${\mathit{u}}_0={\mathit{u}}_{\mathit{i}\mathit{n}\mathit{i}\mathit{t}\mathit{i}\mathit{a}\mathit{l}}\left(\mathit{X},0\right)=0$$

(52)

$${\mathit{u}}_1={\dot{\mathit{u}}}_{\mathit{i}\mathit{n}\mathit{i}\mathit{t}\mathit{i}\mathit{a}\mathit{l}}\left(\mathit{X},0\right)=0$$

(53)

Three kinds of traction at the free end of the beam while using $P\left(x,t\right)=\mathrm{10,000}\mathrm{g}\left(t\right)\mathrm{N}/\mathrm{m}$ are considered in this example: one is a Heaviside step loading, another is a transient loading with a finite decreasing time and the last is a transient loading with a finite increasing time, as shown in Figure 13. Additionally, the $\mathrm{g}\left(t\right)$ is the time-dependent function. The vertical displacement at point A, and the normal stress at points B and C (shown in Figure 11) were computed. To verify the accuracy of the present algorithm, the problem was also analyzed by FEM software ANSYS (Mechanical), where 2425 nodes (27 times more than our method) were adopted.

• Heaviside step loading with an infinite duration

The transient response of the beam subjected to Heaviside step loading with an infinite duration is considered. The loading function is determined by

$$g\left(t\right)=1.0$$

(54)

as shown in Figure 13a. This type of dynamics analysis under impact loading is usually defined as dynamic relaxation [26]. The mixed MLPG5 method was combined with the time-adaptive method and used to obtain the transient response. The displacements $u_y$ of point A with damping coefficients $c=1, 10 \mathrm{and} 20 \mathrm{Ns}/\mathrm{m}$ are plotted in Figure 14, Figure 15 and Figure 16, respectively. It is evident that the response converges to the static deformation ($u_y=10.682 \mathrm{m}$, obtained by ANSYS) once a damping is introduced, and the deformation declines fast with an increase in damping coefficient. All of them are in good agreement with the ANSYS results.

In addition, the vertical displacement fields of the deformed cantilever beam at different time steps (t = 0.005, 0.01, and 0.02 s) are shown in Figure 17, Figure 18 and Figure 19. One can observe that with time, the stress wave propagates from the free end towards the fixed end. Thus, the displacement distribution in elasto-dynamics is quite different from that in static analysis.

The normal stress ${\sigma }_{xx}$ at point B is shown in Figure 20. The damping coefficient is $c=1 \mathrm{Ns}/\mathrm{m}$. It can be seen that the results obtained by Newmark method have larger errors due to the numerical dissipation and dispersion, and the results obtained by our time-adaptive method are in good agreement with those of the reference solution. Thus, our method works very well, and it is more accurate than the Newmark method for the forced vibration analysis.

b.

Transient loading with finite decreasing time

The transient response of the beam subjected to a transient loading with a finite decreasing time is considered. The loading function is defined as

$$g\left(t\right)=\{\begin{array}{c}1-t 0<t<1\\ 0 t>1\end{array},$$

(55)

as shown in Figure 13b.

The displacements $u_y$ of point A with different damping coefficients are plotted in Figure 21, Figure 22 and Figure 23. It can be seen that as the damping coefficient increases, the amplitudes decay faster and the duration of vibration is shorter. Very stable results with different damping coefficients were obtained by our method, and they are in good agreement with the results obtained by ANSYS. The computing time with the mixed MLPG5 time-domain adaptive method and ANSYS (FEM) are included in

Table 3. Computing time of two algorithms.

Numerical Methods	Number of Nodes	Step Size (s)	Computing Time (s)
Mixed MLPG5 time Adaptive Method	85 (17 × 5)	0.001	3.3257
	175 (25 × 7)		10.105
ANSYS(FEM)	85(17 × 5)		10.484
	175 (25 × 7)		21.828

, for different nodal numbers. Additionally, the time step was fixed to 0.001 s. It also can be seen that our method is more efficient than ANSYS (FEM). From these results, our method shows good approximations to the transient responses of different damps with high efficiency.

c.

Time-dependent damping

In this example, the transient response of the beam subjected to a time dependent loading (shown in Figure 13c) and nonlinear damping is considered. The loading function and the damping coefficient are defined as follows:

$$g\left(t\right)=\{\begin{array}{c}2.5t 0<t<0.4\\ 0 t>0.4\end{array}$$

(56)

$$c=1+t$$

(57)

The example was solved by our method using different time step sizes. It can be seen in Figure 24 that the normal stress levels ${\sigma }_{xx}$ at point C with different time step sizes are quite close to each other. For comparison, the normal stress levels ${\sigma }_{xx}$ at point C solved by the Newmark method are plotted in Figure 25. It can be seen that the Newmark method still has large errors due to the numerical dissipation and dispersion. It also cannot accurately capture the stress peaks when a larger time step (∆t = 0.006 s) is adopted.

In addition, the shear stress levels ${\tau }_{xy}$ at different times solved by our method are shown in Figure 26, Figure 27 and Figure 28. It can be clearly seen in Figure 27 and Figure 28 that the stress concentrated on the center of the beam, as a result of the diffraction and reflection of elastic stress waves. This shows that the stress distribution in the dynamic problem is very different from that in a static problem. The results demonstrate that our method works well for the nonlinear forced vibration analysis.

Example 4:

A Perforated Tension Strip

The last example is a perforated strip under axial tension, as shown in Figure 29. This problem has been studied by Kontoni and Beskos, using the dual reciprocity BEM [27]. The strip was subjected to a Heaviside tension step load with initial value P = 20 Pa. The material properties of the strip were taken as: length $L=1.6 \mathrm{m}$, width $D=1.0 \mathrm{m}$, Young’s modulus $E=2\times {10}^3 \mathrm{Pa}$, Poisson ratio $\nu =0.3$, density $\rho =1\mathrm{kg}/{\mathrm{m}}^3$ and damping coefficient $c=1\mathrm{Ns}/\mathrm{m}$. The initial conditions were defined as:

$${\mathit{u}}_0={\mathit{u}}_{\mathit{i}\mathit{n}}\left(\mathit{X},0\right)=0$$

(58)

$${\mathit{u}}_1={\dot{\mathit{u}}}_{\mathit{i}\mathit{n}}\left(\mathit{X},0\right)=0$$

(59)

Symmetry conditions were imposed on the left and right edges, and the inner boundary of the hole had no traction. Regular uniform nodal configurations with nodal distances d = 0.03 m were used in this example, as shown in Figure 30. The time step used in the time-adaptive method was $\Delta t=0.001 \mathrm{s}$. The horizontal displacement of point A (0.00, 0.05) and vertical displacement of point B (0.05, 0.00) are plotted in Figure 31 and Figure 32, respectively. For comparison, solutions for this problem were also obtained using the finite element software ANSYS (Mechanical). It is evident that the results obtained by our method are in very good agreement with those obtained by ANSYS.

It can be observed that the maximum displacement level for point A and point B occurs at t = 0.065 s, and the maximum displacement of point A in the reverse direction occurs at t = 0.105 s. The displacement fields of this perforated tension strip at t = 0.065 s and t = 0.105 s are shown in Figure 33 and Figure 34. The results prove the efficiency and accuracy of the developed meshless time-adaptive method for forced vibration analysis in multiple connected domains.

5. Conclusions

In this paper, a new meshless time-domain adaptive method was presented for vibration analysis through mixed MLPG-FVM (MLPG5). In this method, each variable is interpolated by time series of variable order in the time domain. Thus, more accurate stress and displacement can be obtained, and larger time steps can be used in vibration analysis compared with the Newmark method (when the time step is 0.04, the calculation error of this method is only 1/17 of that of the Newmark method). Furthermore, through the independent interpolation of strain and displacement, the differentiation of the shape function is eliminated and the lower-order polynomial basis can be used in the MLS interpolations. Thus, smaller support sizes ($S_s=0.4-0.7$, and the test sizes $S_t$ were 1.2–1.8) can be used in the MLPG approach. By using the Heaviside function as the weighted function, the domain integral of stiffness matrix is removed and the calculation efficiency is improved. All the numerical results show that the time-domain adaptive method can cooperate well with the meshless method, and the calculation accuracy of the present method is satisfactory with various time step sizes. This high-accuracy time-domain scheme is very attractive for second-order PDE in time as elasto-dynamics equations. Put simply, this method provides a high efficiency and accuracy solution for solving free and forced vibration problems in both simply and multiply-connected domains under large time step sizes without any type of mesh.