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search.filters.applied.f.access: openAccess × Type: Text × Publisher: Amsterdam [u.a.] : Elsevier × Subject: Geometry × WGL Institute: WIAS ×

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    Toward mixed-element meshing based on restricted Voronoi diagrams
    (Amsterdam [u.a.] : Elsevier, 2014) Pellerin, J.; Lévy, B.; Caumon, G.
    In this paper we propose a method to generate mixed-element meshes (tetrahedra, triangular prisms, square pyramids) for B-Rep models. The vertices, edges, facets, and cells of the final volumetric mesh are determined from the combinatorial analysis of the intersections between the model components and the Voronoi diagram of sites distributed to sample the model. Inside the volumetric regions, Delaunay tetrahedra dual of the Voronoi diagram are built. Where the intersections of the Voronoi cells with the model surfaces have a unique connected component, tetrahedra are modified to fit the input triangulated surfaces. Where these intersections are more complicated, a correspondence between the elements of the Voronoi diagram and the elements of the mixedelement mesh is used to build the final volumetric mesh. The method which was motivated by meshing challenges encountered in geological modeling is demonstrated on several 3D synthetic models of subsurface rock volumes.
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    On Tetrahedralisations Containing Knotted and Linked Line Segments
    (Amsterdam [u.a.] : Elsevier, 2017) Si, Hang; Ren, Yuxue; Lei, Na; Gu, Xianfeng
    This paper considers a set of twisted line segments in 3d such that they form a knot (a closed curve) or a link of two closed curves. Such line segments appear on the boundary of a family of 3d indecomposable polyhedra (like the Schönhardt polyhedron) whose interior cannot be tetrahedralised without additional vertices added. On the other hand, a 3d (non-convex) polyhedron whose boundary contains such line segments may still be decomposable as long as the twist is not too large. It is therefore interesting to consider the question: when there exists a tetrahedralisation contains a given set of knotted or linked line segments? In this paper, we studied a simplified question with the assumption that all vertices of the line segments are in convex position. It is straightforward to show that no tetrahedralisation of 6 vertices (the three-line-segments case) can contain a trefoil knot. Things become interesting when the number of line segments increases. Since it is necessary to create new interior edges to form a tetrahedralisation. We provided a detailed analysis for the case of a set of 4 line segments. This leads to a crucial condition on the orientation of pairs of new interior edges which determines whether this set is decomposable or not. We then prove a new theorem about the decomposability for a set of n (n ≥ 3) knotted or linked line segments. This theorem implies that the family of polyhedra generalised from the Schonhardt polyhedron by Rambau [1] are all indecomposable.