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Author: Si, Hang × End date: 2019 × Start date: 2019 × End date: 2019 × Start date: 2010 × Subject: edge flip × WGL Institute: WIAS ×

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    On tetrahedralisations of reduced Chazelle polyhedra with interior Steiner points
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Si, Hang; Goerigk, Nadja
    The polyhedron constructed by Chazelle, known as Chazelle polyhedron [4], is an important example in many partitioning problems. In this paper, we study the problem of tetrahedralising a Chazelle polyhedron without modifying its exterior boundary. It is motivated by a crucial step in 3d finite element mesh generation in which a set of arbitrary boundary constraints (edges or faces) need to be entirely preserved. We first reduce the volume of a Chazelle polyhedron by removing the regions that are tetrahedralisable. This leads to a 3d polyhedron which may not be tetrahedralisable unless extra points, so-called Steiner points, are added. We call it a reduced Chazelle polyhedron. We define a set of interior Steiner points that ensures the existence of a tetrahedralisation of the reduced Chazelle polyhedron. Our proof uses a natural correspondence that any sequence of edge flips converting one triangulation of a convex polygon into another gives a tetrahedralization of a 3d polyhedron which have the two triangulations as its boundary. Finally, we exhibit a larger family of reduced Chazelle polyhedra which includes the same combinatorial structure of the Schönhardt polyhedron. Our placement of interior Steiner points also applies to tetrahedralise polyhedra in this family.
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    A curvature-adapted anisotropic surface remeshing method
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Dassi, Franco; Si, Hang
    We present a new method for remeshing surfaces that respect the intrinsic anisotropy of the surfaces. In particular, we use the normal informations of the surfaces, and embed the surfaces into a higher dimensional space (here we use 6d). This allow us to form an isotropic mesh optimization problem in this embedded space. Starting from an initial mesh of a surface, we optimize the mesh by improving the mesh quality measured in the embedded space. The mesh is optimized by combining common local modifications operations, i.e., edge flip, edge contraction, vertex smoothing, and vertex insertion. All perations are applied directly on the 3d surface mesh. This method results a curvature-adapted mesh of the surface. This method can be easily adapted to mesh multi-patches surfaces, i.e., containing corner singularities and sharp features. The reliability and robustness of the proposed re-meshing technique is provided by a large number of examples including both implicit surfaces and CAD models.