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 unstructured tetrahedral mesh used for such a simulation includes 2d and 1d
substructures embedded in the vertices, edges and faces of the mesh.
The simplification of such tetrahedral 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 unstructured tetrahedral 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
Keywords: tetrahedral meshes, simplification, structure preservation, topology.
F. Vivodtzev, G.-P. Bonneau, S. Hahmann, H. Hagen
Substructure Topology Preserving Simplification of Tetrahedral Meshes,
In Topological Methods in Data Analysis and Visualization, Springer (Ed.) (2011)
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