Topologic and geometric constraint-based hexahedral mesh generation

Hexahedral finite element meshes have historically offered some mathematical benefit over tetrahedral finite element meshes in terms of reduced error and smaller element counts, especially with respect to finite element analyses within highly elastic, and plastic, structural domains. However, because hexahedral finite element mesh generation often requires significant geometric decomposition, generating hexahedral meshes can be extremely difficult to perform and automate and the process often takes several orders of magnitude longer in time to complete than current methods for generating tetrahedral meshes. In this dissertation, we focus on delineating known constraints associated with hexahedral meshes and formulating these constraints utilizing the dual of the hexahedral mesh. Utilizing these constraints, we show that hexahedral mesh generation can be viewed as an optimization problem. We review existing hexahedral algorithms and describe how these algorithms operate to satisfy the hexahedral mesh generation constraints. The concept of a fundamental hexahedral mesh will be introduced and it will be shown how the fundamental mesh relates to a minimal hexahedral mesh for a given geometry. We will demonstrate conversion of existing hexahedral meshes to fundamental hexahedral meshes using hexahedral flipping operations to convert boundary sheets to fundamental sheets. Building on existing algorithms for generating hexahedral meshes from volumetric image data, we will show significant improvement in hexahedral mesh quality through the introduction of a single fundamental sheet into hexahedral meshes generated from isosurfacing techniques. We will outline a method for constructing hexahedral meshes where all hexahedra are convex and have positive volume utilizing triangle meshes of manifold surfaces to guide the placement of fundamental sheets into an existing hexahedral mesh. Finally, we demonstrate construction of hexahedral meshes for multi-surface geometric solids by introducing multiple fundamental sheets to satisfy the hexahedral mesh generation constraints for the geometric solid.