Collaborative Research: Non-smoothness in Meshing and Reconstruction

牵头项目编号：0635008机构：俄亥俄州立大学研究基金会主要研究人员：Dey， Tamal K. & Wenger， R. s .合作项目编号：0635366机构：伊利诺伊大学主要研究人员：Ramos， edgar项目名称：合作研究：网格划分和重建中的非平滑问题网格划分和重建问题在需要数字表示、分析、检查或原型的几何域的科学和工程中普遍存在。尽管这两个问题的输入形式不同，但它们都有一个相似的目标，即生成几何图形的三角形表示。最近在设计和实现这两种可证明算法方面取得了相当大的理论进展。然而，这些算法假定输入域具有某种形式的平滑性。因此，目前的解决方案不够广泛，无法处理科学研究和工程应用中出现的许多几何域。在汽车工业中设计机械零件，在建筑中创建虚拟环境，在科学研究中模拟裂缝和冲击，这些都是将不平滑作为基本障碍的几个例子。本课题通过设计合理的算法和基于这些算法开发鲁棒性软件，研究非光滑性在网格划分和重建中的困难。网格划分产生明确指定的几何域的三角剖分，而重建则对点样本进行相同的剖分。大多数可证明的重建算法利用光滑表面的微分结构，而网格算法虽然允许多面体域，但限制了输入角度不能小。这些限制的结果是，非光滑域，如分段光滑曲面和非流形，不能用完全的一般性来处理。该项目的目标是扩大输入几何的类别，使模型可以在保证精度的情况下进行计算。该项目的研究使用了各种数学学科的概念，如微分几何、微分拓扑和非光滑分析，以及理论计算机科学领域的工具，如计算几何、计算拓扑和数值优化。该项目支持的研究生发展理论计算机科学的技能，尤其是在计算几何和拓扑学以及编写健壮、高效和用户友好的软件方面。

Lead proposalNUMBER: 0635008INSTITUTION: Ohio State University Research FoundationPRINCIPAL INVESTIGATORS: Dey, Tamal K. & Wenger, R. S.Collaborative ProposalNUMBER: 0635366INSTITUTION: University of IllinoisPRINCIPAL INVESTIGATORS: Ramos, EdgarPROJECT TITLE: Collaborative Research: Non-smoothness in Meshing and ReconstructionThe problems of meshing and reconstruction are pervasive in science and engineering where geometric domains need to be digitally represented, analyzed, inspected, or prototyped. Although the input forms to these two problems differ, both have a similar goal in producing a triangular representation of a geometry. Considerable theoretical advances have been recently made in designing and implementing provable algorithms for both. However, these algorithms assume some form of smoothness of the input domain. As a result, current solutions are not broad enough to handle many of the geometric domains which arise in scientific studies and engineering applications. Designing machine parts in automotive industry, creating virtual environments with buildings, simulating cracks and shocks in scientific studies are a few examples with non-smoothness as a basic impediment. This project studies the difficulty of non-smoothness in meshing and reconstruction by designing sound algorithms and by developing robust software based on these algorithms.Meshing produces a triangulation of an explicitly specified geometric domain whereas reconstruction does the same with a point sample. Most of the provable reconstruction algorithms exploit the differential structure of a smooth surface whereas meshing algorithms, though allowing polyhedral domains, restrict the input angles not to be small. The result of these restrictions is that non-smooth domains such as piecewise smooth surfaces and non-manifolds cannot be handled with full generality. The goal of this project is to broaden the class of input geometry for which models can be computed with assurance of accuracy. The research in this project uses concepts from various mathematical disciplines such as differential geometry, differential topology, and non-smooth analysis and also tools from areas of theoretical computer science such as computational geometry, computational topology, and numerical optimization. Graduate students supported by the project develop skills in theoretical computer science, most notably in computational geometry and topology and also in writing robust, efficient and user-friendly software.