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## Interpolating element-free Galerkin method for viscoelasticity problems

Zhang Peng-Xuan, Peng Miao-Juan
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• #### 摘要

基于改进的移动最小二乘插值法, 提出了黏弹性问题的插值型无单元Galerkin方法. 采用改进的移动最小二乘插值法建立形函数, 根据黏弹性问题的Galerkin弱形式建立离散方程, 推导了相应的计算公式. 与无单元Galerkin方法相比, 本文提出的黏弹性问题的插值型无单元Galerkin方法具有直接施加本质边界条件的优点. 通过数值算例讨论了影响域、节点数对计算精确性的影响, 说明了该方法具有较好的收敛性; 将计算结果与无单元Galerkin方法和有限元方法或解析解比较, 说明了该方法具有提高计算效率的优点.

#### Abstract

In this paper, based on the improved interpolating moving least-square (IMLS) approximation, the interpolating element-free Galerkin (IEFG) method for two-dimensional viscoelasticity problems is presented. The shape function constructed by the IMLS approximation can overcome the shortcomings that the shape function of the moving least-squares (MLS) can-not satisfy the property of Kronecker function, so the essential boundary conditions can be directly applied to the IEFG method. Under a similar computational precision, compared with the meshless method based on the MLS approximation, the meshless method using the IMLS approximation has a high computational efficiency. Using the IMLS approximation to form the shape function and adopting the Galerkin weak form of the two-dimensional viscoelasticity problem to obtain the final discretized equation, the formulae for two-dimensional viscoelasticity problem are derived by the IEFG method. The IEFG method has some advantages over the conventional element-free Galerkin (EFG) method, such as the concise formulae and direct application of the essential boundary conditions, For the IEFG method of two-dimensional viscoelasticity problems proposed in this paper, three numerical examples and one engineering example are given. The convergence of the method is analyzed by considering the effects of the scale parameters of influence domains and the node distribution on the computational precision of the solutions. It is shown that when dmax = 1.01−2.00, the method in this paper has a good convergence. The numerical results from the IEFG method are compared with those from the EFG method and from the finite element method or analytical solution. We can see that the IEFG method in this paper is effective. The results of the examples show that the IEFG method has the advantage in improving the computational efficiency of the EFG method under a similar computational accuracy. And the engineering example shows that the IEFG method can not only has higher computational precision, but also improve the computational efficiency.

#### 作者及机构信息

###### 通信作者: 彭妙娟, mjpeng@shu.edu.cn
• 基金项目: 国家自然科学基金(批准号: 11571223) 资助的课题

#### Authors and contacts

###### Corresponding author: Peng Miao-Juan, mjpeng@shu.edu.cn
• Funds: Project supported by the National Natural Science Foundation of China (Grant No. 11571223)

#### 施引文献

• 图 1  受均布荷载的悬臂梁

图 2  不同节点分布下有限元法解的方差

Fig. 2.  The variances of the solutions of FEM under different node distributions.

图 3  不同节点分布下的相对误差

Fig. 3.  The relative error under different node distributions.

图 4  节点布置

Fig. 4.  Node distribution.

图 5  不同影响域比例参数下的相对误差

Fig. 5.  The relative error for different scale parameters of influence domains.

图 6  $t = 20 \;{\rm{s}}$时悬臂梁中轴线上各点的挠度

Fig. 6.  Vertical displacements of nodes on the neutral axis of the beam when $t = 20 \;{\rm{s}}$.

图 7  梁右端中点的挠度随时间的变化

Fig. 7.  Time history of vertical displacement of midpoint in the right end of the beam.

图 8  纯弯曲的梁

Fig. 8.  A beam subjected to simple bending

图 9  节点分布

Fig. 9.  Node distribution.

图 10  $t = 30 \;{\rm{s}}$时梁中轴线上的节点挠度

Fig. 10.  Vertical displacements of nodes on the neutral axis of the beam when $t = 30 \;{\rm{s}}$.

图 11  梁右端中点的挠度随时间$t$的变化

Fig. 11.  Time history of vertical displacement of midpoint in the right end of the beam.

图 12  受均布内压的厚壁圆筒

Fig. 12.  Circular ring under a distributed inner pressure

图 13  受均布内压1/4圆筒

Fig. 13.  A quarter of the circular ring under a distributed inner pressure

图 14  1/4圆筒的节点分布

Fig. 14.  Node distribution of a quarter of the circular ring.

图 15  $t = 30 \;{\rm{s}}$时沿${x_2} = {\rm{0}}$线上节点的位移

Fig. 15.  Radial displacements at${x_2} = {\rm{0}}$ when $t = 30 \;{\rm{s}}$.

图 16  $(2,0)$的径向位移随时间$t$的变化

Fig. 16.  Time history of radial displacement at point $(2,0)$.

图 17  受静水压力的混凝土水坝

Fig. 17.  A concrete dam under hydrostatic pressure.

图 18  混凝土水坝的节点分布

Fig. 18.  Node distribution of a concrete dam.

图 19  $t = 500$ d时沿${x_1} = 15$方向上节点的水平位移

Fig. 19.  Horizontal displacements at ${x_1} = 15$ when $t = 500\; {\rm d}$.

图 20  混凝土坝上点$(15, 50)$的水平位移与时间的关系

Fig. 20.  Time history of horizontal displacement of the point $(15, 50)$.