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DOF‐gathering stable generalized finite element methods for crack problems
Author(s) -
Zhang Qinghui
Publication year - 2020
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22459
Subject(s) - finite element method , extended finite element method , mathematics , heaviside step function , convergence (economics) , mathematical analysis , stability (learning theory) , degrees of freedom (physics and chemistry) , computer science , structural engineering , engineering , physics , quantum mechanics , machine learning , economics , economic growth
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.