求解断裂问题的新型无网格数值流形法

doi:10.13722/j.cnki.jrme.2019.0524

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摘要无网格数值流形法(MLS-NMM)将基于节点近似的移动最小二乘插值法(moving least squares，MLS)引入数值流形法(numerical manifold method，NMM)中，降低了NMM的网格依赖性，提高了计算精度，同时继承了NMM能自然地处理连续与非连续问题的优点。为提高MLS-NMM求解断裂问题的适应性，通过将裂纹尖端附近的数学片整合生成复合片，并在其上定义有限项的位移Williams级数重现裂纹尖端的位移场，同时在裂纹尖端附近采用四叉树局部加密方式提高计算精度，建立求解线弹性断裂问题的新型无网格数值流形法。该方法可直接利用应力强度因子和Williams级数项的关系求解应力强度因子，避免了后处理方法(如交互作用积分法)求解。为模拟裂纹扩展，使用最大周向应力准则判定裂纹扩展方向。最后，通过2个应力强度因子算例和2个裂纹扩展算例的计算模拟，结果显示：本文方法求解线弹性断裂问题是有效的，且建议取9项Williams级数和采用2层局部加密方式。

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	李伟1
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	郑宏4
	王海龙1
	郭宏伟5

关键词 ：数值分析, 数值流形法, 移动最小二乘插值法, 局部加密, 应力强度因子, 裂纹扩展

Abstract：Combined the moving least squares(MLS) based on the approximation of nodes with numerical manifold method(NMM)，the meshfree-numerical manifold method is developed，abbreviated as MLS-NMM. MLS-NMM owns lower mesh-dependency and higher calculate precision than NMM，and naturally solves the continuous and non-continuous problems in a unified way. To increase the adaptability of MLS-NMM about analyzing the linear elastic fracture problems，a new MLS-NMM is proposed. It makes the mathematical patches containing the same crack tip integrate into a complex patch，in which the finite items of Williams’ displacement series are defined as the local approximation in ordering to reproduce the displacement field of the crack tip. And the method of quadtree′s local refinement is adopted to enhance the calculate precision in the vicinity of the crack tip. This method can directly acquire the stress intensity factors(SIFs) using the relationship between SIFs and items of Williams′ displacement series，avoiding post-processing solution such as the interaction integral method. In this paper，the maximum circumferential stress criterion is applied to decide the direction of crack propagation. At last，two examples of SIF and two examples of crack propagation are simulated，the results reveal that the proposed method can effectively deal with the linear elastic fracture problems，and simultaneously suggest that 2 layer local refinement and 9 items of Williams′ displacement series should be used.

Key words： numerical analysis numerical manifold method moving least squares local refinement stress intensity factor crack propagation

引用本文:

李伟1，2，3，郑宏4，王海龙1，郭宏伟5. 求解断裂问题的新型无网格数值流形法[J]. 岩石力学与工程学报, 2020, 39(S1): 2655-2664.
LI Wei1，2，3，ZHENG Hong4，WANG Hailong1，GUO Hongwei5. A new meshfree-numerical manifold method for solving the fracture problem. , 2020, 39(S1): 2655-2664.

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http://rockmech.whrsm.ac.cn/CN/10.13722/j.cnki.jrme.2019.0524 或 http://rockmech.whrsm.ac.cn/CN/Y2020/V39/IS1/2655

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