88‐Element solution to Schneiders' pyramid hex‐meshing problem
Soji Yamakawa, Kenji Shimada · International Journal for Numerical Methods in Biomedical Engineering · 2009
Abstract This paper presents a solution to a hex‐meshing problem known as Schneiders' problem, which was introduced by Robert Schneiders in 1996. Consider a pyramid enclosed by a square and four equilateral triangles. The exterior of the pyramid can be subdivided into an all‐quadrilateral mesh by (1) inserting a node at the middle of each edge and at the center of each face and (2) connecting a node inserted at the center of the face with the nodes inserted on the edges of the face. The goal of Schneiders' problem is to find an all‐hex mesh that conforms to this quadrilateral mesh. The solution should consist of as few elements as possible, and as well‐shaped elements as possible. Since a solution to this problem could contribute to a fully automatic unstructured all‐hex mesh generation scheme, the problem has been studied for years. However, no practically usable solution has been found. The smallest known solution has been a hexahedral mesh consisting of 118 elements. This paper presents a new solution to the problem that consists of 88 elements. All elements included in the mesh are positive, and no two neighboring elements share more than one face. Although it is unknown whether 88‐element solution is the smallest possible solution, it is one step advance from the previously known 118‐element solution. Copyright © 2009 John Wiley & Sons, Ltd.