Generalized or eXtended finite element methods (GFEM/XFEM) for crack problems have been studied extensively. The GFEM/XFEM are called extrinsic if additional functions are enriched at every node in certain domains, while they are called degree of freedom (DOF)‐gathering if the singular enriched functions are gathered using cutoff functions. The DOF‐gathering GFEM/XFEM save the additional DOFs compared with the extrinsic approach. Both extrinsic and DOF‐gathering GFEM/XFEM suffer from difficulties of stabilities in a sense that their scaled condition numbers (SCN) of stiffness matrices could be much larger than those of the standard FEM. A GFEM/XFEM is referred to as stable GFEM (SGFEM) if it reaches optimal convergence orders, and its SCN is of same order as that of FEM. An extrinsic SGFEM was established in Zhang et al for the Poisson crack problems. Objective of this article is to propose the SGFEM for elasticity crack problems; both extrinsic and DOF‐gathering schemes are addressed. The main idea is to modify the enriched functions by subtracting their FE interpolants, which was developed by Babuška and Banerjee. To remove local almost linear dependence introduced by multifold enrichments at one node, we propose a local principal component analysis technique to identify and analyze “contributions” of multifold enrichments at one node. Numerical studies demonstrate that the proposed SGFEM and DOF‐gathering SGFEM are of optimal convergence and have the SCNs of same order as in the FEM.

中文翻译：

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弹性裂纹问题的稳定广义有限元方法

针对裂纹问题的通用或扩展有限元方法（GFEM / XFEM）已得到广泛研究。如果在某些域中的每个节点上都丰富了附加功能，则GFEM / XFEM称为外部，而如果使用截止功能收集了单个丰富功能，则将它们称为自由度（DOF）。与外部方法相比，DOF聚集GFEM / XFEM可以节省额外的DOF。在某种程度上，外在的和自由度聚集的GFEM / XFEM都存在稳定性方面的困难，因为它们的刚度矩阵的标度条件数（SCN）可能比标准FEM大得多。如果GFEM / XFEM达到最佳收敛阶数，并且其SCN与FEM的阶数相同，则称为稳定GFEM（SGFEM）。Zhang等人针对泊松裂纹问题建立了外在的SGFEM。本文的目的是提出用于弹性裂纹问题的SGFEM。外部和自由度聚集方案都可以解决。主要思想是通过减去由Babuška和Banerjee开发的有限元插值来修改丰富函数。为了消除一个节点上多重富集引入的局部几乎线性依赖性，我们提出了一种局部主成分分析技术来识别和分析一个节点上多重富集的“贡献”。数值研究表明，所提出的SGFEM和自由度聚集SGFEM具有最佳收敛性，并且具有与FEM中相同阶的SCN。外部和自由度聚集方案都可以解决。主要思想是通过减去由Babuška和Banerjee开发的有限元插值来修改丰富函数。为了消除一个节点上多重富集引入的局部几乎线性依赖性，我们提出了一种局部主成分分析技术来识别和分析一个节点上多重富集的“贡献”。数值研究表明，所提出的SGFEM和自由度聚集SGFEM具有最佳收敛性，并且具有与FEM中相同阶的SCN。外部和自由度聚集方案都可以解决。主要思想是通过减去由Babuška和Banerjee开发的有限元插值来修改丰富函数。为了消除一个节点上多重富集引入的局部几乎线性依赖性，我们提出了一种局部主成分分析技术来识别和分析一个节点上多重富集的“贡献”。数值研究表明，所提出的SGFEM和自由度聚集SGFEM具有最佳收敛性，并且具有与FEM中相同阶的SCN。我们提出了一种本地主成分分析技术，以识别和分析一个节点上多重富集的“贡献”。数值研究表明，所提出的SGFEM和自由度聚集SGFEM具有最佳收敛性，并且具有与FEM中相同阶的SCN。我们提出了一种本地主成分分析技术，以识别和分析一个节点上多重富集的“贡献”。数值研究表明，所提出的SGFEM和自由度聚集SGFEM具有最佳收敛性，并且具有与FEM中相同阶的SCN。