3.4.7 Local Transformations

The edges in a mesh are locally modified while the mesh points are left unchanged. Such topological transformations of the mesh elements are an important technique to improve the mesh and to get rid of imperfections in an efficient and fairly straightforward manner. In two dimensions transformations are simple and can in fact be restricted to a single type of operation on a set of two triangles. The diagonal edge is flipped which is often also called edge swapping. The same technique applied to surface triangulations in three dimensions is defined more precisely in Section 6.3. Topological changes for volume elements in three dimensions are much more sophisticated [71,72,75]. A transformation generally changes the number of involved tetrahedra and several types of flip operations have to be defined. The basic operation is a 2-3 or 3-2 flip where facet swapping respectively introduces or eliminates a third tetrahedron (Fig. 3.15). More complex operations involve five or more tetrahedra.

Figure 3.15: 3-2 or 2-3 local transformation. The internal facet which is being swapped is drawn shaded.

As can be seen from Fig. 3.15 local transformations can be useful to remove slivers which only exist in three or higher dimensions. It should be noted that an algorithm performing only local transformations may get stuck in local optima.