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采用遺傳算法優(yōu)化加工夾具定位和加緊位置 摘要：工件變形的問(wèn)題可能導(dǎo)致機(jī)械加工中的空間問(wèn)題。支撐和定位器是用于減少工件彈性變形引起的誤差。支撐、定位器的優(yōu)化和夾具定位是最大限度的減少幾何在工件加工中的誤差的一個(gè)關(guān)鍵問(wèn)題。本文應(yīng)用夾具布局優(yōu)化遺傳算法（GAs）來(lái)處理夾具布局優(yōu)化問(wèn)題。遺傳算法的方法是基于一種通過(guò)整合有限的運(yùn)行于批處理模式的每一代的目標(biāo)函數(shù)值的元素代碼的方法，用于來(lái)優(yōu)化夾具布局。給出的個(gè)案研究說(shuō)明已開(kāi)發(fā)的方法的應(yīng)用。采用染色體文庫(kù)方法減少整體解決問(wèn)題的時(shí)間。已開(kāi)發(fā)的遺傳算法保持跟蹤先前的分析設(shè)計(jì)，因此先前的分析功能評(píng)價(jià)的數(shù)量降低大約93%。結(jié)果表明，該方法的夾具布局優(yōu)化問(wèn)題是多模式的問(wèn)題。優(yōu)化設(shè)計(jì)之間沒(méi)有任何明顯的相似之處，雖然它們提供非常相似的表現(xiàn)。 關(guān)鍵詞：夾具設(shè)計(jì)；遺傳算法；優(yōu)化 1.引言 夾具用來(lái)定位和束縛機(jī)械操作中的工件，減少由于對(duì)確保機(jī)械操作準(zhǔn)確性的夾緊方案和切削力造成的工件和夾具的變形。傳統(tǒng)上，加工夾具是通過(guò)反復(fù)試驗(yàn)法來(lái)設(shè)計(jì)和制造的，這是一個(gè)既造價(jià)高又耗時(shí)的制造過(guò)程。為確保工件按規(guī)定尺寸和公差來(lái)制造，工件必須給予適當(dāng)?shù)亩ㄎ缓蛫A緊以確保有必要開(kāi)發(fā)工具來(lái)消除高造價(jià)和耗時(shí)的反復(fù)試驗(yàn)設(shè)計(jì)方法。適當(dāng)?shù)墓ぜㄎ缓蛫A具設(shè)計(jì)對(duì)于產(chǎn)品質(zhì)量的精密度、準(zhǔn)確度和機(jī)制件的完飾是至關(guān)重要的。 從理論上說(shuō)，3-2-1定位原則對(duì)于定位所有的棱柱形零件是很令人滿意的。該方法具有最大的剛性與最少量的夾具元件。從動(dòng)力學(xué)觀點(diǎn)來(lái)看定位零件意味著限制了自由移動(dòng)物體的六自由度（三個(gè)平動(dòng)自由度和三個(gè)旋轉(zhuǎn)自由度）。在零件下部設(shè)置三個(gè)支撐來(lái)建立工件在垂直軸方向的定位。在兩個(gè)外圍邊緣放置定位器旨在建立工件在水平x軸和y軸的定位。正確定位夾具的工件對(duì)于制造過(guò)程的全面準(zhǔn)確性和重復(fù)性是至關(guān)重要的。定位器應(yīng)該盡可能的遠(yuǎn)距離的分開(kāi)放置并且應(yīng)該放在任何可能的加工面上。放置的支撐器通常用來(lái)包圍工件的重力中心并且盡可能的將其分開(kāi)放置以維持其穩(wěn)定性。夾具夾子的首要任務(wù)是固定夾具以抵抗定位器和支撐器。不應(yīng)該要求夾子反抗加工操作中的切削力。 對(duì)于給定數(shù)量的夾具元件，加工夾具合成的問(wèn)題是尋找?jiàn)A具優(yōu)化布局或工件周?chē)鷬A具元件的位置。本篇文章提出一種優(yōu)化夾具布局遺傳算法。優(yōu)化目標(biāo)是研究一個(gè)二維夾具布局使工件不同位置上最大的彈性變形最小化。ANSYS程序以用于計(jì)算工件變形情況下夾緊力和切削力。本文給出兩個(gè)實(shí)例來(lái)說(shuō)明給出的方法。 2.回顧相關(guān)工程結(jié)構(gòu) 最近幾年夾具設(shè)計(jì)問(wèn)題受到越來(lái)越多的重視。然而，很少有注意力集中于優(yōu)化夾具布局設(shè)計(jì)。Menassa和Devries用FEA計(jì)算變形量使設(shè)計(jì)準(zhǔn)則要求的位點(diǎn)的工件變形最小化。設(shè)計(jì)問(wèn)題是確定支撐器位置。Meyer和Liou提出一個(gè)方法就是使用線性編程技術(shù)合成動(dòng)態(tài)編程條件中的夾具。給出了使夾緊力和定位力最小化的解決方案。Li和Melkote用非線性規(guī)劃方法解決布局優(yōu)化問(wèn)題。這個(gè)方法使工件位置誤差最小化歸于工件的局部彈性變形。Roy和Liao開(kāi)發(fā)出一種啟發(fā)式方法來(lái)計(jì)劃最好的支撐和夾緊位置。Tao等人提出一個(gè)幾何推理的方法來(lái)確定最優(yōu)夾緊點(diǎn)和任意形狀工件的夾緊順序。Liao和Hu提出一種夾具結(jié)構(gòu)分析系統(tǒng)這個(gè)系統(tǒng)基于動(dòng)態(tài)模型分析受限于時(shí)變加工負(fù)載的夾具—工件系統(tǒng)。本文也調(diào)查了夾緊位置的影響。Li和Melkote提出夾具布局和夾緊力最優(yōu)合成方法幫我們解釋加工過(guò)程中的工件動(dòng)力學(xué)。本文提出一個(gè)夾具布局和夾緊力優(yōu)化結(jié)合的程序。他們用接觸彈性建模方法解釋工件剛體動(dòng)力學(xué)在加工期間的影響。Amaral等人用ANSYS驗(yàn)證夾具設(shè)計(jì)的完整性。他們用3-2-1方法。ANSYS提出優(yōu)化分析。Tan等人通過(guò)力鎖合、優(yōu)化與有限建模方法描述了建模、優(yōu)化夾具的分析與驗(yàn)證。 以上大部分的研究使用線性和非線性編程方式這通常不會(huì)給出全局最優(yōu)解決方案。所有的夾具布局優(yōu)化程序開(kāi)始于一個(gè)初始可行布局。這些方法給出的解決方案在很大程度上取決于初始夾具布局。他們沒(méi)有考慮到工件夾具布局優(yōu)化對(duì)整體的變形。 GAs已被證明在解決工程中優(yōu)化問(wèn)題是有用的。夾具設(shè)計(jì)具有巨大的解決空間并需要搜索工具找到最好的設(shè)計(jì)。一些研究人員曾使用GAs解決夾具設(shè)計(jì)及夾具布局問(wèn)題。Kumar等人用GAs和神經(jīng)網(wǎng)絡(luò)設(shè)計(jì)夾具。Marcelin已經(jīng)將GAs用于支撐位置的優(yōu)化。Vallapuzha等人提出基于優(yōu)化方法的GA，它采用空間坐標(biāo)來(lái)表示夾具元件的位置。夾具布局優(yōu)化程序設(shè)計(jì)的實(shí)現(xiàn)是使用MATLAB和遺傳算法工具箱。HYPERMESH和MSC / NASTRAN用于FE模型。Vallapuzha等人提出一些結(jié)果關(guān)于一個(gè)廣泛調(diào)查不同優(yōu)化方法的相對(duì)有效性。他們的研究表明連續(xù)遺傳算法提出了最優(yōu)質(zhì)的解決方案。Li和Shiu使用遺傳算法確定了夾具設(shè)計(jì)最優(yōu)配置的金屬片。MSC/NASTRAN已經(jīng)用于適應(yīng)度值評(píng)價(jià)。Liao提出自動(dòng)選擇最佳夾子和夾鉗的數(shù)目以及它們?cè)诮饘倨系膴A具中的最優(yōu)位置。Krishnakumar和Melkote開(kāi)發(fā)了一種夾具布局優(yōu)化技術(shù)，它是利用遺傳算法找到了夾具布局，由于整個(gè)刀具路徑中的夾緊力和加工力使加工表面變形量最小化。通過(guò)節(jié)點(diǎn)編號(hào)使定位器和夾具位置特殊化。一個(gè)內(nèi)置的有限元求解器研制成功。 一些研究沒(méi)考慮到整個(gè)刀具路徑的優(yōu)化布局以及磨屑清除。一些研究采用節(jié)點(diǎn)編號(hào)作為設(shè)計(jì)參數(shù)。 在本研究中，開(kāi)發(fā)GA工具用于尋找在二維工件中的最優(yōu)定位器和夾緊位置。使用參考邊緣的距離作為設(shè)計(jì)參數(shù)而不是用FEA節(jié)點(diǎn)編號(hào)。真正編碼遺傳算法的染色體的健康指數(shù)是從FEA結(jié)果中獲得的。ANSSYS用于FEA計(jì)算。用染色體文庫(kù)的方法是為了減少解決問(wèn)題的時(shí)間。用兩個(gè)問(wèn)題測(cè)試已開(kāi)發(fā)的遺傳算法工具。給出的兩個(gè)實(shí)例說(shuō)明了這個(gè)開(kāi)發(fā)的方法。本論文的主要貢獻(xiàn)可以概括為以下幾個(gè)方面： （1） 開(kāi)發(fā)了遺傳算法編碼結(jié)合商業(yè)有限元素求解； （2） 遺傳算法采用染色體文庫(kù)以降低計(jì)算時(shí)間； （3） 使用真正的設(shè)計(jì)參數(shù)，而不是有限元節(jié)點(diǎn)數(shù)字； （4） 當(dāng)工具在工件中移動(dòng)時(shí)考慮磨屑清除工具。 3.遺傳算法概念 遺傳算法最初由John Holland開(kāi)發(fā)。Goldberg出版了一本書(shū)，解釋了這個(gè)理論和遺傳算法應(yīng)用實(shí)例的詳細(xì)說(shuō)明。遺傳算法是一種隨機(jī)搜索方法，它模擬一些自然演化的機(jī)制。該算法用于種群設(shè)計(jì)。種群從一代到另一代演化，通過(guò)自然選擇逐漸提高了適應(yīng)環(huán)境的能力，更健康的個(gè)體有更好的機(jī)會(huì)，將他們的特征傳給后代。 該算法中，要基于為每個(gè)設(shè)計(jì)計(jì)算適合性，所以人工選擇取代自然環(huán)境選擇。適應(yīng)度值這個(gè)詞用來(lái)指明染色體生存幾率，它在本質(zhì)上是該優(yōu)化問(wèn)題的目標(biāo)函數(shù)。生物定義的特征染色體用代表設(shè)計(jì)變量的字符串中的數(shù)值代替。 被公認(rèn)的遺傳算法與傳統(tǒng)的梯度基礎(chǔ)優(yōu)化技術(shù)的不同主要有如下四種方式： （1） 遺傳算法和問(wèn)題中的一種編碼的設(shè)計(jì)變量和參數(shù)一起工作而不是實(shí)際參數(shù)本身。 （2） 遺傳算法使用種群—類(lèi)型研究。評(píng)價(jià)在每個(gè)重復(fù)中的許多不同的設(shè)計(jì)要點(diǎn)而不是一個(gè)點(diǎn)順序移動(dòng)到下一個(gè)。 （3） 遺傳算法僅僅需要一個(gè)適當(dāng)?shù)幕蚰繕?biāo)函數(shù)值。沒(méi)有衍生品或梯度是必要的。 （4） 遺傳算法以用概率轉(zhuǎn)換規(guī)則來(lái)發(fā)現(xiàn)新設(shè)計(jì)為探索點(diǎn)而不是利用基于梯度信息的確定性規(guī)則來(lái)找到這些新觀點(diǎn)。 4.方法 4.1夾具定位原則 加工過(guò)程中，用夾具來(lái)保持工件處于一個(gè)穩(wěn)定的操作位置。對(duì)于夾具最重要的標(biāo)準(zhǔn)是工件位置精確度和工件變形。一個(gè)良好的夾具設(shè)計(jì)使工件幾何和加工精度誤差最小化。另一個(gè)夾具設(shè)計(jì)的要求是夾具必須限制工件的變形?？紤]切削力以及夾緊力是很重要的。沒(méi)有足夠的夾具支撐，加工操作就不符合設(shè)計(jì)公差。有限元分析在解決這其中的一些問(wèn)題時(shí)是一種很有力的工具。 棱柱形零件常見(jiàn)的定位方法是3-2-1方法。該方法具有最大剛體度以及最小夾具元件數(shù)。在三維中一個(gè)工件可能會(huì)通過(guò)六自由度定位方法快速定位為了限制工件的九個(gè)自由度。其他的三個(gè)自由度通過(guò)夾具元件消除了?；?-2-1定位原理的二位工件布局的例子如圖4。 圖4 3-2-1對(duì)二維棱柱工件定位布局 定位面得數(shù)量不得超過(guò)兩個(gè)避免冗余的位置?；?-2-1的夾具設(shè)計(jì)原則有兩種精確的定位平面包含于兩個(gè)或一個(gè)定位器。因此，在兩邊有最大的夾緊力抵抗每個(gè)定位平面。夾緊力總是指向定位器為了推動(dòng)工件接觸到所有的定位器。定位點(diǎn)對(duì)面應(yīng)定位夾緊點(diǎn)防止工件由于夾緊力而扭曲。因?yàn)榧庸ちρ刂庸っ?，所以有必要確保定位器的反應(yīng)力在所有時(shí)間內(nèi)是正的。任何負(fù)面的反應(yīng)力表示工件從夾具元件中脫離。換句話說(shuō)，當(dāng)反應(yīng)力是負(fù)的時(shí)候，工件和夾具元件之間接觸或分離的損失可能發(fā)生。定位器內(nèi)正的反應(yīng)力確保工件從切削開(kāi)始到結(jié)束都能接觸到所有的定位器。夾緊力應(yīng)該充分束縛和定位工件且不導(dǎo)致工件的變形或損壞。本文不考慮夾緊力的優(yōu)化。 4.2基于夾具布局優(yōu)化方法的遺傳算法 在實(shí)際設(shè)計(jì)問(wèn)題中，設(shè)計(jì)參數(shù)的數(shù)量可能很大并且它們對(duì)目標(biāo)函數(shù)的影響會(huì)是非常復(fù)雜的。目標(biāo)函數(shù)曲線必須是光滑的并且需要一個(gè)程序計(jì)算梯度。遺傳算法在理念上遠(yuǎn)不同于其他的探究方法，它們包括傳統(tǒng)的優(yōu)化方法和其他隨機(jī)方法。通過(guò)運(yùn)用遺傳算法來(lái)對(duì)夾具優(yōu)化布局，可以獲得一個(gè)或一組最優(yōu)的解決方案。 本項(xiàng)研究中，最優(yōu)定位器和夾具定位使用遺傳算法確定。它們是理想的適合夾具布局優(yōu)化問(wèn)題的方法因?yàn)闆](méi)有直接分析的關(guān)系存在于加工誤差和夾具布局中。因?yàn)檫z傳算法僅僅為一個(gè)特別的夾具布局處理設(shè)計(jì)變量和目標(biāo)函數(shù)值，所以不需要梯度或輔助信息。 建議方案流程圖如圖5。 使用開(kāi)發(fā)的命名為GenFix的Delphi語(yǔ)言軟件來(lái)實(shí)現(xiàn)夾具布局優(yōu)化。位移量用ANSYS軟件計(jì)算。通過(guò)WinExec功能在GenFix中運(yùn)行ANSYS很簡(jiǎn)單。GenFix和ANSYS之間相互作用通過(guò)四部實(shí)現(xiàn)： （1） 定位器和夾具位置從二進(jìn)制代碼字符串中提取作為真正的參數(shù)。 （2） 這些參數(shù)和ANSYS輸入批處理文件（建模、解決方案和后置處理）用WinExec功能傳給ANSYS。 （3） 解決后將位移值寫(xiě)成一個(gè)文本文件。 （4） GenFix讀這個(gè)文件并為當(dāng)前定位器和夾緊位置計(jì)算適應(yīng)度值。 為了減少計(jì)算量，染色體與適應(yīng)度值儲(chǔ)存在一個(gè)文庫(kù)里以備進(jìn)一步評(píng)估。GenFix首先檢查是否當(dāng)前的染色體的適應(yīng)度值已經(jīng)在之前被計(jì)算過(guò)。如果沒(méi)有，定位器位置被送到ANSYS，否則從文庫(kù)中取走適應(yīng)度值。在初始種群產(chǎn)生過(guò)程中，檢查每一個(gè)染色體可行與否。如果違反了這個(gè)原則，它就會(huì)出局然后新的染色體就產(chǎn)生了。這個(gè)程序創(chuàng)造了可行的初始種群。這保證了初始種群的每個(gè)染色體在夾緊力和切削力作用下工件的穩(wěn)定性。用兩個(gè)測(cè)試用例來(lái)驗(yàn)證提到的遺傳算法計(jì)劃。第一個(gè)實(shí)例是使用Himmelblau功能。在第二個(gè)測(cè)試用例中，遺傳算法計(jì)劃用來(lái)優(yōu)化均布載荷作用下梁的支撐位置。 圖5 設(shè)計(jì)方法的流程與ANSYS相配合流程 5.夾具布局優(yōu)化的個(gè)案研究 該夾具布局優(yōu)化問(wèn)題的定義是：找到定位器和夾子的位置以使在特定區(qū)工件變形降到最小程度。那么多的定位器和夾子并不是設(shè)計(jì)參數(shù)因?yàn)樗鼈冊(cè)?-2-1方案中是已知的和固定的。因此，設(shè)計(jì)參數(shù)的選擇如同定位器和夾子的位置。本研究中不考慮摩擦力。兩個(gè)實(shí)例研究來(lái)說(shuō)明以提出的方法。 6.結(jié)論 本文提出了一個(gè)夾具布局優(yōu)化的評(píng)價(jià)優(yōu)化技術(shù)。ANSYS用于FE計(jì)算適應(yīng)度值?？梢钥吹剑z傳算法和FE方法的結(jié)合對(duì)當(dāng)今此類(lèi)問(wèn)題似乎是一種強(qiáng)大的方法。遺傳算法特別適合應(yīng)用于解決那些在目標(biāo)函數(shù)和設(shè)計(jì)變量之間不存在一個(gè)定義明確的數(shù)學(xué)關(guān)系的問(wèn)題。結(jié)果證明遺傳算法在夾具布局優(yōu)化問(wèn)題方面的成功應(yīng)用。本項(xiàng)研究中，遺傳算法在夾具布局優(yōu)化應(yīng)用中的主要困難是較高的計(jì)算成本。種群中每個(gè)染色體需要工件的重嚙合。但是，染色體庫(kù)的使用，F(xiàn)E評(píng)價(jià)的數(shù)量從6000下降到415。這就導(dǎo)致了巨大的增益計(jì)算效益。其他減少處理時(shí)間的方法是在局域網(wǎng)內(nèi)使用分布式計(jì)算。 該方法結(jié)果表明，夾具布局優(yōu)化問(wèn)題是多模態(tài)問(wèn)題。優(yōu)化設(shè)計(jì)之間沒(méi)有任何明顯的相似之處盡管他們提供非常相似的表現(xiàn)。結(jié)果表明夾具布局問(wèn)題是多模態(tài)問(wèn)題然而用于夾具設(shè)計(jì)的啟發(fā)式規(guī)則應(yīng)該用于遺傳算法來(lái)選擇最優(yōu)的設(shè)計(jì)。 Machining fixture locating and clamping position optimization using genetic algorithms Necmettin Kaya* Department of Mechanical Engineering, Uludag University, Go¨ru¨kle, Bursa 16059, Turkey Received 8 July 2004; accepted 26 May 2005 Available online 6 September 2005 Abstract Deformation of the workpiece may cause dimensional problems in machining. Supports and locators are used in order to reduce the error caused by elastic deformation of the workpiece. The optimization of support, locator and clamp locations is a critical problem to minimize the geometric error in workpiece machining. In this paper, the application of genetic algorithms (GAs) to the fixture layout optimization is presented to handle fixture layout optimization problem. A genetic algorithm based approach is developed to optimise fixture layout through integrating a finite element code running in batch mode to compute the objective function values for each generation. Case studies are given to illustrate the application of proposed approach. Chromosome library approach is used to decrease the total solution time. Developed GA keeps track of previously analyzed designs; therefore the numbers of function evaluations are decreased about 93%. The results of this approach show that the fixture layout optimization problems are multi-modal problems. Optimized designs do not have any apparent similarities although they provide very similar performances. Keywords: Fixture design; Genetic algorithms; Optimization 1. Introduction Fixtures are used to locate and constrain a workpiece during a machining operation, minimizing workpiece and fixture tooling deflections due to clamping and cutting forces are critical to ensuring accuracy of the machining operation. Traditionally, machining fixtures are designed and manufactured through trial-and-error, which prove to be both expensive and time-consuming to the manufacturing process. To ensure a workpiece is manufactured according to specified dimensions and tolerances, it must be appropriately located and clamped, making it imperative to develop tools that will eliminate costly and time-consuming trial-and-error designs. Proper workpiece location and fixture design are crucial to product quality in terms of precision, accuracy and finish of the machined part. Theoretically, the 3-2-1 locating principle can satisfactorily locate all prismatic shaped workpieces. This method provides the maximum rigidity with the minimum number of fixture elements. To position a part from a kinematic point of view means constraining the six degrees of freedom of a free moving body (three translations and three rotations). Three supports are positioned below the part to establish the location of the workpiece on its vertical axis. Locators are placed on two peripheral edges and intended to establish the location of the workpiece on the x and y horizontal axes. Properly locating the workpiece in the fixture is vital to the overall accuracy and repeatability of the manufacturing process. Locators should be positioned as far apart as possible and should be placed on machined surfaces wherever possible. Supports are usually placed to encompass the center of gravity of a workpiece and positioned as far apart as possible to maintain its stability. The primary responsibility of a clamp in fixture is to secure the part against the locators and supports. Clamps should not be expected to resist the cutting forces generated in the machining operation. For a given number of fixture elements, the machining fixture synthesis problem is the finding optimal layout or positions of the fixture elements around the workpiece. In this paper, a method for fixture layout optimization using genetic algorithms is presented. The optimization objective is to search for a 2D fixture layout that minimizes the maximum elastic deformation at different locations of the workpiece. ANSYS program has been used for calculating the deflection of the part under clamping and cutting forces. Two case studies are given to illustrate the proposed approach. 2. Review of related works Fixture design has received considerable attention in recent years. However, little attention has been focused on the optimum fixture layout design. Menassa and DeVries[1]used FEA for calculating deflections using the minimization of the workpiece 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 linear programming technique to synthesize fixtures for dynamic machining conditions. Solution for the minimum clamping forces and locator forces is given. Li and Melkote[3]used a nonlinear programming method to solve the layout optimization problem. The method minimizes workpiece location errors due to localized elastic deformation of the workpiece. Roy andLiao[4]developed a heuristic method to plan for the best supporting and clamping positions. Tao et al.[5]presented a geometrical reasoning methodology for determining the optimal clamping points and clamping sequence for arbitrarily shaped workpieces. Liao and Hu[6]presented a system for fixture configuration analysis based on a dynamic model which analyses the fixture–workpiece system subject to time-varying machining loads. The influence of clamping placement is also investigated. Li and Melkote[7]presented a fixture layout and clamping force optimal synthesis approach that accounts for workpiece dynamics during machining. A combined fixture layout and clamping force optimization procedure presented.They used the contact elasticity modeling method that accounts for the influence of workpiece rigid body dynamics during machining. Amaral et al. [8] used ANSYS to verify fixture design integrity. They employed 3-2-1 method. The optimization analysis is performed in ANSYS. Tan et al. [9] described the modeling, analysis and verification of optimal fixturing configurations by the methods of force closure, optimization and finite element modeling. Most of the above studies use linear or nonlinear programming methods which often do not give global optimum solution. All of the fixture layout optimization procedures start with an initial feasible layout. Solutions from these methods are depending on the initial fixture layout. They do not consider the fixture layout optimization on overall workpiece deformation. The GAs has been proven to be useful technique in solving optimization problems in engineering [10–12]. Fixture design has a large solution space and requires a search tool to find the best design. Few researchers have used the GAs for fixture design and fixture layout problems. Kumar et al. [13] have applied both GAs and neural networks for designing a fixture. Marcelin [14] has used GAs to the optimization of support positions. Vallapuzha et al. [15] presented GA based optimization method that uses spatial coordinates to represent the locations of fixture elements. Fixture layout optimization procedure was implemented using MATLAB and the genetic algorithm toolbox. HYPERMESH and MSC/NASTRAN were used for FE model. Vallapuzha et al. [16] presented results of an extensive investigation into the relative effectiveness of various optimization methods. They showed that continuous GA yielded the best quality solutions. Li and Shiu [17] determined the optimal fixture configuration design for sheet metal assembly using GA. MSC/NASTRAN has been used for fitness evaluation. Liao [18] presented a method to automatically select the optimal numbers of locators and clamps as well as their optimal positions in sheet metal assembly fixtures. Krishnakumar and Melkote [19] developed a fixture layout optimization technique that uses the GA to find the fixture layout that minimizes the deformation of the machined surface due to clamping and machining forces over the entire tool path. Locator and clamp positions are specified by node numbers. A built-in finite element solver was developed. Some of the studies do not consider the optimization of the layout for entire tool path and chip removal is not taken into account. Some of the studies used node numbers as design parameters. In this study, a GA tool has been developed to find the optimal locator and clamp positions in 2D workpiece. Distances from the reference edges as design parameters are used rather than FEA node numbers. Fitness values of real encoded GA chromosomes are obtained from the results of FEA. ANSYS has been used for FEA calculations. A chromosome library approach is used in order to decrease the solution time. Developed GA tool is tested on two test problems. Two case studies are given to illustrate the developed approach. Main contributions of this paper can be summarized as follows: (1) developed a GA code integrated with a commercial finite element solver; (2) GA uses chromosome library in order to decrease the computation time; (3) real design parameters are used rather than FEA node numbers; (4) chip removal is taken into account while tool forces moving on the workpiece. 3. Genetic algorithm concepts Genetic algorithms were first developed by John Holland. Goldberg [10] published a book explaining the theory and application examples of genetic algorithm in details. A genetic algorithm is a random search technique that mimics some mechanisms of natural evolution. The algorithm works on a population of designs. The population evolves from generation to generation, gradually improving its adaptation to the environment through natural selection; fitter individuals have better chances of transmitting their characteristics to later generations. In the algorithm, the selection of the natural environment is replaced by artificial selection based on a computed fitness for each design. The term fitness is used to designate the chromosome’s chances of survival and it is essentially the objective function of the optimization problem. The chromosomes that define characteristics of biological beings are replaced by strings of numerical values representing the design variables. GA is recognized to be different than traditional gradient based optimization techniques in the following four major ways [10]: 1. GAs work with a coding of the design variables and parameters in the problem, rather than with the actual parameters themselves. 2. GAs makes use of population-type search. Many different design points are evaluated during each iteration instead of sequentially moving from one point to the next. 3. GAs needs only a fitness or objective function value. No derivatives or gradients are necessary. 4. GAs use probabilistic transition rules to find new design points for exploration rather than using deterministic rules based on gradient information to find these new points. 4. Approach 4.1. Fixture positioning principles In machining process, fixtures are used to keep workpieces in a desirable position for operations. The most important criteria for fixturing are workpiece position accuracy and workpiece deformation. A good fixture design minimizes workpiece geometric and machining accuracy errors. Another fixturing requirement is that the fixture must limit deformation of the workpiece. It is important to consider the cutting forces as well 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 of some of these problems [22]. Common locating method for prismatic parts is 3-2-1 method. This method provides the maximum rigidity with the minimum number of fixture elements. A workpiece in 3D may be positively located by means of six points positioned so that they restrict nine degrees of freedom of the workpiece. The other three degrees of freedom are removed by clamp elements. An example layout for 2D workpiece based 3-2-1 locating principle is shown in Fig. 4. Fig. 4. 3-2-1 locating layout for 2D prismatic workpiece The number of locating faces must not exceed two so as to avoid a redundant location. Based on the 3-2-1 fixturing principle there are two locating planes for accurate location containing two and one locators. Therefore, there are maximum of two side clampings against each locating plane. Clamping forces are always directed towards the locators in order to force the workpiece to contact all locators. The clamping point should be positioned opposite the positioning points to prevent the 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 are positive for all the time. Any negative reaction force indicates that the workpiece is free from fixture elements. In other words, loss of contact or the separation between the workpiece and fixture element might happen when the reaction force is negative. Positive reaction forces at the locators ensure that the workpiece maintains contact with all the locators from the beginning of the cut to the end. The clamping forces should be just sufficient to constrain and locate the workpiece without causing distortion or damage to the workpiece. Clamping force optimization is not considered in this paper. 4.2. Genetic algorithm based fixture layout optimization approach In real design problems, the number of design parameters can be very large and their influence on the objective function can be very complicated. The objective function must be smooth and a procedure is needed to compute gradients. Genetic algorithms strongly differ in conception from other search methods, including traditional optimization methods and other stochastic methods [23]. By applying GAs to fixture layout optimization, an optimal or group of sub-optimal solutions can be obtained. In this study, optimum locator and clamp positions are determined using genetic algorithms. They are ideally suited for the fixture layout optimization problem since no direct analytical relationship exists between the machining error and the fixture layout. Since the GA deals with only the design variables and objective function value for a particular fixture layout, no gradient or auxiliary information is needed [19]. The flowchart of the proposed approach is given in Fig. 5. Fixture layout optimization is implemented using developed software written in Delphi language named GenFix. Displacement values are calculated in ANSYS software [24]. The execution of ANSYS in GenFix is simply done by WinExec function in Delphi. The interaction between GenFix and ANSYS is implemented in four steps: (1) Locator and clamp positions are extracted from binary string as real parameters. (2) These parameters and ANSYS input batch file (modeling, solution and post processing commands) are sent to ANSYS using WinExec function. (3) Displacement values are written to a text file after solution. (4) GenFix reads this file and computes fitness value for current locator and clamp positions. In order to reduce the computation time, chromosomes and fitness values are stored in a library for further evaluation. GenFix first checks if current chromosome’s fitness value has been calculated before. If not, locator positions are sent to ANSYS, otherwise fitness values are taken from the library. During generating of the initial population, every chromosome is checked whether it is feasible or not. If the constraint is violated, it is eliminated and new chromosome is created. This process creates entirely feasible initial population. This ensures that workpiece is stable under the action of clamping and cutting forces for every chromosome in the initial population. The written GA program was validated using two test cases. The first test case uses Himmelblau function [21]. In the second test case, the GA program was used to optimise the support positions of a beam under uniform loading. 5. Fixture layout optimization case studies The fixture layout optimization problem is defined as: finding the positions of the locators and clamps, so that workpiece deformation at specific region is minimized. Note that number of locators and clamps are not design parameter, since they are known and fixed for the 3-2-1 locating scheme. Hence, the design parameters are selected as locator and clamp positions. Friction is not considered in this paper. Two case studies are given to illustrate the proposed approach. 6. Conclusion In this paper, an evolutionary optimization technique of fixture layout optimization is presented. ANSYS has been used for FE calculation of fitness values. It is seen that the combined genetic algorithm and FE method approach seems to be a powerful approach for present type problems. GA approach is particularly suited for problems where there does not exist a well-defined mathematical relationship between the objective function and the design variables. The results prove the success of the application of GAs for the fixture layout optimization problems. In this study, the major obstacle for GA application in fixture layout optimization is the high computation cost. Re-meshing of the workpiece is required for every chromosome in the population. But, usages of chromosome library, the number of FE evaluations are decreased from 6000 to 415. This results in a tremendous gain in computational efficiency. The other way to decrease the solution time is to use distributed computation in a local area network. The results of this approach show that the fixture layout optimization problems are multi-modal problems. Optimized designs do not have any apparent similarities although they provide very similar performances. It is shown that fixture layout problems are multi-modal therefore heuristic rules for fixture design should be used in GA to select best design among others. Fig. 5. The flowchart of the proposed methodology and ANSYS interface.