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An accurate quadrature technique for the contact boundary in 3D finite element computations

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Abstract

This paper presents a new numerical integration technique for 3D contact finite element implementations, focusing on a remedy for the inaccurate integration due to discontinuities at the boundary of contact surfaces. The method is based on the adaptive refinement of the integration domain along the boundary of the contact surface, and is accordingly denoted RBQ for refined boundary quadrature. It can be used for common element types of any order, e.g. Lagrange, NURBS, or T-Spline elements. In terms of both computational speed and accuracy, RBQ exhibits great advantages over a naive increase of the number of quadrature points. Also, the RBQ method is shown to remain accurate for large deformations. Furthermore, since the sharp boundary of the contact surface is determined, it can be used for various purposes like the accurate post-processing of the contact pressure. Several examples are presented to illustrate the new technique.

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Notes

  1. In this paper, the phrase ‘contact boundary’ is used to refer to the boundary of the contact surface and not to the contact surface itself, as is often done elsewhere since it is part of the boundary of the contacting bodies.

  2. 2D surface manifold

  3. The hunting grid can be further improved for efficiency by an adaptive schemes e.g. with the quad tree structure.

  4. Although the convergence here is measured in the energy norm, other measures can also be considered.

  5. Later, RBQ segments can be used for an effective re-meshing around the contact boundary.

    Fig. 5
    figure 7

    RBQ partitioning for post-processing: a finite elements and nodes with contact boundary curve \(\mathcal {C}\) detected by \(\alpha _1=2; \alpha _2=1\), b RBQ segments and RBQ vertices obtained by RBQ partitioning along \(\mathcal {C}\) considering quadratic Lagrange segments

    Full size image

  6. This is due to the fact that the two-half-pass algorithm does not enforce the contact traction continuity explicitly for the discretized system, but rather recovers it in the continuum limit, i.e. as the mesh is refined.

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Acknowledgments

The authors are grateful to the German Research Foundation (DFG) for supporting this research under Projects SA1822/5-1 and GSC 111. We also thank Callum Corbett for helpful discussions and support.

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Authors and Affiliations

  1. Aachen Institute for Advanced Study in Computational Engineering Science (AICES), RWTH Aachen University, Templergraben 55, 52056, Aachen, Germany

    Thang X. Duong & Roger A. Sauer

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  1. Thang X. Duong
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  2. Roger A. Sauer
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Correspondence to Roger A. Sauer.

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Duong, T.X., Sauer, R.A. An accurate quadrature technique for the contact boundary in 3D finite element computations. Comput Mech 55, 145–166 (2015). https://doi.org/10.1007/s00466-014-1087-0

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