EAGER: Collaborative Research: Towards Robust and Scalable Hexahedral Meshing

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The first step of computer modeling, simulation, or analysis for objects with complex, 3-dimensional geometry is to represent the object as a mesh of simpler objects: (warped) cubes that share faces are the most natural and support many analysis tools, but getting cube faces to match while fitting the geometry is a challenge that causes many tools to use tetrahedra instead. Tetrahedra can be refined locally (replacing one with four, or a touching pair by three) without changing the other faces, but splitting a cube propagates to other cubes, requiring more global insight. This is what this project aims to provide using the concept of "frame fields," with the benefit for better modeling of material and electromagnetic properties of computer-designed and computer-simulated objects. The PIs also plan to continue their work with under-represented groups and with high school students.Frame fields assign coordinate axes in a smooth manner throughout a space. The PIs have done some pioneering work on turning 2d frame fields into quadrilateral surface meshes, and want to explore the possibility of extension to 3d. This project will look to overcome three significant obstacles: 1. The group of symmetries of the cube is more complex than that of the square -- the PIs will explore a representation based on spherical harmonics. 2. The third dimension requires changing sizes and warping where 2d could use unit frame fields -- the PIs will explore sizing and anisotropy fields to fit object gradients. 3. The topology of face-connectedness cubes constrains the mesh -- the PIs will explore the tradeoffs between smooth frames and topology constraints.