Practical Applications of Swapping
Judy Hill
Geometric Modeling Project
Spring, 1998

Freitag and Ollivier-Gooch (1997) also tested the effect of face and edge swapping. The results of these tests are shown below in Table FE1. In these tests, a random mesh, generated using the Delaunay algorithm with no improvement, was generated containing 5104 tetrahedra. The mesh was then improved using either face swaps only or a combination of face and edge swaps. Several criteria for determining the "quality" of the mesh were examined. Presented here are two of those criteria: minimizing the maximum dihedral angle (minmax angle) and maximizing the minimum sine of the dihedral angle (maxmin sine).

Comparing the improvement of the minimum and maximum dihedral angles among the tetrahedral elements, there appears to be little improvement between the initial and "improved" meshes. However, the percentage of elements with extremely large or small angles is reduced by swapping. Face swapping, alone, reduces the number of distorted elements by, at some degrees, a factor of two. Furthermore, when edge swapping is added, the percentage of distorted elements is significantly reduced. Elements with large dihedral angles are almost non-existent (0.43%). Elements with extremely small angles are very few (1.42 percent, in the worst case).

Thus, as can be seen from these results, local topological improvement of element shape reduces the number of elements with extremely large or small angles. However, it cannot be determined from this analysis is whether the numerical accuracy has been improved.