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Stable generalized finite element methods for elasticity crack problems
Author(s) -
Cui Cu,
Zhang Qinghui
Publication year - 2020
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.6347
Subject(s) - finite element method , extended finite element method , mathematics , mathematical analysis , structural engineering , engineering
Summary 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.

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