On Local Invertibility and Quality of Free-boundary Deformations

Mesh untangling is still a hot topic in applied mathematics. Tangled or folded meshes appear in many applications involving mappings or deformations. Despite the fact that a large number of mesh untangling strategies was proposed during the last decades, this problem still persists. Recently we have proposed a numerical optimization scheme [1] that provably untangles 2d and 3d meshes with inverted elements by partially solving a finite number of unconditional minimization problems. The method is robust for fixed boundary mesh untangling problems, and it can be applied to some extent to free boundary untangling. The problem, however, is that the absence of inverted elements does not guarantee invertibility of the deformation (map). The invertibility is lost if the mesh gets caught in a k-covering trap, i.e. in a local minimum of the deformation energy where all mesh elements are not inverted but total angle around certain vertex is above 2π for 2D and above 4π for 3D. This problem is particularly vexing when partially constrained mesh deformation problems are considered. In this paper we show how to improve the method suggested in [1]. Namely, we show the way to guarantee absence of k-covering folds, and so, the local invertibility is assured. We demonstrate enhanced stability of suggested untangling technique which has a potential to make untangling a routine operation over meshes.