Combined Laplacian and Optimization-based Smoothing for Quadratic Mixed Surface Meshes

Quadratic elements place stringent requirements on a surface mesh smoother. One of the biggest challenges is that a good linear element may become invalid when mid-side nodes are introduced. To help alleviate this problem, a new objective function for optimization-based smoothing is proposed for triangular and quadrilateral elements, linear or quadratic. Unlike the current popular approaches, this objective function makes it possible for a smoothing algorithm to untangle and smooth in a single process. This objective function has higher order continuous derivatives and only one minimum, if any, that make it suitable for optimization techniques. Even though optimization-based smoothing obtains much higher quality results compared to other algorithms, such as constrained Laplacian smoothing, it is also slower than these algorithms. That said, we also present an effective way to limit the number of calls to optimization-based smoothing such that the highest quality mesh is obtained in the least amount of time.