A 66-element mesh of Schneiders' pyramid: bounding the difficulty of hex-meshing problems
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This paper shows that constraint programming techniques can successfully be used to solve challenging hex-meshing problems. Schneiders' pyramid is a square-based pyramid whose facets are subdivided into three or four quadrangles by adding vertices at edge midpoints and facet centroids. In this paper, we prove that Schneiders' pyramid has no hexahedral meshes with fewer than 12 interior vertices and 14 hexahedra, and introduce a valid mesh with 66 hexahedra. We also introduce a parity-changing operator for hexahedral meshes, simpler than the construction by Bern, Eppstein, and Erickson. These results were obtained through a general purpose algorithm that computes the hexahedral meshes conformal to a given quadrilateral surface boundary. The lower bound is obtained by exhaustively listing the hexahedral meshes with up to 11 interior vertices and which have the same boundary as the pyramid. The upper bound is obtained by modifying the previously known smallest solution with 88 hexahedra. The number of elements was reduced to 66 by locally simplifying groups of hexahedra. Given the boundary of such a group, our algorithm is used to find a mesh of its interior that has fewer elements than the initial subdivision. The resulting mesh is untangled to obtain a valid hexahedral mesh.
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