Generalized adaptive refinement for grid-based hexahedral meshing

Due to their nice numerical properties, conforming hexahedral meshes are considered a prominent computational domain for simulation tasks. However, the automatic decomposition of a general 3D volume into a small number of hexahedral elements is very challenging. Methods that create an adaptive Cartesian grid and convert it into a conforming mesh offer superior robustness and are the only ones concretely used in the industry. Topological schemes that permit this conversion can be applied only if precise compatibility conditions among grid elements are observed. Some of these conditions are local, hence easy to formulate; others are not and are much harder to satisfy. State-of-the-art approaches fulfill these conditions by prescribing additional refinement based on special building rules for octrees. These methods operate in a restricted space of solutions and are prone to severely over-refine the input grids, creating a bottleneck in the simulation pipeline. In this article, we introduce a novel approach to transform a general adaptive grid into a new grid meeting hexmeshing criteria, without resorting to tree rules. Our key insight is that we can formulate all compatibility conditions as linear constraints in an integer programming problem by choosing the proper set of unknowns. Since we operate in a broader solution space, we are able to meet topological hexmeshing criteria at a much coarser scale than methods using octrees, also supporting generalized grids of any shape or topology. We demonstrate the superiority of our approach for both traditional grid-based hexmeshing and adaptive polycube-based hexmeshing. In all our experiments, our method never prescribed more refinement than the prior art and, in the average case, it introduced close to half the number of extra cells.