Velocity fields using NURBS with distortion control for structural shape optimization

The paper presents methods for the calculation of design velocity fields and mesh updating in the context of shape optimization. Velocity fields have a fundamental role in the integration of the main conceptual and software components in shape design optimization. Nonuniform rational B-splines are used to parameterize the domain boundary. A Newton/Raphson procedure is used to calculate the curve and surface internal parameters. A preconditioning iterative conjugate gradient method with low precision is used to improve the solution performance of the auxiliar problem in the calculation of the velocity fields. The velocity fields are also used to perturb the finite element mesh and element distortion measures are introduced. Finally, examples of two- and three-dimensional elastic problems are presented to illustrate the application of the algorithms.     

基于NURBS方法的气动外形优化设计

采用NURBS曲线曲面,对钝锥弹头和钝双锥弹体建立参数化曲面模型,取NURBS曲线控制点作为设计参数,应用高超声速面元法求解气动力特性,在给定设计约束下,采用遗传算法进行气动外形优化设计,并对优化结果进行了比较分析。结果表明,采用NURBS方法构造参数化外形,并结合优化技术可方便快速地获得所需最优外形;与应用二次曲线构造参数化外形相比,该方法对弹体形状控制更加灵活,并可局部修改弹头曲线形状。因此,基于NURBS方法发展整套的系统优化设计算法很有现实意义和应用价值。     

Full analytical sensitivities in NURBS based isogeometric shape optimization

Non-uniform rational B-spline (NURBS) has been widely used as an effective shape parameterization technique for structural optimization due to its compact and powerful shape representation capability and its popularity among CAD systems. The advent of NURBS based isogeometric analysis has made it even more advantageous to use NURBS in shape optimization since it can potentially avoid the inaccuracy and labor-tediousness in geometric model conversion from the design model to the analysis model.Although both positions and weights of NURBS control points affect the shape, until very recently, usually only control point positions are used as design variables in shape optimization, thus restricting the design space and limiting the shape representation flexibility.This paper presents an approach for analytically computing the full sensitivities of both the positions and weights of NURBS control points in structural shape optimization. Such analytical formulation allows accurate calculation of sensitivity and has been successfully used in gradient-based shape optimization.The analytical sensitivity for both positions and weights of NURBS control points is especially beneficial for recovering optimal shapes that are conical e.g. ellipses and circles in 2D, cylinders, ellipsoids and spheres in 3D that are otherwise not possible without the weights as design variables.     

Non-numerical modeling techniques in structural optimization

Structural optimization encompasses much more than just solving numerical optimization problems. As computer capabilities increase, the entire process of modeling structural optimization problems must be considered. In particular, the creation, transformation and evaluation of the underlying concepts, rather than just brute numerical power, are becoming a more and more dominant factor in finding safe-guarded solutions. In this paper a layer-based model of the total structural optimization process is presented. Each layer contains individual components, a major number of which are of non-numerical nature.Computerization of the non-numerical components requires new programming paradigms. The selection of an appropriate optimization method, to be discussed as a typical non-numerical problem, is difficult because a wide variety of distinct methods exist. Therefore, automated assistance based upon experience and knowledge gained through current research is of prime interest.     

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.     

Semi-Lagrange method for level-set-based structural topology and shape optimization

In this paper, we introduce a semi-Lagrange scheme to solve the level-set equation in structural topology optimization. The level-set formulation of the problem expresses the optimization process as a solution to a Hamilton–Jacobi partial differential equation. It allows for the use of shape sensitivity to derive a speed function for a descent solution. However, numerical stability condition in the explicit upwind scheme for discrete level-set equation severely restricts the time step, requiring a large number of time steps for a numerical solution. To improve the numerical efficiency, we propose to employ a semi-Lagrange scheme to solve level-set equation. Therefore, a much larger time step can be obtained and a much smaller number of time steps are required. Numerical experiments comparing the semi-Lagrange method with the classical explicit upwind scheme are presented for the problem of mean compliance optimization in two dimensions.     

Weight/shape structural optimization exploiting rigid movement

The paper explores the possibility of performing limited shape/size optimization to obtain a minimum weight design using rigid body movement. A two step approach is adopted where changes in the shape parameters are made separately from changes in the size parameters. The sizing step uses a conventional optimization, STARS, which is available commercially. For simplicity the method is demonstrated using planar arches subject to a variety of load and boundary conditions.     

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     

Performance comparison of SAM and SQP methods for structural shape optimization

This paper presents a numerical performance comparison of a modern version of the well-established sequential quadratic programming (SQP) method and the more recent spherical approximation method (SAM). The comparison is based on the application of these algorithms to examples with nonlinear objective and constraint functions, among others: weight minimization problems in structural shape optimization. The comparison shows that both the SQP and SAM-algorithms are able to converge to accurate minimum weight values. However, because of the lack of a guaranteed convergence property of the SAM method, it exhibits an inability to consistently converge to a fine tolerance. This deficiency is manifested by the appearance of small oscillations in the neighbourhood of the solution.     

Performance comparison of SAM and SQP methods for structural shape optimization

This paper presents a numerical performance comparison of a modern version of the well-established sequential quadratic programming (SQP) method and the more recent spherical approximation method (SAM). The comparison is based on the application of these algorithms to examples with nonlinear objective and constraint functions, among others: weight minimization problems in structural shape optimization. The comparison shows that both the SQP and SAM-algorithms are able to converge to accurate minimum weight values. However, because of the lack of a guaranteed convergence property of the SAM method, it exhibits an inability to consistently converge to a fine tolerance. This deficiency is manifested by the appearance of small oscillations in the neighbourhood of the solution.     

电磁散射目标的NURBS建模

采用非均匀有理B样条(NURBS)对复杂目标进行建模,利用Cox-DeBoor算法将NURBS曲线曲面转化为更适合数值计算的有理Bézier曲线曲面,采用后向面判别法处理不同面元间的遮挡问题,并给出导弹雷达散射截面的计算结果。     

A simple technique for NURBS shape modification

Non-uniform rational B-splines (NURBS) have become a de-facto industry standard primarily because they offer a unified mathematical form for representing both freeform shapes and analytical curves or surfaces. Despite these advantages, designers need software that lets them work with NURBS in a natural way. The paper presents a unified approach to NURBS shape modification that builds on a perspective functional transformation of arbitrary origin and provides a homogeneous interface based on simple, easily understood geometric concepts     

NURBS曲面形状修改的一种改进算法

提出一种修改NURBS曲面形状的新算法。该算法运用约束优化方法,通过改变控制顶点和权因子修改NURBS曲面的形状。实验证明,该算法可以获得令人满意的效果。     

Improved image interpreting and modeling technique for automated structural optimization system

The automated structural optimization system (ASOS) proposed in the previous work incorporates the image-preprocessing techniques, the image-interpreting technique, and the automated shape-optimization modeling techniques to successfully obtain an autonomously integrated topology and shape optimization. However, the characteristic value-based image-interpreting technique used in ASOS is unable to accurately interpret complicated hole shapes, necessitating the use of the hole shape-adjusting strategy in addition to the hole-expanding strategy and the interference analysis in the automated shape-optimization modeling techniques to obtain a viable initial design and side constraints of design variables. In order to solve the above-mentioned problem in ASOS, this paper proposes the improved automated structural optimization system (IASOS) and uses the polygonal image-interpreting technique to replace the characteristic value-based image-interpreting technique used in ASOS. This alteration significantly increases the accuracy of image interpretation. Moreover, it can simplify the process of automated shape-optimization modeling techniques, reduce the design duration, and produce better results.     

最小描述长度优化下的医学图像统计形状建模

统计形状模型(SSM)是有效的图像处理与分析方法。为了建立模型,需要从形状样本集中提取出具有对应关系的轮廓采样点集合,这是决定模型性能的关键。传统的手动标定这些点集来确保对应关系枯燥耗时且带有主观性,更难以向高维拓展。对形状建立逐层的多尺度参数表示,基于最小描述长度(MDL),在粗尺度上建立反映点对应程度的目标函数并最小化,提出首先确保粗尺度上具有最优意义的点对应,同时在精尺度上使用最便捷的弧长参数函数来确定特征点,完成感兴趣目标的快速统计形状建模,进而统计分析以验证模型性能,为后续图像分割或定量分析打下基础。实验对肌肉骨骼核磁共振成像(MRI)中椎骨、椎间盘以及半月板等具有临床意义的结构建立了统计形状模型,验证了本文方法与手动取点相比具有客观可重复性且更加简洁,与单一尺度下的MDL方法相比时间效率更高。基于此模型的图像分割与基于手动建模的分割相比,误差相当或有所降低。     

Foot shape modeling

This study is an attempt to show how a "standard" foot can be parameterized using foot length, foot width, foot height, and a measure of foot curvature so that foot shape can be predicted using these simple anthropometric measures. The prediction model was generated using 40 Hong Kong Chinese men, and the model was validated using a different group of 25 Hong Kong Chinese men. The results show that each individual foot shape may be predicted to a mean accuracy of 2.1 mm for the left foot and 2.4 mm for the right foot. Application of this research includes the potential design and development of custom footwear without the necessity of expensive 3-D scanning of feet.     

A novel method of structural shape optimization coupling BEM with an optimization method based on biological growth

Local stress peaks are the main reason for fatigue failure of structures, and therefore many researchers make every effort to search for effective methods of structural shape optimization. Mattheck and Burkhard (1990) have developed a method by simulating the self-optimization mechanism of biological growth, using FEM. BEM is a very effective method in which only the boundary is divided into elements, and particularly, accurate results at the boundary can be obtained. It is obvious that BEM is a more suitable analysis method for structural shape optimization than FEM. Coupling BEM of three-dimensional thermoelasticity problems with an optimization method based on biological growth, a novel method of structural shape optimization is successfully developed by the authors. Several examples are used to demonstrate that the method is successful and efficient.