Least-squares approximation of affine mappings for sweep mesh generation: functional analysis and applications

A new comparative analysis between classical functional formulations and a new functional presented by the authors is presented and it is proved under which conditions the minimization of the analyzed functionals leads to a full rank linear system.

Abstract

Sweep methods are one of the most robust techniques to generate hexahedral meshes in extrusion volumes. The main issue in sweep algorithms is the projection of cap surface meshes along the sweep path. The most competitive technique to determine this projection is to find a least-squares approximation of an affine mapping. Several functional formulations have been defined to carry out this least-squares approximation. However, these functionals generate unacceptable meshes for several common geometries in CAD models. In this paper we present a new comparative analysis between these classical functional formulations and a new functional presented by the authors. In particular, we prove under which conditions the minimization of the analyzed functionals leads to a full rank linear system. Moreover, we also prove the equivalences between these formulations. These allow us to point out the advantages of the proposed functional. Finally, from this analysis we outline an automatic algorithm to compute the nodes location in the inner layers.