The structured grid generation presented in Chapter 17 is restricted to those cases where the physical domain can be transformed into a computational domain through one-to-one mapping. For irregular geometries, however, such mapping processes may become either inconvenient or impossible to apply. In these cases, the structured grid generation methods are abandoned and we turn to unstructured grids where transformation into the computational domain from the physical domain is not required. Even for the regular geometries, an unstructured grid generation may be preferred for the purpose of adaptive meshing in which the structured grids initially constructed become unstructured as adaptive refinements are carried out.

Finite volume and finite element methods can be applied to unstructured grids. This is because the governing equations in these methods are written in integral form and numerical integration can be carried out directly on the unstructured grid domain in which no coordinate transformation is required. This is contrary to the finite difference methods in which structured grids must be used.

There are two major unstructured grid generation methods: Delaunay-Voronoi methods (DVM) and advancing front methods (AFM) for triangles (2-D) and tetrahedrals (3-D). Numerous other methods for quadrilaterals (2-D) and hexahedrals (3-D) are available (tree methods, paving methods, etc.). We shall discuss these and other topics in this chapter.

DELAUNAY-VORONOI METHODS

A two-dimensional domain may be triangulated as shown in Figure 18.1.1a (light lines). Each side line of the triangles can be bisected in a perpendicular direction such that these three bisectors join a point within the triangle (heavy lines in Figure 18.1.1a), forming a polygon surrounding the vertex of each triangle, known as the Voronoi polygon (diagram) [Voronoi, 1908].