Robust Topological Construction of All-hexahedral Boundary Layer Meshes

We present a robust technique to build a topologically optimal all-hexahedral layer on the boundary of a model with arbitrarily complex ridges and corners. The generated boundary layer mesh strictly respects the geometry of the input surface mesh, and it is optimal in the sense that the hexahedral valences of the boundary edges are as close as possible to their ideal values (local dihedral angle divided by 90°). Starting from a valid watertight surface mesh (all-quad in practice), we build a global optimization integer programming problem to minimize the mismatch between the hexahedral valences of the boundary edges and their ideal values. The formulation of the integer programming problem relies on the duality between boundary hexahedral configurations and triangulations of the disk, which we reframe in terms of integer constraints. The global problem is solved efficiently by performing combinatorial branch-and-bound searches on a series of sub-problems defined in the vicinity of complicated ridges/corners, where the local mesh topology is necessarily irregular because of the inherent constraints in hexahedral meshes. From the integer solution, we build the topology of the all-hexahedral layer, and the mesh geometry is computed by untangling/smoothing. Our approach is fully automated, topologically robust, and fast.