Collaborative Research: Non-smoothness in Meshing and Reconstruction

主要提案编号：0635008研究：俄亥俄州立大学研究基金会PRINCIPAL调查人员：Dey，Tamal K.&Amp；Wenger，R.S.合作建议编号：0635366INSTUTION：伊利诺伊斯大学PRINCIPAL调查人员：Ramos，EdgarPROJECT标题：合作研究：网格和重建中的非光滑问题在科学和工程中普遍存在，其中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，但它们在生成几何图形的三角形表示方面都有相似的目标。最近在设计和实现这两种算法的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光滑性。因此，目前的解决方案不够广泛，不足以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，用建筑物创建虚拟环境，在科学研究中模拟裂缝和冲击，这些都是以不平顺为基本障碍的几个例子。这个项目通过设计合理的算法和开发基于这些算法的健壮软件来研究网格划分和重建中的非光滑性困难。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同样的处理。大多数可证明的重建算法利用了光滑曲面的微分结构，而网格化算法虽然允许多面体区域，但限制了输入角度不能小。这些限制的结果是，诸如分段光滑曲面和非流形之类的非光滑区域不能完全通用地处理。这个项目的目标是扩大输入几何的类别，可以保证计算模型的准确性。这个项目的研究使用了不同数学学科的概念，如微分几何、微分拓扑和非光滑分析，也使用了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。该项目资助的研究生培养理论计算机科学方面的技能，尤其是计算几何和拓扑学，以及编写健壮、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.