高精度几何近似造型的关键技术及其应用研究

WitIn current CAD/CAM systems, there are three contradictions needed to be solved urgently. That is, the contradiction between the increasing requirements of the design accuracy in modern manufacturing and polynomial being a main tool for shape design; the contradiction between the pursuit of high accuracy of algorithms for shape design and the shortage of approximation function for modern math modeling tools; the contradiction between the high degree of approximation rational curves for offset curves while improving the accuracy of the algorithm and the tolerance ability of design system. Therefore, we will do deep research on the key techniques, basic theory of high-accuracy geometric approximation modeling and its applications. First, based on the new tool of constrained dual Bernstein basis, to discuss polynomial approximation with high accuracy of conic sections, preserve geometric characteristics and construct efficient algorithms; Next, to design a brand-new metric to approximate Hausdorff distance, explore polynomial approximation with high accuracy of rational curves with geometric continuity and expand to rational triangular surfaces and NURBS surfaces; Third, creatively introducing Chebyshev rational approximation method, to achieve high-accuracy rational approximation of offset curves and surfaces. Meanwhile, we will study on the applications of high-accuracy approximation techniques in the fields such as CAD, precision machinery, NC machining and railway engineering survey, and so on. This project will provide powerful theoretical support and core algorithm for geometric modeling design and have important theoretical significance and application value.

当前CAD/CAM有三大矛盾亟待解决: 现代制造业对设计精度的要求日益提高, 与外形系统对多项式数模沿用不衰的矛盾; 现代设计对外形算法的高精度追求, 与现有数模工具对几何近似功能不足的矛盾; 以及等距曲线有理高精度近似引发的,近似曲线次数过高与设计系统承受力间的矛盾。为此，申请人将对高精度几何近似造型的关键技术、基础理论及其应用作深入研究: 以新颖的约束对偶基为工具，讨论圆锥曲线曲面的高精度多项式近似表示，保持几何特征，建立高效算法；设计一种全新度量来近似Hausdorff距离，探索保几何连续的有理曲线高精度多项式近似算法，并拓展到有理三角乃至NURBS曲面；创造性地引入Chebyshev有理近似方法，实现等距曲线曲面的高精度有理近似。与此同时,探索其在CAD、精密机械、数控加工和铁路工程测量等领域的应用研究。本项目将为几何造型设计提供强大理论支撑与核心算法,具有重要理论意义和应用价值。

本项目对高精度的几何近似造型的关键技术及其应用进行了研究，为几何造型设计提供了理论支撑和高效算法。研究内容包括：1.对特殊曲线(圆锥曲线、广义Cornu螺线)提出了保几何连续的高精度多项式逼近算法，并推广到二次曲面；2.利用B基的升阶思想，重新参数化方法以及带权重的渐进迭代(WPIA)方法，分别提出了均匀有理B样条、有理Bezier曲线及有理三角曲面片的高精度多项式近似算法；3.提出了保G3连续的两条Bezier曲线的显式拼接逼近算法，并将其推广到多条Bezier曲线的拼接逼近；4.在等几何分析中，从理论上严格证明了基于逼近思想的等几何配置法(IGA-C)的相容性和收敛性。针对等几何分析问题中的四面体网格，提出了基于B样条体的迭代重建算法。本项目的研究将不断丰富和完善几何近似造型技术理论及其应用体系，为计算机辅助几何设计的发展注入活力。