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

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Authors

  • Jia He Lv
  • Yu Yong Jiao
  • Xia Ting Feng
  • Peter Wriggers
  • Xiao Ying Zhuang

Research Organisations

External Research Organisations

  • China University of Geosciences
  • Northeastern University, Shenyang (NEU)
  • Bauhaus-Universität Weimar
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Details

Original languageEnglish
Pages (from-to)38-60
Number of pages23
JournalInternational Journal for Numerical Methods in Engineering
Volume118
Issue number1
Early online date5 Mar 2019
Publication statusPublished - 6 Apr 2019

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

ASJC Scopus subject areas

Cite this

A series of Duffy-distance transformation for integrating 2D and 3D vertex singularities. / Lv, Jia He; Jiao, Yu Yong; Feng, Xia Ting et al.
In: International Journal for Numerical Methods in Engineering, Vol. 118, No. 1, 06.04.2019, p. 38-60.

Research output: Contribution to journalArticleResearchpeer review

Lv JH, Jiao YY, Feng XT, Wriggers P, Zhuang XY, Rabczuk T. A series of Duffy-distance transformation for integrating 2D and 3D vertex singularities. International Journal for Numerical Methods in Engineering. 2019 Apr 6;118(1):38-60. Epub 2019 Mar 5. doi: 10.1002/nme.6016
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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.",
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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).",
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T1 - A series of Duffy-distance transformation for integrating 2D and 3D vertex singularities

AU - Lv, Jia He

AU - Jiao, Yu Yong

AU - Feng, Xia Ting

AU - Wriggers, Peter

AU - Zhuang, Xiao Ying

AU - Rabczuk, Timon

N1 - 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).

PY - 2019/4/6

Y1 - 2019/4/6

N2 - 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.

AB - 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.

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