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Stable generalized finite element method (SGFEM) for three-dimensional crack problems

机译：三维裂纹问题的稳定广义有限元法（SGFEM）

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摘要

This paper proposes a stable generalized finite element method (SGFEM) for the linear 3D elasticity problem with cracked domains. Conventional material-independent branch functions serve as singular enrichments. We prove that the proposed SGFEM with the geometric enrichment scheme yields the optimal order of convergence in the energy norm, O(h), for fully 3D elasticity planar crack problems; h is the mesh parameter. To improve the conditioning of SGFEM, two stability techniques have been employed, namely, (a) a cubic polynomial has been used as the PU (partition of unity), instead of the standard FE hat-functions, to address the possible almost linear dependence between the PU functions and the enrichments, and (b) a local principal component analysis (LPCA) has been implemented to address the local bad conditioning produced by multi-fold enrichments at a node. The scaled condition number for the proposed SGFEM is shown to be O(h(-2)) (same as that of a standard Finite Element Method), for various relative positions of crack surface and mesh. The robustness of the scaled condition number for the proposed SGFEM, with respect to the relative positions of the crack-surface and the element boundaries, has been observed numerically. The numerical experiments for both the planar and fully 3D planar crack problems are presented to show the efficiency of the proposed SGFEM.
机译：该文提出了一种稳定的广义有限元方法（SGFEM）用于裂纹域的线性三维弹性问题。传统的与材料无关的分支功能用作单一的富集。证明了采用几何富集方案提出的SGFEM在全三维弹性平面裂纹问题中产生了能量范数O（h）的最优收敛阶数;h 是网格参数。为了改善SGFEM的调节，采用了两种稳定性技术，即（a）使用三次多项式作为PU（单位划分），而不是标准的FE帽函数，以解决PU函数与富集之间可能存在的几乎线性依赖性，以及（b）实施了局部主成分分析（LPCA）以解决节点处多倍富集产生的局部不良调节。对于裂纹面和网格的不同相对位置，所提出的SGFEM的缩放条件数为O（h（-2））（与标准有限元方法相同）。通过数值观察了所提出的SGFEM的缩放条件数相对于裂纹表面和单元边界的相对位置的鲁棒性。通过对平面和全三维平面裂纹问题的数值实验，验证了所提SGFEM的效率。

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