基于Newmark隐式时间积分方案的裂纹动态扩展的数值计算方法

doi:10.16183/j.cnki.jsjtu.2020.021

上海交通大学学报 ›› 2021, Vol. 55 ›› Issue (6): 689-697.doi: 10.16183/j.cnki.jsjtu.2020.021

所属专题：《上海交通大学学报》2021年12期专题汇总专辑；《上海交通大学学报》2021年“土木建筑工程”专题

基于Newmark隐式时间积分方案的裂纹动态扩展的数值计算方法

郭德平¹^,², 李铮³(), 彭森林⁴, 曾志凯³, 吴岱峰³

1.叙镇铁路有限责任公司, 云南昭通 657900
2.武汉大学土木建筑工程学院, 武汉 430072
3.重庆市城市建设投资(集团)有限公司, 重庆 400023
4.重庆大学土木工程学院, 重庆 400045

收稿日期:2020-01-14 出版日期:2021-06-28 发布日期:2021-06-30
通讯作者: 李铮 E-mail:2756232300@qq.com
作者简介:郭德平(1983-),男,高级工程师,主要从事隧道工程和岩土工程等方面的科研工作
基金资助:
国家自然科学基金重点项目(51679017);国家自然科学基金重点项目(51839009)

Numerical Calculation Method for Crack Dynamic Propagation Based on Newmark Implicit Time Integration Scheme

GUO Deping¹^,², LI Zheng³(), PENG Senlin⁴, ZENG Zhikai³, WU Daifeng³

1. Xuzhen Railway Co., Ltd., Zhaotong 657900, Yunnan, China
2. School of Civil Engineering, Wuhan University, Wuhan 430072, China
3. Chongqing City Construction Investment (Group) Co., Ltd., Chongqing 400023, China
4. Department of Civil Engineering, Chongqing University, Chongqing 400045, China

Received:2020-01-14 Online:2021-06-28 Published:2021-06-30
Contact: LI Zheng E-mail:2756232300@qq.com

摘要/Abstract

摘要：

扩展有限单元法(XFEM)是基于单位分解的思想,在常规有限元的位移模式中加入能够反映裂纹面不连续性的跳跃函数和裂纹尖端的渐近位移场函数,避免了常规有限单元法计算断裂问题时需要对裂纹尖端重新划分网格的不便以及繁重的计算量,并且裂纹的扩展独立于网格.标准有限元在处理时间积分时,在裂纹不断扩展的过程中整体刚度矩阵的自由度也会不断地增大,从而导致迭代计算无法进行.本文基于扩展有限单元法模拟动态裂纹扩展的方法,提出了新的Newmark隐式时间积分方案.此方法将所有节点都富集Heaviside函数和裂纹尖端的渐近位移场函数,即每个节点都有12个自由度,从而使得总体刚度矩阵式保持一致,避免迭代计算无法进行.同时,提出了一种稀疏矩阵技术来解决矩阵所占内存大和计算时间长的问题.

关键词: 扩展有限元, 动态裂纹, Newmark隐式时间积分, 稀疏矩阵技术

Abstract:

Extended finite element method (XFEM) is based on the idea of unit decomposition. The jump function that can reflect the discontinuity of the crack surface and the progressive displacement field function of the crack tip is added to the conventional finite element displacement mode, which avoids the inconvenience of remeshing the crack tip and the heavy calculation. Then the conventional finite element method calculates the fracture problem, and the crack propagation is independent of the mesh. When the standard finite element deals with time integration, the degree of freedom of the overall stiffness matrix will continue to increase in the process of crack propagation, which makes iterative calculation impossible. This paper proposes a novel Newmark implicit time integration scheme based on the XFEM to simulate dynamic crack growth. This method enriches all the nodes with the Heaviside function and the asymptotic displacement field function at the crack tip, that is, each node has 12 degrees of freedom, so that the overall stiffness matrix is consistent without making iterative calculation impossible. At the same time, a sparse matrix technology is proposed to solve the problems of large memory and long calculation time occupied by the matrix.

Key words: extended finite element method (XFEM), dynamic cracks, Newmark implicit time integration scheme, sparse matrix technique

中图分类号:

TU43

郭德平, 李铮, 彭森林, 曾志凯, 吴岱峰. 基于Newmark隐式时间积分方案的裂纹动态扩展的数值计算方法[J]. 上海交通大学学报, 2021, 55(6): 689-697.

GUO Deping, LI Zheng, PENG Senlin, ZENG Zhikai, WU Daifeng. Numerical Calculation Method for Crack Dynamic Propagation Based on Newmark Implicit Time Integration Scheme[J]. Journal of Shanghai Jiao Tong University, 2021, 55(6): 689-697.

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基于Newmark隐式时间积分方案的裂纹动态扩展的数值计算方法

Numerical Calculation Method for Crack Dynamic Propagation Based on Newmark Implicit Time Integration Scheme

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