A robust efficient tracing scheme for triangulating trimmed parametric surfaces

An efficient, robust parametric trimmed surface triangulation method is presented. Efficiency is gained during trimmed curve tracing by minimising the number of cells processed. Key feature is the efficient tracing algorithm, and knowledge of orientation of the trimming curves is not required. The method is applicable to NURBS surfaces and operates on the untrimmed surface, constructing a rectangular parametric grid onto which the trimming curves are traced. This approach also minimises the occurrence of degenerate triangles and copes with holes independently of the grid size.     

隐式裁剪曲面的造型及绘制

隐式曲面没有参数域的概念,故对其裁剪问题的研究,很少有人问津.提出了一种隐式裁剪曲面的造型和绘制方法:先把投影平面变换到xy平面,将该xy平面作正交化网格剖分,由参数表示或隐式表示的裁剪曲线在该xy平面定义裁剪区域,再把裁剪后留下的区域内的网格投影到隐式曲面上,从而实现了隐式裁剪曲面的多边形化绘制.     

A geometric approach to boundary-conformed toolpath generation

Isoparametric or streamline-like toolpath generation is an important method for CNC surface machining. For a trimmed surface, the generation of such a toolpath is not straightforward because by simply following the isoparametric curves of the parametric surface to generate the toolpath, the resulting toolpath may no longer conform to the trimmed boundary of the surface. Various methods have been proposed to solve this problem. Some methods work well for single surface machining, but may not be feasible for multi-patch surface machining. A new geometric approach to the problem of multi-patch machining is proposed in this paper. The new method works by generating bisectors to partition the region into smaller subregions, and generates the toolpath by offsetting the subregion boundary by using a special offset function. As this new method does not rely on processing on the parametric domain of the surface patches, it can be used for both NURBS and subdivision surfaces. Examples are given to demonstrate the capability of the new method, and to show that the new method compares favorably with existing techniques.     

Isogeometric topology optimization using trimmed spline surfaces

In the present work, a new spline based topology optimization using trimmed spline surfaces and the isogeometric analysis is proposed. In the proposed approach, the trimmed surface analysis which can treat topologically complex spline surfaces using trimming information provided by CAD systems is employed for structural response analysis and sensitivity calculation in the topology optimization. The outer and inner boundaries of design models are represented by a spline surface and trimming curves. Design variables used in this approach are the coordinates of control points of a spline surface and those of trimming curves. New sensitivity formulations for the control points in the trimmed surface analysis are proposed and their efficiency and accuracy are verified. The creation of new inner fronts during optimization is allowed for the topological flexibility. An inner front merging algorithm is also presented. The proposed spline based topology optimization is used to solve some benchmarking problems. Design space dependency which is one of serious shortcomings in conventional topology optimization approaches is naturally eliminated by the proposed spline based optimization. Design dependent load problems which are difficult to treat with conventional grid based topology optimization methods are easily dealt with by the proposed one. It is also shown that post-processing effort for converting to CAD model is eliminated by using the same spline information in numerical analysis and design optimization.     

Computing offsets of trimmed NURBS surfaces

Offsetting of trimmed NURBS surfaces is one of the widely used functionalities in the design and manufacture simulations of composite laminates. This paper presents an approach for the offsetting of a trimmed NURBS surface. The approach has been developed mainly to meet the stringent accuracy requirements in the simulation of composite laminate design and manufacturing processes. However, the approach is applicable for the offset of a general trimmed NURBS surface. Though the method is based on known techniques in literature, the practical approach and the treatment of the subject presented is unique and has not been reported earlier. The basic approach consists of offsetting the underlying surface, offsetting of all the trimming loops and the creation of offset trimmed surface using the offset surface and the offset trimming loops. This is a unified offset approach for trimmed surfaces where in the offset of underlying surface and the offset of trimming loops are obtained using the same approach. The approach has better error control and results in less number of control points. Further the approach can be extended to obtain offsets of a general B-Rep. The approach has been used in the creation of offset surfaces of various aircraft components.     

基于解析法的裁剪PDE曲面的生成技术

为解决基于偏微分方程的曲面裁剪问题,研究一种广泛应用于偏微分方程曲面的裁剪方法.首先介绍基于偏微分方程的曲面生成方法,其次由参数域内的曲线在曲面上的投影,得到所求裁剪曲面的边界,然后利用解析法求得裁剪后的PDE曲面,最后列举一系列的实例来说明该裁剪方法的应用并且专门研究多个裁剪区域的问题.     

Robust Approximation of Offsets,Bisectors, and Medial Axes of Plane Curves

Most methods for computing offsets, bisectors, and medial axes of parametric curves in the plane are based on a local formulation of the distance to a curve. As a consequence, the computed objects may contain spurious parts and components, and have to be trimmed. We approach these problems as global optimization problems, and solve them using interval arithmetic, thus generating robust approximations that need not be trimmed.     

Trimmed Surface Triangulation for Aircraft RCS Computation

1 Introduction Radar cross section(RCS)is one of the main parameters that estimate the stealth performance of aircraft. The theory of RCS is based on the scattered field created by the model subject to electromagnetic scattering, and the estimation is made through applying various kinds of computing method and technology to compute the RCS of the model under the circumstances [1]. There are two approaches to analyze RCS at present, one is testing the real model; the other is computing wit…     

一种裁剪参数曲面的有限元网格剖分方法

在板料冲压成形模拟分析中,从CAD系统输入的模具的曲面模型包含大量的裁剪参数曲面,曲面之间的相邻关系复杂,针对这种曲面模型的特点,提出了一种裁剪参数曲面的有限元网格剖分方法,单个裁剪参数曲面采用非约束边界的等参数映射法,各个裁剪参数曲面各自独立地网格剖分产生了网格单元后,再将各个裁剪参数曲面的网格单元合并为单元相容,即单元间无裂缝和覆盖的网格模型,这种方法适合于需要大量裁剪参数曲面拼合的复杂曲面模型,如汽车覆盖件模型的网格剖分。     

Ray-tracing triangular trimmed free-form surfaces

This paper presents a new approach to rendering triangular algebraic free-form surfaces. A hierarchical subdivision of the surface with associated tight bounding volumes provides for quick identification of the surface regions likely to be hit by a ray. For each leaf of the hierarchy, an approximation to the corresponding surface region is stored. The approximation is used to compute a good starting point for the iteration, which ensures rapid convergence. Trimming curves are described by a tree of trimming primitives, such as squares, circles, polygons, and free-form curves, combined with Boolean operations. For trimmed surfaces, an irregular adaptive subdivision is constructed to quickly eliminate all parts outside the trimming curve from consideration during rendering. Cost heuristics are introduced to optimize the rendering time further     

细分曲面求交裁剪算法研究

基于细分曲面的参数化表示,研究了细分曲面的精确求交、裁剪算法。首先对控制网格建立局部坐标系,将细分曲面表示为一系列小的面片,并对每个控制顶点赋予参数值。然后用改进的轮廓删除法细分控制网格,在关联曲面间进行相交性检测,得到近似交点及其参数值,再用迭代法求得精确解。根据用户指定的裁剪区域确定交线的走向,将被裁剪曲面的控制网格面分为保留面、裁剪面和删除面,设置每个裁剪面的裁剪域,从而实现细分曲面的精确裁剪。算例表明,该文的方法简单、有效。     

基于曲面精确表示的距离极值点的计算及在刀具干涉检测中的应用

针对基于曲面精确表示的刚体碰撞检测中裁剪曲面距离极值点的求解问题,提出了 利用平面向量场估计初始曲面距离极值点的方法,避免了曲面过度细分,讨论了距离极值点满足 的微分几何条件,给出了解析曲面/参数曲面、参数曲面/参数曲面、点/参数曲面和曲线/参数曲面 的距离极值点迭代算法。实例验证分析了该算法的高效性和可靠性。     

运用多项式曲线在多项式曲面上裁剪Bézier曲面

在多项式曲面的定义域上,以两多项式曲线及两直线段围成的简单区域作为裁剪区域,运用参数变换将该区域变换到标准正方形区域,以多项式开花为工具,将裁剪区域对应的子曲面片表示成Bézier曲面形式。对于参数平面上的复杂裁剪区域,则分割为若干简单区域来进行。该裁剪算法能处理形状较为复杂的曲面裁剪,方法对任意多项式曲面适用,而且能推广到有理情况。     

Trimming local and global self-intersections in offset curves/surfaces using distance maps

A robust and efficient algorithm for trimming both local and global self-intersections in offset curves and surfaces is presented. Our scheme is based on the derivation of a rational distance map between the original curve or surface and its offset. By solving a bivariate polynomial equation for an offset curve or a system of three polynomial equations for an offset surface, all local and global self-intersection regions in offset curves or surfaces can be detected. The zero-set of polynomial equation(s) corresponds to the self-intersection regions. These regions are trimmed by projecting the zero-set into an appropriate parameter space. The projection operation simplifies the analysis of the zero-set, which makes the proposed algorithm numerically stable and efficient. Furthermore, in a post-processing step, a numerical marching method is employed, which provides a highly precise scheme for self-intersection elimination in both offset curves and surfaces. The effectiveness of our approach is demonstrated using several experimental results.     

Geometric Properties of Bisector Surfaces

This paper studies algebraic and geometric properties of curve–curve, curve–surface, and surface–surface bisectors. The computation is in general difficult since the bisector is determined by solving a system of nonlinear equations. Geometric considerations will help us to determine several distinguished curve and surface pairs which possess elementary computable bisectors. Emphasis is on low-degree rational curves and surfaces, since they are of particular interest in surface modeling.     

Design and recovery of 2-D and 3-D shapes using rational Gaussian curves and surfaces

A new representation for parametric curves and surfaces is introduced here. It is in rational form and uses rational Gaussian bases. This representation allows design of 2-D and 3-D shapes, and makes recovery of shapes from noisy image data possible. The standard deviations of Gaussians in a curve or surface control the smoothness of a recovered shape. The control points of a surface in this representation are not required to form a regular grid and a scattered set of control points is sufficient to reconstruct a surface. Examples of shape design, shape recovery, and image segmentation using the proposed representation are given.     

Trimming of free-form objects for a B-Rep solid modeller

This paper describes a technique for trimming solid objects in a B-Rep solid modelling system. A B-Rep scheme based on the split-edge structure, which is suitable for the representation of trimmed surfaces, is first discussed. Algorithms for graph traversal, graph merging and graph splitting are then given in detail. Based on this B-Rep scheme, a technique for trimming objects is developed. The trimming process involves the evaluation of surface intersections, graph cutting along intersections and merging of appropriate graphs to obtain the required trimmed objects.     

裁剪面的一类参数变换

给出了一类可以保持几何与拓扑信息一致性的裁剪面的参数变换定理及其算法。首先,确定了参数变换对裁剪面表示信息的影响。然后,根据参数变换后几何与拓扑信息的一致性要求,给出了对裁剪面表示信息进行调整的方法。最后,通过建立参数变换的关系,以裁剪球面为例阐述了这类参数变换的具体实现方法。     

Generating cutter swept envelopes in five-axis milling by two-parameter families of spheres

This paper presents an efficient parametric approach of determining the shape of the envelope surface by a generalized cutter that follows five-axis tool path during NC machining. In this approach the cutter is modeled as a canal surface. By considering the tool motions the cutter is decomposed into a set of characteristic and great circles which are generated by two-parameter families of spheres. The center of a sphere from these families is described by two parameters which represent the spine curve and the tool path, and the radius of the sphere is described by one parameter representing the spine curve. Considering the relationship between characteristic and great circles the grazing points on the tool surface are identified. Analytically it is proven for the NC cutter geometries that any point on the envelope surface is located at the intersection of the characteristic and great circles. Then based on the proofs a closed-form solution for computing the grazing points generated by a surface of revolution is presented. The presented methodology is reduced to a simpler parametric form when the NC cutters are described by pipe surfaces.