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附件一 外文翻譯
譯文1 使用響應(yīng)面法和遺傳算法可以高效率優(yōu)化塑料薄殼部件翹曲
摘要
在使用注塑成型生產(chǎn)塑料薄殼部件的過(guò)程時(shí)經(jīng)常遇到的翹曲取決于工藝條件。在這一項(xiàng)研究中，利用調(diào)查研究耦合有限元分析法(FE)，響應(yīng)面方法(RSM)，和遺傳算法(GA)來(lái)有效最小化塑料薄殼部件的翹曲程度。以汽車(chē)吊燈基座作為塑料薄殼部件的例子. 為了達(dá)到翹曲的最小化，最優(yōu)工藝條件參數(shù)是已知的，工藝條件參數(shù)是模具溫度，熔化溫度，保壓壓力，保壓時(shí)間和冷卻時(shí)間。有限單元分析用于統(tǒng)計(jì)三級(jí)完整因素實(shí)驗(yàn)設(shè)計(jì)的組合參數(shù)。 已知基于方差分析法的有限單元分析結(jié)果是影響翹曲的最重要工藝條件參數(shù)。從預(yù)測(cè)性的響應(yīng)面模型得出翹曲值?？梢酝ㄟ^(guò)結(jié)合響應(yīng)面與有效的遺傳算法來(lái)尋找最佳工藝參數(shù)值
1簡(jiǎn)介
生產(chǎn)塑料薄殼部件常采用注塑成型的方法。而薄殼部件翹曲在注射成型過(guò)程中是很常見(jiàn)的。薄殼部分翹曲程度是非常重要的成型工藝條件參數(shù)。是否有智能的方法可以調(diào)節(jié)成型工藝參數(shù)，將翹曲程度降低到一個(gè)可接受的水平。雖然專(zhuān)家們已經(jīng)進(jìn)行了無(wú)數(shù)關(guān)于塑料薄殼部件翹曲的研究。然而，卻很少人致力于這方面的研究。
在這項(xiàng)研究中,利用耦合有限元分析法，響應(yīng)面方法，和遺傳算法來(lái)降低塑料薄殼部件的翹曲程度是一個(gè)有效的優(yōu)化方法.汽車(chē)吊燈基座就是成熟的優(yōu)化方法運(yùn)用在薄殼部件的一個(gè)例子.燈座有限元分析是在優(yōu)化過(guò)程中合理建立在全因子設(shè)計(jì)實(shí)驗(yàn)統(tǒng)計(jì)基礎(chǔ)上的工藝參數(shù).模腔溫度，熔體溫度，保壓力，保壓時(shí)間和冷卻時(shí)間都是影響翹曲的工藝條件參數(shù)?；诜讲罘治龇ǖ挠邢迒卧治鍪怯绊懧N曲的最重要工藝條件參數(shù)。預(yù)測(cè)模型的角度對(duì)翹曲的關(guān)鍵工藝參數(shù)，然后使用創(chuàng)建的RSM 。一個(gè)RS模型再加上一個(gè)有效的遺傳算法尋找最佳工藝參數(shù)值。以下各節(jié)將會(huì)詳細(xì)介紹翹曲最優(yōu)化方案。
2汽車(chē)吊燈基座的有限元模型
用于本研究的薄殼部件的幾何形狀如圖1所示 。該薄殼部件相當(dāng)于一個(gè)汽車(chē)吊燈基座。這個(gè)燈座寬度，長(zhǎng)度和厚度分別為130毫米， 450毫米和2毫米。它是以ABS （ GE ）5500S為原料的,表2 給出了該原料的組成。
一個(gè)燈座的有限元模型是根據(jù)離散幾何中比較簡(jiǎn)單的原理創(chuàng)建的。在圖1給出的有限元模型中包括有14 348個(gè)四面體元素。使用模流分析商業(yè)軟件有限元分析汽車(chē)吊燈基座的的組合工藝參數(shù)如表1所示 。組合工藝參數(shù)所產(chǎn)生的三級(jí)全因子試驗(yàn)設(shè)計(jì)產(chǎn)量35 = 243有限元分析需要進(jìn)行5個(gè)參數(shù)。
在表1中，MoT, MeT, PP, PPT和CT分別表示模腔溫度，熔體溫度，保壓壓力，保壓時(shí)間和冷卻時(shí)間。在表2和表3中分別給出了進(jìn)行有限元分析所使用的ABS材料性及制造條件。每個(gè)分析需要一臺(tái)IBM-P3電腦的CPU花八個(gè)小時(shí)來(lái)處理.
為了保存由觀察所獲得的243個(gè)翹曲值及其工藝參數(shù)的范圍，在圖2中給出了翹曲值的單一圖形。雖然在圖2中沒(méi)有清楚的給出了翹曲值的生產(chǎn)組合工藝參數(shù)，但它足以表示出翹曲計(jì)算值的大小。在圖2中可以注意到汽車(chē)吊燈基座每個(gè)翹曲值中的最大值都會(huì)適應(yīng)的與一個(gè)特定的組合工藝參數(shù)響應(yīng)。
3方差分析翹曲數(shù)據(jù)
進(jìn)行方差分析（方差分析），以確定在本研究中對(duì)于翹曲最重要的工藝參數(shù)。如果某些參數(shù)沒(méi)有明顯影響翹曲，他們可以被制造商確定在從薄殼塑料零件獲取最佳利益的值和被從預(yù)測(cè)模型的產(chǎn)生和優(yōu)化過(guò)程中排除。這將提高優(yōu)化進(jìn)程的效率。方差分析是使用243進(jìn)行了有限元分析結(jié)果在上一節(jié)和下面的信噪比
比率（的S / N ）和均方偏差數(shù)方程：
其中N是一些翹曲數(shù)據(jù)集它等于243 ，Yi是翹曲值對(duì)應(yīng)第i個(gè)的數(shù)據(jù)，MSD是均方偏差,分析結(jié)果列于表4 。
表4，均方偏差的分析結(jié)果
F比對(duì)應(yīng)的95 ％可信度水平的工藝參數(shù)計(jì)算，準(zhǔn)確值是F= 245.364 。它可以從平均均方偏差的平均和的比值中計(jì)算得到。在表4 ， P值說(shuō)明了各個(gè)工藝參數(shù)的重要水平，其百分比
就是重要率。從表4可以看出保壓壓力（PP） ，成型溫度（ MOT ）的，熔體溫度（MET） ， 保壓時(shí)間（PPT） ，和冷卻時(shí)間（ CT ）對(duì)于翹曲的影響分別是 37.39 ％ ， 31.35 ％ ， 26.94 ％ ， 3.65 ％和0.6 ％。這個(gè)百分之?dāng)?shù)字表明，模具溫度，熔體溫度， 和注射壓力有顯著的影響，而保壓時(shí)間和冷卻時(shí)間對(duì)翹曲的影響不大，因此，保壓時(shí)間和冷卻時(shí)間被排除在第4部分中針對(duì)翹曲的響應(yīng)曲面模型的生成實(shí)驗(yàn)外。
4、翹曲的響應(yīng)面模型
昂貴的有限元模型是不適合大量、重復(fù)的需要優(yōu)化過(guò)程的分析。因此,在此研究中,翹曲的有限元模型應(yīng)該用更簡(jiǎn)單的、更多的、有預(yù)測(cè)性的、由反應(yīng)曲面研究分類(lèi)法創(chuàng)建的模型代替。反應(yīng)曲面研究分類(lèi)法是一種基于統(tǒng)計(jì)實(shí)驗(yàn)設(shè)計(jì)和適當(dāng)?shù)淖钚《朔ǖ慕７椒āＳ煞磻?yīng)曲面研究分類(lèi)法創(chuàng)建的現(xiàn)有數(shù)據(jù)的多項(xiàng)式模型設(shè)定如下：
在上式中，aij 、a0、ai是調(diào)整參數(shù)，n是模型參數(shù)（例如工藝參數(shù)）。由反應(yīng)曲面研究分類(lèi)法生成的多項(xiàng)式模型在文獻(xiàn)方面常被稱(chēng)為響應(yīng)面模型。反應(yīng)曲面研究分類(lèi)法最初是由響應(yīng)的模型物理實(shí)驗(yàn)箱發(fā)展而來(lái)隨后在其領(lǐng)域被采納。為了創(chuàng)建響應(yīng)面模型、從而寫(xiě)出了計(jì)算機(jī)的MATLAB程序編程語(yǔ)言。這個(gè)程序的能力是在足夠的數(shù)據(jù)支持下能創(chuàng)建10個(gè)參數(shù)值的響應(yīng)面模型。所有交叉的模型都可以考慮到。響應(yīng)面模型也可以由
相反的參數(shù)得到。如果想得到響應(yīng)面模型，即Xi也可以是被1/xi替換的(即相反的)。一級(jí)到三級(jí)不同順序的響應(yīng)面模型被成熟的程序創(chuàng)造和測(cè)試。 在創(chuàng)建的響應(yīng)面模型中，新的數(shù)據(jù)庫(kù)從243個(gè)數(shù)據(jù)中獲得。新的數(shù)據(jù)庫(kù)包含27個(gè)分析結(jié)果和影響翹曲的三個(gè)重要加工參數(shù)值的組合。因此，響應(yīng)面翹曲模型依照有效的加工參數(shù)（模溫、熔化溫度各注射壓力）描述生成。新的數(shù)據(jù)庫(kù)是分開(kāi)兩部分的：一部分用于創(chuàng)建模型，而另一部分用于檢測(cè)所建模型的準(zhǔn)確性。這個(gè)數(shù)據(jù)庫(kù)如圖5和6所示。
所創(chuàng)建的響應(yīng)面模型的準(zhǔn)確性數(shù)據(jù)如表7。由表7可知，由相反加工參數(shù)組成的三次多項(xiàng)式出錯(cuò)的底限是6.23%，因此，此模型是由基因算法得到的翹曲模型最優(yōu)化，如表5。
5翹曲優(yōu)化與遺傳算法
在本節(jié)中，見(jiàn)表1中給出的最小限度的翹曲變形的工藝參數(shù)的測(cè)定是使用整體最佳化方法的: 就如遺傳演算法(GA) [ 10 ] 。汽車(chē)吊燈基座的翹曲變形最優(yōu)化的問(wèn)題可以出標(biāo)準(zhǔn)的數(shù)學(xué)格式定義如下：
在參數(shù)范圍內(nèi)：
模具溫度(MoT)，熔化溫度(MeT) 范圍的選擇是基于Moldflow用戶(hù)手冊(cè)中建議值[ 8 ] 。保壓壓力(PP)范圍的選定是基于汽車(chē)吊燈基座生產(chǎn)商的經(jīng)驗(yàn)的.
為了有效率地解決均衡器的優(yōu)化問(wèn)題。有效的遺傳演算法(GA)已經(jīng)被寫(xiě)入Matlab程序設(shè)計(jì)語(yǔ)言中。成熟的遺傳演算法(GA)是與響應(yīng)曲面法(RSM)的聯(lián)系得到很好的運(yùn)用如表4 ,而用于翹曲變形全面的最優(yōu)化如圖3 。遺傳算法通過(guò)模擬生物進(jìn)化過(guò)程來(lái)解決最優(yōu)化的問(wèn)題，達(dá)爾文的優(yōu)勝劣汰理論，就如其他的遺傳演算法(GA)。該解決方法由通過(guò)查閱文獻(xiàn)中人類(lèi)染色體所得到的方法中得到的。染色體是用位串的形式隨機(jī)產(chǎn)生的。染色體進(jìn)化經(jīng)過(guò)是長(zhǎng)久時(shí)間的，而在這長(zhǎng)久的時(shí)間進(jìn)里反復(fù)地優(yōu)化選取的。新世代使用了交叉和變異技術(shù)：交叉涉及兩個(gè)染色體分裂,然后與另外的染色體的二分之一結(jié)合然后結(jié)合,一部的染色體轉(zhuǎn)變成其他的。染色體優(yōu)劣由一套適合的標(biāo)準(zhǔn)來(lái)評(píng)價(jià)。最好的保存，而其余的被丟棄。這個(gè)過(guò)程重復(fù)直到有最佳染色體出現(xiàn)。遺傳演算法(GA)中的一些臨界參數(shù)與數(shù)代的人類(lèi)中所發(fā)生的突變率的大小有關(guān)，在這項(xiàng)研究中用的人口為50 ，交叉率為1.0 ，突變率為0.1 ，變量位數(shù)為16，代數(shù)為500.
在研究中的先進(jìn)的遺傳演算法選用基于目標(biāo)值和一定程度的限制行為的染色體。染色體的是硬度值趨向最小目標(biāo)值和交叉階段的最小不可行集。在大多數(shù)文獻(xiàn)中，遺傳演算法通過(guò)處罰函數(shù)，把限制最優(yōu)化的問(wèn)題轉(zhuǎn)化為不可限制最優(yōu)化的問(wèn)題來(lái)尋找解決方法。這樣帶來(lái)了需要用戶(hù)經(jīng)驗(yàn)的問(wèn)題相關(guān)的處罰系數(shù)合理選擇的難題。在本研究的項(xiàng)目開(kāi)發(fā)中，由于不涉及問(wèn)題相關(guān)的系數(shù).這難題是完全可以被免的.
遺傳算法優(yōu)化了減少汽車(chē)吊燈底座翹曲由2.47毫米到1.32毫米大約減少了46 ％。翹曲值相應(yīng)的最佳工藝參數(shù)和初始工藝參數(shù)的比較表8 。各種優(yōu)化工藝參數(shù)對(duì)應(yīng)的汽車(chē)吊燈底座的翹曲分布如圖4所示。最佳重復(fù)優(yōu)化歷史的表現(xiàn)在圖5 。從表8和圖4 ，可以看出從GA預(yù)測(cè)優(yōu)化后的翹曲值1.32和MOLDFLOW得出的1.315相當(dāng)吻合。
6結(jié)語(yǔ)
在這項(xiàng)研究中，在最小化注塑制造的塑料薄殼部件的翹曲度的過(guò)程中引入了響應(yīng)曲面法和遺傳演算法的最優(yōu)化方法論。以汽車(chē)吊燈基座為薄殼塑件的例子，為了實(shí)現(xiàn)最低翹曲，必須使用適當(dāng)?shù)墓に嚄l件參數(shù)。工藝條件參數(shù)包括模具溫度，熔體溫度，保壓壓力，保壓時(shí)間，冷卻時(shí)間。有限單元分析用于統(tǒng)計(jì)三級(jí)完整因素實(shí)驗(yàn)設(shè)計(jì)的參數(shù)組合。已知影響翹曲度的最重要工藝條件參數(shù)（翹曲度的計(jì)算基于方差分析法的有限單元分析結(jié)果）。方差分析法顯示：影響翹曲度的保壓壓力、，模具溫度、熔體溫度、保壓時(shí)間和冷卻時(shí)間分別為37.39%, 31.35%, 26.94%, 3.65%, 和0.6%。依據(jù)最重要的工藝條件參數(shù)：保壓壓力、模具溫度和融化溫度（這些工藝條件參數(shù)應(yīng)用響應(yīng)曲面法來(lái)減少優(yōu)化過(guò)程的計(jì)算成本）來(lái)構(gòu)造翹曲度的預(yù)測(cè)性模型。響應(yīng)曲面模型響應(yīng)面與有效的遺傳算法相結(jié)合來(lái)尋找最優(yōu)工藝條件參數(shù)。遺傳算法顯著地減少了初始模具的翹曲度。翹曲度大約改進(jìn)了46%，這表示，在此研究中優(yōu)化方法還可以用來(lái)改善其他塑料薄殼部件。
原文：
1. Hasan Kurtaran · Tuncay Erzurumlu，Efficient warpage optimization of thin shell plastic partsusing response surface methodology and genetic algorithm，《Int J Adv Manuf Technol》， (2006) 27: 468–472.
Efficient warpage optimization of thin shell plastic parts using response surface methodology and genetic algorithm
Received: 30 April 2004 / Accepted: 2 July 2004 / Published online: 27 April 2005 Springer-Verlag London Limited 2005
Abstract
During the production of thin shell plastic parts by injection molding, warpage depending on the process conditions is often encountered. In this study, efficient minimization of warpage on thin shell plastic parts by integrating finite element (FE) analysis, statistical design of experiment method, response surface methodology (RSM), and genetic algorithm (GA) is investigated. A bus ceiling lamp base is considered as a thin shell plastic part example. To achieve the minimum warpage, optimum process condition parameters are determined. Mold temperature, melt temperature, packing pressure, packing time, and cooling time are considered as process condition parameters. FE analyses are conducted for a combination of process parameters organized using statistical three-level full factorial experimental design. The most important process parameters influencing warpage are determined using FE analysis results based on analysis of variance (ANOVA) method. A predictive response surface model for warpage data is created using RSM. The response surface (RS) model is interfaced with an effective GA to find the optimum process parameter values.
Keywords :Analysis of variance · Design of experiment · Genetic algorithm · Injection molding · Response surface methodology · Thin shell plastic · Warpage optimization
1 Introduction
Injection molding is commonly used in the production of thin shell plastic parts. During the injection molding process, warpage of thin shell parts is often encountered. The level of warpage is highly related to molding process condition parameters. If molding process parameters can be adjusted in an intelligent way, warpage can be minimized towards an acceptable level. Numerous research has been conducted on the warpage of thin shell plastic parts [1–5]. However, very few of them are devoted to the optimization of such parts [6, 7].
In this study, an efficient optimization method by coupling FE analysis, RSM, and GA is introduced to minimize warpage of thin shell plastic parts. The developed optimization method is applied to an example of a thin shell part: bus ceiling lamp base. During the optimization process, FE analyses of the lamp base are conducted for combination of process parameters organized based on statistical full factorial experimental design. Mold temperature, melt temperature, packing pressure, packing time, and cooling time are considered as process condition parameters influencing warpage. The most important process parameters influencing warpage are determined using the ANOVA method. A predictive model for warpage in terms of the critical process parameters is then created using RSM. An RS model is coupled with an effective GA to find the optimum process parameter values. Stages of the warpage optimization are explained in detail in the following sections.
2 FE model of the bus ceiling lamp base
Geometry of the thin shell part utilized in this study is shown in Fig. 1a. The thin shell part corresponds to a bus ceiling lamp base. The lamp base has width, length, and thickness of 130 mm, 450 mm, and 2 mm, respectively. It is made of ABS (GE) 5500S and its material properties are given in Table 2 [8].
An FE model of the lamp base is created by discretizing the geometry into smaller simpler elements. The FE model shown in Fig. 1b includes 14 348 tetrahedron elements. FE analyses of the bus lamp base are performed using commercial software Mold-Flow [8] for the combination of process parameters shown in Table 1. The combination of process parameters generated by three-level full factorial design of experiment yields 35 = 243 FE analyses to be conducted for five parameters [9].
In Table 1, MoT, MeT, PP, PPT and CT indicate mold temperature, melt temperature, packing pressure, packing time and cooling time, respectively. FE analyses are carried out using the Table 1. Low-middle-high values of process parameters in three-level full factorial design of experiment Fig. 1. a Geometry b FE model of the bus ceiling lamp base
MoT (.C) MeT(.C) PP (MPa) PPT (s) CT (s) 40-60-80 230-250-270 30-45-60 4-6-8 15-22.5-30
Table 2. Material properties of ABS (GE) 5500S
Melt density 0.98843 g/cm3
Solid density 1.0541 g/cm3
Eject temperature 85 .C
Maximum shear stress 0.3MPa
Maximum shear rate 50 000 (1/s).
Thermal conductivity 0.23 W/m .C
Elastic module 2600 MPa
Poisson ratio 0.38
material properties of ABS given in Table 2 as well as the manufacturing conditions in Table 3. Each analysis takes about 8 hours of CPU time on an IBM-P3 processor PC.
To save space in visualizing 243 obtained warpage values along with their process parameters, warpage values are given in a single figure in Fig. 2. Although it is not clear in Fig. 2 which combination of process parameters produces which warpage value, it is sufficient to show the magnitude of calculated Table 3. Manufacturing parameters employed in MoldFlow analysis
Injection time 4.74 s Injection pressure 15 MPa Cooling channel diameter 10 mm Between cooling channel’s center distance 105 mm Between cooling channel’s center-parting surface distance 15 mm Upper inlet water temperature 45 .C Lower inlet water temperature 20 .C
Number of gate 6
Fig. 2. Warpage values obtained from FE analyses
warpage values. Note that each warpage value in Fig. 2 corresponds to the maximum warpage value on the bus ceiling lamp base calculated for a particular combination of process parameters.
3 Analysis of variance on warpage data
Analysis of variance (ANOVA) is performed to determine the significance of process parameters on warpage in this study. If some parameters do not significantly affect warpage, they can be fixed to the recommended values of thin shell plastic part manufacturers and excluded in predictive model generation and optimization process. This will increase the efficiency of the optimization process. ANOVA is carried out using 243 FE analysis results in the previous section and the following signal-to-noise ratio (S/N) and mean square deviation (MSD) equations:
N S/N =.10 .log 1 Y 2 (1) i N i=1N1MSD =Y 2 ,.(2)iNi=1
where N is the number of warpage data sets which is equal to 243, Yi is the warpage value for the ith data set, and MSD is the mean square deviation. ANOVA results are shown in Table 4.
Table 4. ANOVA results for warpage Process Variance Percent Sum F ratio P value parameters (%) of squares
PP 29.64 37.39 29.64 5929.2 0.0001
MoT 37.72 31.35 37.72 7545.9 0.0001
MeT 20.52 26.94 20.52 4104.7 0.0001
PPT 1.650 3.66 1.65 231.7 0.0015
CT 0.040 0.66 0.04 8.3 0.0050
F ratio corresponding to 95% confidence level in calculation of process parameters accurately is F = 245.364. It can be computed
from the ratio of the mean sum of squared deviations. In Table 4, the P value reports the significance level, and Percent is significance rate. It is seen from Table 4 that packing pressure (PP), mold temperature (MoT), melt temperature (MeT), packing time (PPT), and cooling time (CT) influence warpage by 37.39%, 31.35%, 26.94%, 3.65%, and 0.6%, respectively. The percent numbers show that mold temperature, melt temperature, and packing pressure have significant effects on warpage while packing time and cooling time have little effect on warpage. Therefore, packing time and cooling time were excluded in the generation of the response surface (RS) model for warpage in Sect. 4.
4 RS model for warpage
A computationally costly FE model is not suitable for a large number of repetitive analyses which are often required in an optimization process. Therefore, in this study, the FE model for warpage is replaced by a simpler and more efficient predictive model created by response surface methodology (RSM). RSM is a model building technique based on statistical design of experiment and least square error fitting. RSM creates polynomial models for the available data set as follows:
n nn f = a0 + aixi + aij xi xj + ...,.(3) i=1 i=1 j=1
where a0, ai,and aij are tuning parameters and n is the number of model parameters (i.e., process parameters). The polynomial models generated by RSM are often referred to as RS models in the literature. RSM was originally developed for the model fitting of physical experiments by Box and Draper [9] and later adopted in other fields. To create RS models, a computer program has been written in MATLAB programming language. The program has the capability of creating RS polynomials up to 10th order if sufficient data exist. All cross terms in the models can be taken into account. RS models can also be generated in terms of 1 inverse of parameters. That is, xi can be replaced as xi (i.e., inversely) in an RS model if desired. RS models of varying orders from first-order to third-order are created and tested with the developed program. In creating the RS models, a new data set obtained from 243 data sets is utilized. The new data set consists of 33 = 27 analysis results and corresponds to the combination of three most important process parameters affecting the warpage. Therefore, RS models generated describe warpage in terms of the effective process parameters (mold temperature, melt temperature, and packing pressure). The new data set is divided into two parts: one part to create the model, and the other part to check the accuracy of the created model. These data sets are shown in Tables 5 and 6.
Table 5. Data set used in creating RS models
Set Mold Melt Packing Warpage number temperature (.C) temperature (.C) pressure (MPa) (mm)
1 40 230 30 5.731 2 40 230 45 5.739 3 40 230 60 5.779 4 60 250 30 4.257 5 60 250 45 4.612 6 80 270 30 3.816 7 80 280 45 3.969 8 80 270 60 4.473 9 60 270 30 3.836 10 60 270 45 3.979 11 60 270 60 2.981 12 80 230 30 3.245 13 80 230 60 3.331 14 40 250 30 2.965 15 40 250 45 2.869 16 40 250 60 2.884 17 80 250 30 1.727 18 80 250 45 1.621 19 80 250 60 1.693 20 40 270 30 2.023 21 40 270 45 2.147 22 40 270 60 2.094 23 60 230 30 1.556 24 60 230 60 1.584
Table 6. Data set used in checking the accuracy of the created RS models
Set Mold Melt Packing Warpage number temperature (.C) temperature (.C) pressure (MPa) (mm) 1 60 250 60 4.722 2 80 230 45 3.316 3 60 230 45 1.584 Table 7. Errorofseveral RS models
Use of the parameters in RS models Error of RS models (%)
MoT MeT PP Linear Parabolic Cubic
Directly Directly Directly 11.09 13.62 7.07 Inversely Directly Directly 10.70 11.17 7.42 Directly Inversely Directly 11.26 13.53 6.84 Directly Directly Inversely 11.14 13.72 6.77 Directly Inversely Inversely 10.74 11.24 7.40 Inversely Directly Inversely 10.87 11.03 7.14 Inversely Inversely Directly 10.93 11.11 6.69 Inversely Inversely Inversely 11.31 13.64 6.23 Accuracy of the created RS models are shown in Table 7. As seen from Table 7, the cubic polynomial, where all process parameters are included inversely, gives the least error of acceptable value (6.23%), and this model is therefore utilized in the warpage optimization with genetic algorithm (GA) in Sect. 5.
5 Warpage optimization with genetic algorithm
In this section, minimum warpage within the range of process parameters given in Table 1 is to be determined using a global optimization method: genetic algorithm (GA) [10]. The warpage optimization problem for the bus ceiling lamp base can be de fined in the standard mathematical format as below: Find: MoT, MeT, PP (4a) To minimize: Warpage(MoT, MeT, PP).(4b) Subjected to constraints: Warpage . 1.5mm (4c) Within parameter ranges: 40 .C . MoT . 80 .C (4d) 230 .C . MeT . 270 .C (4e) 30 MPa . PP . 60 MPa (4f)
Ranges of MoT and MeT have been selected based on the recommended values in the MoldFlow user manual [8]. The range of PP was selected based on the experience of the manufacturer of the bus ceiling lamp base. To solve the optimization problem in Eq. 4 efficiently, an effective GA has been written in MATLAB programming language. The developed GA is coupled with the RS model developed in the Sect. 4 for warpage to yield a global optimum as shown in Fig. 3. The GA solves the optimization problem by simulating the biological evolution process, Darwin’s theory of survival of the fittest, as with other GAs. The solution starts with a set of potential solutions referred to as population or chromosomes in the literature. Chromosomes are in the form of bit strings and are generated randomly. The chromosomes evolve during several generations, and the generations correspond to optimization iterations. New generations are generated using the crossover and mutation technique: crossover involves splitting two chromosomes and then combining one-half of each chromosome with the other pair, and mutation involves flipping a single bit of a chromosome. The chromosomes are then evaluated using a certain fitness criteria. The best ones are kept while the others are discarded. This process repeats until one chromosome has the best fitness. The critical parameters in GAs are the size of the population, mutation rate, number of generations (i.e., itera- RS model, and GA during warpage optimization tions), etc. In this study, population size of 50, crossover rate of 1.0, mutation rate of 0.1, bit number for each variable of 16, and the number of generations of 500 are employed. The developed GA in this study selects chromosomes based
Fig. 3. Interaction of FE software,
on the objective value and the level of constraint violation. Fitness values of the chromosomes are biased towards the minimum objective value and the least infeasible sets in crossover phase. Most of the GAs in the literature converts the constrained optimization problem into an unconstrained optimization problem through penalty functions before the solution. This brings the difficulty of appropriate selection of problem-dependent penalty coefficients that require user experience. In the program developed in this study, this difficulty is fully avoided since no problem dependent coefficient is needed.
GA reduces the warpage of the lamp base from 2.47mm to 1.32 mm by about 46% after optimization. Warpage values corresponding to the optimum process parameters and initial process parameters are compared in Table 8. Warpage distribution on the bus ceiling lamp base with optimum process parameters is shown in Fig. 4, and optimization history in iterations is demonstrated in Fig. 5. From Table 8 and Fig. 4, it is seen that Table 8. Injection molding process parameters before and after optimization process Injection molding process parameters Warpage MoT (.C) MeT(.C) PP (MPa) (mm) Before optimization 60 250 60 2.47 After optimization 80 267.8 60 1.32
Fig. 4. Warpage distribution on the bus ceiling lamp base with optimum
process parameters
Fig. 5. Optimization history with iterations for warpage Fig. 5. Optimization history with iterations for warpage
the optimum warpage value (1.32) predicted from GA correlates very well with that of (1.315) from MoldFlow.
6 Conclusions
In this study, an efficient optimization methodology using response surface methodology (RSM) and genetic algorithm (GA) is introduced in minimizing warpage of thin shell plastic parts manufactured by injection molding. A bus ceiling lamp base is considered as a thin shell plastic part example. To achieve the minimum warpage, the appropriate process condition parameters are determined. Mold temperature, melt temperature, packing pressure, packing time, and cooling time are considered as process condition param