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## Topological Cleanup

The topological cleanup consists of several sets of topological operations that are performed in an appropriate order. The following sets of operations have been implemented

• merge operations (Fig. 8a),
• swap operations (Fig. 8b),
• refine operations (Fig. 8c),
• coarse operations (Fig. 8d),
• split operations (Fig. 8e), and
• transform operations (Fig. 9).

While the merge and transform operations attempt to remove as many triangles as possible, the swap, refine, and coarse operations try to optimize the valence of individual nodes by decreasing the difference between the actual and the ideal valence (see Eq. (14)). The split operations are adopted to remove non-convex quadrilaterals by splitting them into two triangles. A particular topological transformation is accomplished only if violation of the following geometrical criteria does not occur:

• the new quadrilaterals must be convex,
• the new elements must not be inverted,
• the modified or newly created edges must comply (weakly) with the desired mesh size distribution,
• angle between the surface normals at the ends of modified or newly created edges must not exceed .

Theoretically, all triangles could be removed, if the domain is bounded by an even number of edges. In practice, however, individual triangles are topologically so far from each other that the adopted transformation schemes are not capable to remove them.

Note that the nodes newly introduced during the refine operation must satisfy the surface constraint.

Next: Mesh Refinement Up: Mesh Optimization Previous: Laplacian Smoothing

Daniel Rypl
2005-12-03