Skeletal Reduction of Thin Mechanical Components

薄机械部件无处不在，构成了一个数十亿美元的产业；例如汽车和飞机的外壳，大多数塑料部件等。像所有的机械部件一样，它们在生产前都要经过严格的计算分析。由于历史原因，现有的薄组件分析技术是基于薄组件的中表面。不幸的是，对于复杂的对象，中表面常常定义不清。这导致了脆弱和不准确的算法。该项目建议使用固体骨架，放弃“中间表面”方法的概念。骨架，不像中间表面，有很好的定义，并且可以对实体进行唯一的计算。初步研究表明，这将产生鲁棒的算法，包括将康托洛维奇降维原理与骨架表示的独特拓扑和几何特性相结合。如果成功，这项研究将导致一个全新的、统一的薄成分分析理论，包括一个明确的、完整的骨骼还原数学语言，以及相关的代数。这些数学原理也会对从事广泛的边值问题分析领域的研究人员产生兴趣。研究结果的传播将通过研究生课程向威斯康星大学的学生传达，并通过研讨会向其他地方的学生和研究人员传达。它还打算促进随后的理论和算法的技术转移到CAD/CAE行业。该项目对环境的影响是塑料行业的组件几乎都很薄；更好的分析技术有可能显著减少今天使用（和丢弃）的塑料吨位。

Thin mechanical components are ubiquitous and constitute a multi-billion dollar industry; examples include outer-shell of automobiles and aircrafts, most plastic components, etc. Like all mechanical components, they are subject to rigorous computational analysis before production. Existing analysis techniques for thin components, for historical reasons, are based on the mid-surface of a thin component. Unfortunately, the mid-surface is often ill defined for complex objects. This has led to fragile and inaccurate algorithms. This project proposes to use of the skeleton of a solid, abandoning the notion of a "mid-surface" approach. The skeleton, unlike the mid-surface, is well defined and can be computed uniquely for a solid. Initial investigation shows that this would result in robust algorithms, involving the combination of the Kantorovich principle of dimensional reduction with the unique topological and geometrical properties of skeleton representation. If successful, the research would lead to a fundamentally new and unified theory of thin component analysis consisting of an unambiguous and complete mathematical language of skeletal reduction, and associated algebra. The mathematical principles would also be of interest to researchers involved in the broad area of boundary value problem analysis. Dissemination of the findings of the research will be articulated to University of Wisconsin students through a graduate course, and students and researchers elsewhere through seminars. It is also the intention to facilitate a transfer of technology of the ensuing theory and algorithms to the CAD/CAE industry. An environmental impact of the project is in the plastics industry where components are almost exclusively thin; better analysis techniques have the potential of significantly reducing the tonnage of plastics used (and discarded) today.