A 44-element mesh of Schneiders’ pyramid: Bounding the difficulty of hex-meshing problems

Abstract

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 18 interior vertices and 17 hexahedra, and introduce a valid mesh with 44 hexahedra. We also construct the smallest known mesh of the octagonal spindle, with 40 hexahedra and 42 interior vertices. These results were obtained through a general purpose algorithm that computes the hexahedral meshes conformal to a given quadrilateral surface boundary. The lower bound for Schneiders'pyramid is obtained by exhaustively listing the hexahedral meshes with up to 17 interior vertices and which have the same boundary as the pyramid. Our 44-element mesh is obtained by modifying a prior solution with 88 hexahedra. The number of elements was reduced using an algorithm which locally simplifies 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.

Tangled finite element method for handling concave elements in quadrilateral meshes

“…Several extensions of TFEM are possible. The most significant is the 3D extension to implicitly tangled hexahedral meshes 42,43 . The main challenge is computing and integrating over the negative Jacobian region.…”

“…The most significant is the 3D extension to implicitly tangled hexahedral meshes. 42,43 The main challenge is computing and integrating over the negative Jacobian region. To illustrate, consider a 3D tangled hexahedral mesh in Figure 31A with one concave hex element.…”

Hex Me If You Can

“…All the tetrahedral meshes provided by HexMe have been produced from three categories of CAD models, Figure 3: simple models: basic shapes that are assumed to be easily hexmeshable, i.e. the target hexahedral topology is fair, e.g., a cube (s01o_cube.geo), or a cut hemisphere on a cylinder (s10o_cyl_cutsphere.stp). nasty models: academic shapes that are challenging to hex‐mesh, e.g., a pyramid [VPR19] (n09o_pyramid.geo), or a ski jump (n02o_skijump_anti_box_cyl.geo). industrial models: lifelike shapes, which hexahedrization is highly valuable for numerical simulation, e.g., a truck tire (i28o_gc_tire_1218.step from GrabCAD), or an aircraft for CFD (i31o_dlr_f6.brep [VTM∗08,BERF08]).…”

“…nasty models: academic shapes that are challenging to hex‐mesh, e.g., a pyramid [ VPR19 ] (n09o_pyramid.geo), or a ski jump (n02o_skijump_anti_box_cyl.geo).…”