Bézier曲面的广义离散及应用

Ｂéｚｉｅｒ曲面是计算机辅助几何设计（ＣＡＧＤ）中最常用的参数曲面之一,通过引进一些符号和Ｂéｚｉｅｒ曲面的算子表示,并利用参数化技巧,本文推广了张量积Ｂéｚｉｅｒ曲面的离散算法,讨论了Ｂéｚｉｅｒ矩形片沿定义域内一条ｋ次多项式曲线的离散,给出了两种显式的广义离散格式．另外,文中还介绍了广义离散方法在Ｂéｚｉｅｒ曲面几何连续拼接和ｔｒｉｍｍｅｄ曲面参数表示中的应用．     

Bézier曲线/曲面间最近距离的几何裁剪算法

曲线、曲面间距离的计算问题在CAD/CAM、计算机图形学中有着广泛的应用.为了精确计算Bézier曲线/曲面间的最近距离,结合稳定的曲线、曲面分裂技术提出一种基于offset滚动球裁剪的几何算法.首先给出判定条件来裁剪去落在曲面的滚动球外的曲线段,或者落在曲线的滚动球外的曲面片,以摒弃大部分不包含最近点的曲线段或曲面片,为后续可能的Newton方法提供较好的初始点;然后给出判定最近点是否落在曲线的端点或曲面的边界曲线上的条件,将曲线/曲面间的距离计算问题转化为点/曲面或曲线/曲线间的距离计算问题,简化了问题的复杂度,提高了计算效率.实例结果表明,文中算法具有较好的稳定性和较高的效率.     

京剧脸谱分析与合成

京剧脸谱的分析与数字化建模可以使人们深化对传统京剧艺术的认识,对于传统文化的保护与利用具有重要意义.首先分析京剧脸谱纹样形状特征并根据其语义信息将其分类,然后用Bézier曲线构造出矢量化纹样单元库;在合成脸谱阶段,用户只需按照脸谱绘制顺序逐层选取所需纹样,根据创作需要对各层的纹样进行组合得到最终的京剧脸谱图案;还提供一系列变形工具供用户对脸谱局部纹样进行编辑修改,以生成更多的、富有变化的京剧脸谱图案.与手工绘制京剧脸谱相比,文中系统可以用更灵活多变的方式,简单快捷地生成各种复杂的京剧脸谱.该系统在旅游产品开发、书籍装帧、建筑装饰、服饰设计、装潢设计、电子贺卡、京剧教学等领域有着广泛的应用前景.     

采用等几何质点-弹簧模型的布料动态仿真方法

为了高效实时地模拟出布料的动态效果,提出一种基于等几何分析思想的布料动态仿真方法.首先使用张量积Bézier曲面来构建布料几何模型;然后对布料的质点进行受力分析,并直接在该曲面上进行质点-弹簧模型数值求解;最后将更新后的质点的位置和速度变换到等几何布料曲面的控制顶点上,进行实时的碰撞处理.实验结果表明,该方法无需预先对布料进行三角网格剖分,可精确地表示布料几何模型;同时在较少的自由度数目下,可得到高精度的动态仿真效果,提高了仿真效率;可在CPU仿真环境下进行实时的动态仿真,适用于对实时性要求较高的虚拟试衣等工程应用领域.     

复合三角Bézier曲面求交和裁剪的实现

该文利用三角Bézier曲面片的可分割性,解决了迭代收敛、初始交点计算等问题;通过近曲面点、边界点跨越等过程,实现了由一个初始交点将跨越许多曲面片的整条交线跟踪出来的设想.将各交点作为型值点插入曲面中,对三角网格进行三角再划分,以交线为界进行三角网格和型值点的分离,最后重新生成两张复合曲面,实现了裁剪的目的.测试结果显示,此方法简单、可靠,能够满足曲面造型的要求.     

基于法向拟合的微分几何量估计

估计离散曲面的微分几何量在数字几何处理、计算机图形学、计算机视觉领域是一个基础性的课题,在很多应用中估计微分几何量往往作为先决性的一个步骤而存在.目前大部分微分几何量估计方法只是针对规则的、均匀采样的网格数据能取得比较满意的结果,而对于带噪声、不规则采样的点云数据,寻求准确而又鲁棒的估计方法仍是一个研究热点问题.文中针对不规则的离散点云数据,首先通过设计对比实验验证法向在二阶微分量估计中的重要作用,然后据此提出了一种基于法向拟合的微分几何量估计方法,利用单位法向量的偏微分来计算各点的主曲率和主方向.与传统的通过曲面拟合各离散点的位置来估计微分量的方法相比,该方法直接利用法向信息,综合考虑邻域内所有点的法向信息,在很大程度上保证了算法的稳定性和鲁棒性,并能在数据带有很强噪声情况下取得比较精确的估计结果.该方法的一大优点是在一定程度上降低了微分几何量估计结果对建立可靠局部坐标系或者说各点初始法向准确性的依赖,并且能够根据邻域内法向量的分布情况,通过法向拟合修正各点的初始法向,得到更准确的法向信息.     

三角Bézier曲面的几何约束形变

利用三角Bézier曲面的矩阵表达形式,把几何约束下的形状调整算法从曲线和张量积曲面推广到三角Bézier曲面,使得三角Bézier曲面在形变后既能保持外形大致不变,又能满足一系列事先指定的几何约束(点约束和法向约束).利用Lagange乘子法,几何约束形变的条件极值问题被转化为线性方程组的求解问题,以便于快速计算.特别地,三角Bézier曲面在形变前后还可以满足边界曲线在角点处保持(Ca,Cb,Cc)连续.数值实例表明,该算法简单有效,便于CAD(计算机辅助设计)系统进行交互.     

Bézier曲面的函数复合及其应用

目前有两种常用的Bézier曲面片,分别称为三角和四边Bézier曲面片,它们分别用不同的基函数表示.本文通过移位算子和函数复合的方法,得到了两个关于这两种Bézier曲面片的结果.一个是四边Bézier曲面片与一次三角Bézier函数的复合,另一个是三角Bézier曲面片与双线性四边Bézier函数的复合.在每一种情况中,复合所得到的Bézier曲面片的控制顶点是原来Bézier曲面片的控制顶点的线性组合.移位算子的应用使得相应的推导过程变得简洁和直观.这两个结果的应用包括：两种Bézier面片间的转化     

网格的渐进几何压缩

提出一种渐进几何压缩算法.通过对简化算法的改进,网格模型由基网格及多组顶点分裂操作序列表达.当从一层网格向下一层精网格细化时,该组顶点分裂操作序列中的分裂操作顺序是任意的.因此,改进的渐进网格表示可改变每组顶点分裂操作的排序,实现高效率编码.设计了Laplacian几何预测器,通过相邻顶点来预测新增顶点位置,并对位置校正值进行量化及Huffman编码.实验结果表明,该算法可获得高压缩比,适合几何模型的网络渐进传输.     

Bézier曲面的函数复合及其应用(英文)

目前有两种常用的 Bézier曲面片 ,分别称为三角和四边 Bézier曲面片 ,它们分别用不同的基函数表示 .本文通过移位算子和函数复合的方法 ,得到了两个关于这两种 Bézier曲面片的结果 .一个是四边 Bézier曲面片与一次三角 Bézier函数的复合 ,另一个是三角 Bézier曲面片与双线性四边 Bézier函数的复合 .在每一种情况中 ,复合所得到的 Bézier曲面片的控制顶点是原来 Bézier曲面片的控制顶点的线性组合 .移位算子的应用使得相应的推导过程变得简洁和直观 .这两个结果的应用包括 :两种 Bézier面片间的转化、裁剪 Bézier曲面片的精确表示、Bézier曲面片的自然延拓等     

基于分布估计的离散差分演化算法

差分演化(DE)是解决优化问题的非常有效的新兴智能算法,但它主要用于连续优化领域,至今尚不能象解决连续优化问题那样有效的处理组合优化问题.首先提出了离散DE用于组合优化问题,然后在离散DE中引入分布估计算法(EDA)来提高性能,把EDA抽样得到的全局统计信息和离散DE获得的局部演化信息相结合来产生新解,形成基于EDA的离散DE算法.为了保持种群多样性,在提出的算法中引入了位翻转变异操作.实验结果表明,EDA能大大提高离散DE的性能.     

离散三角网格模型顶点法矢量估算

提出一种新的三角网格模型顶点法矢估算方法,采用以三角网格顶点一阶邻域三角形的形状因子与顶点到三角形质心距进行综合加权的方法。同时指出：在同等三角网格曲面,随着三角网格划分精度的提升,网格顶点法矢估算精度有增大趋势;在同等网格划分精度条件下,对于平均曲率小以及平均曲率变化率小的三角网格模型,其网格顶点法矢估算精度也有增大趋势。实例计算和误差分析表明,该方法的计算结果更为精确合理。     

On application of differential geometry to computational mechanics

Developments in the fields of computational science—the finite element method—and mathematical foundations of continuum mechanics result in many new algorithms which give solutions to very complicated, complex, large scaled engineering problems. Recently, the differential geometry, a modern tool of mathematics, has been used more widely in the domain of the finite element method. Its advantage in defining geometry of elements [13–15] or modeling mechanical features of engineering problems under consideration [4–7] is its global character which includes also insight into a local behavior. This fact comes from the nature of a manifold and its bundle structure, which is the main element of the differential geometry.

Manifolds are generalized spaces, topological spaces. By attaching a fiber structure to each base point of a manifold, it locally resembles the usual real vector spaces; e.g. ³. The properties of a differential manifold M are independent of a chosen coordinate system. It is equivalent to say, that there exists smooth or C^r differentiable atlases which are compatible.

In this paper a short survey of applications of differential geometry to engineering problems in the domain of the finite element method is presented together with a few new ideas.

The properties of geodesic curves have been used by Yuan et al. [13–15], in defining distortion measures and inverse mappings for isoparametric quadrilateral hybrid stress four- and eight-node elements in ². The notion of plane or space curves is one of the elementary ones in the theory of differential geometry, because the concept of a manifold comes from the generalization of a curve or a surface in ³.

Further, the real global nature of differential geometry, has been used by Simo et al. [4,6,7]. A geometrically exact beam finite strain formulation is defined. The mechanical basis of such a nonlinear model can be found in the mathematical foundation of elasticity [18]. An abstract infinite dimensional manifold of mappings, a configuration space, is constructed which permits an exact linearization of algorithms, locally. A similar approach is used by Pacoste [5] for beam elements in instability problems.

Special attention is focused on quadrilateral hybrid stress membrane elements with curved boundaries which belong to a series of isoparametric elements developed by Yuan et al. [14]. The distortion measures are redefined for eight-node isoparametric elements in ² for which geodesic coordinates are used as local coordinates.     

As-rigid-as-possible mesh deformation and its application in hexahedral mesh generation

This paper presents an efficient and stable as-rigid-as-possible mesh deformation algorithm for planar shape deformation and hexahedral mesh generation. The deformation algorithm aims to preserve two local geometric properties: scale-invariant intrinsic variables and elastic deformation energy, which are together represented in a quadric energy function. To preserve these properties, the position of each vertex is further adjusted by iteratively minimizing this quadric energy function to meet the position constraint of the controlling points. Experimental results show that the deformation algorithm is efficient, and can obtain physically plausible results, which have the same topology structure with the original mesh. Such a mesh deformation method is useful to project the source surface mesh onto the target surfaces in hexahedral mesh generation based on sweep method, and application results show that the proposed method is feasible to mesh projection not only between similar surface contours but also dissimilar surface contours.     

基于分布估计的离散差分骨干粒子群优化

粒子群优化（PSO）和差分演化（DE）是两种新兴的优化技术,已经成功地应用于连续优化问题,但是它们至今尚不能像解决连续优化问题那样有效地处理组合优化问题。最近,有人提出差分骨干PSO（DBPSO）用于解决连续优化问题。首先提出离散DBPSO用于组合优化问题,然后在离散DBPSO中引入分布估计算法（EDA）来提高性能,把EDA抽样得到的全局统计信息和DBPSO获得的局部演化信息相结合来产生新解,形成基于EDA的离散DBPSO。实验结果表明EDA能大大提高离散DBPSO的性能。     

Practical estimation of high dimensional stochastic differential mixed-effects models

Stochastic differential equations (SDEs) are established tools for modeling physical phenomena whose dynamics are affected by random noise. By estimating parameters of an SDE, intrinsic randomness of a system around its drift can be identified and separated from the drift itself. When it is of interest to model dynamics within a given population, i.e. to model simultaneously the performance of several experiments or subjects, mixed-effects modelling allows for the distinction of between and within experiment variability. A framework for modeling dynamics within a population using SDEs is proposed, representing simultaneously several sources of variation: variability between experiments using a mixed-effects approach and stochasticity in the individual dynamics, using SDEs. These stochastic differential mixed-effects models have applications in e.g. pharmacokinetics/pharmacodynamics and biomedical modelling. A parameter estimation method is proposed and computational guidelines for an efficient implementation are given. Finally the method is evaluated using simulations from standard models like the two-dimensional Ornstein-Uhlenbeck (OU) and the square root models.     

Consistent recursive parameter estimation of partial differential equation models

This article introduces a new residual-based recursive parameter estimation algorithm for linear partial differential equations. The main idea is to replace the unmeasurable noise variables by estimates of the noise and to compute recursively both the model parameters and the noise estimates. It is proven that under some mild assumptions the estimated parameters converge to the true values with probability one. Numerical examples that demonstrate the effectiveness of the proposed approach are also provided.     

Discrete approximation of multivariate densities with application to Bayesian estimation

A procedure is presented for finding a discrete approximation to a continuous multivariate density function. It is based on a previously developed algorithm [2] for determining the L₁ optimal discrete approximation to a univariate density. Results of approximating continuous bivariate density functions, which represent distributions of the parameters of a pharmacokinetic model, show good agreement between the mean and covariance matrix of the approximated and approximating densities. The distribution of a predicted drug conceptration was also calculated using a continuous density and discrete approximations with both 25 and 81 points. The expected values of the predicted concentration, as well as selected percentile points, obtained using each density are in close agreement.     

Creating geometry and mesh models for nuclear reactor core geometries using a lattice hierarchy-based approach

Nuclear reactor cores are constructed as rectangular or hexagonal lattices of assemblies, where each assembly is itself a lattice of fuel, control, and instrumentation pins, surrounded by water or other material that moderates neutron energy and carries away fission heat. We describe a system for generating geometry and mesh for these systems. The method takes advantage of information about repeated structures in both assembly and core lattices to simplify the overall process. The system allows targeted user intervention midway through the process, enabling modification and manipulation of models for meshing or other purposes. Starting from text files describing assemblies and core, the tool can generate geometry and mesh for these models automatically as well. Simple and complex examples of tool operation are given, with the latter demonstrating the generation of meshes with 12 million hexahedral elements in <30?min on a desktop workstation, using about 4?GB of memory. The tool is released as open source software as part of the MeshKit mesh generation library.