Event

The topological structure of a hexahedral mesh is characterized by its singularities, which look like line disclinations in a crystal. In field-based meshing, one first computes a so-called octahedral frame field, which serves as a proxy encoding the alignment and singularities of the mesh. While decoupling the combinatorial component of the meshing problem, this proxy optimization problem is still challenging due to the presence of singularities. I will outline our work on overcoming these “topological barriers” using the tools of convex relaxation, exploring some of the geometry of frame fields and their singularities along the way. Finally, I will discuss some open questions. While field-based meshing has made great advances, it often fails catastrophically when the computed frame field singular structure is not realizable in a mesh. This practical problem reflects a murky distinction between field and mesh topology whose full characterization remains elusive. A better understanding of field and mesh defects would enable reliable automatic hexahedral meshing as well as unlock more efficient optimization approaches through reduced-order modeling. I will outline this story in hopes of a productive exchange with ideas and techniques from condensed matter.