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、机器人等学科中具有重要意义。 本计画旨在发展以物理为基础的计算机辅助几何设计，这是一个结合几何造型与计算物理的新跨学科范例。 该项目的目标是通过创建高级参数和隐式几何模型来显著增强CAGD过程，这些模型具有由物理原理控制的高度直观的行为。本计画将从三个方面来达成其目标：（1）继续进行动态非均匀有理B样条（D-NURBS）的研究奋进;（2）探索并设计新的以物理为基础的造型技术与演算法，其基础是强大的新表示法，包括细分小波、流形样条与隐式函数;（3）充分利用本项目前期的理论成果，开发出一套广泛的基于物理的CAGD的有效算法和交互式软件例程，以供广泛的应用。该项目通过以下教育举措得到进一步加强：（1）开发新的研究生和本科生课程;（2）编写基于物理的计算机辅助几何设计教科书;（3）促进和参与跨学科研究和教育项目;（4）通过指导研究生，指导本科生的高级项目，参与推广和指导计划等;（5）通过与CAGD行业的合作，促进基于物理的CAGD技术在国内企业中的快速传播。 这个项目的结果将产生高度互动的CAGD技术。此外，这些研究和教育的进展将表明，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.