Substructure Topology Preserving Simplification of Tetrahedral Meshes

  • Fabien Vivodtzev
  • Georges-Pierre Bonneau
  • Stefanie Hahmann
  • Hans Hagen
Part of the Mathematics and Visualization book series (MATHVISUAL)


Interdisciplinary efforts in modeling and simulating phenomena have led to complex multi-physics models involving different physical properties and materials in the same system. Within a 3d domain, substructures of lower dimensions appear at the interface between different materials. Correspondingly, an unstructuredtetrahedral mesh used for such a simulation includes 2d and 1d substructures embedded in the vertices, edges and faces of the mesh.The simplification of suchtetrahedral meshes must preserve (1) the geometry and the topology of the 3d domain, (2) the simulated data and (3) the geometry and topology of the embedded substructures. Although intensive research has been conducted on the first two goals, the third objective has received little attention.This paper focuses on the preservation of the topology of 1d and 2d substructures embedded in an unstructuredtetrahedral mesh, during edge collapse simplification. We define these substructures as simplicial sub-complexes of the mesh, which is modeled as an extended simplicial complex. We derive a robust algorithm, based on combinatorial topology results, in order to determine if an edge can be collapsed without changing the topology of both the mesh and all embedded substructures. Based on this algorithm we have developed a system for simplifying scientific datasets defined on irregular tetrahedral meshes with substructures. The implementation of our system is discussed in detail. We demonstrate the power of our system with real world scientific datasets from electromagnetism simulations.


Simplicial Complex Link Condition Tetrahedral Mesh Extended Complex Volume Mesh 
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  1. 1.
    P. Cignoni, D. Costanza, C. Montani, C. Rocchini, and R.Scopigno. Simplification of tetrahedral volume with accurate error evaluation. In Proceedings IEEE Visualization ’00, pages 85–92, 2000.Google Scholar
  2. 2.
    P. Cignoni, L. D. Floriani, P. Lindstrom, V. Pascucci, J. Rossignac, and C. Silva. Multi-resolution modeling, visualization and streaming of volume meshes. In Eurographics 2004, Tutorials 2: Multi-resolution Modeling, Visualization and Streaming of Volume Meshes. INRIA and the Eurographics Association, September 2004.Google Scholar
  3. 3.
    P. Cignoni, C. Montani, E. Puppo, and R. Scopigno. Multiresolution representation and visualization of volume data. IEEE Transactions on Visualization and Computer Graphics, 3(4):352–369, oct–dec 1997.Google Scholar
  4. 4.
    T. Dey, H. Edelsbrunner, S. Guha, and D. Nekhayev. Topology preserving edge contraction. Publications de l’Institut de Mathématiques (BEOGRAD), 66(80):23–45, 1999.Google Scholar
  5. 5.
    L. D. Floriani, P. Magillo, and E. Puppo. The MT (Multi-Tesselation) package. DISI, University of Genova, Italy, 2000.
  6. 6.
    L. D. Floriani, E. Puppo, and P. Magillo. A formal approach to multiresolution modeling. In W. Straer, R. Klein, and R. Rau, editors, Theory and Practice of Geometric Modeling. SpringerVerlag, 1997., 1997.Google Scholar
  7. 7.
    H. Hoppe. Progressive meshes. Computer Graphics, 30(Annual Conference Series):99–108, 1996.Google Scholar
  8. 8.
    H. Hoppe, T. DeRose, T. Duchamp, J. McDonald, and W. Stuetzle. Mesh optimization. Technical Report TR 93-01-01, Dept. of Computer Science and Engineering, University of Washington, Jan. 1993.Google Scholar
  9. 9.
    M. Kraus and T. Ertl. Simplification of nonconvex tetrahedral meshes, 2000.Google Scholar
  10. 10.
    D. Luebke. A developer’s survey of polygonal simplification algorithms. IEEE Computer Graphics and Applications, 21(3):24–35, 2001.CrossRefGoogle Scholar
  11. 11.
    J. Munkres. Elements of algebraic topology. Perseus publishing, Cambridge, 1984.zbMATHGoogle Scholar
  12. 12.
    V. Natarajan and H. Edelsbrunner. Simplification of three-dimensional density maps. IEEE Transactions on Visualization and Computer Graphics, 10(5):587–597, 2004.CrossRefGoogle Scholar
  13. 13.
    W. Schroeder, J. Zarge, and W. Lorensen. Decimation of triangle meshes. Computer Graphics, 26(2):65–70, 1992.CrossRefGoogle Scholar
  14. 14.
    F. Vivodtzev, G.-P. Bonneau, and P. Letexier. Topology preserving simplification of 2d non-manifold meshes with embedded structures. The Visual Computer, 21(8):679–688, 2005.CrossRefGoogle Scholar

Copyright information

© Springer Berlin Heidelberg 2011

Authors and Affiliations

  • Fabien Vivodtzev
    • 1
  • Georges-Pierre Bonneau
    • 2
  • Stefanie Hahmann
    • 2
  • Hans Hagen
    • 3
  1. 1.CEA/CESTA (French Atomic Energy Commission)Le BarpFrance
  2. 2.LJKUniversity of Grenoble and INRIAGrenobleFrance
  3. 3.University of KaiserslauternKaiserslauternGermany

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