Skip to main content
Springer Nature Link
Log in

Geometric Properties of the Adaptive Delaunay Tessellation

  • Conference paper
Mathematical Methods for Curves and Surfaces (MMCS 2008)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5862))

Abstract

Recently, the Adaptive Delaunay Tessellation (Adt) was introduced in the context of computational mechanics as a tool to support Voronoi-based nodal integration schemes in the finite element method. While focusing on applications in mechanical engineering, the former presentation lacked rigorous proofs for the claimed geometric properties of the Adt necessary for the computation of the nodal integration scheme. This paper gives pending proofs for the three main claims which are uniqueness of the Adt, connectedness of the Adt, and coverage of the Voronoi tiles by adjacent Adt tiles. Furthermore, this paper provides a critical assessment of the Adt for arbitrary point sets.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Shewchuk, J.: What is a good linear element? Interpolation, conditioning, and quality measures. In: Proc. Eleventh Int. Meshing Roundtable, Ithaca, New York, pp. 115–126. Sandia National Laboratories (2002)

    Google Scholar 

  2. Lohner, R.: Progress in grid generation via the advancing front technique. Eng. with Computers 12, 186–210 (1996)

    Article  MathSciNet  Google Scholar 

  3. Tournois, J., Alliez, P., Devillers, O.: Interleaving delaunay refinement and optimization for 2D triangle mesh generation. In: Proc. 16th Int. Meshing Roundtable (2007)

    Google Scholar 

  4. Bossen, F., Heckbert, P.: A pliant method for anisotropic mesh generation. In: Proc. 5th Int. Meshing Roundtable, pp. 63–74 (1996)

    Google Scholar 

  5. Antani, L., Delage, C., Alliez, P.: Mesh sizing with additively weighted voronoi diagrams. In: Proc. 16th Int. Meshing Roundtable (2007)

    Google Scholar 

  6. Hjelle, Ø., Dæhlen, M.: Triangulations and Applications. Mathematics and Visualization. Springer, Secaucus (2006)

    Google Scholar 

  7. Constantiniu, A., Steinmann, P., Bobach, T., Farin, G., Umlauf, G.: The adaptive delaunay tessellation: A neighborhood covering meshing technique. Computational Mechanics 42(5), 655–669 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Sukumar, N., Moran, B., Semenov, A., Belikov, V.: Natural neighbour Galerkin methods. Int. J. for Numerical Methods in Engineering 50, 1–27 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen, J., Wu, C., Yoon, S., You, Y.: A stabilized conforming nodal integration for Galerkin mesh-free methods. Int. J. Numer. Methods Eng. 50, 435–466 (2001)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dept. of Computer Science, University Kaiserslautern, Germany

    Tom Bobach & Georg Umlauf

  2. Institute of Applied Mechanics, University of Erlangen-Nuremberg, Germany

    Alexandru Constantiniu & Paul Steinmann

Authors
  1. Tom Bobach
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alexandru Constantiniu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Paul Steinmann
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Georg Umlauf
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Informatics, University of Oslo, Norway

    Morten Dæhlen  & Tom Lyche  & 

  2. Department of Informatics and Centre of Mathematics for Applications, University of Oslo, Norway

    Michael Floater

  3. INSA de Rennes, Rennes, France

    Jean-Louis Merrien

  4. Department of Informatics and Centre of Mathematics for Applications, University of Oslo, Oslo, Norway

    Knut Mørken

  5. Department of Mathematics, Vanderbilt University, 37240, Nashville, TN, USA

    Larry L. Schumaker

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Bobach, T., Constantiniu, A., Steinmann, P., Umlauf, G. (2010). Geometric Properties of the Adaptive Delaunay Tessellation. In: Dæhlen, M., Floater, M., Lyche, T., Merrien, JL., Mørken, K., Schumaker, L.L. (eds) Mathematical Methods for Curves and Surfaces. MMCS 2008. Lecture Notes in Computer Science, vol 5862. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11620-9_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-11620-9_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-11619-3

  • Online ISBN: 978-3-642-11620-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics