Collaborative Research: Non-smoothness in Meshing and Reconstruction

牵头提案编号：0635008机构：俄亥俄州州立大学研究基金会主要负责人：Dey，Tamal K。&温格河S.合作提案编号：0635366机构：伊利诺伊大学校长：拉莫斯，埃德加项目名称：合作研究：网格和重建的非光滑性网格和重建的问题是普遍存在的科学和工程中的几何域需要数字表示，分析，检查，或原型。尽管这两个问题的输入形式不同，但两者在生成几何图形的三角形表示方面都有相似的目标。最近在设计和实现可证明算法方面取得了相当大的理论进展。然而，这些算法假设输入域具有某种形式的平滑性。因此，目前的解决方案是不够广泛的，以处理许多几何领域出现在科学研究和工程应用。汽车工业中的机械零件设计、建筑物的虚拟环境创建、科学研究中的裂缝和冲击模拟都是非光滑性作为基本障碍的几个例子。本计画针对非光顺性在网格化与重建上的困难，设计合理的演算法，并以这些演算法为基础，发展强健的软体。网格化产生一个明确指定几何区域的三角剖分，而重建则产生一个点样本。大多数可证明的重建算法利用光滑表面的微分结构，而网格化算法，虽然允许多面体域，限制输入角度不小。这些限制的结果是，非光滑域，如分片光滑曲面和非流形不能处理与充分的一般性。这个项目的目标是扩大输入几何的类别，模型可以计算的准确性保证。在这个项目中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培养理论计算机科学方面的技能，最显着的是计算几何和拓扑学，以及编写强大，高效和用户友好的软件。

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.