AF: Medium: Theory and Practice of Optimal Meshing

网格划分是有限元模拟的基础。但最近，它已经有了一些应用，每个人都需要在一个领域上定义一个功能，比如图形学、计算机辅助设计、机器人技术，甚至机器学习。算法将被设计成在任何固定的维度上有效地工作，保证输出的大小和质量。目前，即使对于二维情况，也没有理论可以在小输入角存在的情况下制定和产生最佳网格。研究将找到2D和更高维度的有效算法，仅使用简单的不接近180度的角度来生成最佳尺寸的Delaunay三角剖分。机器学习应用程序需要一个在多项式时间内运行的网格划分算法，以在log n维上划分n个点。从历史上看，网格算法返回一组填充空间的简单体。即使是好的纵横比简化也有太小的体积，并且返回一个超多项式大小的网格。因此，将开发新的算法来处理比简单对象体积大得多的原子对象。这个项目的结果非常实用，对科学、工程、制造和机器学习都有广泛的影响。特别是，网格是设计高效风车和汽车以及地震和医疗设备模拟的使能技术。其中一个目标是将这项研究的技术整合到我们的第一代3D代码中，并将其发布在网络上。此外，这些材料将被整合到CMU的课堂教学、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.