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、机器人学等学科中具有重要意义。该项目旨在开发基于物理的CAGD，这是一种将几何建模与计算物理相结合的新的跨学科范式。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，这些模型具有受物理原理控制的高度直观的行为。本项目采取以下三个方面来实现其目标：(1)继续对动态非均匀有理B-Splines(D-NURBS)的研究；(2)探索和设计基于物理的新的建模技术和算法，这些技术和算法基于强大的新表示，包括细分小波、流形样条和隐函数；(3)利用本项目早期阶段的理论进展，开发一套广泛应用的基于物理的CAGD的高效算法和交互软件例程。该项目还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课程；(2)编写基于物理的CAGD教材；(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、参与外联和指导计划等，培养高素质的人才；以及(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，这些研究和教育进展将证明，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.