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單澆口優(yōu)化注塑模 李季泉 李德群 郭智英，呂海原 (塑料成型技術(shù)與裝備研究院,上海交通大學，上海 200030,中國) 郵箱：hutli@163．corn 2006年9月22日收到；2007年3月19日修訂通過 摘要：本文論述了一種單澆口位置優(yōu)化注塑模具的方法。模具設(shè)計應(yīng)盡量減少注塑制品翹曲變形，因為翹曲涉及到質(zhì)量的關(guān)鍵問題，對大多數(shù)塑件而言，都受澆口位置影響。特征翹曲度是用以描述指定特征翹曲程度的，常用于評價塑件的翹曲變形。利用數(shù)值模擬技術(shù)，可以優(yōu)化最佳的澆口位置，其中，模擬退火算法就是用來尋找最佳的澆口位置。最后，利用案例論證了所提出的方法是有效的。 關(guān)鍵詞：注塑模， 澆口位置，結(jié)構(gòu)優(yōu)化，功能翹曲 美國內(nèi)政部：10．163 1／jzus．2007．A1077 文獻標識碼：A中圖分類號：TQ320．66 概述 塑料注塑成型，是一種高效的生產(chǎn)技術(shù)，常用于生產(chǎn)各種塑料產(chǎn)品，尤其是那些精度要求高、形狀復(fù)雜、產(chǎn)量高的產(chǎn)品。注塑模具的質(zhì)量是由塑料材料、零件外形、模具結(jié)果和過程條件決定的。注塑模最重要的基本上是由以下三部分組成：型腔，澆口和澆道，冷卻系統(tǒng)。 Lam 和 Seow （ 2000），Jin和Lain（ 2002）采用不同壁厚的方式達到平衡型腔，使平衡充填過程中內(nèi)部腔形成一個均勻分布的壓力和溫度，可大幅度減少該部的翹曲。型腔平衡是影響零件質(zhì)量的重要因素之一，尤其是當零件有功能要求，其厚度不能頻繁變化。從這個角度注塑設(shè)計模具，澆口是由其尺寸和位置、澆道系統(tǒng)的規(guī)模和布局所決定的，澆口尺寸和澆道布局通常定為常量。相對地，澆口位置和澆道的大小是比較有彈性，能夠影響零件質(zhì)量。因此，他們往往對設(shè)計參數(shù)進行優(yōu)化。 Lee和Kim（1996年）認為流道和澆口的大小能平衡澆道系統(tǒng)。流道維持平衡可以理解為有相同腔的多腔模具的不同入口壓力，在每一個腔每一個熔體流道底部有不同的體積和幾何形狀。該方法已顯示壓力在整個多腔模具成型周期中的單腔里均勻分布。 Zhai等（2005年）認證兩個澆口位置優(yōu)化，它的一個型腔是用一個在壓力梯度的基礎(chǔ)上的高效率的搜索方法（ PGSS） ，為流道多澆口零件定位，通過改變流道尺寸使熔接線在最理想的位置（zhai等， 2006 ）。大體積部分，多澆口需要縮短流徑，與減少相應(yīng)的注射壓力。該方法大可成為設(shè)計多澆口單型腔的澆口和流道。 許多注塑件是只制作一個澆口，無論是單型腔模具或多型腔模具。因此，單澆口的澆口位置是最常見的設(shè)計優(yōu)化參數(shù)。形狀分析方法是由Courbebaisse和Gaarrcia 2002年提出，通過選擇注射成型的最佳澆口位置。后來，他們研制的這種理論進一步研究和應(yīng)用于單一澆口位置優(yōu)化的一個L形例子（Courbebaisse，2005年）。 它易于使用，而不耗費時間，而且它也只厚度均勻的單型腔。 Pandelidis和Zou（1990年）提出的優(yōu)化澆口位置，由間接質(zhì)量相關(guān)引起的翹曲和物質(zhì)降解，這代表著加權(quán)溫度差，摩擦過熱的時間。翹曲是受上述因素的影響， 但它們之間的關(guān)系并不明確。 因此，優(yōu)化效果是受制于質(zhì)量因素。 Lee和Kim（ l996b ）研制出一種自動選擇澆口位置的方法，其中一套由設(shè)計師提出的確定最優(yōu)澆口位置是位于相鄰節(jié)點。因為第一步是基于設(shè)計師的主張， 所以在相當大的程度上，受限于設(shè)計師的經(jīng)驗。 Lam和Jin（2001）開發(fā)了澆口位置優(yōu)化方法，基于減少標準偏差的流徑長度（標準差[大] ）和在成型充填過程中的標準偏差的灌裝時間（標準差[ T ] ）。隨后，沈等人（ 2004 年 ） ，通過最小充氣壓力，灌裝時間區(qū)別不同的水路路徑，溫差變化大，以及過度包裝的百分比優(yōu)化了澆口位置設(shè)計。Zhai等 （ 2005 年）在去年底調(diào)查了最佳澆口位置與評價標準的注射壓力。這些研究人員介紹目標函數(shù)應(yīng)對注塑成型灌裝操作，這對相關(guān)產(chǎn)品的品質(zhì)有益。 但已經(jīng)觀察到它們之間的相關(guān)性是非常復(fù)雜和不清晰。人們還很難選擇適當?shù)募訖?quán)因子為每個函數(shù)。 一個新的目標函數(shù)來評價注塑制品翹曲變形，以優(yōu)化澆口位置。直接衡量零件質(zhì)量，這項調(diào)查定義特征翹曲來評價零件翹曲，就是從"平面翹曲"模擬產(chǎn)出塑料洞察力（電傳等）的軟件。目標函數(shù)最小化，在澆口位置優(yōu)化，以達到最低變形。 模擬退火算法是用來尋找最優(yōu)澆口位置。 給出了一個例子來說明建議優(yōu)化程序的有效性。 質(zhì)量措施：特征翹曲 定義特征翹曲 運用優(yōu)化理論設(shè)計澆口，零件的質(zhì)量措施必須指定在初審。術(shù)語"質(zhì)量"可轉(zhuǎn)介許多產(chǎn)品性能，如力學，熱學， 電子，光學，工效學或幾何性質(zhì)。有兩種零件質(zhì)量測量方法：直接和間接。一個有預(yù)測性的模型，從數(shù)值模擬結(jié)果，可作為一個直接的質(zhì)量測量。相比之下，間接測量的零件質(zhì)量是正相關(guān)目標質(zhì)量，但它并不能提供對其質(zhì)量的直接估計。 翹曲，在相關(guān)工程的間接質(zhì)量測量，是一種注塑成型的方法。這種方法用作分析填充不同流道的時間差，溫度差，過度包裝的比例問題等。這是很明顯的，翹曲是受這些因素的影響，但翹曲和這些因素的關(guān)系是不明確的，而且決定這些因素所占的比重是相當困難的。因此，用上述目標函數(shù)優(yōu)化大概不會減低零件翹曲，甚至是完美的優(yōu)化技術(shù)。有時，不恰當加權(quán)因素，將導(dǎo)致完全錯誤的結(jié)果。 一些統(tǒng)計量計算，節(jié)點位移被定性為直接質(zhì)量測量，以達到最低變形優(yōu)化研究。統(tǒng)計數(shù)量通常是最多節(jié)點位移，平均每年有10%的節(jié)點位移，而且整體平均節(jié)點位移（Lee和Kim， 1995 ; 1996 ） 。這些節(jié)點的位移容易從數(shù)值模擬結(jié)果獲得，統(tǒng)計值，在一定程度上代表著變形。 但統(tǒng)計位移不能有效地描述變形的注塑件。 在工業(yè)方面，設(shè)計者和制造商通常更加注意，部分上翹曲在某些特點上超過整個變形注射模塑件的程度。在這項研究中，特征翹曲是用來形容變形的注塑件。特征翹曲是表面上的最大位移與表面特征的預(yù)計長度之比（見公式1） ： （1） 其中γ是特征翹曲， h是特征表面偏離該參考平臺的最高位移，L是在與參考方向平行的參考平臺上的表面特征的預(yù)計長度。 對于翹曲特征（這里只討論平面特征） ，通常在參考平面內(nèi)分為兩個區(qū)域，它代表一個二維坐標系統(tǒng)： （2） 其中，是特征翹曲在X ， Y方向， ，是表面特征的預(yù)計長度在 X ，Y上的投影。 特征翹曲的評定 與相應(yīng)的參考平面和投影方向結(jié)合起來測定目標特征后，其L的值可以從圖中用解析幾何立即計算出來 。在特定的表面特征和預(yù)測的方向，L是一個常量。 但H的評定比L復(fù)雜得多。 模擬注射成型過程是一種常見的技術(shù)，以預(yù)測質(zhì)量來設(shè)計零件，設(shè)計模具和工藝設(shè)置。結(jié)果翹曲模擬表達為節(jié)點撓度上的X ， Y ， Z分量 ，以及節(jié)點位移W。W是向量長度的矢量總和：+ + ，其中 i，j，k是在X，Y，Z方向上的單位矢量。H是在特征表面上的節(jié)點的最大位移， 這與通常方向的參考平面相同，并能產(chǎn)生結(jié)果的翹曲仿真。 計算h時，節(jié)點的撓度提取如下： 其中是撓度在正常方向參考平面內(nèi)提取節(jié)點; ， ， 是對撓度的X ， Y ， Z分量的提取節(jié)點;α，β，γ是角度的向量參考; A和B是終端節(jié)點，可以預(yù)測方向（圖2 ） ; 和是節(jié)點A和B的撓度： 其中， ，，是對節(jié)點A的撓度在X，Y，Z方向上的分量； ，和是對節(jié)點B的撓度在X ， Y ， Z方向上的分量; 和是終端節(jié)點撓度的加權(quán)因子，計算方法如下： 是提取節(jié)點和節(jié)點A投影間的距離， H是的最大絕對值。 在工業(yè)方面，視察該翹曲借助了一個觸角衡量，被測工件放在一個參考平臺上。 H是一個最大數(shù)值，讀數(shù)在被測工件表面和參考平臺間。 澆口位置優(yōu)化問題的形成 從質(zhì)量來說，“翹曲” ，是指永久變形的部分不是由實用的負載引起的。 它是由整體差動收縮引起，即聚合物流通，包裝，冷卻，結(jié)晶的不平衡。 安置一個澆口，在注射模具整個設(shè)計中是一個最重要的步驟。高質(zhì)量的成型零件受澆口的影響很大，因為它影響塑料流進入型腔的流道。因此，不同的澆口位置會引入不均勻的取向，密度，壓力和溫度分布，因而引入不同的值和分配翹曲。因此，澆口位置，是一個有用的設(shè)計變量，以盡量減少注塑零件翹曲。因為澆口位置和翹曲分布關(guān)系有關(guān)于熔體獨立和模具的溫度，在這項調(diào)查中它是假定該成型條件保持不變。注射成型零件翹曲是量化特征翹曲，其中在上一節(jié)討論了。 因此單一澆口位置優(yōu)化，可以依如下制造 ： 最小化： 主題： 其中γ是特征翹曲變形; p是在澆口位置的注入壓力; 是注入成型機器的可允許注入壓力或被設(shè)計者或制造業(yè)者指定的可允許的注入壓力; x是坐標向量的候選澆口位置; 是節(jié)點有限元網(wǎng)格模型的一部分，為注射成型過程模擬; N是節(jié)點總數(shù)。 在有限元網(wǎng)格模型中，每一個節(jié)點都有可能是一個澆口。 因此，可能是澆口位置的總數(shù) 是一個有關(guān)的總節(jié)點數(shù)目N和總澆口數(shù)n的函數(shù)： 在這項研究中，只對單澆口選址問題進行調(diào)查。 模擬退火算法 模擬退火算法是其中最強大和最流行的以提供良好的實際條件全面化解決辦法。 該算法是基于Metropolis （ 1953 ） ，這原本是用來在原子某一特定溫度找到一個平衡點的方法。這一算法和數(shù)字最小化的聯(lián)系是Pincus（ 1970年）第一個注意到，但是，是Kirkpatrick（ 1983年）等人提議把它形成一項優(yōu)化技術(shù)對組合（或其他）問題。 運用模擬退火法優(yōu)化問題，目標函數(shù)f是用來作為函數(shù)E的能源，而不是找到一個低能源配置，問題就變成尋求近似全局最優(yōu)解。配置的值的設(shè)計變量是替代能源配置本身，控制參數(shù)的過程是取代溫度。一個隨機數(shù)發(fā)生器被用作為設(shè)計變量產(chǎn)生新的值。 這是顯而易見的，該算法只需要將極小化問題列入考慮范圍。因此，在最大化問題上，目標函數(shù)是乘以（ -1 ） 來取得一個可能的數(shù)。 模擬退火算法的主要優(yōu)點是比其他方法更能夠避免局部極小的情況。這種算法采用隨機搜索，而不是只接受變化，即減少目標函數(shù)f ，而且還接受了一些變化來增加它。 后者則是接受一個概率P 其中是f的增量， k是Boltzman常數(shù)， T是一個控制參數(shù)，其中原數(shù)分析是眾所周知的"恒溫"制度 ，并且無視客觀功能參與。 在澆口位置優(yōu)化，實施這一算法的說明圖（圖3），此算法的詳細情況如下： （ 1 ） SA算法開始是從最初的澆口位置，同一個指定值的"溫度"參數(shù)T （"溫度"計數(shù)器K最初定為零） 。 適當控制參數(shù)（ 0 < c < 1 ）給出退火過程與馬爾可夫鏈N。 （ 2 ） SA算法在的旁邊生成一個新的澆口位置來計算目標函數(shù) f（ x ）的值。 （ 3 ）新澆口位置由接受函數(shù)決定接受的概率 一個統(tǒng)一的隨機變量產(chǎn)生[ 0,1 ] ， 如果<， 接受，否則就拒絕。 （ 4 ） 這個過程重復(fù)是的迭代次數(shù)（ ），用這種序列審判澆口位置被稱為馬爾可夫鏈。 （ 5 ）因為減少的"溫度''，生成一個新的馬爾可夫鏈，（在先前的馬爾可夫鏈里，從最后接受的澆口位置生成），這一“溫度”減少的過程將一直持續(xù)直到酸算法結(jié)束。 應(yīng)用與探討 在這一節(jié)，討論復(fù)雜工業(yè)應(yīng)用部分的質(zhì)量測量和優(yōu)化方法。該部分是由一個制造商提供，如圖4所示。在這一部分，對于輪廓精度要求，平坦的基底表面是最重要的。因此 ，翹曲變形特征在基底表面討論，其中參考平臺指定為水平面附于基底表面，縱方向指為預(yù)計參考方向。參數(shù)h是基底面對正常方向的最高偏轉(zhuǎn)即垂直方向，參數(shù)L是基底表面的預(yù)測長度在縱向上的投影。 圖4 制造商提供的工業(yè)產(chǎn)品 該產(chǎn)品的材料是尼龍Zytel 101L（ 30 ％EGP，杜邦工程聚合物）。 在模擬算法中的成型條件列在表1 。 圖5顯示了有限元網(wǎng)格模型的一部分，是受制于數(shù)值模擬。 它有1469個節(jié)點和2492元素。 目標函數(shù)，即特征翹曲，由方程（ 1 ），（ 3 ） ? （ 6 ）定義 。 其中h 是從"流量+流道分析序列中式（ 1 ）里的MPI所得 ，L在該工業(yè)產(chǎn)品中的測量值即L = 20.50毫米。 MPI的是注塑成型模擬使用最廣泛的軟件，它可以向您推薦在流動平衡前提下的最佳澆口位置。對于澆口位置設(shè)計，澆口位置分析是一個有效的工具，但除了實證方法。對于這點，澆口選址分析，MPI認為最佳澆口位置是接近節(jié)點N7459 ，如圖5所示。零件翹曲是模擬在此推薦澆口基礎(chǔ)上，因此，特征翹曲評定： ，這很有價值。 在實際制造中，零件翹曲是可見的在樣品工件上。 這是制造商不能接受的。 表1 在仿真中的成型條件 條件 值 填補時間（秒） 2.5 熔融溫度（ ） 295 模具溫度（ ） 70 包裝時間（秒） 10 包裝壓力（充壓） （ ％ ） 80 在基底表面的最大翹曲，是由不均勻取向分布的玻璃纖維造成的，圖6所示。圖6顯示，玻璃纖維取向的變化，從消極方向到積極方向進行，因為這個澆口位置，尤其是最大的纖維方向轉(zhuǎn)變在這個澆口附近。澆口位置造成的多樣化的纖維取向引起嚴重的差動收縮。因此，特征翹曲是和澆口的位置有關(guān)，必須優(yōu)化，以減少部分翹曲。 優(yōu)化澆口位置，在"模擬退火算法"中的模擬退火， 是適用于這部分的。 最高迭代次數(shù)選定為30至確保精密的優(yōu)化，而且進行多次的隨機試驗，讓每一次迭代中被評為10至跌幅的概率為無效迭代，使之沒有一個重復(fù)的方案。 N7379節(jié)點（圖5） ，是最佳澆口位置。 特征翹曲評定，從翹曲模擬結(jié)果函數(shù)f（X）= γ= 0.97 ％ ，可說是少于MPI建議的澆口。 在實際制造中零件翹曲符合制造商的要求。 圖6b 表明，在模擬纖維取向。它是可見的最優(yōu)澆口位置，取決于玻璃纖維取向，因此，減少收縮差異在垂直方向沿縱向發(fā)展。因此，特征翹曲減少了。 結(jié)論 在這項調(diào)查中，特征翹曲是來描述注塑制品翹曲變形，在數(shù)值模擬軟件MPI的基礎(chǔ)上評定。特征翹曲的評定是為單一澆口位置塑膠注塑模具，基于數(shù)值模擬結(jié)合模擬退火算法優(yōu)化。工業(yè)部分只是作為一個例子來說明所提出的方法。該方法取決于最佳澆口位置，產(chǎn)品是令制造商滿意的。這個方法也適合于其它翹曲最小化的優(yōu)化問題，例如優(yōu)化多澆口位置，流道系統(tǒng)的平衡，并選擇各向異性材料。 參考文獻： Courbebaisse．G．2005．Numerica1 simulation of injectionmoulding process and the pre—m oulding concept． Computational Materials Science， 34(4)：397 405．『doi：1O．1O164．commatsci．2004．11．0041 Courbebaisse，G，Garcia，D．，2002．Shape analysis and injec—tion molding optimization． Computational Materials Science，25(4)：547—553． [doi：lO．1016／S0927—0256(02)00333—6] Jin，S．，Lam，Y．C．，2002．2．5D cavity balancing．Journal ofInjection Molding Technology，6(4)：284—296． Kirkpatrick，S．，Gerlatt，C．D．Jr．，Vecchi，M ．E，1 983．Optimiza一1083tion by simulated annealing．Science，220(4598)：67 1—680．[doi：lO．1126／science 220．4598．671] Lam．Y C．．Seow,L．W 2000．Cavity balance for plastic in—jection molding．Polymer Engineering and Science，40(6)：1273—1280．[doi：1O．1O02／pen 11255] Lam，Y C．，Jin，S．，200 1．Optimization of gate location forplastic injection molding．Journal of Injection MoldingTechnology，5f3)：180一l92． Lee B．H．，Kim ，B．H．，1995．Optimization of part wal1 thick．nesses to reduce warpage of injection—molded parts based on the m odified complex method． Polymer-PlasticsTechnology andEngineering．34(5)：793—8 l1． Lee，B．H．．Kim．B．H．．1 996a．Automated design for the runnersystem of injection molds based on packing simulation． Polymer-Plastics Technology and Engineering，35(1)：147—168． Lee，B．H．，Kim ，B．H．，1 996b．Automated selection of gate1ocation based on desired quality ofinjection molded part．Polymer-Plastics Technology and Engineering，35(2)：253—269． M etropolis，N．，Rosenbluth，A．W ，Rosenbluth，M ．N．，Teller, A．H ．．Teller．E．，1953．Equations of state calculations byfast computing machines．Journal of Chemical Phvs cs，2l(6)：1087—1092．[doi：lO 1063／1 1699114] Pandelidis，I．，Zou，Q．，1 990．Optimization of injection mold—ing design Part I：gate location optimization．Polymer Engineering andScience，30(15)：873—882．[doi：lO．1002／pen．760301 502] Pincus，M ．．1 970．A Monte Carlo method for the approximatesolution of certain types of constrained optimizationproblem s．Operations Research．181225—1228． Shen，C．Y ，Yu，X．R．，Wang，L．X．，Tian，Z．，2004a．Gate 1oca—tion optimization of plastic injection molding．Journal of Chemical Industry and Engineering，55(3)：445—449(inChinese)． Shen，C．Y．，Yu，X．R．，Li，Q．，Li，H．M．，2004b．Gate 1ocationoptimization in injection molding by using modified hill—climbing algorithm． Polymer-Plastics Technologyand Engineering， 43(3)：649—659． [doi：l0．1081／PPT-1 20038056] Zhai，M ．，Lam，L．C．，Au，C．K．，2005a．Algorithm s for two gateoptimization in injection molding．／nternational PolymerProcessing．20(11：14．18． Zhai，M ．，Lam ，L．C．，Au，C．K．，Liu，D．S．，2005b．Automatedselection of gate 1ocation for plastic injection moldingprocessing．Polymer-Plastics Technology and Engineering44(2)：229—242． Zhai，M ．，Lam ，L．C．，Au，C．K．，2006．Runner sizing and weldline positioning for plastics injection moulding withmultiple gates．Engineering with Computers，2l(3)：2 1 8—224．[doi：1 O．1 007／s00366—005—0006—6] 10 Jou rnal of Zhejiang University SCIENCE A ISSN 1 673 565X Print ISSN 1 862 1 775 Online www zju edu cn jzus www springerlink com E mail jzus zju edu ca Lf ef al J Zhejiang Univ SciA 2007 8 7 1077 1083 Single gate optimization for plastic inj ection mold LI Ji quan十 LI De qun GUO Zhi ying LV Hai yuan Department ofPlasticity Technology Shanghai Jiao Tong University Shanghai 200030 China E mail hutli 163 corn Received Nov 22 2006 revision accepted Mar 1 9 2007 1077 Abstract This paper deals with a methodology for single gate location optimization for plastic injection mold The objective of the gate optimization is to minimize the warpage of injection molded parts because warpage is a crucial quality issue for most injection molded parts while it is influenced greatly by the gate location Feature warpage is defined as the ratio of maximum displacement on the feature surface to the projected length of the feature surface to describe part warpage The optimization is combined with the numerical simulation technology to find the optimal gate location in which the simulated annealing algorithm is used to search for the optimum Finally an example is discussed in the paper and it can be concluded that the proposed method is effective Key words Injection mold Gate location Optimization Feature warpage doi 10 163 1 jzus 2007 A1077 Document code A CLC number TQ320 66 INTRODUCTION Plastic injection molding is a widely used com plex but highly efficient technique for producing a large variety of plastic products particularly those with high production requirement tight tolerance and complex shapes The quality ofinjection molded parts is a function of plastic material part geometry mold structure and process conditions The most important part of an injection mold basically is the following three sets of components cavities gates and runners and cooling system Lam and Seow 2000 and Jin and Lain 2002 achieved cavity balancing by varying the wall thick ness of the part A balance filling process within the cavity gives an evenly distributed pressure and tem perature which can drastically reduce the warpage of the part But the cavity balancing is only one of the important influencing factors of part qualities Espe cially the part has its functional requirements and its thicknesses should not be varied usually Project No 50675080 supported by the National Natural Science Foundation of China From the pointview ofthe injection mold design a gate is characterized by its size and location and the runner system by the size and layout The gate size and ruIlller layout are usually determined as constants Relatively gate locations and runner sizes are more flexible which can be varied to influence the quality of the part As a result they are often the design pa rameters for optimization Lee and Kim r l 996a optimized the sizes of runners and gates to balance runner system for mul tiple inj ection cavities The runner balancing was described as the differences of entrance pressures for a multi cavity mold with identical cavities and as differences of pressures at the end of the melt flow path in each cavity for a family mold wim different cavity volumes and geometries The methodology has shown uniform pressure distributions among the cavities during the entire molding cycle of multiple cavities mold Zhai et a1 2005a1 presented the two gate loca tion optimization of one molding cavity by an ef前一 cient search method based on pressure gradient PGSS and subsequently positioned weld lines to the desired locations by varying runner sizes for 維普資訊 1078 L e a1 d Zhejiang Univ SciA 2007 8 7 1077 1083 multi gate parts Zhai et a1 2006 As large volume part multiple gates are needed to shorten the maxi mum flow path with a corresponding decrease in injection pressure The method is promising for de sign of gates and runners for a single cavity wich multiple gates Many of injection molded parts are produced with one gate whether in single cavity mold or in multiple cavities mold Therefore the gate location of a single gate is the most common design parameter for optimization A shape analysis approach was pre sented by COHrbebaisse and Garcia 2002 by which the best gate location of iniection molding was esti mated Subsequently they developed this methodo1 ogy further and applied it to single gate location op timization of an L shape example Courbebaisse 2005 It is easy to use and not time consuming while it only serves the turning of simple flat parts wich uniforii1 thickness Pandelidis and Zou r 1 990 presented the opti mization ofgate location by indirect quality measures relevant to warpage and material degradation which is represented as weighted sum of a temperature dif ferential term an over pack term and a frictional overheating term Warpage is influenced by the above factors but the relationship between them is not clear Therefore the optimization effect is restricted by the determination of the weighting factors Lee and Kim r l 996b developed an automated selection method of gate location in which a set of initial gate locations were proposed by a designer and then the optimal gate was located by the a acent node evaluation method The conclusion to a great extent depends much on the human designer s intuition because the first step of the method is based on the designer s proposition So the result is to a large ex tent limited to the designer s experience Lam and Jin 200 l developed a gate location optimization method based on the minimization of the Standard Deviation of Flow Path Length SD L and Standard Deviation of Filling Time SD T during the molding filling process Subsequently Shen et 口 004a 2004b optimized the gate location design by minimizing the weighted sum of filling pressure filling time difference between different flow paths temperature difference and over pack percentage Zhai et a1 2005b investigated optimal gate location with evaluation criteria of iniection pressure at the end of filling These researchers presented the obi ec tive functions as performances of ini ection molding filling operation which are correlated with product qualities But the correlation between the perforii1 ances and qualities is very complicated and no clear relationship has been observed between them yet It is also difficult to select appropriate weighting factors for eachterm A new objective function is presented here to evaluate the warpage of inj ection molded parts to optimize gate location To measure part quality di rectly this investigation defines feature warpage to evaluate part warpage which is evaluated from the flow plus warpage simulation outputs of Moldflow Plastics Insight MPI software The obj ective func tion is minimized to achieve minimum deformation in gate location optimization Simulated annealing al gorithm is employed to search for the optimal gate location An example is given to illustrate the effec tivity of the proposed optimization procedure QUALITY MEASURES FEATURE WARPGE Definition of feature warpage To apply optimization theory to the gate design quality measures of the part must be specified in the first instance The term quality may be referred to many product properties such as mechanical thermal electrical optical ergonomical or geometrical prop erties There are two types of part quality measures direct and indirect A model that predicts the proper ties from numerical simulation results would be characterized as a direct quality measure In contrast an indirect measure of part quality is correlated with target quality but it cannot provide a direct estimate Ofchat quality For warpage the indirect quality measures in related works are one of performances of iniection molding flowing behavior or weighted sum of those The performances are presented as filling time dif ferential along different flow paths temperature dif ferential over pack percentage and so on It is ob vious that warpage is influenced by these perforii1 ances but the relationship between warpage and these performances is not clear and the determination of these weighting factors is rather difficult Therefore the optimization with the above Obiective function 維普資訊 Lf ef al J Zhejiang Univ SciA 2007 8 1077 1083 probably will not minimize part warpage even with perfect optimization technique Sometimes improper weighting factors will result in absolutely wrong re sults Some statistical quantities calculated from the nodal displacements were characterized as direct quality measures to achieve minimum deformation in related optimization studies The statistical quantities are usually a maximum nodal displacement an av erage oftop 1 0 percentile nodal displacements and an overall average nodal displacement Lee and Kim l 995 I 996b These nodaI displacements are easy to obtain from the simulation results the statistical val ues to some extents representing the deformation But the statistical displacement cannot effectively describe the deformation of the in i ection molded parts In industry designers and manufacturers usually pay more attention to the degree of part warpage on some specific features than the whole deformation of the in iection molded parts In this study feature warpage is defined to describe the deformation of the injection parts The feature warpage is the ratio of the maximum displacement of the feature surface to the proj ected length of the feature surface Fig 1 h 100 L where y is the feature warpage h is the maximum displacement on the feature surface deviating from the reference platform and L is the proj ected length of the feature surface on a reference direction paralleling the reference platform Surface ceplane Fig 1 The definition of feature warpage For complicated features only plane feature discussed here the feature warpage is usually sepa rated into two constituents on the reference plane which are represented on a 2D coordinate system h l0 l0 0o o 1079 where yy are the constituent feature warpages in the X Y direction and Lx L are the projected lengths of the feature surface on Y component Evaluation of feature warpage Atier the determination of target feature com bined with corresponding reference plane and pro iection direction the value of L can be calculated immediately from the part with the calculating method of analytic geometry Fig 2 L is a constant for any part on the specified feature surface and pro iected direction But the evaluation of h is more com plicated than that ofL y Feature Fig 2 The projected length evaluation Simulation of iniection molding process is a common technique to forecast the quality of part de sign mold design and process settings The results of warpage simulation are expressed as the nodal de flections on Z component and the nodal displacement Wis the vector length ofvector sum of and Wz k where k are the unit vectors on Z component The h is the maximum displacement of the nodes on the feature surface which is correlated with the normal orientation of the reference plane and can be derived from the results of warpage simulation To calculate h the deflection of fth node is evaluated firstly as follows COSO cosfl W z cosy一 3 where is the deflection in the normal direction of the reference plane of ith node W WI1 Wiz are the deflections on X Y Z component of ith node 伐 B are the angles ofnormal vector ofthe reference A and B are the terminal nodes of the feature to proiecting direction Fig 2 and are the deflections of nodesA and 維普資訊 1080 Lf ef al dZhefiangUnivSciA 2007 8 1077 1083 f COSa cosfl cosy l COSa cosfl Gz cosy where WA WAy are the deflections on Z component ofnode Wsx and Wez are the de flections on X Y Z component ofnode COiA and lB are the weighting factors of the terminal node deflec tions calculated as follows 2 iA 1一LiA L co 1一 J iA where LiA is the proj ector distance between fth node and node A Ultimately h is the maximum of the absolute value of h max l I I 1 In industry the inspection of the warpage is carried out with the help ofa feeler gauge while the measured part should be placed on a reference plat form The value of h is the maximum numerical reading of the space between the measured part sur face and the reference platform GATE L0CAT10N 0PTIMIZAT10N PR0BLEM FORMATION The quality term warpage means the perma nent deformation of the part which is not caused by an applied load It is caused by differential shrinkage throughout the part due to the imbalance of polymer flow packing cooling and crystallization The placement of a gate in an iniection mold is one of the most important variables of the total mold design The quality of the molded part is greatly af fected by the gate location because it influences the manner that the plastic f1ows into the mold cavity Therefore different gate locations introduce inho mogeneity in orientation density pressure and temperature distribution accordingly introducing different value and distribution of warpage Therefore gate location is a valuable design variable to minimize the injection molded part warpage Because the cor relation between gate location and warpage distribu tion is to a large extent independent of the melt and mold temperature it is assumed that the molding conditions are kept constant in this investigation The inj ection molded part warpage is quantified by the feature warpage which was discussed in the previous section The single gate location optimization can thus be formulated as fo11ows Minimize mini X Subjectto g X P Po一1 0 X X i 1 2 N where is the feature warpage P is the injection pressure at the gate position P0 is the allowable in iection pressure of injection molding machine or the allowable injection pressure specified by the designer or manufacturer X is the coordinate vector of the candidate gate locations is the node on the finite element mesh model of the part for injection molding process simulation N is the total number of nodes In the finite element mesh model of the part every node is a possible candidate for a gate There fore the total number of the possible gate location n is a function of the total number of nodes N and the total number of gate locations to be optimized N N一1 一 一1 In this study only the single gate location problem is investigated SIMULATED ANNEALING ALG0RITHM The simulated annealing algorithm is one of the most powerful and popular meta heuristics to solve optimization problems because of the provision of good global solutions to real world problems The algorithm is based upon that of Metropolis et a1 1 953 which was originally proposed as a means to find an equilibrium configuration of a collection of atoms at a given temperature The connection be tween this algorithm and mathematical minimization was first noted by Pincus r1970 but it was Kirkpatrick et a1 1983 who proposed that it formed the basis of an optimization technique for combina tional and other problems To apply the simulated annealing method to op 維普資訊 Lf ef al d Zhejiang Univ SciA 2007 8 1077 1083 timization problems the obiective function f is used as an energY function E Instead of finding a low energY configuration the problem becomes to seek an approximate global optimal solution The configura tions Of me values of design variables are substituted f or the energy cOnfiguratiOns Of the body and the control parameter for the process is substituted for temperature A random number generator is used as a way ofgenerating new values for the design variables It is obvious that this algorithm iust takes the mini mization problems into account Hence while per forming a maximization problem the obj ective func tion is multiplied by f l 1 to obtain a capable form The maj or advantage of simulated annealing algorithm over omer methods is the ability to avoid being trapped at local minima This algorithm em ploys a random search which not only accepts changes that decrease oNective function but also accepts some changes that increase it The Iatter are accepted with a probability P P e J where is the increase off k is Boltzman s constant and T is a control parameter which by analogy with the original application is known as the system temperature irrespective of the objective function involved In the case of gate location optimization the implementation ofthis algorithm is illustrated in Fig 3 and this algorithm js detailed as follows 1 SA algorithm starts from an initial gate loca tion Xold with an assigned value of the tempera ture parameter T fthe temperature counter k is initially set to zero Proper control parameter 0 c 1 in annealing process and Markov chain enera e are given 2 SA algorithm generates a new gate location Xnew in the neighborhood of Xou and the value of the objective functionflX is calculated 3 The new gate location will be accepted with probability determined bv me acceptance function 唧 min 1 exp 一k f Xn 一f Xo A uniform random variable Punif is generated in 0 1 If Punif 尸accept Xnew is accepted otherwise it is rejected 1081 f4 This process is repeated for a large enough number of iterations enerate for The sequence of trial gate locations generated in this way is known as Markov chain 5 A new Markov chain is then generated starting from the last accepted gate location in the previous Markov chain for a reduced temperature Tk l CTk and the same process continues for de creasing values of temperature until the algorithm stops System initialization f O 朋 O J generate X ld J 1 t 0 1 一 i一 Generate Xnew if一 Fig 3 The flow chart of the simulated annealing algorithm APPLICATION AND DISCUSSION The application to a presented in this section complex industrial part is to illustrate the proposed quality measure and optimization methodology The part is provided by a manufacturer as shown in Fig 4 In mis part the flatness of basal surface is the most important profile precision requirement Therefore the feature warpage is discussed on basal surface in which reference platform is specified as a horizontal plane attached to the basal surface and the longitu dinal direction is specified as pr0iected reference 維普資訊 1082 Lj ef al d Zhejiang Univ SciA 2007 8 1077 1083 direction The parameter h is the maximum basal surface deflection on the normal direction namely the vertical direction and the parameter L is the projected length of the basal surface to the longitudinal direc tion Fig 4 Industrial part provided by the manufacturer The material of the part is Nylon Zytel 101L 30 EGF DuPont Engineering Polymer The molding conditions in the simulation are listed in Table 1 Fig 5 shows the finite element mesh model of the part employed in the numerical simulation It has 1469 nodes and 2492 elements The objective func tion namely feature warpage is evaluated by Eqs 1 3 6 The h is evaluated from the results of Flow Warp Analysis Sequence in MPI by Eq 1 and the is measured on the industrial part immediately L 20 50 mm Table 1 The molding conditions in the simulation Conditions Values Fill time s Melt temperature C Mold temperature C Packing time s Packing pressure offilling pressure 2 5 295 70 10 80 MPI is the most extensive software for the in iection molding simulation which can recommend the best gate location based on balanced flow Gate loca tion analysis is an effective tool for gate location de sign besides empirical method For this part the gate location analysis of MPI recommends that the best gate location is near node N7459 as shown in Fig 5 The part warpage is simulated based on this recom mended gate and thus the feature warpage is evaluated 7 5 15 which is a great value In trial manufactur ing part warpage is visible on the sample work piece This is unacceptable for the manufacturer The great warpage on basal surface is caused by the uneven orientation distribution of the glass fiber as shown in Fig 6a Fig 6a shows that the glass fiber orientation changes from negative direction to posi tive direction because of the location of the gate par ticularly the greatest change of the fiber orientation appears near the gate The great diversifi