一种新的均匀样条曲线曲面设计方法

本文根据均匀B样条基函数的de Boor-Cox递推公式提出了一种新的样条曲线曲面设计方法。该方法从满足正性、局部支柱性和权性的初始基函数出发，可构造出具有高阶低次或低次高阶的多项式样条基函数和多种函数类型的样条函数。给出了设计这种样条曲线曲面的几种方法和实例，并对基函数的连续可微性进行了证明。该样条基函数和样条曲线曲面具有和均匀B样条类似的几何性质，且均匀B样条是其特例，可用于曲线曲面的几何造型和样条插值。     

插值给定对角曲线的能量极小Bézier曲面造型

目的 曲面造型是计算机辅助几何设计中的重要研究内容,张量积类型曲面的对角曲线是衡量曲面性质的重要度量,与曲面的几何形状密切相关。基于输入对角曲线的曲面设计方法在实际应用中具有一定的价值,因此提出一种插值给定对角曲线的能量极小Bézier曲面造型的方法。方法 给定一条对角曲线时,修正用户输入的对角曲线及边界曲线的几何信息,然后运用拉格朗日乘数法,结合曲面内部能量函数,求解待定的内部控制顶点,构造曲面。给定两条对角曲线时,在上述内容基础上加入了两条对角曲线必有交点的考量,增加对对角曲线控制顶点的修正。结果 增加了对角曲线这一约束条件,从对比实验曲线图可以看出,随着横坐标曲面阶数升高,纵坐标修正曲线和用户曲线间的差值越来越小,结果表明曲面阶数越高,修正曲线与用户曲线偏差越小,造型效果越好。结论 该曲面造型方法简单,基于修正后的对角曲线和边界曲线构造的曲面具有极小内部能量,可满足曲面造型方面的相关需求。     

形状调配中带参数的过渡曲线与曲面构造

针对形状调配问题中过渡曲线与曲面构造方法的不足,提出基于带参数的曲线模型构造过渡曲线曲面的方法.首先利用一类带参数的曲线模型构造一种带参数的调配函数;然后研究基于该调配函数的过渡曲线构造问题,并讨论基于能量优化法的最佳过渡曲线构造问题;最后探讨相应的过渡曲面构造问题.实例结果表明,利用带参数的调配函数构造过渡曲线曲面时,2个被过渡曲线曲面可为任意的参数曲线曲面,所构造的过渡曲线曲面不仅在边界处满足拟C2连续,而且还可通过调配函数所带的参数对其形状进行调整.     

三次T-Bézier曲线的光顺延拓

将曲线的最小物理变形能量作为目标函数,提出了一种三次T-Bézier曲线曲面的光顺延拓算法.利用G2连续性作为约束条件,则延拓的曲线具有两个自由度,并取其中的一个自由度为零;基于延拓曲线的最小物理变形能量确定第二个自由度及延拓曲线的控制点,进而确定延拓曲线,重新参数化所延拓的曲线可以与原曲线在拼接点处C3连续拼接;此外,将该方法应用于双三次T-Bézier曲面的延拓.实例表明,该方法构造的曲线曲面具有较好的光顺性.     

构造任意参数曲线曲面的简捷方法

本文探讨了用控制参量形式表示各种参数三次曲线、曲面和二次Ｂｅｚｉｅｒ、二次Ｂ－ｓｐｌｉｎｅ曲线、曲面的基函数统一表达式。采用改变控制参数取值的方法构造所需的各种曲线、曲面，为曲线，曲面造型提供了一种简捷的数学方法，还讨论了参量的不同取值所对应的不同种类曲线、曲面的几何特性。     

一类特殊基函数构造的参数曲线

构造参数曲线曲面一直是计算机辅助几何设计研究的核心内容之一。以Bernstein基函数构造的Bézier曲线是参数曲线造型最基本的方法,B样条曲线和NURBS曲线都是在其基础上发展而来。利用给定的实数节点集,构造一类特殊的基函数,此类基函数是Bernstein基函数的推广。在此基础上,构造了一类新的参数曲线,称为T-Bézier曲线,T-Bézier曲线继承了有理Bézier曲线的若干性质;证明了当节点移动时极限曲线的几何性质,并通过实例进行了验证。     

基于曲线和曲面控制的多边形物体变形反走样

基于参数曲线和曲面控制的空间变形是重要的几何外形编辑和柔性物体动画实现手段．当这两类变形方法的对象是多边形物体时，如何对变形物体进行重采样以得到高质量结果，是计算机动画和几何造型领域中的一个重要问题．该文针对B-样条曲线和曲面控制的空间变形方法，提出了面向多边形物体的空间变形反走样方法．在该方法中，利用等距技术将B-样条曲线或曲面所张成的变形空间近似表示为张量积B-样条参数体，结合作者提出的多边形物体精确B-样条自由变形方法，实现了参数曲线和曲面控制的多边形物体变形反走样．     

多项式混合曲线曲面方法构造

针对混合曲线表示及其求导和求积困难的问题,通过计算构造出一种多项式混合曲线曲面形式.当待混合曲线是多项式时,混合曲线也为多项式形式.该多项式混合公式可以推广得到任意参数连续C(n)和几何连续G(n)的混合曲线曲面.另外,在得到的混合曲线曲面族中构造出了新的更优能量光顺方程,通过设置参数可得到合适的混合曲线曲面.实验结果表明,文中提出的混合曲线曲面造型方法稳定、有效.     

流曲线曲面的概念和构造方法

给出了一种新的曲线曲面－流曲线曲面概念及其构造方法。给出了用多项式和Ｂｅｚｉｅｒ控制网构造流曲线曲面的基本方法和约束条件，这种流曲线曲面方法可用于计算机辅助几何设计和图形学中曲线表示和设计，特别是船体外型的造型和设计。     

基于伸缩因子的代数曲线曲面变形方法

提出了一种基于伸缩因子的代数曲线曲面变形方法。首先构造了一种用于代数曲线曲面变形的伸缩因子,然后将所构造的伸缩因子分别作用于待变形的代数曲线与曲面,通过改变伸缩因子各参数的取值实现对代数曲线曲面的变形。实例表明,该方法操作方便,易于控制,可获得丰富的变形效果,是一种简单有效的代数曲线曲面变形方法。     

Direct manipulations of B-spline and NURBS curves

This paper presents a method for the direct manipulations of B-spline and non-uniform rational B-splines (NURBS) curves using geometric constraints. A deformable model is developed to define the deformation energy functional of B-spline and NURBS curves. The finite element method is used to minimize the deformation energy functional and solve for the deformed shape of curves subjected to constraints. This approach results in a set of linear equations for a B-spline curve and a set of non-linear equations for a NURBS curve. A perspective mapping is used to linearize the NURBS formulations. NURBS curves are first mapped from the 3D Cartesian coordinate space to the 4D homogeneous coordinate space, and transformed to 4D B-spline curves. After the manipulation in the 4D homogeneous coordinate space, the modified NURBS curves are then mapped back to the 3D Cartesian coordinate space. The approach is implemented by a prototype program, which is written in C, and runs under WINDOWS. Several examples are presented to demonstrate the capabilities of this approach.     

Surface Smoothing Procedures in Computational Contact Mechanics

This work addresses the problems arising in the finite element simulation of contact problems undergoing large deformation. The frictional contact problem is formulated in the continuum framework, introducing the interface laws for the normal and tangential stress components in the contact area. The variational formulation is presented, considering different methods to enforce the contact constraints. The spatial discretization within the finite element method is applied, as well as the temporal discretization required to solve the three sources of nonlinearities: geometric, material and frictional contact. The discretization of contact surfaces is discussed in detail, including different surface smoothing procedures. This numerical strategy allows to solve the difficulties associated with the discontinuities in the contact surface geometry introduced by finite element discretization, which leads to nonphysical oscillations of the contact force for large sliding problems. The geometrical accuracy of different interpolation methods is evaluated, paying particular attention to the Nagata patch interpolation recently proposed. In this framework, the Node-to-Nagata contact elements are developed using the augmented Lagrangian method to regularize the variational frictional contact problem. The techniques used to search for contact in case of large deformations are discussed, including self-contact phenomena. Several numerical examples are presented, comprising both the contact between deformable and rigid obstacles and the contact between deformable bodies. The results show that the accuracy and robustness of the numerical simulations is improved when the contact surface is smoothed with Nagata patches.     

The analysis of solids without mesh generation using trimmed patch boundary elements

The tools available to the mechanical engineer—for example, geometric modeling and computer-aided analysis—are individually quite powerful, but they are based on different geometric representations. Hence, they do not always work well together. In this paper an analysis method is presented that operates directly on the geometric modeling representation. Therefore, the time-consuming and error-prone procedure of generating a mesh is skipped. The method is based on boundary integral equations, but unlike previously published methods, the boundary elements aren-sided trimmed patches, the same patches that are used by modern geometric modelers to represent complex solids. The method is made practical by defining shape functions over the trimmed patches in such a way that the number of degrees of freedom can be controlled. This is done by using a concept called virtual nodes. The paper begins by deriving the trimmed patch boundary element. Then its properties are discussed in comparison with existing boundary element and finite element methods, and several examples are given.     

Modeling of shape bifurcation phenomena in manipulations of deformable string objects

A systematic approach to the modeling of deformable string objects is presented. Various string objects such as cords and wires are manipulated in many manufacturing processes. In such processes, it is important for successful manipulation to evaluate their shapes on a computer in advance because their shapes can be changed easily even under the same conditions. We refer to the situation that a shape can be changed into another shape even under the same constraints as shape bifurcation. In this paper, we will develop an analytical model of the shape of string objects including shape bifurcation. First, we will investigate the mechanism of the shape bifurcation phenomena based on the potential energy. Then, we will propose a hypothesis on the mechanism of shape bifurcation based on local minima of the potential energy. Second, the potential energy of a string object and the geometric constraints imposed on it are formulated. The shape of the object can be derived by minimizing the potential energy under the geometric constraints. Thirdly, a procedure to compute the shape of a deformed string object is developed considering the local minima of the energy. Finally, we show some numerical examples with shape bifurcation using our proposed method. From the results, we conclude that our proposed method accurately describes deformed shapes of string objects including shape bifurcation.     

Fast generation of 3-D deformable moving surfaces

Dynamic surface modeling is an important subject of geometric modeling due to their extensive applications in engineering design, entertainment and medical visualization. Many deformable objects in the real world are dynamic objects as their shapes change over time. Traditional geometric modeling methods are mainly concerned with static problems, therefore unsuitable for the representation of dynamic objects. Apart from the definition of a dynamic modeling problem, another key issue is how to solve the problem. Because of the complexity of the representations, currently the finite element method or finite difference method is usually used. Their major shortcoming is the excessive computational cost, hence not ideal for applications requiring real-time performance. We propose a representation of dynamic surface modeling with a set of fourth order dynamic partial differential equations (PDEs). To solve these dynamic PDEs accurately and efficiently, we also develop an effective resolution method. This method is further extended to achieve local deformation and produce n-sided patches. It is demonstrated that this new method is almost as fast and accurate as the analytical closed form resolution method and much more efficient and accurate than the numerical methods.     

Eulerian shape design sensitivity analysis and optimization with a fixed grid

Conventional shape optimization based on the finite element method uses Lagrangian representation in which the finite element mesh moves according to shape change, while modern topology optimization uses Eulerian representation. In this paper, an approach to shape optimization using Eulerian representation such that the mesh distortion problem in the conventional approach can be resolved is proposed. A continuum geometric model is defined on the fixed grid of finite elements. An active set of finite elements that defines the discrete domain is determined using a procedure similar to topology optimization, in which each element has a unique shape density. The shape design parameter that is defined on the geometric model is transformed into the corresponding shape density variation of the boundary elements. Using this transformation, it has been shown that the shape design problem can be treated as a parameter design problem, which is a much easier method than the former. A detailed derivation of how the shape design velocity field can be converted into the shape density variation is presented along with sensitivity calculation. Very efficient sensitivity coefficients are calculated by integrating only those elements that belong to the structural boundary. The accuracy of the sensitivity information is compared with that derived by the finite difference method with excellent agreement. Two design optimization problems are presented to show the feasibility of the proposed design approach.     

Nonlinear analysis of oil pipelines by potential minimization

The problem of finding the deformed shape of a structure can be formulated as finding the minimum of the total potential for the system defined by the structure and the loads acting on it. By defining unknown displacements as free variables and the total potential as objective function in a minimization problem, the displacements can be found by a minimization technique. This method is applied to perform a static analysis of offshore pipelines during installation, which is a geometric nonlinear problem.

The theoretical foundation of this method is briefly described and the relationship to an iterative finite element formulation is pointed out.

Expressions for the total potential of the pipeline system, defined by the strain energy and the potential of acting forces are presented.

Two minimization techniques are used. For this problem, where gradients are available with reasonable effort, a gradient method based upon Fletcher-Powell's search technique has been found faster and more accurate than Powell's direct search technique.

A computer program based upon the method presented is developed and the results are encouraging concerning both convergency and computational effort.     

An improved formulation of space stiffeners

This paper presents a displacement based finite element model for predicting the constraint torsion effect of stiffeners. In structural modelling, the plate/shell and the stiffeners are treated as separate elements where the displacement compatibility transformation between these two types of elements takes into account the constraint torsional warping effect in the stiffeners. The development is based on a general beam theory which includes flexural-torsion coupling, constrained torsion warping, and shear-centre location. The virtual work principle includes the second order terms of finite beam rotations. For finite element analysis, cubic Hermitian polynomials are used as shape functions of the straight space frame element with two nodes. Elastic stiffness and geometric stiffness matrices for an arbitrary cross-section are evaluated in a closed form, and load correction stiffness for eccentric stiffener loads are considered. To demonstrate the importance of torsion warping constraints and to illustrate the accuracy of this formulation, finite element solutions are presented and compared with available solutions.     

Isometric Piecewise Polynomial Curves

The main preoccupations of research in computer-aided geometric design have been on shape-specification techniques for polynomial curves and surfaces, and on the continuity between segments or patches. When modelling with such techniques, curves and surfaces can be compressed or expanded arbitrarily. There has been relatively little work on interacting with direct spatial properties of curves and surfaces, such as their arc length or surface area. As a first step, we derive families of parametric piecewise polynomial curves that satisfy various positional and tangential constraints together with arc-length constraints. We call these curves isometric curves. A space curve is defined as a sequence of polynomial curve segments, each of which is defined by the familiar Hermite or Bézier constraints for cubic polynomials; as well, each segment is constrained to have a specified arc length. We demonstrate that this class of curves is attractive and stable. We also describe the numerical techniques used that are sufficient for achieving real time interaction with these curves on low-end workstations.     

基于形变模型由立体序列图象恢复物体的3D形状

结合立体视觉和形变模型提出了一种新的物体3D形状的恢复方法。采用立体视觉方法导出物体表面的3D坐标;利用光流模型估计物体的3D运动,根据此运动移动形变模型,使其对准物体的表面块;由形变模型将由各幅图象得到的离散的3D点融为一起,得到物体的表面形状。实验结果表明该方法能用于形状复杂的物体恢复。