Quadrilateral and Hexahedral Element Meshes

This chapter explains techniques for the generation of quadrilateral and hexahedral element meshes. Since structured meshes are discussed in detail in other parts of this volume, we focus on the generation of unstructured meshes, with special attention paid to the 3D case. Quadrilateral or hexahedral element meshes are the meshes of choice for many applications, a fact that can be explained empirically more easily than mathematically. An example of a numerical experiment is presented by Benzley [1995], who uses tetrahedral and hexahedral element meshes for bending and torsional analysis of a simple bar, fixed at one end. If elastic material is assumed, second-order tetrahedral elements and first-order hexahedral elements both give good results (first-order tetrahedral elements perform worse). In the case of elastic–plastic material, a hexahedral element mesh is significantly better. A mathematical argument in favor of the hexahedral element is that the volume defined by one element must be represented by at least five tetrahedra. The construction of the system matrix is thus computationally more expensive, in particular if higher order elements are used. Unstructured hex meshes are often used in computational fluid dynamics, where one tries to fill most of the computational domain with a structured grid, allowing irregular nodes but in regions of complicated shape, and for the simulation of processes with plastic deformation, e.g., metal forming processes. In contrast to the favorable numerical quality of quadrilateral and hexahedral element meshes, mesh generation is a very difficult task. A hexahedral element mesh is a very “stiff” structure from a geometrical point of view, a fact that is illustrated by the following observation: Consider a structured grid and a new node that must be inserted by using local modifications only (Figure 21.1). While this can be done in 2D, in the three-dimensional case it is no longer possible! Thus, it is not possible to generate a hexahedral element mesh by point insertion methods, a technique that has proven very powerful for the generation of tetrahedral element meshes (Delaunay–type algorithms, Chapter 16). Many algorithms for the generation of tetrahedral element meshes are advancing front methods (Chapter 17), where a volume is meshed starting from a discretization of its surface and building the volume mesh layer by layer. It is very difficult to use this idea for hex meshing, even for very simple Robert Schneiders