In this paper, we present a solution that uses the least number of hexahedra to build a pyramid, which is the key block required for one type of automatic hex-meshing method to be successful. When the initial result of a hex-meshing program is not appropriate for specific applications, some templates are used for revision. The templates reported thus far are parity-preserving, which means that the parity of the number of hexahedra in a mesh is unchanged after a revision following the templates. We present a parity-changing template that makes the template set integral and more effective. These two findings are obtained by a program that we developed for this study, which is a tool for researchers to observe the characteristics of small hexahedral packings.
Hex meshes are urgently needed in various engineering areas. Due to the conforming requirement shown in figure 2, hex meshing is extremely challenging, and it costs days or even weeks to build a moderate complexity mesh for an experienced engineer. For a half century, researchers have attempted to develop methods to automatically construct a hex mesh [Blacker 2001;Owen 1998;Tautges 2001;Fang et al 2016;Li et al 2011;Nieser et al 2011]. However, there is still a long way to make these methods practical.
(a) (b) (c) Figure 2： The conforming requirement of a hex mesh: if two hexahedra meet at a face, they should meet at the whole face and not part of it. Case (a) is conforming; cases (b) and (c) are nonconforming.
A compromise method of automatic hex meshing is to create a hex-dominant mesh first and then transform it to an all-hex mesh. A hex-dominant mesh needs two other types of elements, a pyramid and tetrahedron, to completely fill a space.
There are different types of hex-dominant meshing methods [Stephenson et al 1992;Tristan et al 2014;Sokolov et al 2016;Xifeng Gao et al 2017], yet we focus on the advancing front approach for discussion here since the transformation stage is the same if all-hex meshes are needed. For a given 3D space, we place hexahedra around the space boundary first and then toward the interior. Due to the conforming requirement, one usually cannot fill the space completely by advancing the front, and a void will be left in the interior, as figure 3 illustrates in the 2D situation. A hex-dominant meshing program will then use the other two types of elements to fill the void. The void is enclosed by quadrilaterals, and from each of the quadrilaterals, a pyramid will grow, as shown in figure 4, which turns the boundary faces of the void into triangles. The void will then be filled with tetrahedrons, and since it is now enclosed by triangles, it is guaranteed to be fulfilled [Bern 1993]. The process mentioned above finishes to fill a space with a hexdominant mesh; one can then transform it to an all-hex mesh by subdividing its elements. Each hexahedron is divided into 8 smaller ones, as shown in figure 5(a), and each tetrahedron is divided into four hexahedra, as shown in figure 5(b). After the subdivision of all the tetrahedrons and hexahedra, the surface of each pyramid will be composed of 16 quadrilaterals, as shown in figure 1(a). How can a pyramid be subdivided into hexahedra that conform to these 16 quadrilaterals? We now face Schneiders’ problem. Schneiders’ problem [Schneiders 1996]: creating a hex mesh of a pyramid that conforms to the surface subdivision shown in figure 1(a). Unlike its first appearance, the problem is very difficult. One must grow hexahedra based on the 16 given quadrilaterals toward the interior and make the pyramid to be filled completely. Obviously, the solution to Schneiders’ problem is a key step for building a hex mesh from a hex-dominant mesh.
To solve the problem, we developed a program, which is actually a tool to study small hexahedra packings and, by using it, we have gotten two findings including the solution to the Schneiders’ problem.
Contributions: We give a 36-element solution to Schneiders’ problem, which is the least element solution to the problem. We report a parity-changing template, which is being expected by the community for years. We developed a program, which can be used to study the characteristics of small hexahedra packings.
Previous works: Yamakawa et al. [2002] presented a solution for the pyramid problem, which has 118 elements. The same authors gave an improved solution in [Yamakawa et al 2010], which has 88 elements, as shown in figure 6. The 88 solution is currently the best result; however, it still has the disadvantage of having too many hexahedra because, for a common engineering problem, one will obtain a rather large number of pyramid elements in a hex-dominant mesh. Multiplying by 88 times will produce an enormous number of hexahedra in the final all-hex mesh, which has the consequence of heavily degrading the performance of its subsequent engineering applications. It is notoriously difficult to improve topology connections of a hex-mesh. Bern et al [2002] studied the basic operations to flip hex-meshes. They claimed a parity-changing template should be existed and it is important theoretically. Yet no parity-changing template has been given so far. The rest of the articles are organized as follows: section 2 gives our method to solve the Schneider’s problem. A program is developed for the purpose. Section 3 uses the program to solve another problem, building a parity-changing template. Section 4 is the discussion and conclusions. The appendix lists the 36element solution and the parity-changing template.
Unlike the 88 and 118 solutions, which are created based on the human imagination, our answer is produced by systematically searching by a computer program.
We first develop a program that adds hexahedra one
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