CAREER: Physics-Based Computer Aided Geometric Design: Theory and Applications

计算机辅助几何设计（CAGD）在几何计算、图形学、可视化、机器人和其他学科中具有重要的基础意义。该项目旨在发展基于物理的CAGD，这是一种将几何建模与计算物理相结合的新型跨学科范式。该项目的目标是通过创建由物理原理控制的具有高度直观行为的高级参数化和隐式几何模型来显着增强CAGD过程。本项目采取以下三方面的方法来实现其目标：(1)继续对动态非均匀有理b样条（D-NURBS）进行研究；(2)探索和设计新的基于物理的建模技术和基于强大的新表示的算法，包括细分小波、流形样条和隐式函数；(3)利用本项目早期阶段的理论进展，开发一套广泛的基于物理的CAGD的高效算法和交互式软件例程，用于广泛的应用。以下教育措施进一步加强了这个项目：(1)开设新的研究生和本科课程；(2)编写一本基于物理的CAGD教材；(3)推动和参与跨学科的研究和教育项目；(4)通过研究生指导、本科生高级项目指导、外联辅导等方式培养高素质人才；(5)通过与CAGD行业的合作，促进基于物理的CAGD技术在国家企业中的快速传播。该项目的结果将产生高度交互的CAGD技术。此外，这些研究和教育的进步将证明，基于物理的CAGD可以作为各种传统设计技术的统一基础。

Computer Aided Geometric Design (CAGD) is of fundamental importance in geometric computing, graphics, visualization, robotics, and other disciplines. This project is to develop Physics-Based CAGD, a new interdisciplinary paradigm which marries geometric modeling with computational physics. The objective of this project is to dramatically enhance the CAGD process by creating advanced parametric and implicit geometric models with highly intuitive behaviors that are governed by physical principles. This project takes the following three-fold approach to achieve its goal: (1) Continue the research endeavor on Dynamic Non-Uniform Rational B-Splines (D-NURBS); (2) Explore and devise novel physics-based modeling techniques and algorithms based on powerful new representations, including subdivision wavelets, manifold splines, and implicit functions; and (3) Exploit the theoretical advances of the earlier stages of this project to develop an extensive set of efficient algorithms and interactive software routines of Physics-Based CAGD for a wide range of applications. This project is further augmented by the following educational initiatives: (1) Develop new graduate and undergraduate courses; (2) Write a textbook on Physics-Based CAGD; (3) Promote and participate in interdisciplinary research and education projects; (4) Produce highly trained personnel by supervising graduate students, directing senior projects of undergraduate students, participating in outreach and mentoring programs, etc; and (5) Promote the rapid dissemination of Physics- Based CAGD technology to national enterprise through collaboration with the CAGD industry. The results of this project will yield highly interactive CAGD techniques. Furthermore, these research and education advances will demonstrate that Physics-Based CAGD can serve as a unified basis for various traditional design techniques.