The geometric element transformation method for tetrahedral mesh smoothing

The geometric element transformation method (GETMe) has been introduced as a new element driven approach to mesh smoothing. It is based on simple geometric transformations, which, if applied iteratively, lead to the regularization of mesh elements. Global mesh smoothing can be accomplished by successively improving the worst elements or by averaging node positions obtained by the simultaneous transformation of all elements. GETMe smoothing has been successfully applied in the case of surface meshes. As shown in this paper, this approach also naturally extends to tetrahedral mesh smoothing without major conceptual modifications. A regularizing transformation for tetrahedra is presented and a combined approach of simultaneous and sequential GETMe smoothing is described. First numerical examples yield high quality meshes superior to those obtained by other geometry-based methods. In fact, the presented results are in a majority of cases at least comparable to those obtained by a state of the art global optimization-based method.