Feature-based decomposition of trimmed surface

A trimmed surface is usually represented by a parametric surface and a set of trimming curves. Because of the complexity in manipulating trimmed surfaces, many CAD processes and algorithms cannot be applied to trimmed surfaces directly. It is thus desirable to represent a trimmed surface by a group of regular surfaces. In this paper, an algorithm for decomposing a trimmed surface is presented. First, bisectors of the Voronoï diagram developed in the parametric space are used to define an isolated region for every trimming curve. Feature points on the trimming curves are extracted by considering curvatures of the curves. Correspondence between feature points and vertices on the bisectors are established by considering the similarity between the trimming curves and the bisectors. Regions of parametric patches are then identified. Finally, a group of regular surfaces are constructed by interpolating a set of sampled surface points on each of the identified regions.     

Isogeometric analysis with trimming technique for problems of arbitrary complex topology

Trimming technique is a powerful and efficacious way of endowing an arbitrary complex topology to CAD files created by using NURBS. In the present work, it is shown that any complex multiply-connected NURBS domain can be described by using trimming curves only. Isogeometric analysis for linear elasticity problems of complex topology described in this way is presented. For fully communicative interaction between CAD and CAE, a specific searching algorithm and an integration scheme of trimmed elements are introduced to utilize the IGES files exported from CAD system for Isogeometric analysis. Schemes for imposing essential and traction boundary conditions on trimming curves are presented. It has been demonstrated that with the presented schemes trimmed cases in any complicated situations can be successfully treated. With the examples of complex topology that could be described by employing trimming curves only, effectiveness and robustness of present method are demonstrated.     

Modifying CAD/CAM surfaces according to displacements prescribed at a finite set of points

This article presents a method for modifying CAD/CAM surfaces automatically in accordance with displacements prescribed at a finite set of points in R³, such as node displacements predicted by finite-element analysis. The method is based on the ‘morphing’ approach introduced by Sederberg and Parry in 1986. The input to the process consists of (a) a CAD/CAM model containing trimmed polynomial B-spline surfaces and (b) a set of points and associated displacement vectors in R³. These points are assumed to be close to, but not necessarily on, the objects of the CAD/CAM model. A rectangular volume, enclosing the CAD/CAM model and the input points in R³, is represented as a volume spline, i.e. a trivariate tensor-product spline. A modified volume spline is computed using (a) a least-squares fit based on the given point displacements, and (b) a smoothing functional. The modified CAD/CAM objects are defined as compositions of the original parametric functions and the modified volume spline (i.e. a morphing). In order to ensure compatibility with standard commercial CAD/CAM systems, the modified surfaces are fitted with appropriate splines using any standard, reasonably shape-preserving, fitting procedure applied in the parameter domains of the original surfaces.     

Integration of topology and shape optimization for design of structural components

This paper presents an integrated approach that supports the topology optimization and CAD-based shape optimization. The main contribution of the paper is using the geometric reconstruction technique that is mathematically sound and error bounded for creating solid models of the topologically optimized structures with smooth geometric boundary. This geometric reconstruction method extends the integration to 3-D applications. In addition, commercial Computer-Aided Design (CAD), finite element analysis (FEA), optimization, and application software tools are incorporated to support the integrated optimization process. The integration is carried out by first converting the geometry of the topologically optimized structure into smooth and parametric B-spline curves and surfaces. The B-spline curves and surfaces are then imported into a parametric CAD environment to build solid models of the structure. The control point movements of the B-spline curves or surfaces are defined as design variables for shape optimization, in which CAD-based design velocity field computations, design sensitivity analysis (DSA), and nonlinear programming are performed. Both 2-D plane stress and 3-D solid examples are presented to demonstrate the proposed approach. Received January 27, 2000 Communicated by J. Sobieski     

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.     

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.     

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.     

B Spline Interpolation Based on Optimization

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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.     

Adaptive CAD model (re-)construction with THB-splines

Computer Aided Design (CAD) software libraries rely on the tensor-product NURBS model as standard spline technology. However, in applications of industrial complexity, this mathematical model does not provide sufficient flexibility as an effective geometric modeling option. In particular, the multivariate tensor-product construction precludes the design of adaptive spline representations that support local refinements. Consequently, many patches and trimming operations are needed in challenging applications. The investigation of generalizations of tensor-product splines that support adaptive refinement has recently gained significant momentum due to the advent of Isogeometric Analysis (IgA) [2], where adaptivity is needed for performing local refinement in numerical simulations. Moreover, traditional CAD models containing many small (and possibly trimmed) patches are not directly usable for IgA. Truncated hierarchical B-splines (THB-splines) provide the possibility of introducing different levels of resolution in an adaptive framework, while simultaneously preserving the main properties of standard B-splines. We demonstrate that surface fitting schemes based on THB-spline representations may lead to significant improvements for the geometric (re-)construction of critical turbine blade parts. Furthermore, the local THB-spline evaluation in terms of B-spline patches can be properly combined with commercial geometric modeling kernels in order to convert the multilevel spline representation into an equivalent – namely, exact – CAD geometry. This software interface fully integrates the adaptive modeling tool into CAD systems that comply with the current NURBS standard. It also paves the way for the introduction of isogeometric simulations into complex real world applications.     

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…     

Correct resolution rendering of trimmed spline surfaces

Current strategies for real-time rendering of trimmed spline surfaces re-approximate the data, pre-process extensively or introduce visual artifacts. This paper presents a new approach to rendering trimmed spline surfaces that guarantees visual accuracy efficiently, even under interactive adjustment of trim curves and spline surfaces. The technique achieves robustness and speed by discretizing at a near-minimal correct resolution based on a tight, low-cost estimate of adaptive domain griding. The algorithm is highly parallel, with each trim curve writing itself into a slim lookup table. Each surface fragment then makes its trim decision robustly by comparing its parameters against the sorted table entries. Adding the table-and-test to the rendering pass of a modern graphics pipeline achieves anti-aliased sub-pixel accuracy at high render-speed, while using little additional memory and fragment shader effort, even during interactive trim manipulation.     

Convolution surfaces for arcs and quadratic curves with a varying kernel

A convolution surface is an isosurface in a scalar field defined by convolving a skeleton, comprising of points, curves, surfaces, or volumes, with a potential function. While convolution surfaces are attractive for modeling natural phenomena and objects of complex evolving topology, the analytical evaluation of integrals of convolution models still poses some open problems. This paper presents some novel analytical convolution solutions for arcs and quadratic spline curves with a varying kernel. In addition, we approximate planar higher-degree polynomial spline curves by optimal arc splines within a prescribed tolerance and sum the potential functions of all the arc primitives to approximate the field for the entire spline curve. Published online: November 20, 2002     

基于约束优化的B样条曲线形状修改

B样条曲线广泛应用于计算机辅助几何设计(CAGD),并且与Bézier曲线等其它著名曲线相比,在形状设计方面有其更独特的性质。对曲线的设计和形状的修改是一个重要的课题,也是计算机图形学、CAD/CAM和数控技术领域最重要的研究主题之一。论文运用约束优化的方法,修改均匀B-样条的控制点,使B样条曲线通过调整的控制点,使修改前后曲线的距离范数达到最小,并给出相应的实例说明算法的有效性。     

基于小波的准均匀B样条曲线曲面变分造型

变分造型是自由曲线曲面交互设计中常用的技术之一，在传统的变分造型技术中，一般将此类问题转化为对控制顶点的优化求解问题，但当基函数多进，B样条的局部性常会导致待求解系统的性态不好，求解效率降低，对此，文中应用小波方法解决准均匀B样条曲线曲面变分造型中的优化求解问题，由于小波基具有多分辨特性，该方法可以大大提高迭代速度。     

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     

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。     

带自由曲面的三视图重建算法研究

针对蕴含自由曲面的三视图,提出了通过恢复边界线和轮廓线进而根据这些恢复 的空间曲线重建自由曲面的算法。首先分析并证明了自由曲面在三视图中的投影性质,从而提 出边界投影和轮廓投影的匹配算法。针对视图中存在的被打断的样条曲线,提出了分段样条曲 线的爬坡算法来解决此类曲线不能匹配的问题。然后基于投影匹配序列重建出自由曲面的边界 线,再由边界投影上的点和轮廓投影端点的对应关系重建出空间轮廓线。由轮廓线等参采样构 造截面线并和边界线一起蒙皮生成最终自由曲面。本文提出的算法扩展了工程图的重建域。     

Patch layout from feature graphs

The structuring of surface meshes is a labor intensive task in reverse engineering. For example, in CAD, scanned triangle meshes must be divided into characteristic/uniform patches to enable conversion into high-level spline surfaces. Typical industrial techniques, like rolling ball blends, are very labor intensive.We provide a novel, robust and quick algorithm for the automatic generation of a patch layout based on a topology consistent feature graph. The graph separates the surface along feature lines into functional and geometric building blocks. Our algorithm then thickens the edges of the feature graph and forms new regions with low varying curvature. Further, these new regions-so-called fillets and node patches-will have highly smooth boundary curves, making the algorithm an ideal preprocessor for a subsequent spline fitting algorithm.