柴油機(jī)汽缸蓋加工工藝及夾具設(shè)計(jì)含6張CAD圖,柴油機(jī),汽缸,加工,工藝,夾具,設(shè)計(jì),cad 附錄1：外文翻譯 薄壁工件夾具設(shè)計(jì)的雙重優(yōu)化模型 摘要 加工過程中必須控制變形，尤其對(duì)于薄壁零件。影響加工變形程度和布局的兩個(gè)主要方面是夾具布局及夾緊力。在本文中，夾具布局和變夾緊力的雙重優(yōu)化模型被運(yùn)用到加工薄壁零件。首先，根據(jù)變形程度和分布考慮最佳夾具布局。然后基于變夾緊力對(duì)上述夾具布局進(jìn)行優(yōu)化。使用有限元法分析工件變形。采用遺傳算法求解優(yōu)化模型。最后通過實(shí)例分析，驗(yàn)證了分層優(yōu)化設(shè)計(jì)方法可以進(jìn)一步減少工件加工變形，提高加工變形均勻度。 1.引言 夾具用于保證機(jī)床中工件被定位和夾緊到正確的位置和方向。設(shè)計(jì)不佳將生產(chǎn)出產(chǎn)生形變的劣品。因此，應(yīng)該合理設(shè)計(jì)出定位元件的位置、夾具及支撐元件，并且能有合理的夾緊力。通常情況下，這些裝置都很大程度上依賴設(shè)計(jì)中的經(jīng)驗(yàn)，據(jù)此選擇夾具等元件安裝位置并計(jì)算夾緊力。因此對(duì)于給定的加工工件，不能保證得出最優(yōu)和接近最優(yōu)的解。因此夾具布局和夾緊力優(yōu)化成為夾具設(shè)計(jì)所需考慮的兩個(gè)重要因素。合理選擇定位元件和夾具，并計(jì)算夾緊力以保證工件變形最小化和均勻。 本文提出雙重優(yōu)化方案用于夾具布局設(shè)計(jì)與變夾緊力的優(yōu)化，目的是使加工元件表面的最大彈性變形得以降低以及最大化的均勻性變形。有限元分析軟件在給定的夾緊力和切割力下計(jì)算工件的變形。隨著遺傳算法的發(fā)展，采用數(shù)學(xué)軟件（MATLAB）直接解決優(yōu)化問題。最后根據(jù)實(shí)例研究說明擬定方案的應(yīng)用。 2.文獻(xiàn)評(píng)論 隨著有關(guān)行業(yè)對(duì)優(yōu)化方案的廣泛運(yùn)用，夾具布局和夾緊力的優(yōu)化近年來取得一些成果。King和Hutter提出了一種使用夾具 - 工件系統(tǒng)的剛體模型進(jìn)行最佳夾具布局設(shè)計(jì)的方法，但考慮到接觸剛度[1]。DeMeter使用剛體模型來分析和合成最佳夾具布局和最小夾緊力[2]。Li和Melkote采用非線性規(guī)劃方法和接觸彈性模型來解決布局優(yōu)化問題[3]。Dengand Melkote [4]提出了一種基于模型的框架，用于確定最小所需的夾緊力，以確保在加工過程中夾具工件的動(dòng)態(tài)穩(wěn)定性。 大多數(shù)上述研究使用非線性規(guī)劃方法，這通常沒有給出最優(yōu)解。夾具設(shè)計(jì)優(yōu)化的問題是非線性的，因?yàn)槟繕?biāo)函數(shù)和設(shè)計(jì)變量之間沒有直接的分析關(guān)系，即加工表面誤差和夾具參數(shù)（定位元件和夾具的位置以及夾緊力）之間。 以前的研究人員已經(jīng)表明，遺傳算法（GA）是解決這些優(yōu)化問題的有用技術(shù)。Vallapuzha等人使用空間坐標(biāo)在基于GA的夾具布局優(yōu)化中進(jìn)行編碼[5]。Krishnakumar等人使用GA找到最小化加工表面變形的夾具布局[6]。Krishnakumar等人 提出了一種迭代算法，通過交替地改變夾具布局和夾緊力，使切割過程中的工件彈性變形最小化[7]。Kaya使用GA和FEM找到2D工件中的最佳定位器和夾緊位置[8]。 Zhou等 提出了一種基于GA的方法，同時(shí)優(yōu)化夾具布局和夾緊力[9]。然而，考慮摩擦和切屑的研究很少，缺乏動(dòng)態(tài)夾緊力優(yōu)化。 3．雙重優(yōu)化模型 薄壁工件受到反作用力，夾緊力和切割力的影響。工件的變形與作用在其上的這些力直接相關(guān)。目標(biāo)函數(shù)表示為最小化加工表面的最大彈性變形，并最大化夾具 - 工件系統(tǒng)中變形的均勻性。 對(duì)于涉及p夾具元件 - 工件接觸和n個(gè)加工載荷階段的夾具，目標(biāo)函數(shù)可以數(shù)學(xué)表達(dá)如下： 其中Δk是加工模擬的第k步的加工區(qū)域的最大彈性變形，Δ是Δk的平均值。 夾具 - 工件系統(tǒng)必須滿足幾個(gè)約束才能有效地執(zhí)行其功能。 優(yōu)化模型中使用的三個(gè)約束描述如下： 其中F ni是第i個(gè)接觸點(diǎn)的法向力，μ是靜摩擦系數(shù)，F(xiàn)τi和Fηi是第i個(gè)接觸點(diǎn)的切向力，pos（i）是第i個(gè)接觸點(diǎn)V（ i）是第i個(gè)接觸點(diǎn)的候選區(qū)域。 首先，確保在加工過程中夾具元件和工件之間沒有滑動(dòng)是重要的，因此在所有接觸點(diǎn)必須滿足公式（3）所寫的庫侖摩擦約束。其次，所有的反作用力必須是正的，以便在加工過程中保持工件與定位器接觸。 這個(gè)約束可以表示為公式（4）。第三，固定件 - 工件接觸點(diǎn)的位置必須在合理的區(qū)域，以確保正確的夾具布局設(shè)計(jì)。 這個(gè)約束可以表示為公式（5）。 通常，施加到工件的夾緊力在加工過程中是固定的，并且通常大于提供夾持穩(wěn)定性所必需的力。其可能導(dǎo)致工件彈性變形過大。 沿刀具路徑的不同位置可能需要不同的力。為了減少工件的變形，在加工過程中應(yīng)提供動(dòng)態(tài)夾緊力并施加在工件上。 對(duì)于夾緊元件 - 工件接觸和n個(gè)加工載荷階段的問題，需要搜索q個(gè)夾緊力的最佳值n次，第j個(gè)搜索模型可以數(shù)學(xué)表達(dá)如下： 無滑動(dòng)約束和接觸約束可以表示為式 （3）和式（4）。 4. 優(yōu)化方案 優(yōu)化過程如圖1所示。 1設(shè)計(jì)可行的夾具布局并優(yōu)化動(dòng)態(tài)夾緊力。 在切割模型中計(jì)算最大切削力，并將力發(fā)送到有限元分析（FEA）模型。 優(yōu)化過程創(chuàng)建一些夾具布局和夾緊力，也可以發(fā)送到FEA模型。 在有限元模塊中，在一定的夾具布局下計(jì)算切削力和夾緊力下的加工變形。 然后將變形發(fā)送到優(yōu)化程序，以搜索最佳夾具布局和動(dòng)態(tài)夾緊力。 圖1 雙重優(yōu)化過程 4.1 GA應(yīng)用于夾具優(yōu)化 GA是模擬生物在自然環(huán)境中的遺傳和進(jìn)化過程而形成的一種自適應(yīng)全局優(yōu)化概率搜索的遺傳算法。遺傳算法首先隨機(jī)產(chǎn)生若干初始群體，利用目標(biāo)函數(shù)構(gòu)造適應(yīng)度函數(shù)，根據(jù)適應(yīng)度函數(shù)計(jì)算個(gè)體適應(yīng)度，較高適應(yīng)度的個(gè)體被 遺傳到下一代群體中的概率較大，通過雜交，變異等遺傳操作作產(chǎn)品生進(jìn)化的下一代群體，如此反復(fù)操作，不斷向更優(yōu)解方向 進(jìn)化，直至獲得問題最優(yōu)解 優(yōu)化夾具設(shè)計(jì)的GA程序?qū)A具布局和夾緊力作為設(shè)計(jì)變量，生成表示不同設(shè)計(jì)的x 1 y 1 z 1 ... f 1 f 2。 并將字符串與自然進(jìn)化的染色體進(jìn)行比較。該GA找到最佳串映射到最佳夾具設(shè)計(jì)。 表1反映GA中有一些主要因素選擇。 表1 選擇GA的參數(shù) 編碼 真實(shí) 縮放 秩 選擇 交叉 突變 控制參數(shù) 余數(shù) 中級(jí) 統(tǒng)一 自適應(yīng) 由于GA很可能會(huì)生成夾具設(shè)計(jì)字符串，在進(jìn)行加工負(fù)載時(shí)不會(huì)完全限制夾具。這些解決方案被認(rèn)為是不可行的，用于驅(qū)動(dòng)GA到可行的解決方案。如果不滿足方程式（2）和（3）中的約束，則設(shè)計(jì)方案被認(rèn)為是不可行的。方案基本上包括將高目標(biāo)函數(shù)值分配給不可行方案，從而在GA的連續(xù)迭代中將其驅(qū)動(dòng)到可行區(qū)域。 對(duì)于約束（4），當(dāng)生成新的個(gè)體時(shí)，需要檢查它們是否滿足條件。為了簡化檢查，使用多邊形表示候選區(qū)域。 數(shù)學(xué)軟件中的多邊形函數(shù)可用于幫助檢查。 基于最佳夾具布局x 1 y 1 z 1 ...，然后通過GA搜索第j個(gè)動(dòng)態(tài)夾緊力（設(shè)計(jì)變量）。 夾緊力根據(jù)確定的順序產(chǎn)生弦，如f j1 f j2 ... f jq。 每個(gè)字符串表示第j步中的一種最佳夾緊力解。 夾緊力來自q組n步，f 11 f 21 ... f n1，f 12 f 22 ...，f 1q f 2q ... f nq是加工過程中的q動(dòng)態(tài)夾緊力。 搜索動(dòng)態(tài)夾緊力的過程與用于確定夾具布局的過程類似。 所需的變化是使夾具布局不變，單獨(dú)添加第j步的切削力，并且只讀取工件的加工變形。 像夾具布局優(yōu)化一樣，應(yīng)該檢查優(yōu)化模型中的限制條件。 4.2． 有限元分析 本研究中有限元分析軟件包用于計(jì)算工件變形。在我們的研究中可以使用半彈性接觸模型和無接觸彈性模型。在半彈性接觸模型中，每個(gè)定位器或支撐件由三個(gè)正交彈簧表示，這些彈簧在X，Y和Z方向上提供約束，每個(gè)夾具與定位器相似，但在正常方向上具有夾緊力。正常方向的彈簧稱為普通彈簧，另外兩個(gè)彈簧稱為切向彈簧。 在這項(xiàng)工作中，考慮到刀具路徑的切屑去除。加工過程中材料的去除會(huì)改變幾何形狀，工件的結(jié)構(gòu)剛度也會(huì)發(fā)生變化。使用元件去除廢屑技術(shù)對(duì)工具運(yùn)動(dòng)和切屑去除分析FEA模型。為了計(jì)算適合度值，對(duì)于每個(gè)加載步驟，存儲(chǔ)位移。然后選擇最大位移作為該夾具設(shè)計(jì)方案的適合度值。 當(dāng)FEA模型中存在大量節(jié)點(diǎn)時(shí)，適應(yīng)度值計(jì)算成本高昂。 因此，有必要加快GA過程的計(jì)算。 隨著一代人的染色體越來越相似。 在這項(xiàng)工作中，計(jì)算的適應(yīng)度值存儲(chǔ)在具有染色體和適應(yīng)度值的SQL Server數(shù)據(jù)庫中。 GA程序首先檢查當(dāng)前染色體的適應(yīng)度值是否已經(jīng)被計(jì)算，如果沒有，則將夾具設(shè)計(jì)方案發(fā)送給有限元分析軟件，否則適合度值取自數(shù)據(jù)庫。 5. 案例分析 幾何和特征如圖1所示。 該工件采用3-2-1夾具布局。 在兩個(gè)表面上添加了四個(gè)對(duì)稱的夾緊力（F3，F(xiàn)4，F(xiàn)5和F6）。 在工件中間的兩個(gè)對(duì)稱位置加上兩個(gè)輔助夾緊力（F 1和F 2），以減少變形。 工件材料為鋁，泊松比為0.3，楊氏模量為71 Gpa。 切割力的最大值為162.235 N（切向）和137.9 N（徑向）。 加工過程由36個(gè)過程模擬，每個(gè)工況中力大小相同。 F 1和F 2是夾具設(shè)計(jì)優(yōu)化的重點(diǎn)。 F 2的位置和值作為優(yōu)化參數(shù)。 優(yōu)化目標(biāo)是在整個(gè)加工過程中最小化加工面上的最大彈性變形f 1，并使σ最小化。 根據(jù)經(jīng)驗(yàn)確定GA的控制參數(shù)。 對(duì)于該示例，P s（群體大?。? 20，P c（交叉的概率）= 0.8，P m（突變的概率）= 0.2，N max（演化的最大數(shù)）= 70。 f 1和σ的懲罰函數(shù)為Φ（f v）= f v + 50，其中f v表示f 1和σ兩者。 當(dāng)最佳適應(yīng)度停滯變化達(dá)到6時(shí)，交叉和突變的概率將變?yōu)?.3和0.6。 圖2. 工件和FEA模型 圖3. 工件最佳適應(yīng)值 最適合值為44.458μm。 設(shè)計(jì)變量和目標(biāo)函數(shù)值如表2所示多目標(biāo)優(yōu)化方法與本例中的經(jīng)驗(yàn)設(shè)計(jì)相比具有優(yōu)勢。最大變形量減少了22.6％，變形均勻度提高了22.1％，最大夾緊力下降了93.7％。 表2.案例優(yōu)化結(jié)果 根據(jù)表2所示的最佳夾具布局搜索動(dòng)態(tài)夾緊力。輔助夾緊力F1和F2的值作為優(yōu)化參數(shù)。優(yōu)化目標(biāo)是最小化當(dāng)前加工區(qū)域的最大彈性變形f1。在GA中，使用以下參數(shù)值：Ps=20，Pc=0.8，Pm = 0.2，Nmax=100。搜索到的動(dòng)態(tài)夾緊力如圖4 。 圖4. 變夾緊力 為了比較夾具解決方案，圖5顯示了加工面的變形分布。由于GA的隨機(jī)性和混合布局與夾緊力之間的耦合，在動(dòng)態(tài)夾緊力下的變形在經(jīng)驗(yàn)設(shè)計(jì)下仍然大于相應(yīng)的變形。在動(dòng)態(tài)夾緊力的作用下，變形分布得到改善。 圖5. 變形比較 6. 結(jié)論 本文提出了夾具布局和變夾緊力優(yōu)化方法。變夾緊力優(yōu)化程序基于最佳夾具布局。 本研究的結(jié)果表明，基于最佳夾具布局的變夾緊力優(yōu)化方法可以最大限度地減少變形，最有效地使變形均勻。對(duì)于NC加工中的變形控制也是有意的。 參考文獻(xiàn) [1]King LS, and Hutter I., Theoretical Approach for Generating Optimal Fixturing Locations for Prismatic Workparts in Automated Assembly, Journal Manufacturing System, 1993, Vol.12, No.5, pp.409-416. [2]De Meter EC., Min-Max Load Model for Optimizing Machine Fixture Performance, ASME- Journal of Engineering Industry, 1995, Vol.117, No.2, pp.183-186. [3] Bo L, Melkote SN., Improved Workpiece Location Accuracy through Fixture Layout Optimization, International Journal of Machine Tools & Manufacture, 1999, Vol.39, No.6, pp.871-883. [4]Deng HY, Melkote SN. Determination of Minimum Clamping Force for Dynamically Stable Fixturing, International Journal of Machine Tools & Manufacture, 2006, Vol.46, No.7-8, pp.847-857. [5]Vallapuzha S, De Meter EC, Choudhuri S, et al. An Investigation into the Use of Spatial Coordinates for the Genetic Algorithm Based Solution of the Fixture Layout Optimization Problem, International Journal of Machine Tools & Manufacture, 2002, Vol.42, No.2, pp.265-275. [6]Kulankara K, Melkote SN., Machining Fixture Layout Optimization Using the Genetic Algorithm, International Journal of Machine Tools & Manufacture, 2000, Vol.40, No.4, pp.579-598. [7]Kulankara K, Satyanarayana S, Melkote SN., Iterative Fixture Layout and Clamping Force Optimization Using the Genetic Algorithm, Journal of Manufacturing Science and Engineering, 2002, Vol.124, No.1, pp.119-125. [8]Kaya N., Machining Fixture Locating and Clamping Position Optimization Using Genetic Algorithms, Computers in Industry, 2006, Vol.57, No.2, pp.112-120. [9]Zhou XL, Zhang WH, Qin GH, On Optimizing Fixture Layout and Clamping Force Simultaneously Using Genetic Algorithm, Mechanical Science and Technology, 2005, Vol.24, No.3, pp.339-342, (in Chinese). 附錄2：外文原文 A Dual Optimization Model of Fixture Design for the Thin-walled Workpiece Weifang Chen, Hua Chen, Lijun Ni and Jianbin Xue Nanjing University of Aeronautics and Astronautics,No.29, Yudao Street, Nanjing 210016 China chenweifang@263.net,meewfchen@nuaa.edu.cn Abstract The deformation must be controlled during machining, especially for the thin-walled workpiece. Fixture layout and clamping force are the major two aspects that influence the degree and distribution of machining deformation. In this paper, a dual optimization model of fixture layout and dynamic clamping force has been established for machining the thin-walled workpiece. First, an optimal fixture layout is generated by considering the deformation degree and distribution. Thereafter, dynamic clamping force are optimized based on the optimal fixture layout. The finite element method is used to analyze the workpiece deformation. A genetic algorithm is developed to solve the optimization model. Finally, an example is used to illustrate that a satisfactory result has been obtained, which is far superior to the experiential one. This optimization method can reduce the machining deformation effectively and improve the distribution condition. 1. Introduction A fixture is used to establish and maintain the required position and orientation of a workpiece in machine tool. A poor design can lead to undesirable workpiece deformation. Consequently, the positions of the locators, clamps and supports should be strategically designed and appropriate clamping forces should be applied. Typically, it relies heavily on the designer’s experience to choose the positions of the fixture elements and to determine the clamping forces. Thus there is no assurance that the resultant solution is optimal or near optimal for a given workpiece. Consequently, the fixture layout and the clamping forces optimization become two important aspects in fixture design. The positions of locators and clamps should be properly selected, and the clamping forces should be calculated so that the workpiece deformation is minimized and uniformed. In this paper, a dual optimization method is presented for the fixture layout design and dynamic clamping forces optimization. The objective is to minimize the maximum  elastic deformation of the machined surfaces and maximize the uniformity of the deformation. The ANSYS software package is used to calculate the deformation of the workpiece under given clamping forces and cutting force. A genetic algorithm is developed and the direct search toolbox of MATLAB is employed to solve the optimization problem. Finally a case study is given to illustrate the application of the proposed approach. 2. Literature review With the wide applications of optimization methods in industry, fixture layout optimization and clamping forces optimization have gained some interesting in recent years. King and Hutter presented a method for optimal fixture layout design using a rigid body model of the fixture-workpiece system but accounting for the contact stiffness [1]. DeMeter used a rigid body model for the analysis and synthesis of optimal fixture layouts and minimum clamping forces [2]. Li and Melkote used a nonlinear programming method and a contact elasticity model to solve the layout optimization problem [3]. Deng and Melkote [4] presented a model-based framework for determining the minimum required clamping forces that ensure the dynamic stability of a fixtured workpiece during machining. Most of the above studies used nonlinear programming methods, which often did not give global or near-global optimum solutions. The problem of fixture design optimization is nonlinear because there is no direct analytical relationship between the objective function and design variables, i.e., between the machined surface error and the fixture parameters (positions of locator and clamp, and clamping forces). Previous researchers had shown that genetic algorithm (GA) was a useful technique in solving such optimization problems. Vallapuzha et al. used spatial coordinates to encode in a GA based fixture layout optimization [5]. Krishnakumar et al. used GA to find the fixture layout that minimized the deformation of the machined surface [6]. Krishnakumar et al. presented an iterative algorithm that minimized the workpiece elastic deformation for the 978 -1-4244-1579-3/07/$25.00 ? 2007 IEEE cutting process by alternatively varying the fixture layout and clamping forces [7]. Kaya used the GA and FEM to find the optimal locator and clamping positions in 2D workpiece [8]. Zhou et al. presented a GA based method that optimizes fixture layout and clamping forces simultaneously [9]. However, there were few studies taking friction and chip removal into account and had lack on dynamic clamping forces optimization. 3. A dual optimization model A thin-walled workpiece is subject to reaction forces, clamping forces and cutting force. The deformation of the workpiece is directly related to these forces acting on it. The objective function is expressed as minimize the maximum elastic deformation of the machined surfaces and maximize the uniformity of the deformation in the fixture–workpiece system. For a fixture involving p fixture element- workpiece contacts and n machining load steps, the objective function can be mathematically stated as follows min(max( 1 , 2 ,?, k ,?, n )) ,k = 1,…,n (1) n 2 = min(σ ) (2) ∑ ( k ? ) min / n k =1 where k refers to maximum elastic deformation at a machining region in the kth step of the machining simulation, is the average of k. A fixture-workpiece system has to satisfy several constraints to effectively perform its functions. Three constraints used in the optimization model are described as follows (3) 2 2 μFni ≥ Fτi + Fηi (4) Fni ≥ 0 pos(i)∈V (i) , i = 1, 2, …, p (5) where Fni is the normal force at the ith contact point, μ is the static coefficient of friction, Fτi and Fηi are the tangential force at the ith contact point, pos(i) is the ith contact point, V(i) is the candidate region of the ith contact point. First, it is important to ensure that there is no slip between the fixture elements and workpiece during machining, so the Coulomb friction constraint written as Eq. (3) must be satisfied at all contact points. Secondly, all of the reaction forces must be positive in order to keep the workpiece in contact with the locators during the machining process. This constraint can be expressed as Eq. (4) . Thirdly, positions of fixture element-workpiece contact points must be in the reasonable regions to ensure proper fixture layout design. This constraint can be expressed as Eq. (5). Typically, the clamping forces applied to the workpiece  are fixed during the machining process and usually larger than necessary to provide fixturing stability. It may cause excessive workpiece elastic deformation. Different forces may be required at different positions along the tool path. In order to decrease the deformation of the workpiece, dynamic clamping forces should be provided and applied on the workpiece during the machining process. For a problem involving q clamping element-workpiece contacts and n machining load steps, the optimal values of q clamping forces need searched n times, and the jth searching model can be mathematically stated as follows min( j ) j=1,2…,n (6) The no slip constraint and the contact constraint can be expressed as Eq. (3) and Eq. (4). 4. Optimization method The optimization process is illustrated in Fig. 1 to design a feasible fixture layout and optimize the dynamic clamping forces. The maximum cutting force is calculated in cutting model and the forces are sent to finite element analysis (FEA) model. Optimization procedure creates some fixture layout and clamping forces which are sent to the FEA model too. In FEA block, machining deformation under the cutting force and clamping forces are calculated under a certain fixture layout. And the deformation is then sent to optimization procedure to search for an optimal fixture layout and dynamic clamping forces. Figure 1. A dual optimization process 4.1. GA applied to fixture optimization GA is robust, stochastic and heuristic optimization method based on biological reproduction processes. Each individual candidate is assigned a fitness value through a fitness function tailored to the specific problem. The GA then uses 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 leads to an increase in the global fitness of the population and the fittest individuals represent the best solutions. The GA procedure to optimize fixture design takes fixture layout and clamping forces as design variables to generate strings x1y1z1 …f1f2 which represent different design. And the strings are compared to chromosomes of natural evolution. The optimal string which GA finds is mapped to the optimal fixture design. There are some main factors in GA which are selected as what is listed in Table 1. Table 1. Selection of GA’s parameters Encoding Real Scaling Rank Selection Remainder Crossover Intermediate Mutation Uniform Control parameter Self-adapting 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 if it does not satisfy the constraints in Eqs. (2) and (3). The penalty method essentially involves assigning a high objective function value to the infeasible scheme, thus driving it to the feasible region in successive iterations of GA. For constraint (4), when new individuals are generated, it is necessary to check up whether they satisfy the condition. The genuine candidate regions are those candidate regions excluding invalid regions. In order to simplify the checking, polygons are used to represent the candidate regions. The polygon function in MATLAB could be used to help the checking. Based on the optimal fixture layout x1y1z1…, the jth dynamic clamping forces(design variables) is then searched by GA. The clamping forces generate strings according to determinate order, such as fj1 fj2…fjq . Each string represents a kind of optimal clamping forces solution in the jth step. Clamping forces got from q groups of n steps, f11 f21…fn 1，f12 f22…，f1q f2q…fnq, are the q dynamic clamping forces in the machining process. The process for searching dynamic clamping forces is similar to which used to determine fixture layout. Changes needed are making fixture layout a constant, adding cutting force of the jth step alone, and only reading out machining deformation of workpiece. Like fixture layout optimization, restriction conditions in optimization model should be checked. 4.2. Finite element analysis The software package of ANSYS is used for calculating the workpiece deformation in this study. The semi-elastic contact model and no contact elastic model can be used in our study. In a semi-elastic contact model, each locator or support is represented by three orthogonal springs that provide restrain 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. 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. 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, displacements are stored for each load step. Then the maximum displacement is selected as fitness value for this fixture design scheme. When there are a large number of nodes in FEA model, the fitness value is costly to compute. Thus it is necessary 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 taken from the database. 5. Case study The geometry and features are shown in Fig. 2. This workpiece uses the 3-2-1 fixture layout. Four symmetrical clamping forces (F3, F4, F5 and F6) are added on the two surfaces. Two assistant clamping forces (F1 and F2) are added on two symmetrical positions of the middle of the workpiece to reduce the deformation. The material of the workpiece is aluminum with a Poisson ration of 0.3 and Young’s modulus of 71 Gpa. The maximum values of cutting force are 162.235 N (tangential) and 137.9 N (radial). The tool path is discretized into 36 load steps and the values of cutting force are same in each step. F1 and F2 are the focuses in the fixture design optimization. The position and value of F2 are taken as optimization parameters. The optimization aim is to minimize the maximum elastic deformation f1 at machining surface in whole machining process and to