Efficient integration of crack singularities in the extended finite element method

Duffy-distance transformation and conformal preconditioning strategy

authored by: Jia He Lv, Yu Yong Jiao, Peter Wriggers, Timon Rabczuk, Xia Ting Feng, Fei Tan
Abstract: For integrals with various orders of vertex singularity encountered in the generalized/extended finite element method, three difficulties should be concerned as: (a) the singularity with respect to local distance r; (b) the near singularity caused by ill-shaped patches; (c) the near singularity during the iso-parametric transformation. In this paper, based on the Duffy transformation, the near singularities caused by ill-shaped patches and element shape distortion are revealed theoretically and numerically, a Duffy-distance transformation is constructed to damp out the near singularity caused by ill-shaped patches, and a conformal preconditioning strategy is developed to fully eliminate the near singularity caused by element shape distortion. Extensive numerical examples, including crack-tip elements with distorted shape, are given to validate the efficiency and robustness of the presented scheme in this paper.
Organisation(s): Institute of Continuum Mechanics
External Organisation(s): China University of Geosciences
Bauhaus-Universität Weimar
Northeastern University, Shenyang (NEU)
Type: Article
Journal: Computer Methods in Applied Mechanics and Engineering
Volume: 340
Pages: 559-576
No. of pages: 18
ISSN: 0045-7825
Publication date: 01.10.2018
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: Computational Mechanics, Mechanics of Materials, Mechanical Engineering, General Physics and Astronomy, Computer Science Applications
Electronic version(s): https://doi.org/10.1016/j.cma.2018.05.028 (Access: Closed)
: Details in the research portal "Research@Leibniz University"

BibTeX

@article{e21b5a099ba34560ad2b082faa61a4e3,
title = "Efficient integration of crack singularities in the extended finite element method: Duffy-distance transformation and conformal preconditioning strategy",
abstract = "For integrals with various orders of vertex singularity encountered in the generalized/extended finite element method, three difficulties should be concerned as: (a) the singularity with respect to local distance r; (b) the near singularity caused by ill-shaped patches; (c) the near singularity during the iso-parametric transformation. In this paper, based on the Duffy transformation, the near singularities caused by ill-shaped patches and element shape distortion are revealed theoretically and numerically, a Duffy-distance transformation is constructed to damp out the near singularity caused by ill-shaped patches, and a conformal preconditioning strategy is developed to fully eliminate the near singularity caused by element shape distortion. Extensive numerical examples, including crack-tip elements with distorted shape, are given to validate the efficiency and robustness of the presented scheme in this paper.",
keywords = "Conformal preconditioning, Duffy transformation, Numerical quadrature, Weak singularity, XFEM",
author = "Lv, {Jia He} and Jiao, {Yu Yong} and Peter Wriggers and Timon Rabczuk and Feng, {Xia Ting} and Fei Tan",
year = "2018",
month = oct,
day = "1",
doi = "10.1016/j.cma.2018.05.028",
language = "English",
volume = "340",
pages = "559--576",
journal = "Computer Methods in Applied Mechanics and Engineering",
issn = "0045-7825",
publisher = "Elsevier",
}