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沈陽理工大學學士學位論文 附錄二 ：中文翻譯 通過夾具布局設計和夾緊力的優(yōu)化控制變形 摘 要 工件變形必須控制在數值控制機械加工過程之中。夾具布局和夾緊力是影響加工變形程度和分布的兩個主要方面。在本文提出了一種多目標模型的建立，以減低變形的程度和增加均勻變形分布。有限元方法應用于分析變形。遺傳算法發(fā)展是為了解決優(yōu)化模型。最后舉了一個例子說明，一個令人滿意的結果被求得， 這是遠優(yōu)于經驗之一的。多目標模型可以減少加工變形有效地改善分布狀況。 關鍵詞：夾具布局；夾緊力； 遺傳算法；有限元方法 1 引言 夾具設計在制造工程中是一項重要的程序。這對于加工精度是至關重要。一個工件應約束在一個帶有夾具元件，如定位元件，夾緊裝置，以及支撐元件的夾具中加工。定位的位置和夾具的支力，應該從戰(zhàn)略的設計，并且適當的夾緊力應適用。該夾具元件可以放在工件表面的任何可選位置。夾緊力必須大到足以進行工件加工。通常情況下，它在很大程度上取決于設計師的經驗，選擇該夾具元件的方案，并確定夾緊力。因此，不能保證由此產生的解決方案是某一特定的工件的最優(yōu)或接近最優(yōu)的方案。因此，夾具布局和夾緊力優(yōu)化成為夾具設計方案的兩個主要方面。 定位和夾緊裝置和夾緊力的值都應適當的選擇和計算，使由于夾緊力和切削力產生的工件變形盡量減少和非正式化。 夾具設計的目的是要找到夾具元件關于工件和最優(yōu)的夾緊力的一個最優(yōu)布局或方案。在這篇論文里， 多目標優(yōu)化方法是代表了夾具布局設計和夾緊力的優(yōu)化的方法。 這個觀點是具有兩面性的。一，是盡量減少加工表面最大的彈性變形； 另一個是盡量均勻變形。 ANSYS軟件包是用來計算工件由于夾緊力和切削力下產生的變形。遺傳算法是MATLAB的發(fā)達且直接的搜索工具箱，并且被應用于解決優(yōu)化問題。最后還給出了一個案例的研究，以闡述對所提算法的應用。 2 文獻回顧 隨著優(yōu)化方法在工業(yè)中的廣泛運用，近幾年夾具設計優(yōu)化已獲得了更多的利益。夾具設計優(yōu)化包括夾具布局優(yōu)化和夾緊力優(yōu)化。King 和 Hutter提出了一種使用剛體模型的夾具-工件系統(tǒng)來優(yōu)化夾具布局設計的方法。DeMeter也用了一個剛性體模型，為最優(yōu)夾具布局和最低的夾緊力進行分析和綜合。他提出了基于支持布局優(yōu)化的程序與計算質量的有限元計算法。李和melkote用了一個非線性編程方法和一個聯(lián)絡彈性模型解決布局優(yōu)化問題。兩年后， 他們提交了一份確定關于多鉗夾具受到準靜態(tài)加工力的夾緊力優(yōu)化的方法。他們還提出了一關于夾具布置和夾緊力的最優(yōu)的合成方法，認為工件在加工過程中處于動態(tài)。相結合的夾具布局和夾緊力優(yōu)化程序被提出，其他研究人員用有限元法進行夾具設計與分析。蔡等對menassa和devries包括合成的夾具布局的金屬板材大會的理論進行了拓展。秦等人建立了一個與夾具和工件之間彈性接觸的模型作為參考物來優(yōu)化夾緊力與，以盡量減少工件的位置誤差。Deng和melkote 提交了一份基于模型的框架以確定所需的最低限度夾緊力，保證了被夾緊工件在加工的動態(tài)穩(wěn)定。 大部分的上述研究使用的是非線性規(guī)劃方法，很少有全面的或近全面的最優(yōu)解決辦法。所有的夾具布局優(yōu)化程序必須從一個可行布局開始。此外，還得到了對這些模型都非常敏感的初步可行夾具布局的解決方案。夾具優(yōu)化設計的問題是非線性的，因為目標的功能和設計變量之間沒有直接分析的關系。例如加工表面誤差和夾具的參數之間（定位、夾具和夾緊力）。 以前的研究表明，遺傳算法（ GA ）在解決這類優(yōu)化問題中是一種有用的技術。吳和陳用遺傳算法確定最穩(wěn)定的靜態(tài)夾具布局。石川和青山應用遺傳算法確定最佳夾緊條件彈性工件。vallapuzha在基于優(yōu)化夾具布局的遺傳算法中使用空間坐標編碼。他們還提出了針對主要競爭夾具優(yōu)化方法相對有效性的廣泛調查的方法和結果。這表明連續(xù)遺傳算法取得最優(yōu)質的解決方案。krishnakumar和melkote 發(fā)展了一個夾具布局優(yōu)化技術，用遺傳算法找到夾具布局，盡量減少由于在整個刀具路徑的夾緊和切削力造成的加工表面的變形。定位器和夾具位置被節(jié)點號碼所指定。krishnakumar等人還提出了一種迭代算法，盡量減少工件在整個切削過程之中由不同的夾具布局和夾緊力造成的彈性變形。Lai等人建成了一個分析模型，認為定位和夾緊裝置為同一夾具布局的要素靈活的一部分。Hamedi 討論了混合學習系統(tǒng)用來非線性有限元分析與支持相結合的人工神經網絡（ ANN ）和GA。人工神經網絡被用來計算工件的最大彈性變形，遺傳算法被用來確定最佳鎖模力。Kumar建議將迭代算法和人工神經網絡結合起來發(fā)展夾具設計系統(tǒng)。Kaya用迭代算法和有限元分析，在二維工件中找到最佳定位和夾緊位置，并且把碎片的效果考慮進去。周等人。提出了基于遺傳算法的方法，認為優(yōu)化夾具布局和夾緊力的同時，一些研究沒有考慮為整個刀具路徑優(yōu)化布局。一些研究使用節(jié)點數目作為設計參數。一些研究解決夾具布局或夾緊力優(yōu)化方法，但不能兩者都同時進行。 有幾項研究摩擦和碎片考慮進去了。 碎片的移動和摩擦接觸的影響對于實現更為現實和準確的工件夾具布局校核分析來說是不可忽視的。因此將碎片的去除效果和摩擦考慮在內以實現更好的加工精度是必須的。 在這篇論文中，將摩擦和碎片移除考慮在內，以達到加工表面在夾緊和切削力下最低程度的變形。一多目標優(yōu)化模型被建立了。一個優(yōu)化的過程中基于GA和有限元法提交找到最佳的布局和夾具夾緊力。最后，結果多目標優(yōu)化模型對低剛度工件而言是比較單一的目標優(yōu)化方法、經驗和方法。 3 多目標優(yōu)化模型夾具設計 一個可行的夾具布局必須滿足三限制。首先，定位和夾緊裝置不能將拉伸勢力應用到工件；第二，庫侖摩擦約束必須施加在所有夾具-工件的接觸點。夾具元件-工件接觸點的位置必須在候選位置。為一個問題涉及夾具元件-工件接觸和加工負荷步驟，優(yōu)化問題可以在數學上仿照如下： 這里的△表示加工區(qū)域在加工當中j次步驟的最高彈性變形。 其中 是△的平均值； 是正常力在i次的接觸點； μ是靜態(tài)摩擦系數； fhi是切向力在i次的接觸點； pos(i)是i次的接觸點； 是可選區(qū)域的i次接觸點； 整體過程如圖1所示，一要設計一套可行的夾具布局和優(yōu)化的夾緊力。最大切削力在切削模型和切削力發(fā)送到有限元分析模型中被計算出來。優(yōu)化程序造成一些夾具布局和夾緊力，同時也是被發(fā)送到有限元模型中。在有限元分析座內，加工變形下，切削力和夾緊力的計算方法采用有限元方法。根據某夾具布局和變形，然后發(fā)送給優(yōu)化程序，以搜索為一優(yōu)化夾具方案。 圖1 夾具布局和夾緊力優(yōu)化過程 4 夾具布局設計和夾緊力的優(yōu)化 4.1 遺傳算法 遺傳算法（ GA ）是基于生物再生產過程的強勁，隨機和啟發(fā)式的優(yōu)化方法。基本思路背后的遺傳算法是模擬“生存的優(yōu)勝劣汰“的現象。每一個人口中的候選個體指派一個健身的價值，通過一個功能的調整，以適應特定的問題。遺傳算法，然后進行復制，交叉和變異過程消除不適宜的個人和人口的演進給下一代。人口足夠數目的演變基于這些經營者引起全球健身人口的增加和優(yōu)勝個體代表全最好的方法。 遺傳算法程序在優(yōu)化夾具設計時需夾具布局和夾緊力作為設計變量，以生成字符串代表不同的布置。字符串相比染色體的自然演變，以及字符串，它和遺傳算法尋找最優(yōu)，是映射到最優(yōu)的夾具設計計劃。在這項研究里，遺傳算法和MATLAB的直接搜索工具箱是被運用的。 收斂性遺傳算法是被人口大小、交叉的概率和概率突變所控制的 。只有當在一個人口中功能最薄弱功能的最優(yōu)值沒有變化時，nchg達到一個預先定義的價值ncmax ，或有多少幾代氮，到達演化的指定數量上限nmax， 沒有遺傳算法停止。有五個主要因素，遺傳算法，編碼，健身功能，遺傳算子，控制參數和制約因素。 在這篇論文中，這些因素都被選出如表1所列。 表1 遺傳算法參數的選擇 由于遺傳算法可能產生夾具設計字符串，當受到加工負荷時不完全限制夾具。這些解決方案被認為是不可行的，且被罰的方法是用來驅動遺傳算法，以實現一個可行的解決辦法。1夾具設計的計劃被認為是不可行的或無約束，如果反應在定位是否定的。在換句話說，它不符合方程（2）和（3）的限制。罰的方法基本上包含指定計劃的高目標函數值時不可行的。因此，驅動它在連續(xù)迭代算法中的可行區(qū)域。對于約束（4），當遺傳算子產生新個體或此個體已經產生，檢查它們是否符合條件是必要的。真正的候選區(qū)域是那些不包括無效的區(qū)域。在為了簡化檢查，多邊形是用來代表候選區(qū)域和無效區(qū)域的。多邊形的頂點是用于檢查?！癷npolygon ”在MATLAB的功能可被用來幫助檢查。 4.2 有限元分析 ANSYS軟件包是用于在這方面的研究有限元分析計算。有限元模型是一個考慮摩擦效應的半彈性接觸模型，如果材料是假定線彈性。如圖2所示，每個位置或支持，是代表三個正交彈簧提供的制約。 圖2 考慮到摩擦的半彈性接觸模型 在x ， y和z 方向和每個夾具類似，但定位夾緊力在正常的方向。彈力在自然的方向即所謂自然彈力，其余兩個彈力即為所謂的切向彈力。接觸彈簧剛度可以根據向赫茲接觸理論計算如下： 隨著夾緊力和夾具布局的變化，接觸剛度也不同，一個合理的線性逼近的接觸剛度可以從適合上述方程的最小二乘法得到。連續(xù)插值，這是用來申請工件的有限元分析模型的邊界條件。在圖3中說明了夾具元件的位置，顯示為黑色界線。每個元素的位置被其它四或六最接近的鄰近節(jié)點所包圍。 圖3 連續(xù)插值 這系列節(jié)點，如黑色正方形所示，是（37，38，31和30 ），（9，10 ，11 ， 18，17號和16號）和（ 26，27 ，34 ， 41，40和33 ）。這一系列彈簧單元，與這些每一個節(jié)點相關聯(lián)。對任何一套節(jié)點，彈簧常數是： 這里， kij 是彈簧剛度在的j -次節(jié)點周圍i次夾具元件， Dij 是i次夾具元件和的J -次節(jié)點周圍之間的距離， ki是彈簧剛度在一次夾具元件位置, ηi 是周圍的i次夾具元素周圍的節(jié)點數量 為每個加工負荷的一步，適當的邊界條件將適用于工件的有限元模型。在這個工作里，正常的彈簧約束在這三個方向（X ， Y ， Z ）的和在切方向切向彈簧約束，（X ， Y ）。夾緊力是適用于正常方向（Z）的夾緊點。整個刀具路徑是模擬為每個夾具設計計劃所產生的遺傳算法應用的高峰期的X ，Y ，z切削力順序到元曲面，其中刀具通行證。在這工作中，從刀具路徑中歐盟和去除碎片已經被考慮進去。在機床改變幾何數值過程中，材料被去除，工件的結構剛度也改變。 因此，這是需要考慮碎片移除的影響。有限元分析模型，分析與重點的工具運動和碎片移除使用的元素死亡技術。在為了計算健身價值，對于給定夾具設計方案，位移存儲為每個負載的一步。那么，最大位移是選定為夾具設計計劃的健身價值。 遺傳算法的程序和ANSYS之間的互動實施如下。定位和夾具的位置以及夾緊力這些參數寫入到一個文本文件。那個輸入批處理文件ANSYS軟件可以讀取這些參數和計算加工表面的變形。 因此， 健身價值觀，在遺傳算法程序，也可以寫到當前夾具設計計劃的一個文本文件。 當有大量的節(jié)點在一個有限元模型時，計算健身價值是很昂貴的。因此，有必要加快計算遺傳算法程序。作為這一代的推移，染色體在人口中取得類似情況。在這項工作中，計算健身價值和染色體存放在一個SQL Server數據庫。遺傳算法的程序，如果目前的染色體的健身價值已計算之前，先檢查；如果不，夾具設計計劃發(fā)送到ANSYS，否則健身價值觀是直接從數據庫中取出。嚙合的工件有限元模型，在每一個計算時間保持不變。每計算模型間的差異是邊界條件，因此，網狀工件的有限元模型可以用來反復“恢復”ANSYS 命令。 5 案例研究 一個關于低剛度工件的銑削夾具設計優(yōu)化問題是被顯示在前面的論文中，并在以下各節(jié)加以表述。 5.1 工件的幾何形狀和性能 工件的幾何形狀和特點顯示在圖4中，空心工件的材料是鋁390與泊松比0.3和71Gpa的楊氏模量。外廓尺寸152.4mm×127mm*76.2mm.該工件頂端內壁的三分之一是經銑削及其刀具軌跡，如圖4 所示。夾具元件中應用到的材料泊松比0.3和楊氏模量的220的合金鋼。 圖4 空心工件 5.2 模擬和加工的運作 舉例將工件進行周邊銑削，加工參數在表2中給出?；谶@些參數，切削力的最高值被作為工件內壁受到的表面載荷而被計算和應用，當工件處于330.94 n（切）、398.11 N （下徑向）和22.84 N （下軸） 的切削位置時。整個刀具路徑被26個工步所分開，切削力的方向被刀具位置所確定 表2加工參數和條件 。 5.3 夾具設計方案 夾具在加工過程中夾緊工件的規(guī)劃如圖5所示。 圖5 定位和夾緊裝置的可選區(qū)域 一般來說， 3-2-1定位原則是夾具設計中常用的。夾具底板限制三個自由度，在側邊控制兩個自由度。這里，在Y=0mm截面上使用了4個定點（L1，L2 ， L3和14 ），以定位工件并限制2自由度；并且在Y=127mm的相反面上，兩個壓板（C1,C2）夾緊工件。在正交面上，需要一個定位元件限制其余的一個自由度，這在優(yōu)化模型中是被忽略的。在表3中給出了定位加緊點的坐標范圍。 表3 設計變量的約束 由于沒有一個簡單的一體化程序確定夾緊力，夾緊力很大部分（6673.2N）在初始階段被假設為每一個夾板上作用的力。且從符合例5的最小二乘法，分別由4.43×107 N/m 和5.47×107 N/m得到了正常切向剛度。 5.4 遺傳控制參數和懲罰函數 在這個例子中，用到了下列參數值：Ps=30, Pc=0.85, Pm=0.01, Nmax=100和Ncmax=20.關于f1和σ的懲罰函數是 這里fv可以被F1或σ代表。當nchg達到6時，交叉和變異的概率將分別改變成0.6和0.1. 5.5 優(yōu)化結果 連續(xù)優(yōu)化的收斂過程如圖6所示。且收斂過程的相應功能（1）和（2）如圖7、圖8所示。優(yōu)化設計方案在表4中給出。 圖6 夾具布局和夾緊力優(yōu)化程序的收斂性遺傳算法 圖7 第一個函數值的收斂 圖8第二個函數值的收斂性 表4 多目標優(yōu)化模型的結果 表5 各種夾具設計方案結果進行比較， 5.6 結果的比較 從單一目標優(yōu)化和經驗設計中得到的夾具設計的設計變量和目標函數值，如表5所示。單一目標優(yōu)化的結果，在論文中引做比較。在例子中，與經驗設計相比較，單一目標優(yōu)化方法有其優(yōu)勢。最高變形減少了57.5 ％，均勻變形增強了60.4 ％。最高夾緊力的值也減少了49.4 ％ 。從多目標優(yōu)化方法和單目標優(yōu)化方法的比較中可以得出什么呢？最大變形減少了50.2％ ，均勻變形量增加了52.9 ％，最高夾緊力的值減少了69.6 ％ 。加工表面沿刀具軌跡的變形分布如圖9所示。很明顯，在三種方法中，多目標優(yōu)化方法產生的變形分布最均勻。 與結果比較，我們確信運用最佳定位點分布和最優(yōu)夾緊力來減少工件的變形。圖10示出了一實例夾具的裝配。 圖9沿刀具軌跡的變形分布 圖10 夾具配置實例 6 結論 本文介紹了基于GA和有限元的夾具布局設計和夾緊力的優(yōu)化程序設計。優(yōu)化程序是多目標的：最大限度地減少加工表面的最高變形和最大限度地均勻變形。ANSYS軟件包已經被用于 健身價值的有限元計算。對于夾具設計優(yōu)化的問題，GA和有限元分析的結合被證明是一種很有用的方法。 在這項研究中，摩擦的影響和碎片移動都被考慮到了。為了減少計算的時間，建立了一個染色體的健身數值的數據庫，且網狀工件的有限元模型是優(yōu)化過程中多次使用的。 傳統(tǒng)的夾具設計方法是單一目標優(yōu)化方法或經驗。此研究結果表明，多目標優(yōu)化方法比起其他兩種方法更有效地減少變形和均勻變形。這對于在數控加工中控制加工變形是很有意義的。 參考文獻 1、 King LS，Hutter（ 1993年） 自動化裝配線上棱柱工件最佳裝夾定位生成的理論方法。De Meter EC (1995) 優(yōu)化機床夾具表現的Min - Max負荷模型。 2、 De Meter EC (1998) 快速支持布局優(yōu)化。Li B, Melkote SN (1999) 通過夾具布局優(yōu)化改善工件的定位精度。 3、 Li B, Melkote SN (2001) 夾具夾緊力的優(yōu)化和其對工件的定位精度的影響。 4、 Li B, Melkote SN (1999) 通過夾具布局優(yōu)化改善工件的定位精度。 5、 Li B, Melkote SN (2001) 夾具夾緊力的優(yōu)化和其對工件定位精度的影響。 6、 Li B, Melkote SN (2001) 最優(yōu)夾具設計計算工件動態(tài)的影響。 7、 Lee JD, Haynes LS (1987) 靈活裝夾系統(tǒng)的有限元分析。 8、 Menassa RJ, DeVries WR (1991) 運用優(yōu)化方法在夾具設計中選擇支位。 9、 Cai W, Hu SJ, Yuan JX (1996) 變形金屬板材的裝夾的原則、算法和模擬。 10、 Qin GH, Zhang WH, Zhou XL (2005) 夾具裝夾方案的建模和優(yōu)化設計。 11、Deng HY, Melkote SN (2006) 動態(tài)穩(wěn)定裝夾中夾緊力最小值的確定。 12、Wu NH, Chan KC (1996) 基于遺傳算法的夾具優(yōu)化配置方法。 13、Ishikawa Y, Aoyama T(1996) 借助遺傳算法對裝夾條件的優(yōu)化。 14、Vallapuzha S, De Meter EC, Choudhuri S, et al (2002) 一項關于空間坐標對基于遺傳算法的夾具優(yōu)化問題的作用的調查。 15、Vallapuzha S, De Meter EC, Choudhuri S, et al (2002) 夾具布局優(yōu)化方法成效的調查。 16、Kulankara K, Melkote SN (2000) 利用遺傳算法優(yōu)化加工夾具的布局。 17、Kulankara K, Satyanarayana S, Melkote SN (2002) 利用遺傳算法優(yōu)化夾緊布局和夾緊力。 18、Lai XM, Luo LJ, Lin ZQ (2004) 基于遺傳算法的柔性裝配夾具布局的建模與優(yōu)化。 19、Hamedi M (2005) 通過一種人工神經網絡和遺傳算法混合的系統(tǒng)設計智能夾具。 20、Kumar AS, Subramaniam V, Seow KC (2001) 采用遺傳算法固定裝置的概念設計。 21、Kaya N (2006) 利用遺傳算法優(yōu)化加工夾具的定位和夾緊點。 22、Zhou XL, Zhang WH, Qin GH (2005) 遺傳算法用于優(yōu)化夾具布局和夾緊力。 23、Kaya N, ?ztürk F (2003) 碎片位移和摩擦接觸的運用對工件夾具布局的校核。 62 ORIGINAL ARTICLE Deformation control through fixture layout design and clamping force optimization Weifang Chen Δ 2 jj; :::; Δ j C12 C12 C12 C12 ; :::; Δ n jj C0C1 t s ; j ? 1; 2; :::; n e1T Subject to m F ni jjC21 ?????????????????? F 2 ti t F 2 hi q e2T F ni C21 0 e3T pos ieT2VieT; i ? 1; 2; :::; p e4T where Δ j refers to the maximum elastic deformation at a machining region in the j-th step of the machining operation, σ? ?????????????????????????????????????? X n j?1 Δj C0Δ C0 C16C17 2 C30 n v u u t Δ is the average of Δ j F ni is the normal force at the i-th contact point μ is the static coefficient of friction F ti ; F hi are the tangential forces at the i-th contact point pos(i) is the i-th contact point V(i) is the candidate region of the i-th contact point. The overall process is illustrated in Fig. 1 to design a feasible fixture layout and to optimize the clamping force. The maximal cutting force is calculated in cutting model and the force is sent to finite element analysis (FEA) model. Optimization procedure creates some fixture layout and clamping force which are sent to the FEA model too. In FEA block, machining deformation under the cutting force and the clamping force is calculated using finite element method under a certain fixture layout, and the deformation is then sent to optimization procedure to search for an optimal fixture scheme. 4 Fixture layout design and clamping force optimization 4.1 A genetic algorithm Genetic algorithms (GA) are robust, stochastic and heuristic optimization methods based on biological reproduction processes. The basic idea behind GA is to simulate “survival of the fittest” phenomena. Each individual candidate in the population is assigned a fitness value through a fitness function tailored to the specific problem. The GA then conducts reproduction, crossover and mutation processes to eliminate unfit individuals and the population evolves to the next generation. Sufficient number of evolutions of the population based on these operators lead to an increase in the global fitness of the population and the fittest individual represents the best solution. The GA procedure to optimize fixture design takes fixture layout and clamping force as design variables to generate strings which represent different layouts. The strings are compared to the chromosomes of natural evolution, and the string, which GA find optimal, is mapped to the optimal fixture design scheme. In this study, the genetic algorithm and direct search toolbox of MATLAB are employed. The convergence of GA is controlled by the population size (P s ), the probability of crossover (P c )andthe probability of mutations (P m ). Only when no change in the best value of fitness function in a population, N chg , reaches a pre-defined value NC max , or the number of generations, N, reaches the specified maximum number of evolutions, N max ., did the GA stop. There are five main factors in GA, encoding, fitness function, genetic operators, control parameters and con- straints. In this paper, these factors are selected as what is listed in Table 1. Since GA is likely to generate fixture design strings that do not completely restrain the fixture when subjected to machining loads. These solutions are considered infeasible and the penalty method is used to drive the GA to a feasible solution. A fixture design scheme is considered infeasible or unconstrained if the reactions at the locators are negative, in other words, it does not satisfy the constraints in equations (2)and(3). The penalty method essentially involves Machining Process Model FEA Optimization procedure cutting forces fitness Optimization result Fixture layout and clamping force Fig. 1 Fixture layout and clamp- ing force optimization process Table 1 Selection of GA’s parameters Factors Description Encoding Real Scaling Rank Selection Remainder Crossover Intermediate Mutation Uniform Control parameter Self-adapting Int J Adv Manuf Technol assigning a high objective function value to the scheme that is infeasible, thus driving it to the feasible region in successive iterations of GA. For constraint (4), when new individuals are generated by genetic operators or the initial generation is generated, it is necessary to check up whether they satisfy the conditions. The genuine candidate regions are those excluding invalid regions. In order to simplify the checking, polygons are used to represent the candidate regions and invalid regions. The vertex of the polygons are used for the checking. The “inpolygon” function in MATLAB could be used to help the checking. 4.2 Finite element analysis The software package of ANSYS is used for FEA calculations in this study. The finite element model is a semi-elastic contact model considering friction effect, where the materials are assumed linearly elastic. As shown in Fig. 2, each locator or support is represented by three orthogonal springs that provide restrains in the X, Y and Z directions and each clamp is similar to locator but clamping force in normal direction. The spring in normal direction is called normal spring and the other two springs are called tangential springs. The contact spring stiffness can be calculated according to the Herz contact theory [8] as follows k iz ? 16R C3 i E C32 i 9 C16C171 3 f iz 1 3 k iz ? k iy ? 6 E C3 i 2C0v fi G fi t 2C0v wi G wi C16C17 C01 C1 k iz 8 > : e5T where k iz , k ix , k iy are the tangential and normal contact stiffness, 1 R C3 i ? 1 R wi t 1 R fi is the nominal contact radius, 1 E C3 i ? 1C0 V 2 wi E wi t 1C0 V 2 fi E fi is the nominal contact elastic modulus, R wi , R fi are radius of the i-th workpiece and fixture element, E wi , E fi are Young’s moduli for the i-th workpiece and fixture materials, ν wi , ν fi are Poisson ratios for the i-th workpiece and fixture materials, G wi , G fi are shear moduli for the i-th workpiece and fixture materials and f iz is the reaction force at the i-th contact point in the Z direction. Contact stiffness varies with the change of clamping force and fixture layout. A reasonable linear approximation of the contact stiffness can be obtained from a least-squares fit to the above equation. The continuous interpolation, which is used to apply boundary conditions to the workpiece FEA model, is Fig. 2 Semi-elastic contact model taking friction into account Spring position Fixture element position 1234567 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 Fig. 3 Continuous interpolation Fig. 4 A hollow workpiece Table 2 Machining parameters and conditions Parameter Description Type of operation End milling Cutter diameter 25.4 mm Number of flutes 4 Cutter RPM 500 Feed 0.1016 mm/tooth Radial depth of cut 2.54 mm Axial depth of cut 25.4 mm Radial rake angle 10 Helix angle 30 Projection length 92.07 mm Int J Adv Manuf Technol illustrated in Fig. 3. Three fixture element locations are shown as black circles. Each element location is surrounded by its four or six nearest neighboring nodes. These sets of nodes, which are illustrated by black squares, are {37, 38, 31 and 30}, {9, 10, 11, 18, 17 and 16} and {26, 27, 34, 41, 40 and 33}. A set of spring elements are attached to each of these nodes. For any set of nodes, the spring constant is k ij ? d ij P k2h i d ik k i e6T where k ij is the spring stiffness at the j-th node surrounding the i-th fixture element, d ij is the distance between the i-th fixture element and the j-th node surrounding it, k i is the spring stiffness at the i-th fixture element location. η i is the number of nodes surrounding the i-th fixture element location. For each machining load step, appropriate boundary conditions have to be applied to the finite element model of the workpiece. In this work, the normal springs are constrained in the three directions (X, Y, Z)andthe tangential springs are constrained in the tangential direc- tions (X, Y). Clamping forces are applied in the normal direction (Z) at the clamp nodes. The entire tool path is simulated for each fixture design scheme generated by the GA by applying the peak X, Y, Z cutting forces sequentially to the element surfaces over which the cutter passes [23]. In this work, chip removal from the tool path is taken into account. The removal of the material during machining alters the geometry, so does the structural stiffness of the workpiece. Thus, it is necessary to consider chip removal affects. The FEA model is analyzed with respect to tool movement and chip removal using the element death technique. In order to calculate the fitness value for a given fixture design scheme, displacements are stored for each load step. Then the maximum displacement is selected as fitness value for this fixture design scheme. The interaction between GA procedure and ANSYS is implemented as follows. Both the positions of locators and clamps, and the clamping force are extracted from real strings. These parameters are written to a text file. The input batch file of ANSYS could read these parameters and calculate the deformation of machined surfaces. Thus the fitness values in GA procedure can also be written to a text file for current fixture design scheme. It is costly to compute the fitness value when there are a largenumberofnodesinanFEMmodel.Thusitisnecessary to speed up the computation for GA procedure. As the generation goes by, chromosomes in the population are getting similar. In this work, calculated fitness values are stored in a SQL Server database with the chromosomes and fitness values. GA procedure first checks if current chromosome’s fitness value has been calculated before, if not, fixture design scheme are sent to ANSYS, otherwise fitness values are directly taken from the database. The meshing of workpiece FEA model keeps same in every calculating time. The difference among every calculating model is the boundary conditions. Thus, the meshed workpiece FEA model could be used repeatedly by the “resume” command in ANSYS. 5 Case study An example of milling fixture design optimization problem for a low rigidity workpiece displayed in previous research papers [16, 18, 22] is presented in the following sections. Fig. 5 Candidate regions for the locators and clamps Table 3 Bound of design variables Minimum Maximum X /mm Z /mm X /mm Z /mm L 1 0 0 76.2 38.1 L 2 76.2 0 152.4 38.1 L 3 0 38.1 76.2 76.2 L 4 76.2 38.1 152.4 76.2 C 1 0 0 76.2 76.2 C 2 76.2 0 152.4 76.2 F 1 /N 0 6673.2 F 2 /N 0 6673.2 Int J Adv Manuf Technol 5.1 Workpiece geometry and properties The geometry and features of the workpiece are shown in Fig. 4. The material of the hollow workpiece is aluminum 390 with a Poisson ration of 0.3 and Young’s modulus of 71 Gpa. The outline dimensions are 152.4 mm×127 mm× 76.2 mm. The one third top inner wall of the workpiece is undergoing an end-milling process and its cutter path is also shown in Fig. 4. The material of the employed fixture elements is alloy steel with a Poisson ration of 0.3 and Young’s modulus of 220 Gpa. 5.2 Simulating and machining operation A peripheral end milling operation is carried out on the example workpiece. The machining parameters of the operation are given in Table 2. Based on these parameters, the maximum values of cutting forces that are calculated and applied as element surface loads on the inner wall of the workpiece at the cutter position are 330.94 N (tangential), 398.11 N (radial) and 22.84 N (axial). The entire tool path is discretized into 26 load steps and cutting force directions are determined by the cutter position. 5.3 Fixture design plan The fixture plan for holding the workpiece in the machining operation is shown in Fig. 5.Generally,the3–2–1 locator principleisusedinfixturedesign.Thebasecontrols3degrees. One side controls two degrees, and another orthogonal side controlsonedegree.Here,itusesfourlocators(L1,L2,L3and L4) on the Y=0 mm face to locate the workpiece controlling two degrees, and two clamps (C1, C2) on the opposite face where Y=127 mm, to hold it. On the orthogonal side, one locator is needed to control the remaining degree, which is neglectedintheoptimalmodel.Thecoordinateboundsforthe locating/clamping regions are given in Table 3. Since there is no simple rule-of-thumb procedure for determining the clamping force, a large value of the clamping force of 6673.2 N was initially assumed to act at each clamp, and the normal and tangential contact stiffness obtained from a least-squares fit to Eq. (5) are 4.43×10 7 N/m and 5.47×10 7 N/m separately. 5.4 Genetic control parameters and penalty function The control parameters of the GA are determined empiri- cally. For this example, the following parameter values are Fig. 6 Convergence of GA for fixture layout and clamping force optimization procedure Fig. 7 Convergence of the first function values Fig. 8 Convergence of the second function values Table 4 Result of the multi-objective optimization model Multi-objective optimization X /mm Z /mm L 1 17.102 30.641 L 2 108.169 25.855 L 3 21.315 56.948 L 4 127.846 60.202 C 1 22.989 62.659 C 2 117.615 25.360 F 1 /N 167.614 F 2 /N 382.435 f 1 /mm 0.006568 σ/mm 0.002683 Int J Adv Manuf Technol used: P s =30, P c =0.85, P m =0.01, N max =100 and N cmax = 20. The penalty function for f 1 and σ is φ f v eT?f v t 50 Here f v can be represented by f 1 or σ. When N chg reaches 6 the probability of crossover and mutation will be change into 0.6 and 0.1 separately. 5.5 Optimization result The convergence behavior for the successive optimization steps is shown in Fig. 6, and the convergence behaviors of corresponding functions (1) and (2) are shown in Fig. 7 and Fig. 8. The optimal design scheme is given in Table 4. 5.6 Comparison of the results The design variables and objective function values of fixture plans obtained from single objective optimization and from that designed by experience are shown in Table 5. The single objective optimization result in the paper [22]is quoted for comparison. The single objective optimization method has its preponderance comparing with that designed by experience in this example case. The maximum deformation has reduced by 57.5%, the uniformity of the deformation has enhanced by 60.4% and the maximum clamping force value has degraded by 49.4%. What could be drawn from the comparison between the multi-objective optimization method and the single objective optimization method is that the maximum deformation has reduced by 50.2%, the uniformity of the deformation has enhanced by 52.9% and the maximum clamping force value has degraded by 69.6%.The deformation distribution of the machined surfaces along cutter path is shown in Fig. 9. Obviously, the deformation from that of multi-objective optimization method distributes most uniformly in the deformations among three methods. With the result of comparison, we are sure to apply the optimal locators distribution and the optimal clamping force to reduce the deformation of workpiece. Figure 10 shows the configuration of a real-case fixture. 6 Conclusions This paper presented a fixture layout design and clamping force optimization procedure based on the GA and FEM. The optimization procedure is multi-objective: minimizing the maximum deformation of the machined surfaces and maximizing the uniformity of the deformation. The ANSYS software package has been used for FEM calculation of fitness values. The combinationof GAand FEM isproven to be a powerful approach for fixture design optimization problems. In this study, both friction effects and chip removal effects are considered. In order to reduce the computation time, a database is established for the chromosomes and fitness values, and the meshed workpiece FEA model is repeatedly used in the optimization process. Table 5 Comparison of the results of various fixture design schemes Experimental optimization Single objective optimization X/mm Z/mm X/mm Z/mm L 1 12.700 12.700 16.720 34.070 L 2 139.7 12.700 145.360 17.070 L 3 12.700 63.500 18.400 57.120 L 4 139.700 63.500 146.260 58.590 C 1 12.700 38.100 5.830 56.010 C 2 139.700 38.100 104.400 22.740 F 1 /N 2482 444.88 F 2 /N 2482 1256.13 f 1 /mm 0.031012 0.013178 σ/mm 0.014377 0.005696 Fig. 9 Distribution of the deformation along cutter path Fig. 10 A real case fixture configuration Int J Adv Manuf Technol Thetraditionalfixturedesignmethodsaresingleobjective optimization method or by experience. The results of this study show that the multi-objective optimization method is more effective in minimizing the deformation and uniform- ing the deformation than other two methods. It is meaningful for machining deformation control in NC machining. References 1. 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