复杂曲面产品设计中外形度量近似关键技术研究及其应用

Shape metric approximation modeling technology of curves and surfaces directly influences computer aided design system's shape control, effectiveness of the algorithm, data exchange and transmission,etc. Therefore, shape metric approximation modeling technology is one of the key factors for improving the complex surface product design system's efficiency and accuracy. There are four aspects in our research. First, we will present a method to approximate NURBS curves/surfaces with polynomial B-spline curves/surfaces and prove the convergence of the algorithm. The approximation method will be very simple and guarantees the convergence of the approximation. At the same time, we also consider an simplified approximation method with high accuracy for NURBS curves/surfaces. Second, we intend to optimize the upper bounds on NURBS curves and surfaces'arbitrary degree derivatives, using own properties of B-spline functions and knot insertion algorithm. Third, we intend to investigate the approximation algorithm for the minimum distance between NURBS curves/surfaces, which will improve the algorithm's accuracy and efficiency by simplifying the minimum distance computation equation system and designing some suitable geometric methods. Fourth,based on the above theorectical algorithms, we will apply our simple and high-efficiency shape metric approximation modeling methods to CAD systems and related fields. Our research will make the theorectical system of approximation modeling further perfect, hopefully cultivate the new growing point of interdisciplinary including mathematics, computer graphics and computational fluid dynamics, and finally promote the complex surface product design system's function enhancements and optimization.

曲线曲面的外形度量近似造型方法因直接关系到计算机辅助设计系统的形状控制、算法的有效性、数据的交换和传递等，故成为提高复杂曲面产品设计系统效率和精度的一个关键因素。本项目拟研究内容包括以下四个方面：(1) 设计NURBS形式简单、保收敛的B样条近似简化算法并证明其收敛性，以及研究NURBS曲线曲面高精度的近似简化算法；(2)利用B样条函数自身的性质和节点插入技术研究NURBS曲线曲面任意阶导矢界的优化算法；(3) 研究NURBS曲线曲面间最短距离计算的近似算法，从近似简化最短距离计算方程组和设计适当的几何方法来提高该算法的精度和效率；（4）在理论算法基础之上，将NURBS简单、高效的外形度量近似造型方法应用于CAD系统及相关领域。本项目的研究，将有利于进一步完善近似造型方法的理论体系，有望培育数学、计算机图形学和计算流体力学等交叉学科一个新的生长点，促进复杂曲面产品设计系统的功能强化与优化。

曲线曲面的外形度量近似造型方法因直接关系到计算机辅助设计系统的形状控制、算法的有效性、数据的交换和传递等，故成为提高复杂曲面产品设计系统效率和精度的一个关键因素。本项目主要研究了简单、高效的外形度量近似造型方法及其在CAD系统与相应领域的应用方面：(1) 基于升阶技术，结合不等式技巧和无穷小分析技术解决了多项式B样条曲面逼近NURBS曲面的近似简化新算法；(2) 辅以有理和恒等式、特殊函数逼近的导数性质，基于离散B样条、Dir函数以及B样条函数化积为和的公式，推导了NURBS曲线的导矢界公式，进而研究了NURBS曲面导矢界的上限公式；(3) 结合细分方法设计了类滚动球裁剪思想，研究了两条NURBS曲线间的最短距离计算问题；(4) 研究了简单、高效的NURBS曲线曲面外形度量近似造型方法在CAD系统及其相应领域的应用。本项目的研究，进一步完善了近似造型方法的理论体系，并有效促进了复杂曲面产品设计系统的功能强化与优化。