椭球体易拉罐形状和尺寸的优化设计模型

从材料最省、外观最美、放置稳定的角度设计了"椭球体"易拉罐的形状,然后通过建立非线性规划模型Ⅰ,求出了"椭球体"易拉罐的中腰半径、底面半径和高度.然后从手感舒适的角度建立了优化模型Ⅱ,从而确定了两种型号的"椭球体"易拉罐的最优尺寸.     

易拉罐形状和尺寸的最优设计模型

用游标卡尺测量了当前流行的几种易拉罐饮料尺寸参数,然后建立微分方程模型和规划模型,借助MATLAB 6.5,LINGO8.0编程求解出了易拉罐为正圆柱体、圆台和圆柱体的组合体时的最优设计.最后综合经济、美观、实用等因素,运用非线性规划和层次分析法得出设想中易拉罐的最佳设计,对2006"高教社杯"全国数学建模竞赛C题中的各问题作出了完整的解答.     

易拉罐形状和尺寸的最优设计

从用料最省的角度研究了易拉罐的形状和尺寸的优化设计问题,首先通过多次测量取平均值的方法得到了题目所需的数据．然后就问题二和问题三分别建立了优化模型,并借助数学软件进行了求解,得到了最优设计的尺寸．最后设计出了椭球形状的易拉罐作为自己的最优设计．     

易拉罐的优化设计

本文讲述了"易拉罐形状和尺寸的最优设计"问题的命题、建模和求解,评述了学生递交的论文中的优缺点,提出了若干建议.     

易拉罐的设计方案

在日常生活中,我们总会买些易拉罐装的饮料或食品.殊不知,这易拉罐的设计便包含了一定的数学道理. 对易拉罐的设计,经营者总是考虑让成本最低.如:设计一个体积固定为V的圆柱形易拉罐,什么样的设计方案最优? 要比较易拉罐优劣的标准,有下面两种不同的标准进行考虑. 第一种标准:根据制造过程中消耗铁皮的多少来判别优劣,即最优易拉罐应该具有的最小表面S. 分析设易拉罐的高为h,底面半径为r.由圆柱的体积公式V=πr2h,得h=V/πr2.又易拉罐的表面积S=2πr2 2πrh ①     

考虑不确定因素影响的沉淀池优化设计

沉淀池是污水处理流程中一个必不可少的重要环节,设计中存在着大量的不确定因素.本文在沉淀池设计中引入了不确定性模糊非线性规划模型.该模型利用处理效率与沉淀池尺寸之间的非线性关系,求解过程引入区间数和模糊算子,尝试在沉淀池设计中考虑不确定性因素,并取得满意的结果.     

铁电颗粒中的尺寸效应

利用Landau唯象理论研究了球形铁电颗粒的表面效应和尺寸效应.计算了不同尺寸下二级相变和一级相变铁电体的Curie温度和自发极化强度.给出了铁电颗粒的外推长度与尺寸的关系.理论计算结果与实验符合.     

基于修正偶应力理论的Timoshenko微梁模型和尺寸效应研究

基于修正偶应力理论,将Timoshenko微梁的应力、偶应力、应变、曲率等基本变量,描述为位移分量偏导数的表达式.根据最小势能原理,推导了决定Timoshenko微梁位移场的位移场控微分方程.利用级数法求解了任意载荷作用下Timoshenko简支微梁的位移场控微分方程,得到了反映尺寸效应的挠度、转角及应力的偶应力理论解.通过对承受余弦分布载荷Timoshenko简支微梁的数值计算,研究了Timoshenko微梁的挠度、转角和应力的尺寸效应,分析了Poisson比对Timoshenko微梁力学行为及其尺寸效应的影响.结果表明:当截面高度与材料特征长度的比值小于5时,Timoshenko微梁的刚度和强度均随着截面高度的减小而显著提高,表现出明显的尺寸效应;当截面高度与材料特征长度的比值大于10时,Timoshenko微梁的刚度与强度均趋于稳定,尺寸效应可以忽略;材料Poisson比是影响Timoshenko微梁力学行为及尺寸效应的重要因素,Poisson比越大Timoshenko微梁刚度和强度的尺寸效应越显著.该文建立的Timoshenko微梁模型,能有效描述Timoshenko微梁的力学行为及尺寸效应,可为微电子机械系统（MEMS）中的微结构设计与分析提供理论基础和技术参考.     

有尺寸的同型机分批排序问题的近似算法

对工件有不同到达时间、不同加工时间和尺寸的同型机分批排序问题寻找近似算法.对于大工件(工件的体积严格大于机器容量的÷)的加工时间不小于小工件(工件的体积小于或等于机器容量的÷)的加工时间的特定情形,利用动态规划的方法和拆分的技巧,我们设计了近似算法并分析了其最差性能比.     

基于传递矩阵法的周期性圆柱壳体振动特性分析

基于声子晶体理论和Love壳体理论,建立了圆柱壳体的轴对称波动微分方程,数值分析了周期性圆柱壳的能带特性.利用传递矩阵法建立了相邻胞元间的传递矩阵,推导了周期性圆柱壳各胞元环径向轴对称波动动态刚度矩阵.结合数值算例分析了弹性模量变化和几何尺寸变化对圆柱壳体波的传播特性的影响,数值结果表明:振动波在传播过程中存在禁带域和通带域,长度比的变化对周期性圆柱壳体禁带的幅值、宽度和个数影响显著,因此可以通过调整结构尺寸参数改变结构中波的传播特性,该文的研究可以为结构的抗震设计、减振控制提供一种新思路.     

The shape optimization of the arterial graft design by level set methods

The goal of the arterial graft design problem is to find an optimal graft built on an occluded artery, which can be mathematically modeled by a fluid based shape optimization problem. The smoothness of the graft is one of the important aspects in the arterial graft design problem since it affects the flow of the blood significantly. As an attractive design tool for this problem, level set methods are quite efficient for obtaining better shape of the graft. In this paper, a cubic spline level set method and a radial basis function level set method are designed to solve the arterial graft design problem. In both approaches, the shape of the arterial graft is implicitly tracked by the zero-level contour of a level set function and a high level of smoothness of the graft is achieved. Numerical results show the efficiency of the algorithms in the arterial graft design.     

Using artificial reaction force to design compliant mechanism with multiple equality displacement constraints

Monolithic compliant mechanisms are elastic workpieces which transmit force and displacement from an input position to an output position. Continuum topology optimization is suitable to generate the optimized topology, shape and size of such compliant mechanisms. The optimization strategy for a single input single output compliant mechanism under volume constraint is known to be best implemented using an optimality criteria or similar mathematical programming method. In this standard form, the method appears unsuitable for the design of compliant mechanisms which are subject to multiple outputs and multiple constraints. Therefore an optimization model that is subject to multiple design constraints is required. With regard to the design problem of compliant mechanisms subject to multiple equality displacement constraints and an area constraint, we here present a unified sensitivity analysis procedure based on artificial reaction forces, in which the key idea is built upon the Lagrange multiplier method. Because the resultant sensitivity expression obtained by this procedure already compromises the effects of all the equality displacement constraints, a simple optimization method, such as the optimality criteria method, can then be used to implement an area constraint. Mesh adaptation and anisotropic filtering method are used to obtain clearly defined monolithic compliant mechanisms without obvious hinges. Numerical examples in 2D and 3D based on linear small deformation analysis are presented to illustrate the success of the method.     

Optimality criteria coupled adaptive mesh method for optimal shape design of Stokes flow

An adaptive mesh method combined with the optimality criteria algorithm is applied to optimal shape design problems of fluid dynamics. The shape sensitivity analysis of the cost functional is derived. The optimization problem is solved by a simple but robust optimality criteria algorithm, and an automatic local adaptive mesh refinement method is proposed. The mesh adaptation, with an indicator based on the material distribution information, is itself shown as a shape or topology optimization problem. Taking advantages of this algorithm, the optimal shape design problem concerning fluid flow can be solved with higher resolution of the interface and a minimum of additional expense. Details on the optimization procedure are provided. Numerical results for two benchmark topology optimization problems are provided and compared with those obtained by other methods. Copyright © 2016 John Wiley & Sons, Ltd.     

Direct method for the design of optimal three-dimensional aerodynamic shapes

A direct optimization method for a broad class of three-dimensional aerodynamic shapes based on the approximation of the desired geometry by Bernstein-Bézier surfaces is developed. The high efficiency of the method is demonstrated by applying it to the design of an optimal supersonic section of an axisymmetric maximum-thrust de Laval nozzle. The method is also tested as applied to the design of a three-dimensional supersonic nozzle section in a dense multi-nozzle setup. In addition to three-dimensional supersonic nozzle sections with a circular throat, nozzles with a varying throat shape are considered. The results suggest that the method can be applied to various problems of 3D shape optimization.     

织物疵点图像的特征参数提取

疵点图像的特征提取是识别织物疵点的重要依据,它直接影响疵点识别的效率和准确率.基于特征参数应具有灵敏度高、独立性强、运算快的特点,提取出疵点的区域面积、边界周长、中心点坐标、长短轴长度、形状参数F和离心率E作为疵点的特征参数.实验证明它能够识别出大部分常见疵点,并为疵点定位标识提供充分依据.     

Shape functional optimization with restrictions boosted with machine learning techniques

Shape optimization is a widely used technique in the design phase of a product. Current ongoing improvement policies require a product to fulfill a series of conditions from the perspective of mechanical resistance, fatigue, natural frequency, impact resistance, etc. All these conditions are translated into equality or inequality restrictions which must be satisfied during the optimization process that is necessary in order to determine the optimal shape. This article describes a new method for shape optimization that considers any regular shape as a possible shape, thereby improving on traditional methods limited to straight profiles or profiles established a priori. Our focus is based on using functional techniques and this approach is, based on representing the shape of the object by means of functions belonging to a finite-dimension functional space. In order to resolve this problem, the article proposes an optimization method that uses machine learning techniques for functional data in order to represent the perimeter of the set of feasible functions and to speed up the process of evaluating the restrictions in each iteration of the algorithm. The results demonstrate that the functional approach produces better results in the shape optimization process and that speeding up the algorithm using machine learning techniques ensures that this approach does not negatively affect design process response times.     

Advances in concrete arch dams shape optimization

This paper presents an efficient methodology to find the optimum shape of arch dams. In order to create the geometry of arch dams a new algorithm based on Hermit Splines is proposed. A finite element based shape sensitivity analysis for design-dependent loadings involving body force, hydrostatic pressure and earthquake loadings is implemented. The sensitivity analysis is performed using the concept of mesh design velocity. In order to consider the practical requirements in the optimization model such as construction stages, many geometrical and behavioral constrains are included in the model in comparison with previous researches. The optimization problem is solved via the sequential quadratic programming (SQP) method. The proposed methods are applied successfully to an Iranian arch dam, and good results are achieved. By using such methodology, efficient software for shape optimization of concrete arch dams for practical and reliable design now is available.     

A staggered approach to structural shape and topology optimization

This contribution is concerned with the design and evaluation of a staggered approach to computational shape and topology optimization. The main idea is to interrelate a classical boundary variation scheme based on the results from shape sensitivity analysis and an evolutionary-type element removal procedure that relies on the topological sensitivity as a rejection criterion. The key ingredient in this perspective is to consider an advancing front algorithm that is gradually removing, from the evolving boundary of the domain, a number of finite elements based on the evaluation of the topological sensitivity. This process is repeated until the minimum topological sensitivity is no longer encountered at the boundary but in the interior of the domain, such that a topological change is to be considered for the current design layout. However, we first aim to establish a more accurate approximation of the true optimal shape of the newly established (zig-zag representation) of the design boundary. This is achieved by the evaluation of the shape sensitivity and the accompanying boundary variations obtained by a basic descent algorithm. Only then, we create a hole in the interior of the domain by removing all cells that are adjacent to the nodal point that exhibits the minimum topological sensitivity and resume the advancing front algorithm to alter the newly established design boundary. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical instabilities in structural optimization – analogy between topology & shape design problems

The formulation of structural optimization problems on the basis of the finite–element–method often leads to numerical instabilities resulting in non–optimal designs, which turn out to be difficult to realize from the engineering point of view. In the case of topology optimization problems the formation of designs characterized by oscillating density distributions such as the well–known “checkerboard–patterns” can be observed, whereas the solution of shape optimization problems often results in unfavourable designs with non–smooth boundary shapes caused by high–frequency oscillations of the boundary shape functions. Furthermore a strong dependence of the obtained designs on the finite–element–mesh can be observed in both cases. In this context we have already shown, that the topology design problem can be regularized by penalizing spatial oscillations of the density function by means of a penalty–approach based on the density gradient. In the present paper we apply the idea of problem regularization by penalizing oscillations of the design variable to overcome the numerical difficulties related to the shape design problem, where an analogous approach restricting the boundary surface can be introduced. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Reconstruction of curves with minimal energy using a blending interpolator

A weighted blending interpolator, a kind of smooth rational spline based only on function values, is constructed using a rational cubic spline and a polynomial spline. In order to meet the needs of practical design, a new control method is employed to control the shape of curves. The advantage of the method is that it can be applied to modify the local shape of an interpolating curve by selecting suitable parameters and weight coefficients simply. Also, when the weight coefficient is in [0,1], the error estimation formula of this interpolator is obtained. Copyright © 2012 John Wiley & Sons, Ltd.