A series of Duffy-distance transformation for integrating 2D and 3D vertex singularities

authored by: Jia He Lv, Yu Yong Jiao, Xia Ting Feng, Peter Wriggers, Xiao Ying Zhuang, Timon Rabczuk
Abstract: With the development of the generalized/extended finite element method for fracture problems, the accurate and efficient integration of singular enrichment functions has been an open issue, especially for the 3D case. In this paper, we reveal the near singularities caused by distorted integral patch/cell shape numerically and theoretically during the implementation of generalized Duffy transformation, and the Duffy-distance transformation is developed step by step for the 2D and 3D vertex singularities. Meanwhile, the 3D conformal preconditioning strategy is constructed to eliminate the near singularity caused by element shape distortion during the iso-parametric transformation, which enables the Duffy-distance transformation to be applicable for arbitrary shaped tetrahedral elements. As a result, the near singularities can be fully or partly canceled depending on the order of singularity. The implementation of the proposed scheme in existing codes is straightforward. Numerous numerical examples for arbitrary shaped triangles and tetrahedrons are presented to demonstrate its robustness and efficiency, along with comparisons to the generalized Duffy transformation.
Organisation(s): Institute of Continuum Mechanics
External Organisation(s): China University of Geosciences
Northeastern University, Shenyang (NEU)
Bauhaus-Universität Weimar
Type: Article
Journal: International Journal for Numerical Methods in Engineering
Volume: 118
Pages: 38-60
No. of pages: 23
ISSN: 0029-5981
Publication date: 06.04.2019
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: Numerical Analysis, General Engineering, Applied Mathematics
Electronic version(s): https://doi.org/10.1002/nme.6016 (Access: Closed)
: Details in the research portal "Research@Leibniz University"

BibTeX

@article{b8f1b98b552649be9055c61cfe0d098c,
title = "A series of Duffy-distance transformation for integrating 2D and 3D vertex singularities",
abstract = "With the development of the generalized/extended finite element method for fracture problems, the accurate and efficient integration of singular enrichment functions has been an open issue, especially for the 3D case. In this paper, we reveal the near singularities caused by distorted integral patch/cell shape numerically and theoretically during the implementation of generalized Duffy transformation, and the Duffy-distance transformation is developed step by step for the 2D and 3D vertex singularities. Meanwhile, the 3D conformal preconditioning strategy is constructed to eliminate the near singularity caused by element shape distortion during the iso-parametric transformation, which enables the Duffy-distance transformation to be applicable for arbitrary shaped tetrahedral elements. As a result, the near singularities can be fully or partly canceled depending on the order of singularity. The implementation of the proposed scheme in existing codes is straightforward. Numerous numerical examples for arbitrary shaped triangles and tetrahedrons are presented to demonstrate its robustness and efficiency, along with comparisons to the generalized Duffy transformation.",
keywords = "distance transformation, Duffy transformation, near singularity, numerical quadrature, vertex singularity",
author = "Lv, {Jia He} and Jiao, {Yu Yong} and Feng, {Xia Ting} and Peter Wriggers and Zhuang, {Xiao Ying} and Timon Rabczuk",
note = "Funding information: The authors acknowledge the financial support from the National Natural Science Foundation of China (grants 51621006, 51509240, 41731284, and 11672360) and the State Key Research Development Program of China (grant 2016YFC0600707).",
year = "2019",
month = apr,
day = "6",
doi = "10.1002/nme.6016",
language = "English",
volume = "118",
pages = "38--60",
journal = "International Journal for Numerical Methods in Engineering",
issn = "0029-5981",
publisher = "John Wiley and Sons Ltd",
number = "1",
}