Mesh Smoothing Schemes Based on Optimal Delaunay Triangulations

We present several mesh smoothing schemes based on the concept of optimal Delaunay triangulations. We define the optimal Delaunay triangulation (ODT) as the triangulation that minimizes the interpolation error among all triangulations with the same number of vertices. ODTs aim to equidistribute the edge length under a new metric related to the Hessian matrix of the approximated function. Therefore we define the interpolation error as the mesh quality and move each node to a new location, in its local patch, that reduces the interpolation error. With several formulas for the interpolation error, we derive a suitable set of mesh smoothers among which Laplacian smoothing is a special case. The computational cost of proposed new mesh smoothing schemes in the isotropic case is as low as Laplacian smoothing while the error-based mesh quality is provably improved. Our mesh smoothing schemes also work well in the anisotropic case.