Shape optimization of curves and surfaces considering fairness metrics and elastic stiffness

A method is presented for generating round curves and surfaces allowing discontinuities in tangent vectors and curvatures. The distance of the center of curvature from the specified point is used for formulating the objective function that is a continuous function of the design variables through convex and concave shapes. It is shown that a shell with and without ribs can be generated within the same problem formulation if the minimization problem is converted into a maximization problem and the parameter region where integration is to be carried out is restricted in view of the sign of the curvature. Optimal shapes are also found under constraints on the compliance against static loads. A multiobjective optimization problem is solved by the constraint method to generate a trade-off design between roundness and mechanical performance. The online version of the original article can be found at:     

基于专家系统法的异形件最小包络矩形求解

异形件布局问题广泛存在于社会各领域.目前,直接处理异形件布局问题的算法设计难度和计算量较大,而矩形件布局已有一些成熟算法.因此,将异形件转化为矩形件后处理布局能简化问题,并保证良好布局效果.本文基于专家系统法处理异形件,使其转换为易解决布局的包络矩形.     

基于混合灵敏度优化的鲁棒特征结构配置

将H_∞理论的混合灵敏度指标融入到特征结构配置中, 将混合灵敏度指标转换为特征结构设计参数的函数关系, 建立了特征结构配置的频率域指标约束. 这两种方法的结合各自利用了其在时域和频率域的设计优势. 通过优化算法所获得的鲁棒特征结构配置, 既保证了闭环特征值/特征向量, 又使反馈回路具有一定的频率域特性. 计算实例表明这种综合设计方法的有效性.     

空间曲线基于内在几何量的高质量采样和B样条拟合

为解决均匀参数采样在许多情况下得到质量不高的采样点,进而生成不理想的B样条拟合曲线,提出空间曲线基于内在几何量的均匀采样方法,以获得给定总数且具有代表性的采样点.首先定义基于弧长、曲率和挠率加权组合的特征函数,通过调整组合参数更好匹配不同的曲线形状;然后提出空间曲线基于内在几何量的自适应采样方法,迭代生成满足给定距离阈值的采样点.采用最大绝对误差和均方根误差作为评价指标,与均匀弧长采样方法和基于弧长和曲率平均的均匀采样方法进行对比,并通过实例进行验证.结果表明,文中方法在采样质量和B样条拟合结果上获得明显改善.     

A non-parametric free-form optimization method for shell structures

This paper presents a numerical shape optimization method for the optimum free-form design of shell structures. It is assumed that the shell is varied in the out-of-plane direction to the surface to determine the optimal free-form. A compliance minimization problem subject to a volume constraint is treated here as an example of free-form design problem of shell structures. This problem is formulated as a distributed-parameter, or non-parametric, shape optimization problem. The shape gradient function and the optimality conditions are theoretically derived using the material derivative formulae, the Lagrange multiplier method and the adjoint variable method. The negative shape gradient function is applied to the shell surface as a fictitious distributed traction force to vary the shell. Mathematically, this method is a gradient method with a Laplacian smoother in the Hilbert space. Therefore, this shape variation makes it possible both to reduce the objective functional and to maintain the mesh regularity simultaneously. With this method, the optimal smooth curvature distribution of a shell structure can be determined without shape parameterization. The calculated results show the effectiveness of the proposed method for the optimum free-form design of shell structures.     

Generation of Smooth Surfaces by Controlling Curvature Variation

To satisfy a designer's intention for constructing aesthetic shapes such as automotive bodies, we propose a surface generation method. In the surface design process, designers determine shapes according to their great concern for the reflected images of vehicle surroundings, shade lines and highlight lines. Since reflection and shading are affected by changes of surface normal, the curvature variation of the surface, which represents the change of the surface normal, should be smooth and distributed as designers want. The proposed method controls curvature distribution directly by determining a surface shape from an evolute, which is a locus of the curvature center of the generatrix and moves along directrices to form the surface. It first generates evolutes of boundary curves to be generatrices as rational Bezier curves, then interpolates their shapes with the Bezier polygons, and locates the interpolated shape to the corresponding position of the directrices. By applying this method, we have confirmed that a smooth shape is generated from four boundary curves.     

Shape and topology optimization of acoustic lens system using phase field method

A layout optimization method for a two-dimensional acoustic lens system used in underwater imaging is presented. To this end, a shape and topology optimization is formulated for the design problem of a lens system for the first time. The layout of a lens system to be optimized includes the number of lenses, shape of lens surfaces, distances between lenses, and lens materials. A phase field function is employed to implicitly parameterize the boundaries of the lenses, which move according to design sensitivities during optimization. Multiple lenses with different materials are optimized using a single phase field function. Because the ratio of the acoustic wavelength with respect to lens dimensions is large, diffraction effects should be taken into account. Accordingly, the performance of a lens system should be analyzed using wave acoustics and not the ray tracing method. The optimization problem is formulated to remove the aberrations of coma and field curvature. The validity of the proposed optimization method is demonstrated by solving benchmark design problems including a lens system with a large field of view.     

Polyhedral Surface Modeling with a Diffusion System

This paper presents a method of generating polyhedral surfaces by using a diffusion system that calculates the positional and normal vectors on their vertices. The system generates smooth shapes that satisfy the minimum norm property, and can be extended to imitate the shape controls of curvature continuous surfaces with bias and tension parameters. The shape of a surface is determined by the stable state of nonlinear and local calculations between vertices, and is easily controlled by adding constraints on arbitrary vertices. Such bottom-up calculation of surfaces enhances flexibility in the interactive design of complicated free-form shapes.     

On the optimization problem of fillets and holes in plates with curvature constraints

The focus of this study is two-dimensional optimization problem of fillets and holes in plates considering curvature constraints, defined with the goal of minimizing stress concentration factor. The optimality criteria of uniform energy density are extended to shape optimization with curvature constraint assuming that a good shape design can be obtained when constant energy density along the segment of the designed boundary is achieved, except for the section that has to satisfy the geometry constraint. Feasible solutions are sought out under the assumption that the minimum curvature radius is constant on the last part of the designed boundary, and some interesting features of optimal shape of fillets and holes with prescribed minimum curvature radii are revealed. A finite-element-based method in conjunction with a gradientless algorithm is developed to obtain the optimal shape with curvature constraint. Numerical examples of optimal fillets and holes in flat plates are presented to validate the proposed assumption.     

Fast and effective retrieval of medical tumor shapes

Investigates the problem of retrieving similar shapes from a large database; in particular, we focus on medical tumor shapes (finding tumors that are similar to a given pattern). We use a natural similarity function for shape matching, based on concepts from mathematical morphology, and we show how it can be lower-bounded by a set of shape features for safely pruning candidates, thus giving fast and correct output. These features can be organized in a spatial access method, leading to fast indexing for range queries and nearest-neighbor queries. In addition to the lower-bounding, our second contribution is the design of a fast algorithm for nearest-neighbor searching, achieving significant speedup while provably guaranteeing correctness. Our experiments demonstrate that roughly 90% of the candidates can be pruned using these techniques, resulting in up to 27 times better performance compared to sequential scanning     

Curvature and the fairness of curves and surfaces

The use of curvature plots for the design of curves that have to meet aesthetic requirements is discussed. The aim is to emphasize the usefulness of curvature as a measure for curve fairness. A local method to optimize the curvature plot of a given cubic spline curve is presented. It automatically determines where the curve is to be faired and can be applied repeatedly. A straightforward generalization to surfaces is easy to formulate     

Non-parametric free-form optimization method for frame structures

In this paper, we propose a parameter-free shape optimization method based on the variational method for designing the smooth optimal free-form of a spatial frame structure. A stiffness design problem where the compliance is minimized under a volume constraint is solved as an example of shape design problems of frame structures. The optimum design problem is formulated as a distributed-parameter shape optimization problem under the assumptions that each member is varied in the out-of-plane direction to the centroidal axis and that the cross section is prismatic. The shape gradient function and the optimality conditions are then theoretically derived. The optimal curvature distribution is determined by applying the derived shape gradient function to each member as a fictitious distributed force both to vary the member in the optimum direction and to minimize the objective functional without shape parametrization, while maintaining the members’ smoothness. The validity and practical utility of this method were verified through several design examples. It was confirmed that axial-force-carrying structures were obtained by this method.     

Robust mask-layout and process synthesis

A method for automated mask-layout and process synthesis for MEMS is presented. The synthesis problem is approached by use of a genetic algorithm. For a given desired device shape, and several fabrication process choices, this synthesis method will produce one or more mask-layouts and associated fabrication process sequences (which when used can generate shapes close to the desired one). Given complicated device shapes and wide range of fabrication process possibilities, the designer may encounter difficulty producing the right mask-layout and fabrication procedure by experience and trial and error. An automated synthesis tool like this will be helpful to the designer by increasing the efficiency and accuracy of the design of MEMS devices.     

Numerical investigation of the effect of spiral curvature on the flow field in a spiral channel viscous micropump

This paper investigates the effect of spiral curvature on the flow field within a newly introduced spiral channel viscous micropump. Increasing the spiral curvature increases the angle between the channel axis and the direction of upper plate motion, and more fluid is dragged normal to the axis of the channel, which changes the flow characteristics. A number of 3D models for the pump geometry with different design parameters are built and analyzed using the finite volume method. Numerical visualization of the flow field contours through the spiral channel is presented. It has been found numerically that the flow rate varies linearly with both the pressure difference and boundary velocity for a wide range of design parameters, which supports the validity of the linear lubrication model for this problem for the full range of spiral curvatures studied. Further, it is found that the error resulting from ignoring the spiral curvature exceeds 5% for spiral curvature ratio above 2. Obtained results depict a complete representation of the effect of spiral channel width, height, length and curvature on the flow field through a spiral-grooved micropump.     

A univariate method for plane elastic curves

The problem to interpolate Hermite-type data (i.e., two points with attached tangent vectors) with elastic curves of prescribed tension is known to have multiple solutions. A method is presented that finds all solutions of length not exceeding one period of its curvature function. The algorithm is based on algebraic relations between discrete curvature information which allow to transform the problem into a univariate one. The method operates with curves that by construction interpolate a subset of the given data. Hereby the objective function of the problem is drastically simplified. A bound on the maximum curvature value is established that provides an interval containing all solutions.     

基于曲率模态振型的刚架结构损伤检测

结构损伤前后动力特性的变化来快速、直接、方便地判定损伤的存在、程度及位置．本文采用曲率模态对刚架结构的损伤检测进行了研究．首先用有限元法计算出结构的位移模态振型,然后用差分法计算出曲率模态振型．数值模拟结果表明：曲率模态振型对结构的损伤敏感,可同时确定结构损伤的存在、程度和位置,并且可以用于结构多位置损伤的检测．实验结果证实了数值模拟结论．     

Direct search of feasible region and application to a crashworthy helicopter seat

The paper proposes a novel approach to identify the feasible region for a constrained optimisation problem. In engineering applications the search for the feasible region turns out to be extremely useful in the understanding of the problem as the feasible region defines the portion of the domain where design parameters can be ranged to fulfil the constraints imposed on performances, manufacturing and regulations. The search for the feasible region is not a trivial task as non-convex, irregular and disjointed shapes can be found. The algorithm presented in this paper moves from the above considerations and proposes a recursive feasible-infeasible segment bisection algorithm combined with Support Vector Machine (SVM) techniques to reduce the overall computational effort. The method is discussed and then illustrated by means of three simple analytical test cases in the first part of the paper. A real-world application is finally presented: the search for the survivability zone of a crashworthy helicopter seat under different crash conditions. A finite element model, including an anthropomorphic dummy, is adopted to simulate impacts that are characterised by different deceleration pulses and the proposed algorithm is used to investigate the influence of pulse shape on impact survivability.     

Algorithms for optimal partial matching of free-form objects with scaling effects

A free-form object matching problem is addressed in this paper. Two methods are proposed to solve a partial matching problem with scaling effects and no prior information on correspondence or the rigid body transformation involved. The first method uses umbilical points, which behave as fingerprints of a surface and their qualitative properties can be used for matching purposes. The second method uses an optimization scheme based on the extension of the KH curvature matching method [Comput. Aided Design 35 (2003) 913], first introduced in the context of a matching problem without scaling effects. Two types of curvatures, the Gaussian and the mean curvatures, are used to establish correspondences between two objects. The curvature matching method is formulated in terms of minimization of an objective function depending on the unknown scaling factor, and the rigid body transformation parameters. The accuracy and complexity of the proposed methods as well as the convergence for the optimization approach are analyzed. Examples illustrate the two methods.     

Geometric Heat Equation and Nonlinear Diffusion of Shapes and Images

Visual tasks often require a hierarchical representation of shapes and images in scales ranging from coarse to fine. A variety of linear and nonlinear smoothing techniques, such as Gaussian smoothing, anisotropic diffusion, regularization, etc., have been proposed, leading to scalespace representations. We propose ageometricsmoothing method based on local curvature for shapes and images. The deformation by curvature, or the geometric heat equation, is a special case of thereaction–diffusionframework proposed in [41]. For shapes, the approach is analogous to the classical heat equation smoothing, but with a renormalization by arc-length at each infinitesimal step. For images, the smoothing is similar to anisotropic diffusion in that, since the component of diffusion in the direction of the brightness gradient is nil, edge location is left intact. Curvature deformation smoothing for shape has a number of desirable properties: it preserves inclusion order, annihilates extrema and inflection points without creating new ones, decreases total curvature, satisfies the semigroup property allowing for local iterative computations, etc. Curvature deformation smoothing of an image is based on viewing it as a collection of iso-intensity level sets, each of which is smoothed by curvature. The reassembly of these smoothed level sets into a smoothed image follows a number of mathematical properties; it is shown that the extension from smoothing shapes to smoothing images is mathematically sound due to a number of recent results [21]. A generalization of these results [14] justifies the extension of the entireentropy scale spacefor shapes [42] to one for images, where each iso-intensity level curve is deformed by a combination of constant and curvature deformation. The scheme has been implemented and is illustrated for several medical, aerial, and range images.     

Detection and reconstruction of freeform sweeps

We study the difficult problem of deciding if parts of a freeform surface can be generated, or approximately generated, by the motion of a planar profile through space. While this task is basic for understanding the geometry of shapes as well as highly relevant for manufacturing and building construction, previous approaches were confined to special cases like kinematic surfaces or “moulding” surfaces. The general case remained unsolved so far. We approach this problem by a combination of local and global methods: curve analysis with regard to “movability”, curve comparison by common substring search in curvature plots, an exhaustive search through all planar cuts enhanced by quick rejection procedures, the ordering of candidate profiles and finally, global optimization. The main applications of our method are digital reconstruction of CAD models exhibiting sweep patches, and aiding in manufacturing freeform surfaces by pointing out those parts which can be approximated by sweeps.