TABLE OF CONTENTS Unstructured grid generation was originally developed to support the use of the finite element method for the modeling of structures [Thompson et al., 1985]. Unstructured grids are now widely used in most areas of simulation; they may be easier to generate and may possess higher quality for some applications on particular domain geometry. The popularity of these meshes creates a strong demand for efficient methods of enhancement; extensions of structured methods do not generally result in approaches that are effective for unstructured meshes. However, some of these extensions were explored in the previous chapter.
There are many effective methods to enhance unstructured meshes. In Chapter 1, it was suggested that methods of this type may be arranged into three general categories: 1) geometric methods targeted at meshes comprised of triangular and tetrahedral elements, 2) methods based on optimization of geometric or other cell or element criteria, and 3) methods based on the solution of systems of partial differential equations (PDEs). The scope of this chapter is limited to the last category, and is further restricted to considering only approaches based on the Laplace-Beltrami system using a target metric. The Laplace-Beltrami system was developed in Chapter 7. The development of this approach was motivated by previous work on structured meshes, which was summarized in the previous chapter (Chapter 8).
There have been several studies of PDE-based methods that are applicable to unstructured meshes. For enhancement of unstructured two-dimensional grids, Knupp [Knupp, 1998] applied the Winslow elliptic smoother using the finite...
