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 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.     

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.     

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.     

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.     

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.     

基于参数曲面的增材制造保形晶格结构生成方法

晶格结构因其具备特殊的机械性能,已成为增材制造复杂结构设计和制造的重要研究领域之一.针对传统的模型晶格结构生成方法需要通过对参数化建模的晶格结构网格进行裁剪或对其网格进行保形变形实现,且生成效率较低的问题,提出一种基于参数曲面的增材制造保形晶格结构生成方法,实现晶格结构对曲面空间的适应和高效生成.首先,基于一种矩阵方法...     

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…     

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.     

Approximate development of trimmed patches for surface tessellation

This paper presents a method for constructing an auxiliary planar domain of triangulation for tessellating trimmed parametric surface patches. By minimizing a mapping error function, an approximate locally isometric mapping between a given trimmed parametric surface patch and its triangulation domain is constructed. In this way the shape of triangular elements on the triangulation domain is approximately preserved when mapped into three-dimensional space. We also provide an efficient method to achieve a good initial guess for the minimization of the mapping error function. Furthermore, our proposed method guarantees a homeomorphism between a triangulation domain and parametric space/given surface patch by robustly removing the possibility of self-intersection on the developed surface net. Practical application of the proposed algorithm can include the formation of ship hulls, ducts, shoes, clothing and automobile parts as well as the surface meshing procedure.     

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     

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.     

Spline-based meshfree method with extended basis

In this work, an extension has been performed on the analysis basis of spline-based meshfree method (SBMFM) to stabilize its solution. The potential weakness of the SBMFM is its numerical instability from using regular grid background mesh. That is, if an extremely small trimmed element is produced by the trimming curves that represent boundaries of the analysis domain, it can induce an excessively large condition number in global system matrix. To resolve the instability problem, the extension technique of the weighted extended B-spline (WEB-spline) is implemented in the SBMFM. The basis functions with very small trimmed supports are extrapolated by neighboring basis functions with some special scheme so that those basis functions can be condensed in the solution process. In order to impose essential boundary conditions in the SBMFM with extended basis, Nitsche's method is implemented. Using numerical examples, the presented SBMFM with extended basis is shown to be valid and effective. Moreover, the condition number of the system is well-managed guaranteeing the stability of the numerical analysis.     

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。     

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。     

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.     

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。     

Efficient and accurate B-rep generation of low degree sculptured solids using exact arithmetic: II—computation

We present efficient algorithms for exact boundary computation on low degree sculptured CSG solids using exact arithmetic. These include algorithms for computing the intersection curves of low-degree trimmed parametric surfaces, decomposing them into multiple components for efficient point location queries inside the trimmed regions, and computing the boundary of the resulting solid using topological information and component classification tests. We also employ a number of previously developed algorithms such as algebraic curve classification and multivariate Sturm sequences. We present some results from a preliminary implementation of our approach. This paper follows a previous paper which described the representations used in our approach.     

多裁剪自由曲面生成有限元网格的实现

论述了多裁剪自由曲面生成有限元曲面网格的几个关键技术.采用了推进波前法生成曲面网格,给出了核心算法;在曲面算法中运用了介于参数法与直接法之间的新方法.针对求解曲面上最优点的参数域反算问题,引入了切矢逆求方法,可使迭代次数大为降低.测试表明,该算法快速、稳定.对大型的多裁剪自由曲面生成的曲面有限元网格,可直接用于有限元计算.