Geometric Skeletons for Topologically Evolving Domains

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。该奖项的目的是开发一种全新的方法来构建二维和三维几何形状的几何骨架家族，这些几何形状可以是刚性的，也可以是剧烈的拓扑变形。我们的方法依赖于计算建设性表示的形状与R-功能，操作实值半空间的逻辑运算，可以生成新的，更稳定的骨架，并在本质上支持本地化和并行骨架计算域与刚性或不断变化的边界。该研究计划的目标是开发新的理论基础的几何骨架的几何和拓扑描述符的形状，以及一个计算框架，能够有效地计算局部变化引起的局部变化的骨架域的边界。如果成功，这项研究将导致强大的通用算法计算家庭的几何骨架的复杂空间环境。反过来，这可以通过提供新的和有效的形状描述符来改变所有几何密集的科学领域，包括几乎所有的工程学科。将受益于新方法的应用包括工程工件的几何建模，用于工程分析的全自动网格生成，工程模型的特征识别和失效，以及具有自适应几何约束的自动驾驶车辆和机床的实时轨迹规划。此外，这个新的框架将刺激跨学科研究的关键新途径，涉及工程，生物学，计算机科学和人机交互。该计划将通过康涅狄格州的几个大学外展计划，对K-12学生，教师和当地学区服务团体进行有针对性的外展，这些团体传统上在工程学科中代表性不足。将几何推理和算法设计融入工程课程将有助于培养新一代工程师，他们将能够利用现代几何算法的能力与传统的机械工程知识相结合。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The objective of this award is to develop a fundamentally new approach to construct families of geometric skeletons for 2- and 3-dimensional geometric shapes that can be either rigid or undergoing drastic topological deformations. Our approach relies on computing constructive representations of shapes with R-functions that operate on real-valued half-spaces as logic operations, can generate novel, more stable skeletons, and intrinsically supports localized and parallel skeleton computations for domains with rigid or evolving boundaries. The goals of the research program are to develop new theoretical foundations for geometric skeletons as geometric and topological descriptors of shape, as well as a computational framework capable of efficiently computing local changes to the skeleton induced by local changes to the boundary of the domain.If successful, this research will lead to powerful generic algorithms for computing families of geometric skeletons for complex spatial environments. This, in turn, could transform all geometrically intensive areas of science, including almost all engineering disciplines, by providing access to new and potent descriptors of shape. Applications that would benefit from the new approach include geometric modeling of engineering artifacts, fully automated mesh generation for engineering analysis, feature recognition and defeaturing of engineering models, and real-time trajectory planning of autonomous vehicles and machine tools with adaptive geometric constraints. Furthermore, this new framework will stimulate critical new avenues of interdisciplinary research involving engineering, biology, computer science, and human-computer interaction. This program will perform targeted outreach to K-12 students, teachers and local school district serving groups that have traditionally been underrepresented in the engineering disciplines via several University of Connecticut outreach programs. The integration of geometric reasoning and algorithmic design into the engineering curriculum will help develop a new generation of engineers that will be able to exploit the capabilities of modern geometric algorithms in conjunction with traditional mechanical engineering knowledge.