Generalized or eXtended finite element methods (GFEM/XFEM) have been studied extensively for crack problems. Most of the studies were concentrated on localized enrichment schemes where nodes around the crack tip are enriched by products of singular and finite element shape functions. To attain the optimal convergence rate O(h) (h is the mesh‐size), nodes in a fixed domain containing the tip have to be enriched. This results in many extra degrees of freedom (DOF) and stability issues. A so‐called DOF‐gathering GFEM/XFEM can avoid the increase of DOF, by collecting the singular enriched DOF together. Various novel modifications were designed for the DOF‐gathering GFEM/XFEM to get the optimal convergence O(h). However, they could not improve the stability, namely, condition numbers of stiffness matrices of the DOF‐gathering GFEM/XFEM could be much larger than that of the standard FEM. Motivated from the idea of stable GFEM, we propose in this paper a DOF‐gathering stable GFEM (d.g.SGFEM) for the Poisson problem with crack singularities. The main idea is to modify the singular and Heaviside enrichments by subtracting their finite element interpolants. The optimal convergence O(h) of the proposed d.g.SGFEM is proven theoretically. Moreover, the condition number of stiffness matrices of d.g.SGFEM, utilizing a local orthgonalization technique, is shown to be of same order as that of the standard FEM. Two kinds of commonly used cut‐off functions used to gather the DOF are analyzed in a unified approach. Theoretical convergence and the conditioning results of d.g.SGFEM are verified by numerical experiments.

中文翻译：

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裂纹问题的自由度聚集稳定广义有限元方法

对于裂纹问题，已经广泛研究了通用或扩展的有限元方法（GFEM / XFEM）。大多数研究集中在局部富集方案上，其中裂纹尖端周围的节点被奇异和有限元形状函数的乘积所富集。为了获得最佳收敛速度O（h）（h是网格大小），必须充实包含尖端的固定域中的节点。这导致许多额外的自由度（DOF）和稳定性问题。所谓的DOF聚集GFEM / XFEM通过将奇异富集的DOF收集在一起，可以避免DOF的增加。针对DOF聚集GFEM / XFEM设计了各种新颖的修改，以获得最佳收敛O（h）。但是，它们不能提高稳定性，即，自由度聚集GFEM / XFEM的刚度矩阵的条件数可能比标准FEM的条件数大得多。出于稳定GFEM的想法，我们在本文中提出了一种具有裂纹奇异性的Poisson问题的自由度聚集稳定GFEM（dgSGFEM）。主要思想是通过减去有限元插值来修改奇异和Heaviside富集。最优收敛O（h）的建议dgSGFEM在理论上得到了证明。此外，使用局部正交化技术显示的dgSGFEM刚度矩阵的条件数与标准FEM的阶数相同。统一分析了两种用于收集自由度的常用截止函数。数值实验验证了dgSGFEM的理论收敛性和调节结果。