AF: Medium: Theory and Practice of Optimal Meshing

网格划分是有限元法模拟的基础。 但最近，它已经应用于需要在某个领域定义函数的地方，例如图形学，计算机辅助设计，机器人技术，甚至机器学习。网格将被设计成在任何固定尺寸下有效地工作，并保证输出尺寸和质量。目前，即使对于2D情况，也没有理论在小输入角的情况下制定和产生最佳网格。 这项研究将找到有效的算法2D和更高的维度，生成最佳大小的Delaunay三角剖分，只使用单纯形，没有接近180度的角度。机器学习应用程序需要一个网格化算法，在多项式时间内运行网格化n个点在log n维。历史上，网格化算法返回一组空间填充单纯形。 即使是好的长宽比单纯形也有太小的体积，并返回一个超级多项式大小的网格。 因此，新的出租车将被开发，处理原子物体有比单纯体大得多的体积。从这个项目的结果是非常实用的，并在科学，工程，制造和机器学习产生了广泛的影响。 特别是，网格化是一种设计高效风车和汽车，以及模拟地震和医疗设备的技术。 一个目标是将这项研究的技术结合到我们的第一代3D代码中，我们在网上提供。此外，这些材料将被编入CMU的课程、讲座和会议论文中。

Meshing has been a cornerstone for simulations using the finiteelement method. But more recently it has had applications whereverone needs to define a function over a domain, such as, graphics,computer aided design, robotics, and even machine learning. Algorithmswill be designed to efficiently work in any fixed dimension withguarantees on output size and quality.At present, no theory exists to formulate and produce optimal meshesin the presence of small input angles even for the 2D case. Theresearch will find efficient algorithms 2D and higher dimension thatgenerate optimal size Delaunay triangulations using only simpliceswith no angles approaching 180 degrees.Machine learning applications need a meshing algorithm that runs inpolynomial time for meshing n points in log n dimensions.Historically, meshing algorithms return a set of space-fillingsimplices. Even good aspect ratio simplices have too small a volumeand return a mesh that is of super polynomial size. Thus, newalgorithms will be developed that handle atomic objects that are havemuch larger volume than simplices.The results from this project are eminently practical and have broadimpact on the Sciences, Engineering, Manufacturing, and MachineLearning. In particular, meshing is an enabling technology fordesigning efficient windmills and cars, and simulations of earthquakes and medial devices. One goal is to incorporated techniquesfrom this research into our first generation 3D code that we madeavailable on the web. In addition, this material will be incorporatedinto classes taught at CMU, lectures, and papers presented atconferences.