景永强 彭妙娟

摘要：为提高断裂弹性动力学问题数值计算的精度，避免出现病态或奇异方程组，基于改进的移动最小二乘法建立三维弹性动力学问题的积分弱形式，采用罚函数法施加位移边界条件，引入隐式时间积分并且结合三维断裂力学的形函数考虑裂纹尖端的奇异性，探究将改进的无单元Galerkin（improved element-free Galerkin， IEFG）法用于断裂弹性动力学问题的数值计算。通过悬臂梁、柱和矩形板等3个算例，討论节点分布、影响域比例参数、罚因子和时间步长等参数对计算精度的影响，证明IEFG法用于求解三维断裂弹性动力学问题的正确性和有效性。

关键词：

弹性动力学; 断裂力学; 裂纹; 改进的移动最小二乘法; 形函数; 隐式时间积分

中图分类号：O241.82; TB115.1

文献标志码：B

Improved element-free Galerkin method for

three-dimensional fracture elasto-dynamics

JING Yongqiang， PENG Miaojuan

（Department of Civil Engineering， School of Mechanics and Engineering Science， Shanghai University， Shanghai 200444， China）

Abstract：

To improve the accuracy of the numerical calculation of fracture elasto-dynamics， and to avoid the ill conditioned or singular equations， the improved element-free Galerkin（IEFG） method is applied to the numerical calculation of fracture elasto-dynamics. Based on the improved moving least square method， the integral weak form of the three-dimensional elasto-dynamics problem is established. The penalty function method is used to set the displacement boundary conditions. The singularity of crack tip is considered by introducing the implicit time integration and the shape function of three-dimensional fracture mechanics. Three examples（include the cantilever beam， the column and the square plate） are selected to analyze the influence of computing parameters（include the node distribution， the influence domain scale parameter， the penalty factor and the time step） on the calculation accuracy. It is proved that the IEFG method is correct and effective for solving three-dimensional fracture elasto-dynamics problems.

Key words：

elasto-dynamics; fracture mechanics; crack; improved moving least square method; shape function; implicit time integration

0 引 言

断裂弹性动力学问题是断裂力学与弹性动力学耦合的问题，传统的解析方法难以求解，目前可采用的数值方法主要有有限元法和无单元法等。采用有限元法求解断裂力学问题时，在裂纹的发生扩展过程中需要重新划分网格，计算精度较低。无单元法无须划分网格，只需要节点信息。对于含有裂纹的构件，为提高计算精度，无单元法可以在裂纹处布置加密节点，避免对网格和单元依赖的问题，在求解断裂弹性动力学问题时具有一定的优势。

BELYTSCHKO等[1]采用无单元Galerkin法研究三维几何和材料非线性的动力学问题。CHEN等[2]采用无单元Galerkin法研究材料高速撞击发生弹塑性大变形的问题。LEE等[3]采用无单元Galerkin法对裂纹的奇点进行不连续性建模，提出一种改进的裂纹分析技术。……