NURBS曲在G^1光滑拼接算法

非均匀有理Ｂ样条（ＮＵＲＢＳ）曲线、曲面造型方法，是当前ＣＡＤ／ＣＡＭ领域研究热点之一，大量的基于ＮＵＲＢＳ的衫和造系统得到发展，对ＮＵＲＢＳ而言，虽然具有参数连续性，但为了实用需要，仍需构造具有一光滑程度的合成曲面，满足局部设计和修改的目的，本文给出了实用的具有二次公共边界曲线的ＮＵＲＢＳ曲面片Ｇ＾１光滑拼接条件，得到相应控制顶点、权系数的具体算法；对于一个已知ＮＵＲＢＳ曲面，构造另一人ＮＵＲＢ     

NURBS曲面G~1光滑拼接算法

非均匀有理Ｂ样条（ＮＵＲＢＳ）曲线、曲面造型方法，是当前ＣＡＤ／ＣＡＭ领域研究热点之一，大量的基于ＮＵＲＢＳ的实用造型系统得到发展。对ＮＵＲＢＳ而言，虽然具有参数连续性，但为了实用需要，仍需构造具有一定光滑程度的合成曲面，满足局部设计和修改的目的。本文给出了实用的具有二次公共边界曲线的ＮＵＲＢＳ曲面片Ｇ１光滑拼接条件，得到了相应控制顶点、权系数的具体算法；对于一个已知ＮＵＲＢＳ曲面，构造另一个ＮＵＲＢＳ曲面，使其达到Ｇ１拼接是简单易行的。     

圆弧曲线的三次NURBS表示

本文首次提出三次ＮＵＲＢＳ曲线精确地表示圆弧的充要条件，解决了两方面的问题：一是已知三次ＮＵＲＢＳ曲线，如何判断它是否是圆弧，二是已知一圆弧曲线，怎样用三次ＮＵＲＢＳ曲线精确地表示，给出了圆弧曲线的三次ＮＵＲＢＳ表示的几何构造算法，均匀有理Ｂ样条曲线和有理Ｂｅｚｉｅｒ曲线精确地表示圆弧曲线的充要条件可作为ＮＵＲＢＳ曲线的特殊情形得到，这些研究结果为ＮＵＲＢＳ应用于ＣＡＧＤ，ＣＡＤ／ＣＡＭ提供了一个     

任意形状容器装液不定常晃动的边界元模型

任意形状容器装液不定常晃动的边界元模型冯振兴，李正秀（武汉大学）ＬＡＧＲＡＮＧＩＡＮＢＥＭＦＯＲＡＮＵＮＳＴＥＡＤＹＳＬＯＳＨＩＮＧＭＯＤＥＬＳＵＩＴＡＢＬＥＴＯＡＲＢＩＴＲＡＲＹＴＡＮＫＳ￥ＦｅｎｇＺｈｅｎ－ｘｉｎｇ；ＬｉＺｈｅｎ－ｘｉｕ（Ｗｕｈａ...     

用三次NURBS表示圆弧与整圆的算法研究

提出了一种实用的三次ＮＵＲＢＳ曲线表示圆弧及整圆的方法，并得出了各种情况下现成可用的结果。该方法用４个控制顶点所决定的一段ＮＵＲＢＳ曲线来表示一段小于１８０°的圆弧段；对于大于１８０°的圆弧则采用两段三次ＮＵＲＢＳ曲线来表示，当圆心角为３６０°时，则得到了整圆的表示。文中所述的方法统一、简单、符合对圆弧ＮＵＲＢＳ表示的要求。     

2-D SHAPE BLENDING OF NURBS CURVE SHAPES

２－ＤＳＨＡＰＥＢＬＥＮＤＩＮＧＯＦＮＵＲＢＳＣＵＲＶＥＳＨＡＰＥＳＪｉｎＸｉａｏｇａｎｇ；ＢａｏＨｕｊｕｎ；ＰｅｎｇＱｕｎｓｈｅｎｇ２－ＤＳＨＡＰＥＢＬＥＮＤＩＮＧＯＦＮＵＲＢＳＣＵＲＶＥＳＨＡＰＥＳ￥ＪｉｎＸｉａｏｇａｎｇ；ＢａｏＨｕｊｕｎ；Ｐｅ...     

LOCAL INTERPOLATING BLENDED B－SPLINE SURFACE AND ITS CONVERSION TO THE NURBS SURFACE

ＬＯＣＡＬＩＮＴＥＲＰＯＬＡＴＩＮＧＢＬＥＮＤＥＤＢ－ＳＰＬＩＮＥＳＵＲＦＡＣＥＡＮＤＩＴＳＣＯＮＶＥＲＳＩＯＮＴＯＴＨＥＮＵＲＢＳＳＵＲＦＡＣＥＷａｎｇＬａｚｈｕ；ＺｈｕＸｉｎｘｉｏｎｇＬＯＣＡＬＩＮＴＥＲＰＯＬＡＴＩＮＧＢＬＥＮＤＥＤＢ－ＳＰＬＩ...     

NURBS表示圆弧曲线的实用方法

根据推导得到二次和三次ＮＵＲＢＳ表示圆弧曲线的实用方法,给出计算控制顶点及其权值的表式,同时对该方法的适用范围进行了详细探讨,澄清了以往应用三次ＮＵＲＢＳ表示圆弧曲线的一点错误认识。     

NURBS曲面在工程曲面设计中的应用

本文探讨了工程曲面设计中应用ＮＵＲＢＳ（非均匀有理Ｂ样条）曲面的方法，包括所设计曲面的预处理，控制顶点与节点的设计，以及通过调整权重修改曲面的方法等。     

用NURBS表示的几何实体的纹理映射方法

本文实现了一个用于基于ＮＵＲＢＳ表示的几何造型系统的纹理映射算法，该算法采用曲线的弧长作为基本的纹理映射参数，减少了纹理映射中的图形混淆。该系统能对用ＮＵＲＢＳ表示的各种几何实体使用统一的算法进行纹理映射，算法简单、占用存储空间小。     

D-NURBS: a physics-based framework for geometric design

Presents dynamic non-uniform rational B-splines (D-NURBS), a physics-based generalization of NURBS. NURBS have become a de facto standard in commercial modeling systems. Traditionally, however, NURBS have been viewed as purely geometric primitives, which require the designer to interactively adjust many degrees of freedom-control points and associated weights-to achieve the desired shapes. The conventional shape modification process can often be clumsy and laborious. D-NURBS are physics-based models that incorporate physical quantities into the NURBS geometric substrate. Their dynamic behavior, resulting from the numerical integration of a set of nonlinear differential equations, produces physically meaningful, and hence intuitive shape variation. Consequently, a modeler can interactively sculpt complex shapes to required specifications not only in the traditional indirect fashion, by adjusting control points and setting weights, but also through direct physical manipulation, by applying simulated forces and local and global shape constraints. We use Lagrangian mechanics to formulate the equations of motion for D-NURBS curves, tensor-product D-NURBS surfaces, swung D-NURBS surfaces and triangular D-NURBS surfaces. We apply finite element analysis to reduce these equations to efficient numerical algorithms computable at interactive rates on common graphics workstations. We implement a prototype modeling environment based on D-NURBS and demonstrate that D-NURBS can be effective tools in a wide range of computer-aided geometric design (CAGD) applications     

Full analytical sensitivities in NURBS based isogeometric shape optimization

Non-uniform rational B-spline (NURBS) has been widely used as an effective shape parameterization technique for structural optimization due to its compact and powerful shape representation capability and its popularity among CAD systems. The advent of NURBS based isogeometric analysis has made it even more advantageous to use NURBS in shape optimization since it can potentially avoid the inaccuracy and labor-tediousness in geometric model conversion from the design model to the analysis model.Although both positions and weights of NURBS control points affect the shape, until very recently, usually only control point positions are used as design variables in shape optimization, thus restricting the design space and limiting the shape representation flexibility.This paper presents an approach for analytically computing the full sensitivities of both the positions and weights of NURBS control points in structural shape optimization. Such analytical formulation allows accurate calculation of sensitivity and has been successfully used in gradient-based shape optimization.The analytical sensitivity for both positions and weights of NURBS control points is especially beneficial for recovering optimal shapes that are conical e.g. ellipses and circles in 2D, cylinders, ellipsoids and spheres in 3D that are otherwise not possible without the weights as design variables.     

A New Approach for Direct Manipulation of Free-Form Curve

There is an increasing demand for more intuitive methods for creating and modifying free-form curves and surfaces in CAD modeling systems. The methods should be based not only on the change of the mathematical parameters, such as control points, knots, and weights, but also on the user's specified constraints and shapes. This paper presents a new approach for directly manipulating the shape of a free-form curve, leading to a better control of the curve deformation and a more intuitive CAD modeling interface. The user's intended deformation of a curve is automatically converted into the modification of the corresponding NURBS control points and knot sequence of the curve. The algorithm for this approach includes curve elevation, knot refinement, control point repositioning, and knot removal. Several examples shown in this paper demonstrate that the proposed method can be used to deform a NURBS curve into the desired shape. Currently, the algorithm concentrates on the purely geometric consideration. Further work will include the effect of material properties.     

Partial shape-preserving splines

A complex geometric shape is often a composition of a set of simple ones, which may differ from each other in terms of their mathematical representations and the ways in which they are constructed. One of the necessary requirements in combining these simple shapes is that their original shapes can be preserved as much as possible. In this paper, a set of partial shape-preserving (PSP) spline basis functions is introduced to smoothly combine a collection of shape primitives with flexible blending range control. These spline basis functions can be considered as a kind of generalization of traditional B-spline basis functions, where the shape primitives used are control points or control polygons. The PSP-spline basis functions have all the advantages of the conventional B-spline technique in the sense that they are nonnegative, piecewise polynomial and of property of partition of unity. However, PSP-spline is a more powerful freeform geometric shape design technique in the sense that it is also a kind of shape-preserving spline. In addition, the PSP-spline technique implicitly integrates the weights of shape control primitives into its basis functions, which allows users to design a required geometric shape based on weighted control primitives. Though its basis functions are simply piecewise polynomial functions, it has the same shape design strengths as the rational piecewise polynomial based spline techniques such as NURBS. In particular, when control shape primitives are specified as a set of control points, PSP-spline behaves like a polygon smoother, with which a shape can be designed to approximate the specified control polygon or control mesh smoothly with any required precision. Consequently, a richer set of geometric shapes can be built using a relatively smaller set of control points.     

Dynamic modeling of butterfly subdivision surfaces

The authors develop integrated techniques that unify physics based modeling with geometric subdivision methodology and present a scheme for dynamic manipulation of the smooth limit surface generated by the (modified) butterfly scheme using physics based “force” tools. This procedure based surface model obtained through butterfly subdivision does not have a closed form analytic formulation (unlike other well known spline based models), and hence poses challenging problems to incorporate mass and damping distributions, internal deformation energy, forces, and other physical quantities required to develop a physics based model. Our primary contributions to computer graphics and geometric modeling include: (1) a new hierarchical formulation for locally parameterizing the butterfly subdivision surface over its initial control polyhedron, (2) formulation of dynamic butterfly subdivision surface as a set of novel finite elements, and (3) approximation of this new type of finite elements by a collection of existing finite elements subject to implicit geometric constraints. Our new physics based model can be sculpted directly by applying synthesized forces and its equilibrium is characterized by the minimum of a deformation energy subject to the imposed constraints. We demonstrate that this novel dynamic framework not only provides a direct and natural means of manipulating geometric shapes, but also facilitates hierarchical shape and nonrigid motion estimation from large range and volumetric data sets using very few degrees of freedom (control vertices that define the initial polyhedron)     

A Geometric Representation Scheme Suitable for Shape Optimization

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一种参数化可调的NURBS曲面G~2光滑拼接算法

提出一种用于NURBS曲面G2光滑拼接算法。在创建拼接曲面时,采用"参数可调"的思想,用拼接函数和指重参数先统一两基曲面的参数,计算出拼接曲面上的插值点,并以这些插值点为参考点根据G2连续的几何性质对拼接曲面的内部控制点进行修正。此算法适用于各类曲面的拼接,通过调整平衡因子和指重参数可以得到在满足G2连续的前提下各种曲率的拼接曲面,简化曲面拼接的计算过程。     

Interactive Rendering of Deforming NURBS Surfaces

Non-uniform rational B-splines (NURBS) has been widely accepted as a standard tool for geometry representation and design. Its rich geometric properties allow it to represent both analytic shapes and free-form curves and surfaces precisely. Moreover, a set of tools is available for shape modification or more implicitly, object deformation. Existing NURBS rendering methods include de Boor algorithm, Oslo algorithm, Shantz’s adaptive forward differencing algorithm and Silbermann’s high speed implementation of NURBS. However, these methods consider only speeding up the rendering process of individual frames. Recently, Kumar et al. proposed an incremental method for rendering NURBS surfaces, but it is still limited to static surfaces. In real-time applications such as virtual reality, interactive display is needed. If a virtual environment contains a lot of deforming objects, these methods cannot provide a good solution. In this paper, we propose an efficient method for interactive rendering of deformable objects by maintaining a polygon model of each deforming NURBS surface and adaptively refining the resolution of the polygon model. We also look at how this method may be applied to multi-resolution modelling.     

Discontinuous Galerkin Based Isogeometric Analysis for Geometric Flows and Applications in Geometric Modeling

We propose a method which combines isogeometric analysis with the discontinuous Galerkin (DG) method for second and fourth order geometric flows to generate fairing surfaces, which are composed of multiple patches. This technique can be used to tackle a challenging problem in geometric modeling–gluing multi-patches together smoothly to create complex models. Non-uniform rational B-splines (NURBS), the most popular representations of geometric models developed in Computer Aided Design, are employed to describe the geometry and represent the numerical solution. Since NURBS basis functions over two different patches are independent, DG methods can be appropriately applied to glue the multiple patches together to obtain smooth solutions. We present semi-discrete DG schemes to solve the problem, and \(\mathcal {L}^{2}\)-stability is proved for the proposed schemes. Our method enjoys the following advantages. Firstly, the geometric flexibility of NURBS basis functions, especially the use of multiple patches, enable us to construct surface models with complex geometry and topology. Secondly, the constructed geometry is fair. Thirdly, since only the control points of the NURBS patches evolve in accordance with the geometric flows, and their number (degrees of freedom) is very small, our algorithm is very efficient. Finally, this method can be easily formulated and implemented. We apply the method in mean curvature flows and in quasi surface diffusion flows to solve various geometric modeling problems, such as minimal surface generation, surface blending and hole filling, etc. Examples are provided to illustrate the effectiveness of our method.     

二次NURBS 曲线的退化曲线

NURBS 曲线是几何造型中广泛使用的曲线拟合工具。当某一权因子趋向于无穷 时,NURBS 曲线趋于相应的控制顶点,当所有权因子趋向于无穷时,其极限曲线的几何性质 目前还没有结论。利用NURBS 曲线的节点插入算法,将NURBS 曲线转化为分段有理Bézier 曲线,结合有理Bézier 曲线的退化理论,得到当所有权因子趋向于无穷时其退化曲线的几何 结构。