EAGER: Collaborative Research: Towards Robust and Scalable Hexahedral Meshing

对具有复杂三维几何结构的对象进行计算机建模、仿真或分析的第一步是将对象表示为简单对象的网格：（扭曲的）共享面的立方体是最自然的，并且支持许多分析工具，但是在拟合几何结构的同时使立方体面匹配是一个挑战，导致许多工具使用四面体代替。四面体可以在不改变其他面的情况下局部细化（用四个替换一个，或者用三个替换一个接触的对），但是分割一个立方体会传播到其他立方体，这需要更多的全局洞察力。这就是本项目旨在利用“框架场”的概念提供的，它有利于更好地对计算机设计和计算机模拟对象的材料和电磁特性进行建模。pi还计划继续在代表性不足的群体和高中生中开展工作。帧域在整个空间中以平滑的方式分配坐标轴。pi在将二维帧场转换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。该项目将努力克服三个重大障碍：1。立方体的对称组比正方形的对称组更复杂——pi将探索基于球面谐波的表示。2. 第三个维度需要改变尺寸和翘曲，而2d可以使用单位帧场——pi将探索尺寸和各向异性场来适应对象梯度。3. 面连通立方体的拓扑结构约束了网格——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.