有理B样条曲线的端点性质及形状控制方法

一、引言在有理B样条曲线曲面的表示中,由于引进了对形状具有可控作用的权因子,并通过Piegl L等人在文[1]～[4]的系列文献中,对权因子ω_i的几何意义的揭示,使非均匀有理B样条(NURBS)在很多计算机几何造型系统中得到了进一步的应用,国际标准组织已把UNRBS方法列为STEP国际标准。尽管如此,对该样条曲线中权因子的几何意义的研究与实际应用仍存在很大差距。作者认为研究有理B样条曲线的权因子与     

基于最佳圆弧样条逼近的快速距离曲面计算

距离曲面是一种常用的隐式曲面,它在几何造型和计算机动画中具有重要的应用价值,但以往往在对距离曲面进行多边形化时速较慢,为了提高点到曲线最近距离计算的效率,提出了一种基于最佳圆弧样条逼近的快速线骨架距离曲面计算方法,该算法对于一条任意的二维NURBS曲线,在用户给定的误差范围内,先用最少量的圆弧样条来逼近给定的曲线,从而把点到NURBS曲线最近距离的计算问题转化为点到圆弧样条最近距离的计算问题,由于在对曲面进行多边形化时,需要大量的点到曲线最近距离的计算,而该处可以将点到圆弧样条最近距离很少的计算量来解析求得,故该算法效率很高,该实验表明,算法简单实用,具有很大的应用价值。     

三元Box样条构造方法的实现

在计算机辅助几何设计（CAGD），计算机辅助设计及制造（CAD／CAM）等领域中，样条函数已经成为不可缺少的工具。该文研究的目的在于提高样条空间中高维多面体的计算效率。文中按照Box样条的向量组定义的理论基础，通过一套自定义的从向量定义的角度来实现空间的点线面体类库的方法，最后实现了数学理论设计出来的空间多面体的向量组的一个三元Box样条函数。通过对高维多面体算法的时间复杂度分析，可以得到结论，文中设计采用的方法确实能够高效率地实现高维样条空间多面体的样条函数。     

基于重节点非均匀B样条技术的汽车车身计算机辅助造型系统

本文根据作者用重节点非均匀B样条方法进行汽车车身计算机辅助几何造型的初步实践，介绍了实现车身CAGD的主要技术即车身表面曲线曲面的构造方式，并给出了基于该技术的造型实例。由于B样条的局部性、直观性和保凸性等优点，并且引入了重节点技巧，使作者基于该方法所研制的软件具有较强的曲面造型功能。     

带切向控制的多结点曲线造型方法

在普通的多结点样条中加入相当于导数条件的可控参数，通过调节这些参数控制插值曲线在各型值点的切向量，从而达到满意的曲线造型效果．该方法保持了多结点样条的优越性（基数型，局部性），因此可以只对插值曲线作局部调整而不影响整体，有助于计算机辅助几何设计领域的工程人员设计、调整曲线的形状．     

改变B样条曲面与NURBS曲面的节点

通过改变k×h阶B样条曲面和NURBS(Non UniformRationalB spline)曲面的若干节点,分别产生一个B样条曲面族和NURBS曲面族,并指出:曲面族的包络是用相同控制顶点定义的(k -a)×(h -b)阶B样条曲面和NURBS曲面,其中a ,b分别是两个方向上所改变的节点的重数 对于B样条曲面来说,曲面族与其包络的任意阶相同偏微分之间只相差一个因子,文中所得结果可以作为计算机辅助设计系统中曲面造型和形状修改的理论参考     

基于C—B样条的Catmull—Clark细分曲面

为了解决Catumull-Clark细分曲面在工程上难以推广的问题，给出了一种基于C－B样条的Catumull-Clark细分曲面的算法，C－B样条曲线是B样条曲线的拓广，但它们的形状依赖于参数α，由于新的曲面细分方法充分利用C－B样条能够精确表示圆，椭圆等规则形体的特性，因而使通过此方法生成的细分曲面，除了在奇异点处能保持二阶导数连续外，还能够像C－B样条曲线，曲面一样，精确地表示圆柱等常规曲面，统一工程曲面等的造型，同时它仍然保持细分曲面的造型特点，即能够解决NURBS曲面难以处理的任意拓扑结构的造型问题，另外，还可依赖控制参数α的调节作用来增加造型的自由度，而且当α→0时，它们就退化成Catmul-Clark细分曲面，在工程图形上的应用实例表明，这种算法简单，有效。     

基于NURBS的自由曲面体三维几何特性计算

阐述了非均匀有理B样条（NURBS:Non-Uniform Rational B-Spline）曲面造型的发展背景及三维物体的几何特性计算原理,结合Gauss定理及参数曲面积分方法,给出了基于NURBS曲面的自由曲面体的几何特性计算原理与方法,并进行了实际应用。     

带形状参数的指数均匀B样条模型

本文给出了k(k≥2)阶带形状参数指数多项式的均匀B样条模型.该类模型具有很多与B样条模型相同的性质,并且具有一个可调节的形状参数.由该模型构造的曲线,通过改变形状参数的取值,可以调整曲线接近其控制多边形的程度.该模型可以应用于CAD/CAM领域,作为几何造型一种新的有效模型.     

结点插值新算法

结点插值算法广泛应用开发系统样条曲线、曲面的生成表示和求交分类。本文给出的结点插值新算法不仅可以统一表示已知的Boehm算法和Oslo算法,而且算法效率上优于它们。本算法已用于三维几何造型系统GEMS中。     

Signed algebraic level sets on NURBS surfaces and implicit Boolean compositions for isogeometric CAD–CAE integration

Boundary-representation (B-rep) geometrical models, often mathematically represented using Non-Uniform Rational B-Spline (NURBS) surfaces, are the starting point for complex downstream product life-cycle evaluations including Computer-Aided Engineering (CAE). Boolean operations during B-rep model generation require surface intersection computations to describe the composed entity. However, for parametric NURBS surfaces, intersection operations are non-trivial and typically carried out numerically. The numerical intersection computations introduce challenges relating to the accuracy of the resulting representation, efficiency with which the computation is carried out, and robustness of the result to small variations in geometry. Often, for downstream CAE evaluations, an implicit, procedural knowledge of the Boolean operations between the composed objects that can resolve point containment queries (exact to the original NURBS bounding surfaces) maybe sufficient during quadrature. However, common point containment queries on B-rep models are numerical, iterative and relatively expensive. Thus, the first goal of the present paper is to describe a purely algebraic, and therefore non-iterative, approach to carrying out point containment queries on complex B-rep models built using low-degree NURBS surfaces. For CAE operations, the boundary representation of B-rep solids is, in general, not convenient and as a result, the B-rep model is converted to a meshed volumetric approximation. The major challenges to such a conversion include capturing the geometric features accurately when constructing the secondary (meshed) representation, apart from the efficiency of carrying out such a mesh generation step repeatedly as the geometric shape evolves. Thus, an ideal analysis procedure would operate directly on B-rep CAD models, without needing a secondary mesh, and would procedurally unify the geometric operations during CAD as well as CAE stages. Therefore, the second and broader goal of the present paper is to demonstrate CAD–CAE integration using signed algebraic level set operations directly on B-rep models by embedding or immersing the bounding surfaces within a discretized domain while preserving the geometric accuracy of the surfaces exact to the original NURBS representation during analysis.     

CAE/CAD/CAM curricula implementation - experience at the National University of Singapore

A step-by-step approach to implement CAE/CAD/CAM curricula for undergraduate engineering courses at the National University of Singapore is presented. As the student workload is already quite heavy, each course is examined in detail and the CAE/CAD/CAM contents are carefully blended with the existing course structure. Some students are also assigned CAE/CAD/CAM specific projects to allow them to develop expertise in developing new software packages and building interfaces to existing packages. Experience indicates that CAE/CAD/CAM programmes at the undergraduate level should be implemented in various phases. It is also necessary to consider the state of the local industries in the adoption of the CAE/CAD/CAM technology so that the effort by the university is well synchronized with industry.     

Object representation and recognition using mathematical morphology model

The integration of representation and recognition of rigid solid objects is becoming increasingly important in computer-aided design (CAD), computer-aided manufacturing (CAM), computer graphics, computer vision, and other fields that deal with spatial phenomena. The mathematical framework used for modeling solid objects is mathematical morphology, which is based on set-theoretic concept. The mathematical characteristics of these operators are investigated in order to achieve a formal theory. Using mathematical morphology as a tool, our theoretical research aims at studying the representation schemes for the dimension and tolerance of the geometric structure. Object features can be also extracted by using the mathematical morphology approach. Through a distance transformation, we can obtain the shape number, significant points database, and skeleton. We have also developed the object recognition, localization, and corner and circle detection algorithms.     

An algorithm of determining T-spline classification

T-splines are a new surface modeling technology, whose theoretical framework is not still well founded. Aiming at the problem that how to classify T-splines this paper gives a mathematical analysis, and a sufficient condition of standard T-splines is also given. Then this paper presents an algorithm of determining the T-spline classification, and we can get the inherently mathematical properties of T-splines by the algorithm. At last, the experimental results verify the effectiveness of our algorithm. The results in this paper play an important role in the research on T-spline theory and T-spline modeling algorithms.     

Numerical method for shape optimization using T-spline based isogeometric method

Numerical methods for shape design sensitivity analysis and optimization have been developed for several decades. However, the finite-element-based shape design sensitivity analysis and optimization have experienced some bottleneck problems such as design parameterization and design remodeling during optimization. In this paper, as a remedy for these problems, an isogeometric-based shape design sensitivity analysis and optimization methods are developed incorporating with T-spline basis. In the shape design sensitivity analysis and optimization procedure using a standard finite element approach, the design boundary should be parameterized for the smooth variation of the boundary using a separate geometric modeler, such as a CAD system. Otherwise, the optimal design usually tends to fall into an undesirable irregular shape. In an isogeometric approach, the NURBS basis function that is used in representing the geometric model in the CAD system is directly used in the response analysis, and the design boundary is expressed by the same NURBS function as used in the analysis. Moreover, the smoothness of the NURBS can allow the large perturbation of the design boundary without a severe mesh distortion. Thus, the isogeometric shape design sensitivity analysis is free from remeshing during the optimization process. In addition, the use of T-spline basis instead of NURBS can reduce the number of degrees of freedom, so that the optimal solution can be obtained more efficiently while yielding the same optimum design shape.     

Modified T-splines

T-splines are a generalization of NURBS surfaces, the control meshes of which allow T-junctions. T-splines can significantly reduce the number of superfluous control points in NURBS surfaces, and provide valuable operations such as local refinement and merging of several B-splines surfaces in a consistent framework. In this paper, we propose a variant of T-splines called Modified T-splines. The basic idea is to construct a set of basis functions for a given T-mesh that have the following nice properties: non-negativity, linear independence, partition of unity and compact support. Due to the good properties of the basis functions, the Modified T-splines are favorable both in adaptive geometric modeling and isogeometric analysis.     

GMCADS系统的研究及应用

当今,机械CAD向CAD/CAM/CAE集成化方向发展,机械系统运动学/动力学分析软件是集成化CAD/CAM/CAE不可缺少的支撑软件之一。我们以多刚体系统动力学为基本理论,研制了面向问题的、交互式的通用机械计算机辅助设计系统GMCADS。本文介绍了GMCADS系统的分析原理、图形输出技术,并给出了应用实例。     

基于T-Spline的全自动几何拓扑修复方法

从高质量曲面网格生成的需求出发,提出了一种基于T-Spline的全自动几何拓扑修复方法.本文方法创新性主要可归纳为:1）对原有计算机辅助设计（Computer aided design,CAD）几何模型不进行任何修改保留其本真,自动识别CAD几何模型中常见不必要的几何特征,成功解决了CAD几何模型中存在的几何瑕疵,如短边、窄面、退化边、退化面、非连续光滑边界及尖锐特征等,利用新生成的"虚边"、"虚面"处理几何瑕疵,同时通过虚拓扑重构CAD几何模型的B-Rep;2）开发了一套CAD/CAE集成系统,统一了几何模型与计算分析模型,实现计算机辅助工程（Computer aided engineering,CAE）与CAD两者的无缝集成,所有拓扑修复操作及后续CAE分析计算均在同一环境下进行,避免了几何模型在CAE与CAD系统间进行转换时造成的数据丢失.该方法能够对复杂实体实现全自动几何拓扑修复及网格生成,实验表明,在保证不失真的前提下,修复后的几何模型能够生成质量良好的网格且能降低网格的生成规模,验证了本文方法的实用性和有效性,以满足工程实际分析的需要.     

计算机辅助结构设计与分析的集成框架研究

针对计算机辅助结构设计和分析的集成问题,设计了一种基于统一模型库 方式的组件式CAD/CAE 集成框架,以结构CAD/CAE 集成模型库为框架底层,以几何造型、 可视化交互、第三方CAD、第三方CAE 等组件为中间组件层,以空间结构设计子系统、空 间结构分析子系统为上层应用层,提供空间实体建模、工程模型管理、多种有限元模型分析 与计算等功能,并应用于港口码头的结构设计及有限元分析。