Mesh Smoothing and Untangling

1. Mesh Smoothing and Untangling Optimization of vertex locations in simplicial meshes. Sean Mauch Caltech October, 2003

2. Optimizing Vertex Locations • One can improve the quality of the simplices in a triangle or tetrahedral mesh by moving the vertices. (We consider moving only the internal vertices.) • We iterate over the vertices and consider the local mesh of the simplices incident to a single vertex. • In this Gauss-Seidel iteration, we move each vertex to optimize the local mesh quality. • One can use geometric quality measures, like the minimum dihedral angle, in the optimization. Lori Freitag implemented this approach with the Opt-MS package. One can also use algebraic quality measures defined in terms of the Jacobian matrix of the transformation from the equilateral simplex. (“Simultaneous untangling and smoothing of tetrahedral meshes” by Escobar et. al.) We have implemented the latter approach. 2

3. An Example of Smoothing Initial mesh. Tangled and distorted mesh. One smoothing iteration. 3 Two smoothing iterations. Three smoothing iterations.

4. Preliminary Results • We have implemented mesh smoothing with algebraic quality measures as a C++ class library. • The problem dimension is a template parameter. The code will smooth 2-D triangle meshes, 3-D tetrahedral meshes and higher dimensional simplicial meshes. • Initial tests indicate that the approach may be useful in optimizing input meshes and in repairing meshes during the course of a simulation. 4

5. Future Work • We may implement additional quality measures and compare their performance. • We will implement constrained optimization of the vertices on the boundary. • If the boundary faces have poor quality, so will some of the simplices. • Move the boundary vertices to optimize the quality of the simplices subject to maintaining the shape of the boundary. • We will also implement topological optimization. Flip edges/faces to increase simplex quality. 5