夹具类外文翻译-采用遗传算法优化加工夹具定位和加紧位置【中文4477字】【PDF+中文WORD】

夹具类外文翻译-采用遗传算法优化加工夹具定位和加紧位置【中文4477字】【PDF+中文WORD】,中文4477字,PDF+中文WORD,夹具,外文,翻译,采用,遗传,算法,优化,加工,定位,加紧,位置,中文,4477,PDF,WORD 采用遗传算法优化加工夹具定位和加紧位置 摘要：工件变形的问题可能导致机械加工中的空间问题。支撑和定位器是用于减少工件弹性变形引起的误差。支撑、定位器的优化和夹具定位是最大限度的减少几何在工件加工中的误差的一个关键问题。本文应用夹具布局优化遗传算法（GAs）来处理夹具布局优化问题。遗传算法的方法是基于一种通过整合有限的运行于批处理模式的每一代的目标函数值的元素代码的方法，用于来优化夹具布局。给出的个案研究说明已开发的方法的应用。采用染色体文库方法减少整体解决问题的时间。已开发的遗传算法保持跟踪先前的分析设计，因此先前的分析功能评价的数量降低大约93%。结果表明，该方法的夹具布局优化问题是多模式的问题。优化设计之间没有任何明显的相似之处，虽然它们提供非常相似的表现。 关键词：夹具设计；遗传算法；优化 1.引言 夹具用来定位和束缚机械操作中的工件，减少由于对确保机械操作准确性的夹紧方案和切削力造成的工件和夹具的变形。传统上，加工夹具是通过反复试验法来设计和制造的，这是一个既造价高又耗时的制造过程。为确保工件按规定尺寸和公差来制造，工件必须给予适当的定位和夹紧以确保有必要开发工具来消除高造价和耗时的反复试验设计方法。适当的工件定位和夹具设计对于产品质量的精密度、准确度和机制件的完饰是至关重要的。 从理论上说，3-2-1定位原则对于定位所有的棱柱形零件是很令人满意的。该方法具有最大的刚性与最少量的夹具元件。从动力学观点来看定位零件意味着限制了自由移动物体的六自由度（三个平动自由度和三个旋转自由度）。在零件下部设置三个支撑来建立工件在垂直轴方向的定位。在两个外围边缘放置定位器旨在建立工件在水平x轴和y轴的定位。正确定位夹具的工件对于制造过程的全面准确性和重复性是至关重要的。定位器应该尽可能的远距离的分开放置并且应该放在任何可能的加工面上。放置的支撑器通常用来包围工件的重力中心并且尽可能的将其分开放置以维持其稳定性。夹具夹子的首要任务是固定夹具以抵抗定位器和支撑器。不应该要求夹子反抗加工操作中的切削力。 对于给定数量的夹具元件，加工夹具合成的问题是寻找夹具优化布局或工件周围夹具元件的位置。本篇文章提出一种优化夹具布局遗传算法。优化目标是研究一个二维夹具布局使工件不同位置上最大的弹性变形最小化。ANSYS程序以用于计算工件变形情况下夹紧力和切削力。本文给出两个实例来说明给出的方法。 2.回顾相关工程结构 最近几年夹具设计问题受到越来越多的重视。然而，很少有注意力集中于优化夹具布局设计。Menassa和Devries用FEA计算变形量使设计准则要求的位点的工件变形最小化。设计问题是确定支撑器位置。Meyer和Liou提出一个方法就是使用线性编程技术合成动态编程条件中的夹具。给出了使夹紧力和定位力最小化的解决方案。Li和Melkote用非线性规划方法解决布局优化问题。这个方法使工件位置误差最小化归于工件的局部弹性变形。Roy和Liao开发出一种启发式方法来计划最好的支撑和夹紧位置。Tao等人提出一个几何推理的方法来确定最优夹紧点和任意形状工件的夹紧顺序。Liao和Hu提出一种夹具结构分析系统这个系统基于动态模型分析受限于时变加工负载的夹具—工件系统。本文也调查了夹紧位置的影响。Li和Melkote提出夹具布局和夹紧力最优合成方法帮我们解释加工过程中的工件动力学。本文提出一个夹具布局和夹紧力优化结合的程序。他们用接触弹性建模方法解释工件刚体动力学在加工期间的影响。Amaral等人用ANSYS验证夹具设计的完整性。他们用3-2-1方法。ANSYS提出优化分析。Tan等人通过力锁合、优化与有限建模方法描述了建模、优化夹具的分析与验证。 以上大部分的研究使用线性和非线性编程方式这通常不会给出全局最优解决方案。所有的夹具布局优化程序开始于一个初始可行布局。这些方法给出的解决方案在很大程度上取决于初始夹具布局。他们没有考虑到工件夹具布局优化对整体的变形。 GAs已被证明在解决工程中优化问题是有用的。夹具设计具有巨大的解决空间并需要搜索工具找到最好的设计。一些研究人员曾使用GAs解决夹具设计及夹具布局问题。Kumar等人用GAs和神经网络设计夹具。Marcelin已经将GAs用于支撑位置的优化。Vallapuzha等人提出基于优化方法的GA，它采用空间坐标来表示夹具元件的位置。夹具布局优化程序设计的实现是使用MATLAB和遗传算法工具箱。HYPERMESH和MSC / NASTRAN用于FE模型。Vallapuzha等人提出一些结果关于一个广泛调查不同优化方法的相对有效性。他们的研究表明连续遗传算法提出了最优质的解决方案。Li和Shiu使用遗传算法确定了夹具设计最优配置的金属片。MSC/NASTRAN已经用于适应度值评价。Liao提出自动选择最佳夹子和夹钳的数目以及它们在金属片整合的夹具中的最优位置。Krishnakumar和Melkote开发了一种夹具布局优化技术，它是利用遗传算法找到了夹具布局，由于整个刀具路径中的夹紧力和加工力使加工表面变形量最小化。通过节点编号使定位器和夹具位置特殊化。一个内置的有限元求解器研制成功。 一些研究没考虑到整个刀具路径的优化布局以及磨屑清除。一些研究采用节点编号作为设计参数。 在本研究中，开发GA工具用于寻找在二维工件中的最优定位器和夹紧位置。使用参考边缘的距离作为设计参数而不是用FEA节点编号。真正编码遗传算法的染色体的健康指数是从FEA结果中获得的。ANSSYS用于FEA计算。用染色体文库的方法是为了减少解决问题的时间。用两个问题测试已开发的遗传算法工具。给出的两个实例说明了这个开发的方法。本论文的主要贡献可以概括为以下几个方面： （1） 开发了遗传算法编码结合商业有限元素求解； （2） 遗传算法采用染色体文库以降低计算时间； （3） 使用真正的设计参数，而不是有限元节点数字； （4） 当工具在工件中移动时考虑磨屑清除工具。 3.遗传算法概念 遗传算法最初由John Holland开发。Goldberg出版了一本书，解释了这个理论和遗传算法应用实例的详细说明。遗传算法是一种随机搜索方法，它模拟一些自然演化的机制。该算法用于种群设计。种群从一代到另一代演化，通过自然选择逐渐提高了适应环境的能力，更健康的个体有更好的机会，将他们的特征传给后代。 该算法中，要基于为每个设计计算适合性，所以人工选择取代自然环境选择。适应度值这个词用来指明染色体生存几率，它在本质上是该优化问题的目标函数。生物定义的特征染色体用代表设计变量的字符串中的数值代替。 被公认的遗传算法与传统的梯度基础优化技术的不同主要有如下四种方式： （1） 遗传算法和问题中的一种编码的设计变量和参数一起工作而不是实际参数本身。 （2） 遗传算法使用种群—类型研究。评价在每个重复中的许多不同的设计要点而不是一个点顺序移动到下一个。 （3） 遗传算法仅仅需要一个适当的或目标函数值。没有衍生品或梯度是必要的。 （4） 遗传算法以用概率转换规则来发现新设计为探索点而不是利用基于梯度信息的确定性规则来找到这些新观点。 4.方法 4.1夹具定位原则 加工过程中，用夹具来保持工件处于一个稳定的操作位置。对于夹具最重要的标准是工件位置精确度和工件变形。一个良好的夹具设计使工件几何和加工精度误差最小化。另一个夹具设计的要求是夹具必须限制工件的变形。考虑切削力以及夹紧力是很重要的。没有足够的夹具支撑，加工操作就不符合设计公差。有限元分析在解决这其中的一些问题时是一种很有力的工具。 棱柱形零件常见的定位方法是3-2-1方法。该方法具有最大刚体度以及最小夹具元件数。在三维中一个工件可能会通过六自由度定位方法快速定位为了限制工件的九个自由度。其他的三个自由度通过夹具元件消除了。基于3-2-1定位原理的二位工件布局的例子如图4。 图4 3-2-1对二维棱柱工件定位布局 定位面得数量不得超过两个避免冗余的位置。基于3-2-1的夹具设计原则有两种精确的定位平面包含于两个或一个定位器。因此，在两边有最大的夹紧力抵抗每个定位平面。夹紧力总是指向定位器为了推动工件接触到所有的定位器。定位点对面应定位夹紧点防止工件由于夹紧力而扭曲。因为加工力沿着加工面，所以有必要确保定位器的反应力在所有时间内是正的。任何负面的反应力表示工件从夹具元件中脱离。换句话说，当反应力是负的时候，工件和夹具元件之间接触或分离的损失可能发生。定位器内正的反应力确保工件从切削开始到结束都能接触到所有的定位器。夹紧力应该充分束缚和定位工件且不导致工件的变形或损坏。本文不考虑夹紧力的优化。 4.2基于夹具布局优化方法的遗传算法 在实际设计问题中，设计参数的数量可能很大并且它们对目标函数的影响会是非常复杂的。目标函数曲线必须是光滑的并且需要一个程序计算梯度。遗传算法在理念上远不同于其他的探究方法，它们包括传统的优化方法和其他随机方法。通过运用遗传算法来对夹具优化布局，可以获得一个或一组最优的解决方案。 本项研究中，最优定位器和夹具定位使用遗传算法确定。它们是理想的适合夹具布局优化问题的方法因为没有直接分析的关系存在于加工误差和夹具布局中。因为遗传算法仅仅为一个特别的夹具布局处理设计变量和目标函数值，所以不需要梯度或辅助信息。 建议方案流程图如图5。 使用开发的命名为GenFix的Delphi语言软件来实现夹具布局优化。位移量用ANSYS软件计算。通过WinExec功能在GenFix中运行ANSYS很简单。GenFix和ANSYS之间相互作用通过四部实现： （1） 定位器和夹具位置从二进制代码字符串中提取作为真正的参数。 （2） 这些参数和ANSYS输入批处理文件（建模、解决方案和后置处理）用WinExec功能传给ANSYS。 （3） 解决后将位移值写成一个文本文件。 （4） GenFix读这个文件并为当前定位器和夹紧位置计算适应度值。 为了减少计算量，染色体与适应度值储存在一个文库里以备进一步评估。GenFix首先检查是否当前的染色体的适应度值已经在之前被计算过。如果没有，定位器位置被送到ANSYS，否则从文库中取走适应度值。在初始种群产生过程中，检查每一个染色体可行与否。如果违反了这个原则，它就会出局然后新的染色体就产生了。这个程序创造了可行的初始种群。这保证了初始种群的每个染色体在夹紧力和切削力作用下工件的稳定性。用两个测试用例来验证提到的遗传算法计划。第一个实例是使用Himmelblau功能。在第二个测试用例中，遗传算法计划用来优化均布载荷作用下梁的支撑位置。 图5 设计方法的流程与ANSYS相配合流程 5.夹具布局优化的个案研究 该夹具布局优化问题的定义是：找到定位器和夹子的位置以使在特定区工件变形降到最小程度。那么多的定位器和夹子并不是设计参数因为它们在3-2-1方案中是已知的和固定的。因此，设计参数的选择如同定位器和夹子的位置。本研究中不考虑摩擦力。两个实例研究来说明以提出的方法。 6.结论 本文提出了一个夹具布局优化的评价优化技术。ANSYS用于FE计算适应度值。可以看到，遗传算法和FE方法的结合对当今此类问题似乎是一种强大的方法。遗传算法特别适合应用于解决那些在目标函数和设计变量之间不存在一个定义明确的数学关系的问题。结果证明遗传算法在夹具布局优化问题方面的成功应用。本项研究中，遗传算法在夹具布局优化应用中的主要困难是较高的计算成本。种群中每个染色体需要工件的重啮合。但是，染色体库的使用，FE评价的数量从6000下降到415。这就导致了巨大的增益计算效益。其他减少处理时间的方法是在局域网内使用分布式计算。 该方法结果表明，夹具布局优化问题是多模态问题。优化设计之间没有任何明显的相似之处尽管他们提供非常相似的表现。结果表明夹具布局问题是多模态问题然而用于夹具设计的启发式规则应该用于遗传算法来选择最优的设计。 Machining fixture locating and clamping position optimizationusing genetic algorithmsNecmettin Kaya*Department of Mechanical Engineering,Uludag University,Go ru kle,Bursa 16059,TurkeyReceived 8 July 2004;accepted 26 May 2005Available online 6 September 2005AbstractDeformationoftheworkpiecemaycausedimensionalproblemsinmachining.Supportsandlocatorsareusedinordertoreducetheerrorcausedby elastic deformation of the workpiece.The optimization of support,locator and clamp locations is a critical problem to minimize the geometricerror in workpiece machining.In this paper,the application of genetic algorithms(GAs)to the fixture layout optimization is presented to handlefixture layout optimization problem.A genetic algorithm based approach is developed to optimise fixture layout through integrating a finiteelement code running in batch mode to compute the objective function values for each generation.Case studies are given to illustrate theapplicationofproposedapproach.Chromosomelibraryapproachisusedtodecreasethetotalsolutiontime.DevelopedGAkeepstrackofprevioslyanalyzed designs,therefore the number of function evaulations are decreased about 93%.The results of this approach show that the fixture layoutoptimization problems are multi-modal problems.Optimized designs do not have any apparent similarities although they provide very similarperformances.#2005 Elsevier B.V.All rights reserved.Keywords:Fixture design;Genetic algorithms;Optimization1.IntroductionFixtures are used to locate and constrain a workpiece duringa machining operation,minimizing workpiece and fixturetooling deflections due to clamping and cutting forces arecritical to ensuring accuracy of the machining operation.Traditionally,machining fixtures are designed and manufac-tured through trial-and-error,which prove to be both expensiveand time-consuming to the manufacturing process.To ensure aworkpiece is manufactured according to specified dimensionsand tolerances,it must be appropriately located and clamped,making it imperative to develop tools that will eliminate costlyand time-consuming trial-and-error designs.Proper workpiecelocation and fixture design are crucial to product quality interms of precision,accuracy and finish of the machined part.Theoretically,the 3-2-1 locating principle can satisfactorilylocate all prismatic shaped workpieces.This method providesthe maximum rigidity with the minimum number of fixtureelements.To position a part from a kinematic point of viewmeans constraining the six degrees of freedom of a free movingbody(three translations and three rotations).Three supports arepositioned below the part to establish the location of theworkpiece on its vertical axis.Locators are placed on twoperipheral edges and intended to establish the location of theworkpiece on the x and y horizontal axes.Properly locating theworkpiece in the fixture is vital to the overall accuracy andrepeatability of the manufacturing process.Locators should bepositioned as far apart as possible and should be placed onmachined surfaces wherever possible.Supports are usuallyplaced to encompass the center of gravity of a workpiece andpositioned as far apart as possible to maintain its stability.Theprimary responsibility of a clamp in fixture is to secure the partagainstthelocatorsandsupports.Clampsshouldnotbeexpectedto resist the cutting forces generated in the machining operation.For a given number of fixture elements,the machiningfixture synthesis problem is the finding optimal layout orpositions of the fixture elements around the workpiece.In thispaper,a method for fixture layout optimization using geneticalgorithms is presented.The optimization objective is to searchfor a 2D fixture layout that minimizes the maximum elasticdeformation at different locations of the workpiece.ANSYSprogram has been used for calculating the deflection of the in Industry 57(2006)112120*Tel.:+90 224 4428176;fax:+90 224 4428021.E-mail address:necmiuludag.edu.tr.0166-3615/$see front matter#2005 Elsevier B.V.All rights reserved.doi:10.1016/pind.2005.05.001under clamping and cutting forces.Two case studies are givento illustrate the proposed approach.2.Review of related worksFixture design has received considerable attention in recentyears.However,little attention has been focused on theoptimum fixture layout design.Menassa and DeVries 1 usedFEA for calculating deflections using the minimization of theworkpiece deflection at selected points as the design criterion.The design problem was to determine the position of supports.Meyer and Liou 2 presented an approach that uses linearprogramming technique to synthesize fixtures for dynamicmachining conditions.Solution for the minimum clampingforces and locator forces is given.Li and Melkote 3 used anonlinear programming method to solve the layout optimiza-tion problem.The method minimizes workpiece location errorsdue to localized elastic deformation of the workpiece.Roy andLiao 4 developed a heuristic method to plan for the bestsupporting and clamping positions.Tao et al.5 presented ageometricalreasoning methodologyfor determining theoptimal clamping points and clamping sequence for arbitrarilyshaped workpieces.Liao and Hu 6 presented a system forfixture configuration analysis based on a dynamic model whichanalyses the fixtureworkpiece system subject to time-varyingmachining loads.The influence of clamping placement is alsoinvestigated.Li and Melkote 7 presented a fixture layout andclamping force optimal synthesis approach that accounts forworkpiece dynamics during machining.A combined fixturelayout and clamping force optimization procedure presented.They used the contact elasticity modeling method that accountsfor the influence of workpiece rigid body dynamics duringmachining.Amaral et al.8 used ANSYS to verify fixturedesign integrity.They employed 3-2-1 method.The optimiza-tion analysis is performed in ANSYS.Tan et al.9 describedthe modeling,analysis and verification of optimal fixturingconfigurations by the methods of force closure,optimizationand finite element modeling.Mostoftheabovestudiesuselinearornonlinearprogramming methods which often do not giveglobal optimumsolution.All of the fixture layout optimization procedures startwith an initial feasible layout.Solutionsfrom these methods aredepend on the initial fixture layout.They do not consider thefixture layout optimization on overall workpiece deformation.The GAs have been proven to be useful technique in solvingoptimization problems in engineering 1012.Fixture designhas a large solution space and requires a search tool to find thebest design.Few researchers have used the GAs for fixturedesign and fixture layout problems.Kumar et al.13 haveapplied both GAs and neural networks for designing a fixture.Marcelin 14 has used GAs to the optimization of supportpositions.Vallapuzhaetal.15presentedGAbasedoptimization method that uses spatial coordinates to representthe locations of fixture elements.Fixture layout optimizationprocedure was implemented using MATLAB and the geneticalgorithm toolbox.HYPERMESH and MSC/NASTRAN wereusedforFEmodel.Vallapuzhaetal.16 presentedresults ofanextensive investigation into the relative effectiveness of variousoptimization methods.They showed that continuous GAyielded the best quality solutions.Li and Shiu 17 determinedthe optimal fixture configuration design for sheet metalassembly using GA.MSC/NASTRAN has been used forfitness evaulation.Liao 18 presented a method to auto-matically select the optimal numbers of locators and clamps aswell as their optimal positions in sheet metal assembly fixtures.Krishnakumar and Melkote 19 developed a fixture layoutoptimization technique that uses the GA to find the fixturelayout that minimizes the deformation of the machined surfacedue to clamping and machining forces over the entire tool path.Locator and clamp positions specified by node numbers.Abuilt-in finite element solver was developed.Some of the studies do not consider the optimization of thelayout for entire tool path and chip removal is not taken intoaccount.Some of the studies used node numbers as designparameters.In this study,a GA tool has been developed to find theoptimal locator and clamp positions in 2D workpiece.Distances from the reference edges as design parameters areused rather than FEA node numbers.Fitness values of realencoded GA chromosomes are obtained from the results ofFEA.ANSYS has been used for FEA calculations.Achromosome library approach is used in order to decreasethe solution time.Developed GA tool is tested on two testproblems.Two case studies are givento illustrate the developedapproach.Main contributions of this paper can be summarizedas follows:(1)developed a GA code integrated with a commercial finiteelement solver;(2)GA uses chromosome library in order to decrease thecomputation time;(3)real design parameters are used rather than FEA nodenumbers;(4)chip removal is taken into account while tool forces movingon the workpiece.3.Genetic algorithm conceptsGenetic algorithms were first developed by John Holland.Goldberg 10 published a book explaining the theory andapplication examples of genetic algorithm in details.A geneticalgorithm is a random search technique that mimics somemechanisms of natural evolution.The algorithm works on apopulation of designs.The population evolves from generationto generation,gradually improving its adaptation to theenvironment through natural selection,fitter individuals havebetter chances of transmitting their characteristics to latergenerations.In the algorithm,the selection of the natural environment isreplaced by artificial selection based on a computed fitness foreach design.The term fitness is used to designate thechromosomes chances of survival and it is essentially theobjective function of the optimization problem.The chromo-somes that define characteristics of biological beings areN.Kaya/Computers in Industry 57(2006)112120113replaced by strings of numerical values representing the designvariables.GA is recognized to be different than traditional gradient-basedoptimizationtechniquesinthefollowingfour major ways10:1.GAs work with a coding of the design variables andparameters in the problem,rather than with the actualparameters themselves.2.GAs make use of population-type search.Many differentdesign points are evaluated during each iteration instead ofsequentially moving from one point to the next.3.GAs need only a fitness or objective function value.Noderivatives or gradients are necessary.4.GAs use probabilistic transition rules to find new designpoints for exploration rather than using deterministic rulesbased on gradient information to find these new points.Algorithm of the basic GA is given as follows:1.Initial population:Generate random population of chromo-somes.2.Fitness:Evaluate the fitness of each chromosome in thepopulation.3.Test:If the end condition is satisfied,stop,and return the bestsolution in current population.4.New population:Create a new population by repeatingfollowing steps until the new population is complete.Reproduction:Select two parent chromosomes from thepopulation according to their fitness.Crossover:With a crossover probability,crossover theparents to form a new offspring(children).If no crossoverwas performed,offspring is an exact copy of parents.Mutation:With a mutation probability,mutate new offspringat each locus(position in chromosome).5.Replace:Use new generated population for a further run ofalgorithm.6.Loop:Go to step 2.3.1.Individual representationThe first andmostimportantstep in preparing anoptimization problem for a GA solution is that of defining aparticular coding of the design variables and their arrangementinto a string of numerical values to be used as the chromosomeby the GA.In most GAs,finite length binary coded strings of ones andzeros are used to describe the parameters for each solution.In amultiparameter optimization problem,individual parametercoding are usually concatenated into a complete string which isshown in Fig.1.In this paper,real representation of binary string is used.Thelength of the string depends on the required precision.Themapping from a binary string to a real number is completed intwo steps:Step 1:Find code length for xi(i=1,.,n):c xmaxi?xmini?rwhere r is the required precision(101,102,103,.).Code length for xiis as follows:lxi n 1where,2nc2n1Total string length is given by:l Xni1lxiStep 2:Mapping from a binary string to a real number:xi xminixmaxi?xmini2n?1Xnj1qij2j?1where qij2 0,1.In order to generate the chromosomes,the length of thechromosome is calculated first.Then random numbers in therange of 0,1aregenerated toform the chromosome.Randomfunction is used in Delphi programming language as a randomnumber generator.3.2.Genetic operatorsEstablishing the GA parameters is very crucial in anoptimization problem because there are no guidelines 20.Thegenetic algorithms contains several operators,e.g.reproduc-tion,crossover,mutation,etc.3.2.1.ReproductionThe reproduction operator allows individual strings to becopied for possible inclusion in the next generation.Afterassesingthefitness valuefor eachstringinthe initialpopulation,only a few strings with high fitness value are considered in thereproduction.There are many different types of reproductionoperatorswhichareproportionalselection,tournamentselection,ranking selection,etc.In this study,tournament selection isselected,since it has better convergence and computational timecomparedtoanyotherreproductionoperator11.Intournamentselection,two individuals are choosen from the population atrandom.Then the string which has best fitness value is selected.This procedure is continued until the size of the reproductionpopulation is equal to the size of the population.3.2.2.CrossoverCrossoveristhenextoperationinthegeneticalgorithm.Thisoperation partially exchanges information between any twoN.Kaya/Computers in Industry 57(2006)112120114Fig.1.Binary representation in GA.selected individuals.Crossover selects genes from parentchromosomes and creates new offsprings.Like reproductionoperator,thereexistanumberofcrossoveroperatorsinGA.Inasingle-point crossoveroperator which is used in this paper,bothstrings are cut at an arbitrary place and the right-side portion ofboth strings are swapped among themselves to create two newstrings,as illustrated in Fig.2.In order to carry out the crossover operation,two individualsare selected from the population at random.Then a randomnumber in the range of 0,1 is generated.If this randomnumber is less than the probability of crossover then theseindividuals are subjected to crossover,otherwise they arecopiedtonewpopulationastheyare.Alsothecrossoverpointisselected at random.Probability of crossover(Pc)is selectedgenerally between 0.6 and 0.9.3.2.3.MutationThis is the process of randomly modifying the string withsmall probability.Mutation operator changes 10 and viceversa with a small probability of mutation(Pm).The need formutation is to keep diversity in the population 11.This is toprevent falling all solutions in population into a local optimumof solved problem.Fig.3 illustrates the mutation operation atseventh bit position.In order to determine whether a chromoseme is to besubjectedtomutation,arandomnumberintherangeof0,1isgenerated.If this random number is less than the probability ofmutation,selected chromosome will be mutated.Probability ofmutation should be selected very low as a high mutation willdestroy fit chromosomes and degenerate the GA into a randomwalk.Pmshould be selected between 0.02 and 0.06 21.3.2.4.Constraint handlingIn most application of GAs to constrained optimizationproblems,the penalty function method has been used.In thisstudy a method proposed by Deb 12 is used.Although apenalty term is added to the objective function,this methoddiffers from conventional GA implementations.The methodproposes to use a tournament selection operator,where twosolutions are compared at a time and the following criteria arealways enforced:-Any feasible solution is preferred to any infeasible solution.-Among two feasible solutions,the one having better fitnessvalue is preferred.-Among two infeasible solutions,the one having smallerconstraint violation is preferred.3.2.5.Elitist strategyIn this strategy,some of the best individuals are copied intothe next generation without applying any genetic operators.Elitist strategy always clones the best individuals of the currentgeneration into the next generation.This guarantees that thebest found design is never lost in future generations.4.Approach4.1.Fixture positioning principlesIn machining process,fixtures are used to keep workpiecesin a desirable position for operations.The most importantcriteria for fixturing are workpiece position accuracy andworkpiece deformation.A good fixture design minimizesworkpiece geometric and machining accuracy errors.Anotherfixturing requirement is that the fixture must limit deformationoftheworkpiece.Itisimportanttoconsiderthecuttingforcesaswell as the clamping forces.Without adequate fixture support,machining operations do not conform to designed tolerances.Finite element analysis is a powerful tool in the resolution ofsome of these problems 22.Common locating method for prismatic parts is 3-2-1method.This method provides the maximum rigidity with theminimum number of fixture elements.Aworkpiece in 3D maybe positively located by means of six points positioned so thatthey restrict nine degrees of freedom of the workpiece.Theother three degrees offreedom are removed by clamp elements.An example layout for 2D workpiece based 3-2-1 locatingprinciple is shown in Fig.4.The number of locating faces must not exceed two so as toavoid a redundant location.Based on the 3-2-1 fixturingprinciple there are two locating planes for accurate locationcontainingtwoand onelocators.Therefore,thereare maximumof two side clampings against each locating plane.Clampingforces are always directed towards the locators in order to forcethe workpiece to contact all locators.The clamping pointN.Kaya/Computers in Industry 57(2006)112120115Fig.2.Illustration of crossover operator.Fig.3.Illustration of mutation operator.Fig.4.3-2-1 locating layout for 2D prismatic workpiece.should be positioned opposite the positioning points to preventthe workpiece from being distorted by the clamping force.Since the machining forces travel along the machining area,it is necessary to ensure that the reaction forces at locators arepositive for all the time.Any negative reaction force indicatesthat theworkpiece is free from fixture elements.In other words,loss of contact or the separation between the workpiece andfixture element might happen when the reaction force isnegative.Positive reaction forces at the locators ensure that theworkpiece maintains contact with all the locators from thebeginning of the cut to the end.The clamping forces should bejust sufficient to constrain and locate the workpiece withoutcausing distortion or damage to the workpiece.Clamping forceoptimization is not considered in this paper.4.2.Genetic algorithm based fixture layout optimizationapproachIn real design problems,the number of design parameterscan be very large and their influence on the objective functioncan be very complicated.The objective function must besmooth and a procedure is needed to compute gradients.Genetic algorithms strongly differ in conception from othersearch methods,including traditional optimization methodsand other stochastic methods 23.By applying GAs to fixturelayout optimization,an optimal or group of sub-optimalsolutions can be obtained.In this study,optimum locator and clamp positions aredetermined using genetic algorithms.They are ideally suitedfor the fixture layout optimization problem since no directanalytical relationship exist between the machining error andthe fixture layout.Since the GA deals with only the designvariables and objective function value for a particular fixturelayout,no gradient or auxiliary information is needed 19