EAGER: Collaborative Research: Towards Robust and Scalable Hexahedral Meshing

对具有复杂三维几何体的对象进行计算机建模、模拟或分析的第一步是将对象表示为简单对象的网格：共享面的(扭曲的)立方体是最自然的，并支持许多分析工具，但在拟合几何体的同时使立方体面匹配是一项挑战，这导致许多工具改用四面体。四面体可以在不更改其他面的情况下进行局部细化(用四个替换一个，或者用三个替换一个接触对)，但分割立方体会传播到其他立方体，需要更多的全局洞察力。这就是本项目旨在使用“框架场”的概念提供的，其好处是更好地对计算机设计和计算机模拟的对象的材料和电磁特性进行建模。PI还计划继续与代表性不足的群体和高中生一起工作。框架字段在整个空间中以平稳的方式分配坐标轴。PI在将二维框架场转化为四边形曲面网格方面做了一些开创性的工作，并希望探索扩展到三维的可能性。这个项目将寻求克服三个重要的障碍：1.立方体的对称性组比正方形的对称性组更复杂--PI将探索基于球谐的表示。2.第三维需要更改大小和扭曲，其中2D可以使用单位框架场-PI将探索大小和各向异性场以适应对象梯度。3.面连接立方体的拓扑约束网格--PI将探索平滑框架和拓扑约束之间的权衡。

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.