# 2010 Research Summary

## Aggresive Tetrahedral Mesh Improvement

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Bryan Klingner and Jonathan Shewchuk

National Science Foundation CCF-0635381 and National Science Foundation IIS-0915462

We developed a tetrahedral mesh improvement schedule that usually creates meshes whose worst tetrahedra have a level of quality substantially better than those produced by any previous method for tetrahedral mesh generation or mesh clean-up. Our goal is to aggressively optimize the worst tetrahedra, with speed a secondary consideration. Mesh optimization methods often get stuck in bad local optima (poor-quality meshes) because their repertoire of mesh transformations is weak. We employ a broader palette of operations than any previous mesh improvement software. Alongside the best traditional topological and smoothing operations, we introduce a topological transformation that inserts a new vertex (sometimes deleting others at the same time). We describe a schedule for applying and composing these operations that rarely gets stuck in a bad optimum. We demonstrate that all three techniques--smoothing, vertex insertion, and traditional transformations--are substantially more effective than any two alone. Our implementation usually improves meshes so that all dihedral angles are between 31 and 149 degrees, or (with a different objective function) between 23 and 136 degrees.

Figure 1: A stretched input mesh and four output meshes optimized with different quality measures as objective functions. The histograms tabulate, from top to bottom, dihedral angles, radius ratios (times 3), and volume divided by the cube of the root-mean-squared edge length.