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。该奖项的目标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具有r函数的形状的构造表示，这些函数作为逻辑运算在实值半空间上操作，可以生成新颖，更稳定的骨架，并且本质上支持具有刚性或演化边界的域的局部和并行骨架计算。本研究计划的目标是发展几何骨架作为几何和拓扑形状描述符的新理论基础，以及能够有效计算由区域边界局部变化引起的骨架局部变化的计算框架。如果成功，这项研究将导致复杂空间环境中计算几何骨架族的强大通用算法。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。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特征化，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，这个新框架将激发涉及工程、生物学、计算机科学和人机交互的跨学科研究的关键新途径。该项目将通过康涅狄格大学的几个外展项目，对传统上在工程学科中代表性不足的K-12学生、教师和当地学区服务团体进行有针对性的外展。将几何推理和算法设计整合到工程课程中，将有助于培养新一代的工程师，他们将能够利用现代几何算法与传统机械工程知识相结合的能力。

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.