Abstract

In this work, we describe a method to partition domains adapted to the generation of quadrilateral and hexahedral meshes. Given a domain D, the proposed approach proceeds in two steps: first, a frame field is defined on a background simplicial mesh of D. Then, singular elements of this field are extracted to create a skeleton that partitions D. The key element of this approach is the use of frames: at a point P of D, a frame allows to orient the quadrilateral or the hexahedral. Thus, we propose a complete study of frames and frame fields. We describe the proposed method both in dimension two and three. The main difference between the two is in the way frame fields are generated. In dimension 2, we solve a non-linear PDE, while in 2D, a heuristic is applied that uses initiallu an advancing front algorithm that takes the stability of the field into account, before using a smoothing algorithm. In both cases, a skeleton is extracted from the frame field. In dimension two, this skeleton always leads to a partition of D into quadrilateral-shaped blocks, and singularities are all of degree three and five. In dimension three, the obtained frame field leads to a partition allowing for hexahedral meshing in numerous cases. Many examples are showcased and compared to results obtained by existing methods.