🍩 Database of Original & Non-Theoretical Uses of Topology

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  1. Substructure Topology Preserving Simplification of Tetrahedral Meshes (2011)

    Fabien Vivodtzev, Georges-Pierre Bonneau, Stefanie Hahmann, Hans Hagen
    Abstract 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.