Efficient integration of crack singularities in the extended finite element method: Duffy-distance transformation and conformal preconditioning strategy

Jia He Lv; Yu Yong Jiao; Peter Wriggers; Timon Rabczuk; Xia Ting Feng; Fei Tan

doi:10.1016/j.cma.2018.05.028

Efficient integration of crack singularities in the extended finite element method: Duffy-distance transformation and conformal preconditioning strategy

Jia He Lv, Yu Yong Jiao, Peter Wriggers, Timon Rabczuk, Xia Ting Feng, Fei Tan

Research output: Contribution to journal › Article › peer-review

29 Citations (Scopus)

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.

Original language	English
Pages (from-to)	559-576
Number of pages	18
Journal	Computer Methods in Applied Mechanics and Engineering
Volume	340
DOIs	https://doi.org/10.1016/j.cma.2018.05.028
Publication status	Published - 2018 Oct 1
Externally published	Yes

Bibliographical note

Funding Information:
The authors acknowledge the financial support from the National Natural Science Foundation of China (Nos. 51621006 , 51509240 , 41731284 and 11672360 ).

Publisher Copyright:
© 2018 Elsevier B.V.

Keywords

Conformal preconditioning
Duffy transformation
Numerical quadrature
Weak singularity
XFEM

ASJC Scopus subject areas

Computational Mechanics
Mechanics of Materials
Mechanical Engineering
General Physics and Astronomy
Computer Science Applications

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10.1016/j.cma.2018.05.028

Cite this

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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",

note = "Funding Information: The authors acknowledge the financial support from the National Natural Science Foundation of China (Nos. 51621006 , 51509240 , 41731284 and 11672360 ). Publisher Copyright: {\textcopyright} 2018 Elsevier B.V.",

year = "2018",

month = oct,

day = "1",

doi = "10.1016/j.cma.2018.05.028",

language = "English",

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journal = "Computer Methods in Applied Mechanics and Engineering",

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T1 - Efficient integration of crack singularities in the extended finite element method

T2 - Duffy-distance transformation and conformal preconditioning strategy

AU - Lv, Jia He

AU - Jiao, Yu Yong

AU - Wriggers, Peter

AU - Rabczuk, Timon

AU - Feng, Xia Ting

AU - Tan, Fei

N1 - Funding Information: The authors acknowledge the financial support from the National Natural Science Foundation of China (Nos. 51621006 , 51509240 , 41731284 and 11672360 ). Publisher Copyright: © 2018 Elsevier B.V.

PY - 2018/10/1

Y1 - 2018/10/1

N2 - 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.

AB - 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.

KW - Conformal preconditioning

KW - Duffy transformation

KW - Numerical quadrature

KW - Weak singularity

KW - XFEM

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M3 - Article

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SN - 0045-7825

VL - 340

SP - 559

EP - 576

JO - Computer Methods in Applied Mechanics and Engineering

JF - Computer Methods in Applied Mechanics and Engineering

ER -

Efficient integration of crack singularities in the extended finite element method: Duffy-distance transformation and conformal preconditioning strategy

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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