A general algorithm for numerical integration of three-dimensional crack singularities in PU-based numerical methods

authored by: Jia He Lv, Yu Yong Jiao, Timon Rabczuk, Xiaoying Zhuang, Xia Ting Feng, Fei Tan
Abstract: With the development of PU-based numerical methods for crack problems, the evaluation of various orders of vertex/edge singularity has been one of the most critical issues, which restrains the computational efficiency of PU-based methods, especially for 3D crack problems. In this paper, based on the conventional Duffy transformation, a general algorithm for numerical integration of three-dimensional crack singularities is proposed for the vertex/edge singularity problems, which takes the integration cell shape into full consideration. Besides, the corresponding 3D conformal preconditioning strategy is constructed to fully eliminate the shape influence of tetrahedron elements. Extensive numerical examples, including ill-shaped integration cells and crack-front tetrahedron elements with parallel/nonparallel crack front, are given to validate the feasibility and accuracy of the proposed method. As a result, for each crack-front element, several hundreds of Gauss points are sufficient to achieve the precision of 10⁻⁶ for both kernels 1∕r and 1∕r, in sharp contrast with around ten thousands of Gauss points using the conventional Duffy transformation.
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: 363
ISSN: 0045-7825
Publication date: 19.02.2020
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.2020.112908 (Access: Closed)
: Details in the research portal "Research@Leibniz University"

BibTeX

@article{0c1852d2e83c4750a87bd28dfdb5d4cb,
title = "A general algorithm for numerical integration of three-dimensional crack singularities in PU-based numerical methods",
abstract = "With the development of PU-based numerical methods for crack problems, the evaluation of various orders of vertex/edge singularity has been one of the most critical issues, which restrains the computational efficiency of PU-based methods, especially for 3D crack problems. In this paper, based on the conventional Duffy transformation, a general algorithm for numerical integration of three-dimensional crack singularities is proposed for the vertex/edge singularity problems, which takes the integration cell shape into full consideration. Besides, the corresponding 3D conformal preconditioning strategy is constructed to fully eliminate the shape influence of tetrahedron elements. Extensive numerical examples, including ill-shaped integration cells and crack-front tetrahedron elements with parallel/nonparallel crack front, are given to validate the feasibility and accuracy of the proposed method. As a result, for each crack-front element, several hundreds of Gauss points are sufficient to achieve the precision of 10−6 for both kernels 1∕r and 1∕r, in sharp contrast with around ten thousands of Gauss points using the conventional Duffy transformation.",
keywords = "Distance transformation, Duffy transformation, Numerical quadrature, PU-based numerical methods, Singular integrals",
author = "Lv, {Jia He} and Jiao, {Yu Yong} and Timon Rabczuk and Xiaoying Zhuang 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. 51979253 , 41731284 , 41920104007 , 11672360 , 51621006 and 51879245 ).",
year = "2020",
month = feb,
day = "19",
doi = "10.1016/j.cma.2020.112908",
language = "English",
volume = "363",
journal = "Computer Methods in Applied Mechanics and Engineering",
issn = "0045-7825",
publisher = "Elsevier",
}