Practical Applications of Mesh Smoothing
Judy Hill
Geometric Modeling Project
Spring, 1998

Three different examples of mesh smoothing are presented below. Freitag (1997) and Freitag and Ollivier-Gooch (1997) compared a Laplacian smoothing method and an optimization-based method. Amezua, et.al. (1995) tested the length constraint method. The results of the three tests are presented below.

Freitag initially tested a two-dimensional mesh generated by Delaunay triangulation. The mesh was generated from 500 random points. Figure S1 below illustrates the initial mesh and the final mesh after both Laplacian smoothing and optimization-based smoothing.

Figure S1. A mesh of 580 points (a) after Delaunay triangulation, (b) after Laplacian smoothing,
(c) after optimization-based smoothing. (Freitag 1997)

From Figure S1, it is evident that both the Laplacian and optimization-based smoothing significantly improved element distribution and shape. In Table S1 below, a summary of Freitag's results for Laplacian and optimization-based smoothing (after 3 passes) is presented.

It is immediately evident that the optimization-based smoothing technique provides a better mesh, based on maximizing the minimum and minimizing the maximum angles in the mesh. However, it is five times more costly, in terms of computational time, compared with Laplacian smoothing.

Freitag and Ollivier-Gooch (1997) tested Laplacian smoothing and optimization-based smoothing on more practical applications. The example shown in Figure S2 is a surface wireframe of a tire incinerator.

A significant improvement of the mesh quality was achieved by using both Laplacian and optimization-based smoothing. A combination of both methods, though more costly than Laplacian smoothing alone, provided a better final mesh than was achieved by Laplacian smoothing alone. In addition, Freitag and Ollivier-Gooch found that combining smoothing techniques with local mesh improvement (face and edge swapping) produced an even better mesh.

Amezua, et.al (1995) presented the results of several different meshes improved by the length constraint smoothing. Figure S3 below shows an example of one of their tests.

Figure S3. A mesh of 924 elements (a) after triangulation, (b) after length-constraint smoothing. (Amezua, et.al. 1995)

Figure S3b shows a mesh after four smoothing iterations. Note that the mesh has only been changed in very localized regions. For example, the aspect ratio of some of the triangular elements has been improved. No error function was reported to provide a quantitative comparison between the initial and final meshes.