Complexity and modeling aspects of mesh refinement into quadrilaterals

  • Rolf H. Möhring
  • Matthias Müller-Hannemann
Session 6A
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1350)


We investigate the following mesh refinement problem: Given a mesh of polygons in three-dimensional space, find a decomposition into strictly convex quadrilaterals such that the resulting mesh is conforming and satisfies prescribed local density constraints.

The conformal mesh refinement problem is shown to be feasible if and only if a certain system of linear equations over GF(2) has a solution. To improve mesh quality with respect to optimization criteria such as density, angles and regularity, we introduce a reduction to a minimum cost bidirected flow problem. However, this model is only applicable, if the mesh does not contain branching edges, that is, edges incident to more than two polygons. The general case with branchings, however, turns out to be strongly MP-hard. To enhance the mesh quality for meshes with branchings, we introduce a two-stage approach which first decomposes the whole mesh into components without branchings, and then uses minimum cost bidirected flows on the components in a second phase.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [AMO93]
    R. K. Ahuja, T. L. Magnanti, and J. B. Orlin, Network flows, Prentice Hall, 1993.Google Scholar
  2. [BE95]
    M. Bern and D. Eppstein, Mesh generation and optimal triangulation, Computing in Euclidean Geometry, 2nd Edition (D.-Z. Du and F. Hwang, eds.), World Scientific, Singapore, 1995, pp. 47–123.Google Scholar
  3. [Bra93]
    J. R. Brauer, ed., What every engineer should know about finite element analysis, Marcel Decker Inc., 1993.Google Scholar
  4. [Der88]
    U. Derigs, Programming in networks and graphs, Lecture Notes in Economics and Mathematical Systems, vol. 300, Springer-Verlag, Berlin, 1988.Google Scholar
  5. [Edm67]
    J. Edmonds, An introduction to matching, Lecture notes, University of Michigan, Ann Arbor, 1967.Google Scholar
  6. [Ho88]
    K. Ho-Le, Finite element mesh generation methods: a review and classification, Computer-Aided Design 20 (1988), 27–38.Google Scholar
  7. [Joe95]
    B. Joe, Quadrilateral mesh generation in polygonal regions, Computer-Aided Design 27 (1995), 209–222.Google Scholar
  8. [MH97]
    M. Mülller-Hannemann, High quality quadrilateral surface meshing without template restrictions: A new approach based on network flow techniques, to appear in Proceedings of the Sixth International Meshing Roundtable, Park City, Utah, 1997.Google Scholar
  9. [MMW96]
    R. H. Möhring, M. Müller-Hannemann, and K. Weihe, Mesh refinement via bidirected flows: Modeling, complexity, and computational results, Technical report No. 520/1996, Fachbereich Mathematik, Technische Universität Berlin, 1996, to appear in the Journal of the ACM, vol. 23, no. 2, 1997.Google Scholar
  10. [MW97]
    M. Müller-Hannemann and K. Weihe, Minimum strictly convex quadrangulations of convex polygons, Proceedings of the 13th Annual ACM Symposium on Computational Geometry, Nice, France, ACM, 1997, pp. 193–202.Google Scholar
  11. [TA93]
    T. K. H. Tam and C. G. Armstrong, Finite element mesh control by integer programming, Int. J. Numer. Methods in Eng. 36 (1993), 2581–2605.Google Scholar
  12. [ZT89]
    O. C. Zienkiewicz and R. L. Taylor, The finite element method, McGraw Hill, London 1989.Google Scholar
  13. [ZZHW91]
    J. Z. Zhu, O. C. Zienkiewicz, E. Hinton, and J. Wu, A new approach to the development of automatic quadrilateral mesh generation, Int. J. Numer. Methods in Eng. 32 (1991), 849–866.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Rolf H. Möhring
    • 1
  • Matthias Müller-Hannemann
    • 1
  1. 1.Fachbereich MathematikTechnische Universität BerlinGermany

Personalised recommendations