3.2.2 Geometrical Operations

Figure 3.5: Laplacian smoothing

The triangles after the Laplacian smoothing operation, especially those near the boundary, are less skinny and have better quality (cf. Section 2.2.2).

Geometrical operations, or vertex smoothing algorithms, are operations which only move vertices but do not insert or delete any elements. Vertex smoothing algorithms are very popular algorithms for improving mesh element quality, because they do not alter the mesh topology. Since they do not delete or insert any new elements, they are also easy to handle from the data structure point of view.

Simple movement algorithms, like Laplacian smoothing visualized in Figure 3.5, have been shown to improve the overall mesh quality in many applications [41]. Laplacian smoothing also works well for non-simplex meshes [116]. However, to ensure geometry-conformity, the vertex smoothing algorithms have to be modified by restricting vertex movements of boundary vertices [124]. More complex smoothing algorithms, for example vertex smoothing based on the optimal Delaunay triangulation [78] or using local quality measure optimization [95], have also been proposed.