A surface reconstruction algorithm for topology optimization

For mechanical structural design, topology optimization is often utilized. During this process, a topologically optimized model must be converted into a parametric CAD solid model. The key point of conversion is that a discretized shape of a topologically optimized model must be smoothed, but features such as creases and corners must be retained. Thus, a surface reconstruction algorithm to produce the parametric CAD solid model from a topologically optimized model is proposed in this paper. Our presented algorithm consists of three parts: (1) an enclosed isosurface geometry from which the topologically optimized model is generated, (2) features detected and (3) the parametric CAD solid model reconstructed as biquartic surface splines. In order to generate an enclosed isosurface model effectively, we propose an algorithm based upon the marching cubes method to detect elements intersected by an isosurface. After generating an enclosed isosurface model, we produce biquartic surface splines. By applying our algorithm to an enclosed isosurface model, it is possible to produce smoothed biquartic surface splines with features retained. Some examples are shown and the effectiveness of our algorithm is discussed in this paper.     

An open platform of shape design optimization for shell structure

A general platform built on a computer-aided design (CAD) system is developed for parameterized shape design optimization of shell structure. Within the platform, parameterized surface modeling and computer-aided engineering (CAE) applications are embedded and seamlessly integrated with the CAD system through its application programming interface (API). Firstly, instead of the CAD system inherent surface modeling, a parameterized surface modeling for shell structure is fulfilled through integrating with parametric solid modeling of the CAD system. Thus, any dimensions for parametric solid modeling can be used to control shape modification of shell structure and serve as design variables for shape design optimization. Secondly, seamless integration of geometry modeling and finite-element modeling for shell structure is implemented. Finally, with integrated procedures of finite-element analysis and optimization algorithms, a general platform for parameterized shape optimization of shell structure is realized. Numerical examples are presented, and the results validate the effectiveness and efficiency of the platform. A shorten version of this paper was presented to the 7th World Congress of Computation Mechanics (WCCM 2006), July 16–22, 2006, Los Angeles, CA, USA.     

Integration of design and manufacturing for structural shape optimization

This paper presents an integrated design and manufacturing approach that supports shape optimization of structural components. The approach starts from a primitive concept stage, where boundary and loading conditions of the structural component are given to the designer. Topology optimization is conducted for an initial structural layout. The discretized structural layout is smoothed using parametric B-Spline surfaces. The B-Spline surfaces are imported into a CAD system to construct parametric solid models for shape optimization. Virtual manufacturing (VM) techniques are employed to ensure that the optimized shape can be manufactured at a reasonable cost. The solid freeform fabrication (SFF) system fabricates physical prototypes of the structure for design verification. Finally, a computer numerical control (CNC) machine is employed to fabricate functional parts as well as mold or die for mass production of the structural component. The main contribution of the paper is incorporating manufacturing into the design process, where manufacturing cost is considered for design. In addition, the overall design process starts from a primitive stage and ends with functional parts. A 3D tracked vehicle roadarm is employed throughout this paper to illustrate the overall design process and various techniques involved.     

形状可调的5次组合样条及其参数选择

目的 为了克服3次参数B样条在形状调整与局部性方面的不足,提出带参数的5次多项式组合样条。方法 首先构造一组带参数的5次多项式基函数;然后采用与3次B样条曲线相同的组合方式定义带参数的5次多项式组合样条曲线,并讨论基于能量优化法的5次组合样条曲线参数最佳取值问题;最后定义相应的组合样条曲面,并研究利用粒子群算法求解曲面的最佳参数取值。结果 5次组合样条不仅继承了3次B样条的诸多性质,而且还比3次B样条具有更强的局部性及形状可调性。由于5次组合样条仍为多项式模型,因此方程结构相对较为简单,符合实际工程的需要。利用能量优化法可获得光顺的5次组合样条曲线与曲面。结论 所提出5次多项式组合样条克服了3次参数B样条在形状调整与局部性方面的不足,是一种实用的自由曲线曲面造型方法。     

Isogeometric shape optimization of photonic crystals via Coons patches

In this paper, we present an approach that extends isogeometric shape optimization from optimization of rectangular-like NURBS patches to the optimization of topologically complex geometries. We have successfully applied this approach in designing photonic crystals where complex geometries have been optimized to maximize the band gaps.Salient features of this approach include the following: (1) multi-patch Coons representation of design geometry. The design geometry is represented as a collection of Coons patches where the four boundaries of each patch are represented as NURBS curves. The use of multiple patches is motivated by the need for representing topologically complex geometries. The Coons patches are used as a design representation so that designers do not need to specify interior control points and they provide a mechanism to compute analytical sensitivities for internal nodes in shape optimization, (2) exact boundary conversion to the analysis geometry with guaranteed mesh injectivity. The analysis geometry is a collection of NURBS patches that are converted from the multi-patch Coons representation with geometric exactness in patch boundaries. The internal NURBS control points are embedded in the parametric domain of the Coons patches with a built-in mesh rectifier to ensure the injectivity of the resulting B-spline geometry, i.e. every point in the physical domain is mapped to one point in the parametric domain, (3) analytical sensitivities. Sensitivities of objective functions and constraints with respect to design variables are derived through nodal sensitivities. The nodal sensitivities for the boundary control points are directly determined by the design parameters and those for internal nodes are obtained via the corresponding Coons patches.     

Integration of CAD and boundary element analysis through subdivision methods

Subdivision methods have been mainly used in computer graphics. This paper extends their applications to mechanical design and boundary element analysis (BEA), and fulfills the seamless integration of CAD and BEA in the model and representation.Traditionally, geometric design and BEA are treated as separate modules requiring different representations and models, which include continuous parametric models and discrete models. Due to the incompatibility of the involved representations and models, the post-processing in geometric design or the pre-processing in BEA is essential. The transition from geometric design to BEA requires substantial effort and errors are inevitably introduced during the transition. In this paper, a framework of realizing the integration of CAD and BEA was first presented based on subdivision methods. A common model or a unified representation for geometric design and BEA was created with subdivision surfaces. For general 3D structures, automatic mesh generation for geometric design and BEA was fulfilled through subdivision methods. The seamless integration improves the accuracy of numerical analysis and shortens the cycle of geometric design and BEA.     

A CAD-based design parameterization for shape optimization of elastic solids

In this paper a CAD-based design sensitivity analysis (DSA) and optimization method using Pro/ENGINEER for shape design of structural components is presented. The CAD-based design model is critically important for multidisciplinary shape design optimization. Only when each discipline can compute the design sensitivity coefficients of the CAD-based design model, can a true multidisciplinary what-if study, trade-off analysis, and design optimization be carried out. The proposed method will allow the design engineer to compute design sensitivity coefficients of structural performance measures such. as stress and displacement, evaluated using existing finite element analysis (FEA) tools, both h- and p-versions, with respect to design variables defined in the parameterized CAD model. The proposed method consists of (i) a CAD-based design parameterization technique that ties the structural DSA and optimization to a CAD tool; (ii) a design velocity field computation that defines material point movement due to design change in CAD geometry, satisfies linearity and regularity requirements, and supports both hand p-version FEA meshed using existing mesh generators; and (iii) a design optimization method that supports structural geometric and finite element model updates in Pro/ENGINEER during the optimization process.     

Industrial geometry: recent advances and applications in CAD

Industrial Geometry aims at unifying existing and developing new methods and algorithms for a variety of application areas with a strong geometric component. These include CAD, CAM, Geometric Modelling, Robotics, Computer Vision and Image Processing, Computer Graphics and Scientific Visualization. In this paper, Industrial Geometry is illustrated via the fruitful interplay of the areas indicated above in the context of novel solutions of CAD related, geometric optimization problems involving distance functions: approximation with general B-spline curves and surfaces or with subdivision surfaces, approximation with special surfaces for applications in architecture or manufacturing, approximate conversion from implicit to parametric (NURBS) representation, and registration problems for industrial inspection and 3D model generation from measurement data. Moreover, we describe a ‘feature sensitive’ metric on surfaces, whose definition relies on the concept of an image manifold, introduced into Computer Vision and Image Processing by Kimmel, Malladi and Sochen. This metric is sensitive to features such as smoothed edges, which are characterized by a significant deviation of the two principal curvatures. We illustrate its applications at hand of feature sensitive curve design on surfaces and local neighborhood definition and region growing as an aid in the segmentation process for reverse engineering of geometric objects.     

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。     

Surface shape control using constrained optimization on the B-spline representation

Shape control for surfaces is an important problem in CAD/CAM design as well as in other areas. Until recently, shape control has required the use of skilled engineering labor. This paper presents methods that will help to make this shape control process automatic. The particular problem considered is that of controlling the curvature of isoparametric lines for parametric tensor product spline surfaces. Conditions for curvature control are presented in terms of the B-spline coefficients. These conditions allow the problem to be stated as a constrained optimization problem and to be solved using modern optimization techniques.     

Cross-sectional design with curvature constraints

A practical example of B-spline curve control points manipulation for the geometric construction of a free form shape is presented. Elements of a cross-sectional design methodology are used in conjunction with a skinning type operator for the definition of a B-spline surface. Skinning process is well established in the CAD community, but further difficulties arise in producing smooth surfaces under constraints. This paper attempts to overcome the fairness problem by choosing an appropriate solution where the execution time has to be reasonably short. Main results include an industrial application in a preliminary aerodynamic design cycle where manufacturing tolerances defined by smoothness criteria are maintained.     

An approximation-concepts approach to shape optimal design

In the past, much of the work done in structural optimization consisted in resizing the members of fixed configuration models. In that case, a powerful design procedure has now emerged, which is based on the coordinate use of explicit high-quality approximations of the behavior constraints and dual methods of mathematical programming.There is, however, a large class of problems for which the main degrees of freedom for the designer correspond to the shape of the structure itself.The main objectives of this paper are to recall briefly a convenient geometric representation, in which the boundaries of the structure are represented by Bezier or B-spline curves, and then to discuss the choice of optimization algorithm. It is shown that cost-efficient methods for structural sizing may be advantageously extended to shape optimal design problems. Different approximation schemes are tested and a new general optimization algorithm is presented that combines mixed approximations and dual methods. Many large-scale applications are treated to demonstrate the generality and the efficiency of the new formulation. Finally, considerations are given about an integrated approach including CAD computer codes and finite element optimization software.     

SplineLearner: Generative learning system of design constraints for models represented using B-spline surfaces

Product design involves a computer-aided design (CAD) model with its design (dimensional) parameters. A generative design (GD) system can then be utilized to generate new designs by modifying these parameters. There is a need for a GD system to determine the visual validity of a design that is obtained after parametric modification. In this context, this paper introduces an approach to learn visual (i.e., design) constraints of a CAD model (represented using B-spline surfaces) by means of user feedbacks. A deformation technique (utilizing modification and limit curves) for B-spline surfaces is first introduced, which involves a few design (deformation) parameters. Via a generative learning process, the proposed system, SplineLearner, generates random designs, which are shown to user(s) for visual validity classifications. In a machine learning step, a mathematical model is computed that can perform prediction for a design to be valid or not. The mathematical model is also integrated into SplineLearner (after some user interactions) to prevent imbalances between the numbers of valid and invalid designs. As a proof of concept, B-spline surface models of a car body parts (hood, roof, side and trunk) are utilized, and two user studies are conducted to demonstrate the efficacy of the proposed method.     

Feature-based surface design and machining

A feature-based method for designing and representing functional surfaces such as automobile inner panels that lets a user assemble and present complicated, multifeatured surfaces using known, generally simpler component surfaces and information about feature shape is reviewed. It is shown that, using this method, CAD users can generate numerically controlled (NC) tool paths and use them to automatically machine 3-D surface geometries with various cutting tools such as ball-nose or spherical-end cutters and toroidal or flat-end cutters. The method was tested on models involving simple explicit primary and secondary surfaces as well as more complicated B-spline parametric surfaces. Results indicate that the tool-center-generation algorithm is accurate, robust, and computationally efficient     

Free-form geometric modeling by integrating parametric and implicit PDEs

Parametric PDE techniques, which use partial differential equations (PDEs) defined over a 2D or 3D parametric domain to model graphical objects and processes, can unify geometric attributes and functional constraints of the models. PDEs can also model implicit shapes defined by level sets of scalar intensity fields. In this paper, we present an approach that integrates parametric and implicit trivariate PDEs to define geometric solid models containing both geometric information and intensity distribution subject to flexible boundary conditions. The integrated formulation of second-order or fourth-order elliptic PDEs permits designers to manipulate PDE objects of complex geometry and/or arbitrary topology through direct sculpting and free-form modeling. We developed a PDE-based geometric modeling system for shape design and manipulation of PDE objects. The integration of implicit PDEs with parametric geometry offers more general and arbitrary shape blending and free-form modeling for objects with intensity attributes than pure geometric models     

Alternative representation for parametric cubic curves and surfaces

A representation for parametric cubic curves and surfaces is presented which incorporates the polygonal approach popularized by the Bézier and B-spline schemes. Since the curves more closely mimic the polygon than their counterparts employing the Bézier or B-spline schemes, the method is potentially useful for creating and manipulating geometric models. In connection with the logical extension of the approach from curves to surfaces, a discussion of the relationship between continuity and redundant data storage is included.     

Fitting unorganized point clouds with active implicit B-spline curves

In computer-aided geometric design and computer graphics, fitting point clouds with a smooth curve (known as curve reconstruction) is a widely investigated problem. In this paper, we propose an active model to solve the curve reconstruction problem, where the point clouds are approximated by an implicit B-spline curve, i.e., the zero set of a bivariate tensor-product B-spline function. We minimize the geometric distance between the point clouds and the implicit B-spline curve and an energy term (or smooth term) which helps to extrude the possible extra branches of the implicit curve. In each step of the iteration, the trust region algorithm in optimization theory is applied to solve the corresponding minimization problem. We also discuss the proper choice of the initial shape of the approximation curve. Examples are provided to illustrate the effectiveness and robustness of our algorithm. The examples show that the proposed algorithm is capable of handling point clouds with complicated topologies.     

Interactive Shape Control of Interpolating B-splines

This paper presents a new methodfor providing interactive shape control of interpolating B-splines. The CAD designer can directly interact with geometric entities defined on the B-spline at any interpolated data point; shape adjustments can be performed either globally or locally. Our approach is based on Bλ-splines of order k (λ,k ≥1), i.e. λ-reparametrized, classical B-splines. The method presented can be easily generalised to surfaces defined either as tensor products or by using the skinning technique; interactive shape control can be provided in both surface parametric directions.