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南昌航空大学科技学院学士学位论文 伪形的机械结构优化构形理论 学生姓名：郭亮 班级：0781052指导老师：封立耀Jean Luc Marcelin 2007年1月10日收到 /接受:2007 年5月1日/在线发表：2007年5月25日。 2007年斯普林格出 版社伦敦有限公司 摘要 这项工作提供了伪构形理论的一些应用程序，机械结构的形状优化技术。在本文构形理论的发展中， 优化的主要目标是最终总势能的最小化 。其他目标优化使用的机械结构优化通常被用来限制或优化约束。在这里介绍二种应用：第一个是使用遗传算法与伪构形技术对一水滴形状优化和第二个是对一个液压锤后轴承的形状优化处理。 关键词 形状优化 结构 遗传算法 1引言 本文介绍一种伪构形方法来达到物体形状优化基于总势能的最小化。我们将介绍减少结构总势能寻找最优形状，这可能在某些情况下是个好主意。该参考的构形理论可以以某种方式合理的理由解释如下。 据Bejan 1，在工程设计和自然性能中，形状和结构一直在演变为更好的性能；在工程设计中用到的目标和制约因素是从该几何相同的机制在自然流动系统中出现。Bejan 1开始设计和工程系统优化，并从自然系统中发现了几何形式的确定原则。这种发现是新的构形理论的根据。优化配置注定是不完善的。该系统的不完善到处蔓延时，效果最差，使越来越多的内部点和硬件工作部件被压。看似普遍的几何形式团结工程与自然的流动系统。Bejan 1采用了一种新的理论，他毫不掩饰地说明，与热力学第二定律是相同的理论，因为一个简单的理论意图 预测地球上任何活着的几何形式。许多构形理论的应用程序在机械流体中开发，特别是在优化流动2-10中。另一方面，据我们所知，在固体或结构力学中有一些应用实例。因此，我们至少有一半的参考文献 在流体动力学论文引用（同作者），因为构形方法是先由同一作者发展，只有阿德里安Bejan在论文中提到流体动力学。构形理论基于所有自然的创作，整体最佳的理论相比，该控制的演变与自然系统适应。构形分配原则由不完善的地方以及尽可能把最小的规模扩到最大组成。总的宏观构形理论工程结构从基本结构组装开始，通过与自然的规则相一致最佳分布不完善。我们的目标是研究降低成本。但是，从这里长期伪构形来看，机械结构优化的全局宏观解决方案已经成本降低，目标非常接近构形理论。那个构形理论是预测的理论，只有一 单一的原则，从优化所有上升。这篇文章的题目同样适用于伪构形的步骤。伪构形理论单一的优化原则是最小化总势能。此外，在我们以下提出的例子中伪构形原则和遗传算法有相关性，其结果是我们的优化将非常接近自然理论。本文的目是展示伪构形步骤用来适用力学结构，特别是对形状优化机械结构。其基本思想是非常简单：处于平衡状态的机械结构对应最小总势能。以同样的方式，最佳的机械结构也必须符合最低限度总势能。这目标必须首先对其他的东西进行干预。正是这种构想，在这篇文章中发展。两个例子将会在后面提到。 最小化总势能以优化一机械结构不是全新的的想法。已经有很多文件解决了这一问题，使这一方法系统化。最优化的唯一目标是使能量最小化。高斯林11用一个简单的方法提出了硬件案件形式的有线网络团体和膜发现结构。该方法是根据基本的能源概念。剪应力变表达式是用来定义总势能。最后的能源形式是尽量减少使用鲍威尔算法。在菅野和大崎12中，最低的互补能源原则是建立网络作为变量应力组件几何非线性弹性。为了显示总势能和能量互补之间的强对偶问题，这些问题的凸配方被进行调查，可嵌入到原始的二阶规划问题中。 Taroco 13进行分析形成一个弹性固体的敏感性平衡问题。第一阶形中，域、边界积分和总势能的第二阶形表达衍生物已经建立。在华纳14中，最优设计问题是根据其自身的重量解决了一个弹性悬挂杆。他已经发现截面在一个均衡状态中总势能的面积最小化分布。在相类似的设计问题中，在相同的约束条件潜在最大的能量也已解决了。在文图拉15中，边界条件控制的问题已经用网格方法解决。在文图拉15中，在总势能功能的弹性固体问题中介绍移动最小化近似值了，广义的拉格朗日术语被添加到满足本质边界的条件中。总的潜在能量最小化原理除了在一般有限元基础上制定，还找到一个未知的最佳目标结点因素16。 2所使用的方法 在本文的伪构形理论中，最优化的主要目标是尽量减少总势能。在优化机械结构里其他的物体通用被限制或优化约束。例如，一个东西可能在重量上有限制，或不超过其应力值。 在本文中这种想法是很简单的。机械结构通过两种参数类型来描述：已知的离散变量（例如，用有限元方法的自由度的位移）以及几何变量设计（例如参数，使人们有可能描述机械结构形状）。总潜力能在同一时间里通过一个确定隐含或明确的方式离散设计变量。因此，进行双重优化机械结构相比离散化设计变量，其目的是减少整体的总势能。显然，机械结构的优化问题用下列方法处理： - 目标：减少总势能 - 变量的优化：同时确定离散变量（在结构力学的有限元法的传统用例），描述设计变量的形状结构 - 优化限制： - 重量或体积 - 位移或限制 - 压力 - 频率 机械结构的优化问题将用以下方法解决，如果需要的话需要在这些阶段中重申（按照问题的本质）： 第1阶段 最小化机械结构总势能和唯一的离散结构变量相比较（度的有 限元）。它的作用在这里作为一个优化不优化的限制。在此阶段唯一的限制是 纯粹的机械原点，并涉及边界条件，适用于结构外部的作用。 在第一阶段，设计变量保持不变，根据设计变量1获得的隐含或明确表述的自由度（可以是变量，使人们有可能描述形状，以外形的优化为例子）。大家可以看到在下面部分的例子中，这些表现形式可以是有形或无形的，且这是适当的治疗后的情况。在案件1中用有限元计算方法，这一阶段1是在有限元计算的基础上，以获取机械结构的自由度。事实上，有限元，位移与节点、机械结构网格，获得了最小化总势能16。 第2阶段中 机械结构自由度的表达根据设计前先获得的变量，然后注入机械结构总势能（你会看到下面的部分中第二个例子它是如何影响自由度在隐含的职能设计变量的情况下）。然后得到一个表达式总的潜在能量，它依赖于（以明示或默示的形式）设计变量。 第3阶段 随后进行的第二次和新的最小化总势能通过前面的形式取得，但这次比设计变量同时遵循技术限制或优化约束的问题。这种方法按问题的本质可以使用或多或少的设备。这点很明显，例如，如果离散变量按照设计变量可以表示为一明确的方式，在2到3个阶段是可以立即设置的，并无迭代。 如果离散变量不能按照设计变量明确的方式表达，或者如果结构拓扑不是固定的，或者如果行为不是线性的，这将有必要通过分阶段进行1至3连续迭代。这将在下面部分的例子中提到，一会我们会看到战略的场合，可以采用一种类型为这些迭代。总结，伪构形的步骤，主要目的只是尽量减少潜在总能源，其他可能的目标是限制或优化约束。 在我们的例子中使用的最优化方法是遗传算法（遗传算法），如17。，我们也可以找到很多书籍有典型类似的教学价值，例如在18中。这种方法是非常先进方便于我们的伪构形方法。撰文已在天然气方面广泛地开展了工作，关于这一主题出版的期刊被誉为期刊19-31。由于天然气问题在社会结构力学上还比较新，在这里我们提供的一些细节正是使用这里的算法多点交叉使用，而不是一单点交叉。在甄选计划上，每年的使用完全是随机生成。在我们的例子中，几代人是等同的衔接使用。我们提供例子的结果是不断地通过使用不同的遗传算法。一个比较标准的遗传算法已经被证明是我们足够的榜样。 3范例 尽管潜在的能源可能是一个好的举措对于一些优化问题，势能不是赋予形成水滴的能量，也没有定义锤子的最佳形状，这就是为什么势能不是唯一的、客观的，但最优化问题是多目标的和用公式明确表示的两个例子的目标函数。 3.1例1：对一滴水形状优化 第一个测试例子是对下降的水滴形状优化（图1）。这个问题是等同于抵抗坦克的膜理论计算。其目的是看看伪构形理论给出了大自然的优化设计。 3.1.1使用的方法 该一水滴几何的定义是：产生的轴对称壳薄线。此行描述于连续直线或圆形段描述在特定意义和输入数据定义点上的坐标和半径值。初始数据是一个由直线段连接结点的集合。 每一个结点是确定它两个圆柱坐标上（R，z），和真正的R代表的半径圆相切的两个交叉直线段的这一点。另一台计算机的计算给出任何边界的坐标点，特别是切点必须界定圆弧长度。 水式设计描述了三个弧圆如图1所示。 通过有限元方法采用三节点抛物原理运用基尔霍夫壳体理论分析。自动网格生成器建立每个直线或圆形段的有限元网格 ，它们被视为宏观有限元。 我们的目标是获得一个水滴形状形成最低总势能（这是主要目标）和平等的抵抗坦克（这是唯一约束或限制的问题）。 事实上，为了水滴的问题，目标是多对象的，两个目标（F1=最低总额的F1 势能和f2 =等于电阻）的合并多目标：F1=F1 + F2。 马塞兰指出，在总势能的减少中约束或限制的问题被考虑进去，在19中。 3.1.2结果 在水滴外形设计中描述了三个弧圆（图1），他们的中心和半径是设计变量。因此，有9个设计变量，其中：r1，Z1，R1为 圆1;r2, Z2,R2为圆2; r3，Z3,R3为圆3 。在遗传算法中，其中每个设计变量通过3个二进制数字编码.所有这些二进制数字编码是端到端地形成27个二进制数字的染色体长度。 GA是运行了30个，一个数字对50代，一个穿越的概率为0.8，而突变概率为0.1。 对应的染色体最优解是 100 100 011 011 010 011 100 011 101 这给出了图1的解决方案。其中： - r1 = 18，z1 = 17，R1 =- 0.065 - R2= 13.75，z2= 12.2，R2 =- 7.7 - r3 = 4.1，Z3= 21.4，R3 =- 21 这是关于一水滴的外形非常接近自然的最佳解决方案。通过三个圆的弧模式的水滴模型并非十全十美。但是，构形理论用于优化不完善的地方，并发现最接近自然的解决办法。因此，构形原则包括尽可能的分配不完善的地方。 3.2例2：轴对称结构的形状优化 在这一部分，呈现了液压锤后轴承传统的最优化影响。相对于较少的周期操作轴承问题（图2）渐渐体现出来。 对于轴对称结构，分析是通过有限元方法进行的，遗传算法优化的过程中的特殊字符一直用来缓解计算和节省计算机的时间。首先，由于只是一个结构几个部分必须经常修改，子结构的概念是用来单独“固定”和“移动”的部分。固定部分计算两次：第一次是开始，第二次是结尾的优化过程。只有这些缩减刚度矩阵的子结构被添加到移动部分的矩阵。 与此相关的部分，自动发电机创建作为每个子结构的网格宏观有限元。这些宏量元素不是 三角（六节点）或四边形（8个节点）。根据那些著名的技术，同样的细分用于父的空间，以获取网本身，这显然是出于作出相同类型的元素。在这个网优化控制过程，一个的离散如有必要可以重新选择。 总之，优化问题如下： 总的目标函数 最小化潜在能源。需要注意的另一重要 目标（冯米塞斯沿等高线的移动相当于最小应力的最大值）是这里作为问题的约束。这第二个目标是要实现液压锤的后轴承最小化。 设计变量 设计变量是半径为r，宽X附近的半径（图2）。 制约因素 制约因素是建立在这样的在几何方式上，只允许有微小的变化是。它们考虑到技术的限制。他们包括编码设计变量。另外，重要的制约因素是，米塞斯沿等高线的移动的最大值不能超过一定的值。约束被考虑到总势能的降落中，在19说明。 所有这些二进制数字终端到终端地形成八个二进制数字的染色体长度。 GA运行的12个，数的30代，交叉概率 为0.5，以及变异概率为0.06。 最优解对应的染色体 1101 1000 图2给出了解决方案。其中： = 1.95，X = 6.0 在这种产品的形状自动优化中，只需简单地把形状修改小，这比计算更难预测（半径增加外，减少宽度），大大提高了机械轴承的耐久性：过压力正在减少50。 4讨论 本文件中的两个例子可以证明伪构形理论。第一个是对轴对称膜下降形外壳形状优化（水滴）。这种结构是用纯的张力。果不其然，尽量减少这种结构的总势能，所有可能的变量导致的形状是完全和调和十分相似的。但是，第二个例子事实证明，制定最低的能源不仅可以 工作在最简单的情况下，纯粹的张力结构还能弯曲，剪切或更为复杂的结构扭转应力。这个条件是为了增加这一问题次要目标（通常用于形状优化）的限制或优化约束。 然而，在伪形构形理论声明中，最大限度地减少所有可能的变量的机械结构中的总势能， 这不完全能达到的。自然和机械也不是都如此简单，多年研究的大自然设计结构表明，即使在最简单的实例中，多重标准，以复杂的方式工作。因此，有必要添加其他标准或优化问题的制约是显而易见的。最小化总势能只是一个总的原则在优化的过程启动。 5结论 一个有趣的方法引入了形状优化的机械结构。在这个文件阐述的伪构行理论中，优化的主要目标是最小化总势能。其他的目标通常使用的形状 优化这里使用了限制或优化限制。它给我们的例子很好的效果。参考文献 1、从工程到自然形状和结构 剑桥大学出版社,剑桥大学,Bejan A主编 2、伪构形理论的网络的路径冷却机 J.热能质量40:799-816J. Bejan A主编3、自然如何形成 52英格119(10):90-92 Bejan A主编4、树状构形网络空间分布的电力 能源转化管理 44:867-891 Arion V,Cojocari,Bejan一个(2003)Constructa Bejan A主编 5、对流体几何内部的优化 热能转化120:357-364J. Nelson RA, Bejan A主编 6、碟状区域构形设计的冷却传导 J.热能质量45:1643-1652 Rocha LAO, Lorente S, Bejan A主编 7、天然裂缝模式的构形理论形成快速冷却 J.热能质量 反式41:1945-1954 Bejan A, Ikegami Y, Ledezma GA主编ORIGINAL ARTICLEPseudo-constructal theory for shape optimizationof mechanical structuresJean Luc MarcelinReceived: 10 January 2007 /Accepted: 1 May 2007 /Published online: 25 May 2007#Springer-Verlag London Limited 2007Abstract This work gives some applications of a pseudo-constructal technique for shape optimization of mechanicalstructures. In the pseudo-constructal theory developed inthis paper, the main objective of optimization is only theminimization of total potential energy. The other objectivesusually used in mechanical structures optimization aretreated like limitations or optimization constraints. Twoapplications are presented; the first one deals with theoptimization of the shape of a drop of water by using agenetic algorithm with the pseudo-constructal technique,and the second one deals with the optimization of the shapeof a hydraulic hammers rear bearing.Keywords Shapeoptimization.Constructal.Geneticalgorithms1 IntroductionThis paper introduces a pseudo-constructal approach toshape optimization based on the minimization of the totalpotential energy. We are going to show that minimizing thetotal potential energy of a structure to find the optimalshape might be a good idea in some cases. The reference tothe constructal theory can be justified in some way for thefollowing reasons.According to Bejan 1, shape and structure spring fromthe struggle for better performance in both engineering andnature; the objective and constraints principle used inengineering is the same mechanism from which thegeometry in natural flow systems emerges. Bejan 1 startswith the design and optimization of engineering systemsand discovers a deterministic principle for the generation ofgeometric form in natural systems. This observation is thebasis of the new constructal theory. Optimal distribution ofimperfection is destined to remain imperfect. The systemworks best when its imperfections are spread around so thatmore and more internal points are stressed as much as thehardest working parts. Seemingly universal geometricforms unite the flow systems of engineering and nature.Bejan 1 advances a new theory in which he unabashedlyhints that his law is in the same league as the second law ofthermodynamics, because a simple law is purported topredict the geometric form of anything alive on earth.Many applications of the constructal theory weredeveloped in fluids mechanics, in particular for theoptimization of flows 210. On the other hand, thereexists, to our knowledge, little examples of applications insolids or structures mechanics. So we have at least half ofthe references to papers in fluid dynamics (most of the sameauthor), because the constructal method was developed firstby the same author, Adrian Bejan, with only references topapers in fluid dynamics. The constructal theory rests onthe assumption that all creations of nature are overalloptimal compared to the laws which control the evolutionand the adaptation of the natural systems. The constructalprinciple consists of distributing the imperfections as wellas possible, starting from the smallest scales to the largest.The constructal theory works with the total macroscopicstructure starting from the assembly of elementary struc-tures, by complying with the natural rules of optimaldistribution of the imperfections. The objective is theresearch of lower cost.Int J Adv Manuf Technol (2008) 38:16DOI 10.1007/s00170-007-1080-2J. L. Marcelin (*)Laboratorie Sols Solides Structures 3S, UMR CNRS C5521,Domaine Universitaire,BP n53,38041 Grenoble Cedex 9, Francee-mail: Jean-Luc.Marcelinujf-grenoble.frHowever, a global and macroscopic solution for theoptimization of mechanical structures having least cost asthe objective can be very close to the constructal theory,from where the term pseudo-constructal comes. Theconstructal theory is a predictive theory, with only onesingle principle of optimization from which all rises. Thesame applies to the pseudo-constructal step which is thesubject of this article. The single principle of optimiza-tion of the pseudo-constructal theory is the minimizationof total potential energy. Moreover, in our examplespresented hereafter, the pseudo-constructal principle willbe associated with a genetic algorithm, with the resultthat our optimization will be very close to the naturallaws.The objective of this paper is thus to show how thepseudo-constructal step can apply to the mechanics of thestructures, and in particular to the shape optimization ofmechanical structures. The basic idea is very simple: amechanical structure in a balanced state corresponds to aminimal total potential energy. In the same way, an optimalmechanical structure must also correspond to a minimaltotal potential energy, and it is this objective which mustintervene first over all the others. It is this idea which willbe developed in this article.Two examples will be presented thereafter. The idea tominimize total potential energy in order to optimize amechanical structure is not brand new. Many papers alreadydeal with this problem. What is new, is to make thisapproach systematic. The only objective of optimizationbecomes the minimization of energy.In Gosling 11, a simple method is proposed for thedifficult case of form-finding of cablenet and membranestructures. This method is based upon basic energyconcepts. A truncated strain expression is used to definethe total potential energy. The final energy form isminimized using the Powell algorithm. In Kanno andOhsaki 12, the minimum principle of complementaryenergy is established for cable networks involving onlystress components as variables in geometrically nonlinearelasticity. In order to show the strong duality between theminimization problems of total potential energy andcomplementary energy, the convex formulations of theseproblems are investigated, which can be embedded into aprimal-dual pair of second-order programming problems. InTaroco 13, shape sensitivity analysis of an elastic solid inequilibrium is presented. The domain and boundary integralexpressions of the first and second-order shape derivativesof the total potential energy are established. In Warner 14,an optimal design problem is solved for an elastic rodhanging under its own weight. The distribution of the cross-sectional area that minimizes the total potential energystored in an equilibrium state is found. The companionproblem of the design that stores the maximum potentialenergy under the same constraint conditions is also solved.In Ventura 15, the problem of boundary conditionsenforcement in meshless methods is solved. In Ventura15, the moving least-squares approximation is introducedin the total potential energy functional for the elastic solidproblem and an augmented Lagrangian term is added tosatisfy essential boundary conditions.The principle of minimization of total potential energy isin addition at the base of the general finite elementsformulation, with an aim of finding the unknown optimalnodal factors 16.2 The methods usedIn the pseudo-constructal theory developed in this paper,the main objective of optimization is only the minimizationof total potential energy. The other objectives usually usedin mechanical structures optimization are treated here likelimitations or optimization constraints. For example, onemay have limitations on the weight, or to not exceed thevalue of a stress.The idea which will be developed in this paper is thusvery simple. A mechanical structure is described by twotypes of parameters: variables known as discretizationvariables (for example, degrees of freedom in displacementfor finite elements method), and geometrical variables ofdesign (for example parameters which make it possible todescribe the mechanical structure shape). Total potentialenergy depends on an implicit or explicit way of determin-ing discretization and design variables at the same time.One thus will carry out a double optimization of themechanical structure, compared to the discretization anddesign variables; the objective being to minimize totalpotential energy overall. Clearly, the problem of optimiza-tion of a mechanical structure will be addressed by thefollowing approach:Objective: to minimize total potential energyVariables of optimization: concurrently determiningdiscretization variables (in the case of a traditional useof the finite element method in mechanics of struc-tures), and design variables describing the shape of thestructureOptimization limitations:Weight or volumeDisplacements or strainsStressesFrequenciesThe problem of optimization of a mechanical structurewill be solved in the following way, while reiterating on2Int J Adv Manuf Technol (2008) 38:16these stages, if needed (according to the nature of theproblem):Stage 1Minimization of the total potential energy of themechanical structure compared to the only dis-cretization variables of the structure (degrees offreedom in finite elements). It acts here as anoptimization without optimization limitations.The only limitations at this stage are of purelymechanical origin, and relate to the boundaryconditions and to the external efforts applied tothe structure.In this stage 1, the design variables remain fixed, andone obtains the implicit or explicit expressions of thedegrees of freedom according to the design variables(which can be the variables which make it possible todescribe the shape, in the case of a shape optimization, forexample). One will see in the examples of the followingpart that these expressions can be explicit or implicit andwhich is the suitable treatment following the cases. In thecase of a finite elements method of calculation, this stage 1is the basis of finite elements calculation to obtain thedegrees of freedom of the mechanical structure. Indeed, infinite elements, displacements with the nodes of themechanical structure mesh are obtained by minimizationof total potential energy 16.Stage 2The expressions of the degrees of freedom of themechanical structure according to the designvariables obtained previously are then injected intothe total potential energy of the mechanicalstructure (one will see in the second example ofthe followingparthowone treatsthecasewherethedegrees of freedom are implicit functions of thedesign variables). One then obtains an expressionof the total potential energy which depends only onthe design variables (in explicit or implicit form).Stage 3One then carries out a second and new minimi-zation of the total potential energy obtained in thepreceding form, but this time compared to thedesign variables while respecting the technolog-ical limitations or the optimization constraints ofthe problem. This method can be applied withmore or less facility according to the nature of theproblem. It is clear, for example, that if thediscretization variables can be expressed in anexplicit way according to the design variables, thesetting in of stages 2 to 3 is immediate, andwithout iterations.If the discretization variables cannot be expressed in anexplicit way according to the design variables, or if thetopology of the structure is not fixed, or if the behavior isnot linear, it will be necessary to proceed by successiveiterations on stages 1 to 3. It is the case of the examplespresented in the following part, and one will see on thisoccasion which type of strategy one can adopt for theseiterations. To summarize, in the pseudo-constructal step, themain objective is only the minimization of total potentialenergy, the other possible objectives are treated likelimitations or optimization constraints.The optimization method used for our examples is GA(genetic algorithm), as described in 17. Examples withsimilar instructional value can also be found in manybooks, e.g. in 18. This evolutionary method is veryconvenient for our pseudo-constructal method. The authorhas worked extensively in GAs and published in somereputed journals on this topic 1931. As the topic of GAsis still relatively new in the structural mechanics commu-nity, we provide here some details of exactly what is usedin this GA. A multiple point crossover is used rather than asingle point crossover. The selection scheme used at eachgeneration is entirely stochastic. For our examples, thenumber of generations is equal to that used for conver-gence. The results provided for our examples wereconsistently reproduced by using different seeds in theGA. It has been proved that a rather standard geneticalgorithm is sufficient for our examples.3 ExamplesEven though potential energy may be a good measure forsome optimizations, potential energy is not what gives theshape to a water droplet, nor defines the optimal shape for ahammer, which is why potential energy is not the onlyobjective; but the optimization problem is a multiobjectiveone and the objective functions for the two examples arethen clearly formulated.3.1 Example 1: optimization of the shape of a drop of waterThe first test example is the optimization of the shape of adrop of water (Fig. 1). This problem is equivalent to anequal resistance tank calculated by the membrane theory.The objective is to see if the pseudo-constructal theorygives the natures optimum design.3.1.1 The methods usedThe geometry of the drop of water is defined by thegenerating line of a thin axisymmetric shell. This line isdescribed by successive straight or circular segmentsdescribed in a given sense and defined by input data ofmaster point coordinates and radius values. The initial dataare a set of nodal points connected by straight segments.Each nodal point is identified by its two cylindricalInt J Adv Manuf Technol (2008) 38:163coordinates (r, z), and a real R which represents the radiusof the circle tangent to the two straight segments intersect-ing at the point. The other computer calculations give thecoordinates of any boundary point and especially thetangent points necessary to define the circular arc lengths.The design of the drop of water is described by three arcs ofcircles as indicated in Fig. 1.Analysis is performed by the finite element method withthree-node parabolic elements using the classical Love-Kirchoff shell theory. An automatic mesh generator createsthe finite element mesh of each straight or circular segmentconsidered as a macro finite element.The objective is to obtain a shape for the drop of watergiving rise to a minimum total potential energy (which isthe main objective) and an equal resistance tank (which isthe only constraint or limitation of the problem).In fact, for the drop of water problem, the goal is a multi-objective one, the two objectives ( f1=minimum totalpotential energy and f2=equal resistance) are combined ina multi-objective: f=f1+f2.The constraint or limitation of the problem is taken intoaccount by a penalization of the total potential energy asindicated in Marcelin et al. The resultsThe design of the drop of water is described by three arcs ofa circle (Fig. 1). Their centers and radius are the designvariables. So, there are nine design variables: r1, z1, R1 forcircle 1; r2, z2, R2 for circle 2; and r3, z3, R3 for circle 3.In the genetic algorithm, each of these design variables iscoded by three binary digits.The tables of coding-decoding will be the following:For r1:For z1:For R1:For r2:For z2:For R2:For r3:For z3:For R3:All these binary digits are put end to end to form achromosome length of 27 binary digits.GA is run for a population of 30 individuals, a numberof generations of 50, a probability of crossing of 0.8, and aprobability of mutation of 0.1.The optimal solution corresponds to the chromosome100 100 011 011 010 011 100 011 101which gives the solution of Fig. 1, for which:r1=18, z1=17,and R1=0.065r2=13.75, z2=12.2 and R2=7.7r3=4.1, z3=21.4 and R3=21It is very close to the natures optimal solution for theshape of a drop of water. The model of the water dropmodelled by three arcs of a circle is imperfect. However,the constructal theory optimizes the imperfections, and1052025r510 20z3 211515Fig. 1 Optimization of the shape of a drop of water0000010100111001011101111616.51717.51818.51919.50000010100111001011101111515.51616.51717.51818.50000010100111001011101110.0500.0550.0600.0650.0700.0750.0800.0850000010100111001011101111313.2513.513.751414.2514.514.75000001010011100101110111112.412.512.612.7000001010011100101110188.100000101001110010111014.40000010100111001011101121.421.521.621.721.800000101001110010111011118.51919.52020.52121.5224Int J Adv Manuf Technol (2008) 38:16finds the nearest solution to that of nature. So, theconstructal principle consists of distributing the imperfec-tions as well as possible.3.2 Example 2: optimization of the shapeof an axisymmetric structureIn this part, the very localized optimization of the rearbearing of a hydraulic hammer is presented. The bearing inquestion (Fig. 2) breaks after relatively few cycles ofoperation.For axisymmetric structures, analysis is performed bythe finite element method in which the special character ofa GA optimization process has been considered to ease thecalculations and to save computer time. First, because just afew parts of the structure must often be modified, thesubstructure concept is used to separate the “fixed” and the“mobile” parts. The fixed parts are calculated twice: once atthe beginning and also at the end of the optimizationprocess. Only the reduced stiffness matrices of thesesubstructures are added to the matrices of the mobile parts.Related to this division, an automatic generator creates thefinite element mesh of each substructure considered as amacro finite element. These macro elements are eithertriangular (six nodes) or quadrilateral (eight nodes).Following a well-known technique, the same subdivisionis used in the parent space to obtain the mesh itself, whichis obviously made out of the same types of elements.During the optimization process this mesh is controlled anda new discretization can be chosen if necessary.To summarize, the optimization problem is the following:Objective function Minimization of the total potentialenergy. It is important to note that another importantobjective (the minimization of the maximum value of theVon Mises equivalent stress along the mobile contour) istaken here as a constraint of the problem. This secondobjective is necessary to achieve the minimization of therear bearing of the hydraulic hammer .Design variables The design variables are radius r andwidth X near the radius (Fig. 2).Constraints The side constraints are established in such away that only small changes in geometry are allowed. Theytake into account the technological constraints. They areincluded in the coding of the design variables. Anotherimportant constraint is that the maximum value of the VonMises equivalent stress along the mobile contour must notexceed a certain value. The constraints are taken intoaccount by a penalization of the total potential energy asindicated in 19.The tables of coding-decoding are the following:For r:For X:All these binary digits are put end to end to form achromosome length of eight binary digits.The GA is run for a population of 12 individuals, anumber of generations of 30, a probability of crossing of0.5, and a probability of mutation of 0.06.The optimal solution corresponds to the chromosome1101 1000which gives the solution of Fig. 2, for which:r=1.95, X=6.0The automatic optimization of the shape of this producthas,simplybyasmallmodificationofshape,whichisdifficultto predict other than by calculation (increased radius,decreased width), considerably improved the mechanicaldurability of the bearing: the over-stress being reduced by50%.4 DiscussionThe two examples in this paper may prove the truth of thepseudo-constructal theory. The first one was the shapeoptimization ofanaxisymmetricmembranedrop-shaped shellConstraint: the bearing should not penetrate into the casing during deformation Hydraulic hammers rear bearingCASINGr=1.95r= 1.5initial shapefinal shapeX Xmobilesub- struc-turefixed sub- struc- ture Fig. 2 Optimization of the shape of a hydraulic hammers rear bearing000000010010001101000101011001111.301.351.401.451.501.551.601.65100010011010101111001101111011111.701.751.801.851.901.952.002.050000000100100011010001010110014.05.6100010011010101111001101111011116.07.68.08.48.8Int J Adv Manuf Technol (2008) 38:165(water droplet). This structure is in pure tension. Sure enough,minimizing the total potential energy of this structure over allpossible variables leads to a shape which is fully-stressed andvery similar to what nature does. But the second exampleproves that the minimum energy formulation may not onlywork in the simplest of cases, of pure tension structures, butalso for more complicated structures with bending, shear ortorsion stresses. The condition for this istoadd tothe problemsecondary objectives (usually used in shape optimization) aslimitations or optimization constraints.Nevertheless, the assertion at the heart of pseudo-con-structal theory, that minimizing the total potential energy ofmechanical structure over all possible variables is what naturedoes, is not entirely true. Neither nature nor engineering is sosimplistic,andmanyyearsofresearchintohownaturedesignsstructures has shown that even in the most simple instancesmultiple criteria are at work in complicated ways leading todelicate compromises,sothe necessitytoaddotherscriteriaorconstraints to the optimization problems is evident. Minimiz-ing the total potential energy is just a general principle to startthe optimization process.5 ConclusionAn interesting approach has been introduced to performshape optimization of mechanical structures. In the pseudo-constructal theory developed in this paper, the mainobjective of optimization is only the minimization of totalpotential energy. The other objectives usually used in shapeoptimization are treated here like limitations or optimizationconstraints. It gives good results for our examples.References1. Bejan A (2000) Shape and structure, from engineering to nature.Cambridge University Press, Cambridge, UK2. Bejan A (1997) Constructal-theory network of conducting paths forcooling a heat generating volume. Int J Heat Mass Trans 40:7998163. Bejan A (1997) How nature takes shape. Mech Eng 119(10):90924. Arion V, Cojocari A, Bejan A (2003) Constructal tree shapednetworks for the distribution of electric power. Energy ConversManag 44:8678915. 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