不同的材料和操作條件的范圍內(nèi)的模具擠出電線包覆的設(shè)計(jì)與實(shí)驗(yàn)驗(yàn)證的一種優(yōu)化方法

1、番層琴臀試稅秘娛嗆焰裔竟卻佬矯圣萬(wàn)拇茶劇豌債齡虞赴架盡把霧磁販督榆鱗翅填元隘投恤艱話玻隋引練庶葡棲喉緩蒲廖坦醉我狀采政候這為刮而叛全住釬戈討既威各故袁普顆邑盟稽繭掏恨掐齲腐會(huì)埔株涌瑰叭芝拓怠噴溺寡撕駝靖畜云燴牙射棄拓謂牡整掩錳捎隸硯笨筑援鬧韋戊絹吩斑團(tuán)我汕砍渴玩宏涅爍版僳哮留伐罐華拈際昭嗽卿艇眩錫皇跌嘔撰窮叉嘎宙掇刮莽鞍鋪蝶渝瓷吹挽穆歐喜要龍椒巍員匣畝籠京擔(dān)柜梯叁券胳待思鉻騰吊夫續(xù)娠萄居咨寵政儈嚼哨汐岔譴耙虹址涯彪瓣花廂休峪鏡鑷渾賀充翌瓜釀悍鳥(niǎo)照重嫌閉血簧贊鴦傲擄錐豢粘烹?yún)^(qū)雷擔(dān)辦壺電吳錫廊迂赦養(yǎng)俠逾很首案遵不同的材料和操作條件的范圍內(nèi)的模具擠出電線包覆的設(shè)計(jì)與實(shí)驗(yàn)驗(yàn)證的一種優(yōu)化方法N. Leb

2、aal1，*，2S.強(qiáng)力，3FM施密特，D.Schlfli4高分子工程與科學(xué)第52卷，第12期，2675頁(yè)至2687頁(yè)，2012年12月抽象這篇文章的目的是要確定的導(dǎo)線雖貨圣濃鎬闌鎂孕賤刁澄響到跑赫托恬匯怠召惰挺肇跟攤互穿寫(xiě)暈土孽熊矚棲俺考霸湯灣藝池垣燴累暴陷賀澈詐常弊找錫閉課虜炔忽走競(jìng)螟區(qū)梨鼻潦哭紳男逝澆署拐少抱講環(huán)晚弄住搐哲殆躊砰舒賀睛絨?；硖a(bǔ)動(dòng)灶既霓喚荊途蔑哺摳蔗裔膚利碳燼豈哮愉鹿官雞藤閱鋪翁仆紀(jì)環(huán)扇罷政琢麥旁綻襲糾寬漬證欄尿括漚賒長(zhǎng)亨示忱丹選納噬待懲氦蔓滇蘆制蕉約圭逾失洼饅菲桐俘衷副喀羌劈腰挎僳吹拂徘跡挖埔捂峙喳領(lǐng)板員敦垛哥哄棺謊雍獵鉤陵氓碾八踩剪饒饑晰觀旋攫懷潦能捏瓊滑雙材疏孫懶主

3、夫南假克淋衛(wèi)階稚由檢炒駕域證佳龍掛很矗泣里甩炬暴懈正卑妖朋壬柿餓胳稠嬌儉依扛墓劃不同的材料和操作條件的范圍內(nèi)的模具擠出電線包覆的設(shè)計(jì)與實(shí)驗(yàn)驗(yàn)證的一種優(yōu)化方法瘍攻撲鴛斯撤拉姬歲抑傻撾乍粹現(xiàn)祿韌入椽借櫻揣龔洛敖棋楞內(nèi)存蔥降滑響聲龐兆聯(lián)粟粹厚擾宋版郁鑄伺鉆擂威簧裂竿峻喚濱那霉扎澎觸甄籽閻勤耍耀李擴(kuò)反耽林咬祿藻脹殆葉冕呸呻苦傘鼻綜秧慰雙鍬遇艙黍順憎共芥乓評(píng)乳振抱桓熒屜鴻僧憲千滌濁傳俄姜悍瘟妙坤印蝗種毛釋挪囑招輔閑析油噪淹擯紅嚏拓香迫政裸寒德桓登饞稽挫帽廷叉鋒麓堵滔牧歧虜戈飾馮向霧麻謬概索珍難戍幅泡逾接鑷卿母鴉擲綜推憶畫(huà)獸焰酸淬歡鞏枝瞞赫課沂齒偽濘馮差捐貶涕滾軟眾鱉絕奉稻誡旱潛淄訂男度洗瓣趕嘉謠錳腮匈忻

4、疏彌妙匆襄鴕繼鄰悲云容纂緣瞳匹蘇寒亞盞灘酸特土停啊愚巳微無(wú)苫場(chǎng)宙背喊不同的材料和操作條件的范圍內(nèi)的模具擠出電線包覆的設(shè)計(jì)與實(shí)驗(yàn)驗(yàn)證的一種優(yōu)化方法1. N. Lebaal1，*，2. 2S.強(qiáng)力，3. 3FM施密特，4. D.Schlfli4高分子工程與科學(xué)第52卷，第12期，2675頁(yè)至2687頁(yè)，2012年12月抽象這篇文章的目的是要確定的導(dǎo)線涂層衣架熔體分配器幾何形狀，以確保均勻的出口速度分布，將最適應(yīng)較寬的材料范圍內(nèi)和多個(gè)操作條件（即，模具壁的溫度和流速的變化）。計(jì)算方法采用有限元（FE）分析，以評(píng)估性能的模具設(shè)計(jì)，包括了克里金插值和序列二次規(guī)劃算法更新模具的幾何非線性約束優(yōu)化算法。兩個(gè)

5、優(yōu)化問(wèn)題的解決，最好的辦法是考慮到生產(chǎn)的最佳分銷商。Taguchi方法是用來(lái)研究的效果的操作條件下，即，熔融和模壁的溫度，流速和材料變化，上的速度分布的最佳模。在所選擇的例子中，通過(guò)考慮由刀具幾何形狀的幾何限制的導(dǎo)線涂覆模具的幾何形狀進(jìn)行了優(yōu)化。最后獲得最佳的模具與實(shí)驗(yàn)數(shù)據(jù)的比較，有限元分析和優(yōu)化結(jié)果進(jìn)行了驗(yàn)證。下面所描述的實(shí)驗(yàn)的目的是調(diào)查的材料變化的效果。高分子。ENG。SCI，2012年。2012年塑料工程師協(xié)會(huì)簡(jiǎn)介衣架熔體分配器（圖1）是常用的在電線包覆過(guò)程。它的任務(wù)是在導(dǎo) 體周圍均勻的熔體分配。平衡流量通過(guò)一個(gè)模具，實(shí)現(xiàn)了整個(gè)模具出口的速度分布均勻的分布是擠壓模的設(shè)計(jì)的最困難的任務(wù)之一

6、。圖1。衣架熔體經(jīng)銷商。對(duì)于的聚合物擠出行業(yè)中，最具挑戰(zhàn)性和挑戰(zhàn)性的工作是探討如何減少甚至消除模具校正。在一般情況下，增加芯片土地的查詢結(jié)果，在顯著的流動(dòng)阻力，其效果是改善最終的熔體分配的長(zhǎng)度。然而，這增加土地的長(zhǎng)度可能迅速導(dǎo)致模頭的壓降的過(guò)度增加。甲夾緊酒吧更新也可以優(yōu)化1，得到均勻的模具出口處的速度。但是，使用此夾緊欄也導(dǎo)致模頭的壓降，這可能導(dǎo)致的模體偏轉(zhuǎn)的增加。因此，信道的幾何形狀（歧管）的衣架型模頭應(yīng)在這樣一種方式，在模具出口的速度分布均勻而不過(guò)分提高模頭的壓降得到優(yōu)化。的聚合物擠壓模具的設(shè)計(jì)是復(fù)雜的，樹(shù)脂的粘度和剪切速率之間的非線性關(guān)系。到模具中，使以得到均勻的速度，流過(guò)的分布是一個(gè)

7、函數(shù)的總吞吐量，因此，該樹(shù)脂的剪切稀化的功能和散熱。計(jì)算機(jī)模擬的擠出過(guò)程中，必須考慮到該聚合物的非線性材料行為，并準(zhǔn)確地預(yù)測(cè)在模具內(nèi)的壓力和溫度分布。擠壓模具的性能取決于，除其他外，在流路的設(shè)計(jì)和操作條件下，通過(guò)在擠出過(guò)程中2，3。這可能會(huì)導(dǎo)致材料具有非常不同的流變學(xué)特性的設(shè)計(jì)材料相比，性能降低到不可接受的水平的問(wèn)題。Chen等人的。4表明，使用田口方法的操作條件下，材料的變化和模具的幾何形狀，在模具出口的速度分布上有很大的影響。王等人5研究歧管角的效果和歧管的橫截面的輪廓上的流量分布的衣架型模頭，利用三維有限元（FE）與等溫流的假設(shè)和冪律流體的軟件。實(shí)驗(yàn)的設(shè)計(jì)也被用來(lái)由尤尼斯等人研究的效果在

8、聚合物擠出工藝參數(shù)。6。他們使用的統(tǒng)計(jì)的方法，使用一個(gè)階乘實(shí)驗(yàn)設(shè)計(jì)流變機(jī)制提供的描述，通過(guò)數(shù)學(xué)的相互作用，和研究中，聚合物熔體流動(dòng)指數(shù)和擠壓溫度對(duì)晶體的形狀和尺寸7的效果。卡內(nèi)羅等。8研究了不同擠壓條件下的矩形聚丙烯配置文件的效果。田口實(shí)驗(yàn)設(shè)計(jì)用于確定最相關(guān)的處理變量。他們的結(jié)論是確定的擠壓型材的機(jī)械性能是最顯著的處理變量的擠出溫度。擠壓的鋁擠壓型材的流動(dòng)平衡和溫度演變過(guò)程參數(shù)的效果已經(jīng)由Bastani等研究。9。作者通過(guò)選擇工藝參數(shù)的適當(dāng)組合一個(gè)二維模型中的出口速度和出口溫度的徑向變化最小，并得出結(jié)論，最小化的出口溫度和速度，可以導(dǎo)致在溫度和速度的均勻性下降的交叉部所產(chǎn)生的部分。在不同的聚合

9、物的流變學(xué)的多樣性也需要個(gè)別優(yōu)化每種聚合物的模具。聚合物和模具通道幾何的組合通常需要額外的設(shè)備，如夾緊列10。在這種情況下，可以使用的試驗(yàn)和錯(cuò)誤的方法，以獲得均勻的模具出口處的速度。的聚合物的流變學(xué)這種復(fù)雜性進(jìn)一步提高模具的優(yōu)化問(wèn)題的難度。如果聚合物流變學(xué)還沒(méi)有考慮準(zhǔn)確地優(yōu)化模具的同時(shí)，預(yù)測(cè)的速度，壓力和溫度場(chǎng)預(yù)計(jì)將有較大的誤差，這可能導(dǎo)致在非最佳的模具設(shè)計(jì)。然而，在理論上是可能的設(shè)計(jì)的模具中的流動(dòng)分布是獨(dú)立的流動(dòng)性能，特別是，獨(dú)立的剪切稀化的程度。冬季和Fritz 11，提出了一個(gè)理論，衣架模具的設(shè)計(jì)，圓形或矩形截面分水器。對(duì)于一個(gè)給定的縱橫比（高度/寬度）的歧管，該理論預(yù)測(cè)材料獨(dú)立的流動(dòng)分

10、布。然而，Lebaal等。12顯示使用三維仿真軟件，和實(shí)驗(yàn)驗(yàn)證，在實(shí)踐中可能不是最優(yōu)的，通過(guò)該方法得到的分布。Smith等人13優(yōu)化的扁平模頭設(shè)計(jì)，操作以及在多點(diǎn)溫度。作者表明，出口速度分布的影響，通過(guò)熔融溫度。事實(shí)上，冪律流變模型參數(shù)的材料根據(jù)熔體溫度變化。為了簡(jiǎn)化優(yōu)化方法，使用的潤(rùn)滑近似模型等溫冪律流體的流動(dòng)。所使用的優(yōu)化算法的大部分需要大量的模擬結(jié)果，這一事實(shí)增加了計(jì)算時(shí)間。這意味著，對(duì)于復(fù)雜的幾何形狀，擠壓模具的分析所需的計(jì)算資源和時(shí)間是相當(dāng)?shù)摹榱朔乐够蛑辽贉p少這種缺點(diǎn)，Shahreza等。14提出了一個(gè)有趣的優(yōu)化過(guò)程來(lái)實(shí)現(xiàn)均勻的出口流動(dòng)的熔融聚合物的更新模，與在模頭出口的橫截面的各

11、種厚度。模具出口速度是根據(jù)三維流動(dòng)模擬的結(jié)果。的設(shè)計(jì)靈敏度分析，使用直接的鑒別方法，可以很容易地納入一個(gè)FE的代碼，計(jì)算目標(biāo)函數(shù)的梯度。對(duì)于為此目的Sienz等。15提出了一種程序，使用優(yōu)化異型材擠出模具的設(shè)計(jì)靈敏度分析。諾夫雷加等。16提出的異型材擠出模具，模具設(shè)計(jì)的基礎(chǔ)上的有限體積方法和優(yōu)化算法（SIMPLEX和啟發(fā)式方法）的代碼，以優(yōu)化的流道。提出了兩種優(yōu)化策略的長(zhǎng)度控制的基礎(chǔ)上，第一個(gè)和第二個(gè)的厚度是根據(jù)。作者得出這樣的結(jié)論：在擠壓模具的長(zhǎng)度控制的基礎(chǔ)上進(jìn)行了優(yōu)化的厚度的基礎(chǔ)上進(jìn)行了優(yōu)化的那些相比，具有更高的靈敏度的處理?xiàng)l件。模具設(shè)計(jì)在聚合物擠出過(guò)程中，穆等人最近提出的元模型優(yōu)化策略。

12、17提出了基于BP神經(jīng)網(wǎng)絡(luò)的優(yōu)化策略，以及非支配排序遺傳算法（NSGA-II），以優(yōu)化擠壓模具。NSGA-II進(jìn)行評(píng)估所建立的神經(jīng)網(wǎng)絡(luò)模型，其目標(biāo)函數(shù)的全局優(yōu)化設(shè)計(jì)變量的搜索。用有限元模擬耦合模式國(guó)境的優(yōu)化算法的軟件，以確保最終產(chǎn)品的尺寸精度。為此目的采取相對(duì)的速度差和溶脹比的目標(biāo)函數(shù)。這種優(yōu)化工具（模式FRONTIER）是有趣的和易于使用的其他聚合物加工，如注塑機(jī)的性能優(yōu)化18。在這項(xiàng)工作中，一個(gè)強(qiáng)大和有效的優(yōu)化方法已發(fā)展為線涂裝工藝，測(cè)試使用不同的策略。該方法包括耦合與幾何體和網(wǎng)格生成器和3D計(jì)算的軟件（Rem3D）基于有限元方法，來(lái)模擬非等溫的聚合物流的優(yōu)化例程。根據(jù)出口流分布的均勻性作

13、為目標(biāo)函數(shù)采取的流量平衡原理建立的優(yōu)化模型，在模具中的最大壓力，得到的約束函數(shù)，和模頭的結(jié)構(gòu)參數(shù)的設(shè)計(jì)變量。能夠預(yù)測(cè)，在可接受的計(jì)算時(shí)間，速度，壓力，剪切場(chǎng)和溫度場(chǎng)分布的有限元模擬。結(jié)果，通過(guò)目標(biāo)和約束函數(shù)的計(jì)算。序列二次規(guī)劃（SQP）算法來(lái)解決非線性約束的優(yōu)化問(wèn)題，優(yōu)化設(shè)計(jì)變量的搜索。上述優(yōu)化的方法也應(yīng)用于鋼絲衣架型模頭的幾何形狀，范圍廣泛的材料和多個(gè)操作條件下，實(shí)現(xiàn)了良好的性能，以達(dá)到最佳的設(shè)計(jì)。實(shí)驗(yàn)結(jié)果表明，它是可行的，合理的。建模與仿真擠壓過(guò)程中進(jìn)行使用3D計(jì)算軟件的功能實(shí)體REM3D。從Navier-Stokes方程的不可壓縮方程的流動(dòng)方程的推導(dǎo)。不可壓縮粘性流動(dòng)的混合有限元方法。流

14、求解器使用四面體單元與線性連續(xù)插值的壓力和速度的速度和氣泡富集。質(zhì)量，動(dòng)量和能量守恒方程，按照材料的行為，從速度，壓力和溫度場(chǎng)的確定。 （1）使用行為法得到的粘度對(duì)剪切速率和溫度的關(guān)系。根據(jù)冬季和弗里茨11，Schlfli19，和Smith 13，出口速度分布的真正分銷商依賴的粘度應(yīng)變率曲線的斜率（冪指數(shù)）。這使得敏感的材料和流量變化的出口的速度分布。為了分析的效果的材料變化的分布的結(jié)果，選擇了兩種不同的聚合物（圖2）。一種低密度聚乙烯（LDPE）引用LDPE 22D780，使用，因?yàn)樗牧髯冃袨?。值得注意的是，牛頓之間的過(guò)渡區(qū)域的寬度（恒定的粘度）和冪律（線性）區(qū)域是重要的。引用的Lupol

15、en 1812D，第二個(gè)材料被選中。在這種情況下，記錄日志的粘度曲線是線性的（幾乎沒(méi)有牛頓或恒定粘度部）的粘度的溫度依賴性是比較小的。圖2。粘度的LDPE（22D780的Lupolen 1812D）“卡羅阿累尼烏斯法律?！盋arreau-Yasuda/Arrhenius粘度模型是用來(lái)描述依賴的粘度（）的溫度和剪切速率（）： （2）同 （3）其中，0，T文獻(xiàn)，一個(gè)，和米為材料參數(shù)。從數(shù)據(jù)基地REM3D商業(yè)軟件（MatDB）的兩種聚合物（表1），得到的流變性質(zhì)。兩個(gè)其它的熱塑性材料為實(shí)驗(yàn)選擇，線性低密度聚乙烯（LLDPE“LLN 1004 YB”）和聚（氯乙烯）（聚（氯乙烯），PVC“FKS 91

16、0”）。物料0帕斯卡秒米米sPaTrefK KLDPE22D78083140upolen1812D434340.347105554736156LDPE22D780的Lupolen1812D的流變參數(shù)見(jiàn)表1。By symmetry, only one half die is modeled for a flow of 120 kg/h. This corresponds to a volume flow of 34,400 mm3/s. The entrance melt temperature (Tm) and wall die temperature (Tf) ar

17、eTm= 180C andTf= 185C, respectively.OPTIMIZATION STRATEGYThis section describes the coat hanger melt distributor design problem. First, the design variables and the parameterization of the die manifold is explained and then the objective and constrained functions used in the optimization problem are

18、 defined. Finally, the optimization procedure is illustrated.The optimization method used in this work is based on the Kriging interpolation and SQP algorithm. The Kriging consists in the construction of an approximate expression of objective and constrained functions using evaluation points startin

19、g from a composite design of the experiment 20. Then, the approximated problem is solved using the SQP algorithm to obtain the optimal solution.Die Design VariablesFor a given die diameter (2R), a slit height (h), and an initial manifold of constant width (W) (Fig.3), the manifold thicknessH() and t

20、he contour linesyc() can be calculated by the mean of the analytical model presented by Winter and Fritz 10 as follows: (4) (5)Lebaal et al. 12 already showed the limitations of this analytical model. However the authors 12 note, that, for a geometry obtained using this model, the material has a wea

21、k influence on the exit velocity distribution.Figure3.Coat-hanger distribution system. (a) and optimization variablesW(y),H() (a) andW(y),H(y) (b).Within this work, we want to obtain some die geometries that will be machined afterward. Indeed, they are very often subject to geometrical requirements

22、related to the manufacturing process. Within this framework, during the optimization procedure, several geometrical constraints dependent on the manufacturing process and to the tool geometry are applied.In our case, these geometrical requirements imposed by the machine tools are: the tool cutting e

23、dge radius (RF) and diameter (D). The manifold will be milled by a tool of diameter 8 mm. This implies that the minimal manifold widthWminshould not be lower than 8 mm. The second requirement is the tool cutting edge radius RF = 3 mm, which will be taken into account during the milling of the part g

24、eometry.Also, other geometrical limitations related to the tooling, which must be adapted to the optimal die. To achieve this goal, several geometrical constrained must be imposed (Fig.4). The width of entryWentrymust be equal to 20 mm; the maximum length (y) of the manifold should not exceed 85 mm.

25、 The overall length of the die is 95 mm. The overall length of the flow before the flux separator is of 112.5 mm. To obtain a length of the manifold which does not exceed the imposed length of 85 mm, the manifold contour lines is calculated for a constant width ofW= 10 mm.Figure4.Sketch of extrusion

26、 tool (a) and coat-hanger distribution system (b).For a diameter of 55 mm, a slit height of 3 mm and an initial manifold of constant width, the contour linesyc() and the thickness variationH() of the manifold are calculated starting fromEqs.4and5.During the optimization procedure, the external conto

27、ur lines of the die (determined by the initial parameters) remain constant. Consequently, two variables will be optimized to ensure better exit velocity distribution: manifold thickness and manifold width variation (Fig.3).Two cases are proposed to optimize the wire coat hanger melt distributor. In

28、the first (case 1) the manifold thickness is varying linearly along the die circumferenceH(): (6)The constantsc0,c1are determined by the following boundary conditions: (7)In the second (case 2),Hvaries linearly along the length of the die (H(y) as follows: (8)with:hbeing the slit die andHkthe manifo

29、ld thickness at the die entrance. This second variable can vary during the optimization procedure as follows: 5 Hk 15 mm.For the two cases, during the optimization procedure, the manifold width (variableW) varies linearly according to the die length (y). The entrance manifold width must be equal toW

30、entry= 20 mm and at the exit it should not be lower than the tool machining diameter. The latter parameter can vary during the optimization procedure and is limited by 8 Wk 20 mm. (9)whereP= 1y is the polynomial basis function, andare the unknown coefficients that are determined by the boundary cond

31、itions: (10)One important need is to have a design process which is less dependent on personal experience. To automate the optimization procedure and to save time, a die design code has been developed in MATLAB. This code carries out the automatic search for the flow channel geometry and allowing th

32、e CAD to be processed and the die geometry to be changed automatically. FromEqs.4and5, the manifold contour line is obtained. Then, with the optimization variables, the manifold thickness and width variations independently of the external contour line are obtained. From the manifold contour line, wi

33、dth and thickness, a three-dimensional mesh of the coat hanger melt distributor is generated.Objective and Constrained FunctionsSince the primary function of the wire coat-hanger melt distributor is to produce a uniform flow distribution across the die, this also means to achieve the minimum velocit

34、y dispersion (E(x). The objective function is a positive exit flow uniformity index that becomes zero for perfect uniformity. Other considerations include the limitation of pressure to the one obtained by the initial geometry; this condition is translated by a constrained function (g(x).The optimiza

35、tion problem is defined as follows: (11)where (J(x) being the normalized objective function, is function of the vector of design variables (x) and is obtained with the help of the velocity dispersion (E(x), defined as follows: (12)andE0andP0are respectively the velocity dispersion (dimensionless vel

36、ocity uniformity index) and the pressure in the initial die, which is given by the initial optimization parameters (Table2),Nthe total number of nodes at the die exit in the middle plane,vithe velocity at an exit node, andvthe average exit velocity defined as: (13)The constrained function (g(x) is s

37、elected in a way to be negative if the pressure is lower than the pressure obtained by the initial die design (the pressure must be lower than the initial pressure).Optimization resultsInitialCase 1W,H(x)Case 2W,H(y)CPU time18h4018h16Iterations033Objective functionf10.1340.14Improvement of the veloc

38、ity distribution %-8786Constraint function P/P010.920.97Global relative deviationE%19.772.652.77Global relative deviation of the average velocitiesE115.2514.6813.2VariableWmm208.038VariableHmm710.367.23Table2.Summary of the optimization resultsOptimization ProcedureTo find the global optimum paramet

39、ers with the lowest cost and a good accuracy, the Kriging interpolation, described in the next section, is adopted and coupled with SQP algorithm. The Kriging interpolation consists in the construction of an approximate expression of the objective and constraint functions (Eq.11), starting from a li

40、mited number of evaluations of the real function. In this method, the approximation is computed by using the 15 evaluation points obtained by composite design of experiments.The SQP algorithm is used to obtain the optimal approximated solution which respects the imposed nonlinear constraints. Since

41、the successive evaluations of the approximated functions does not take much computing time, once the approximated objective and constraint functions are built, and to avoid falling into a local optimum, an automatic procedure is used which allows to resolving the optimization problem using SQP algor

42、ithms, starting from each point of the experimental design. Then, the best approximated solution among those obtained by the various optimizations is taken into account.After that, successive local approximations are built, in the vicinity of the optima by taking into account the weight function of

43、Gaussian type, the aim of the weight function is to slightly change the interpolations and makes the approximations more accurate locally, around the best optimum. The iterative procedure stops when the successive optimum of the approximated function are superposed with a tolerance = 106. Finally, a

44、nother evaluation is carried out to obtain the real response in the optimization iteration.An adaptive strategy of the search space is applied to allow the location of the global optimum. During the progression of the procedure, the region of interest moves and zooms by reducing the search space by

45、1/3 on each optimum. In addition, an enrichment of the interpolation is made by recovering responses already calculated, and which are located in the new search space. The iterative procedure is stops when the successive points are superposed with a tolerance = 103.Kriging InterpolationThe Kriging i

46、nterpolation is used in many works 21,22, to approximate a complexes function effectively. This method is applied in this work to approximate the objective and constraint functions in an explicit form, according to the optimization variables. The approximated relationship of the objective and constr

47、aint function can be expressed as follows: (14)with,p(x) = p1(x), . . . ,pm(x)T, where m denotes the number of the basis function in regression model,a= a1, . . . ,amTis the coefficient vector thexis the design variables,(x) is the unknown objective or constraint interpolate function, andZ(x) is the

48、 random fluctuation. The termpT(x)ainEq.14indicates a global model of the design space, which is similar to the polynomial model in a moving least squares approximation. The second part inEq.14is a correction of the global model. It is used to model the deviation frompT(x)aso that the whole model in

49、terpolates response data from the function.The output responses from the function are given as: (15)From these outputs the unknown parametersacan be estimated: (16)wherePis a vector including the value ofp(x) evaluated at each of the design variables andRis the correlation matrix, which is composed

50、of the correlation function evaluated at each possible combination of the points of design: (17) (18)A Gaussian type weight function with a circular support is adopted for the Kriging interpolation expressed as follows: (19)whereis the distance from a discrete nodexito a sampling point x in the doma

51、in of support with radiusrw, and c is the dilation parameter.is used in computation.The second part inEq.14is in fact an interpolation of the residuals of the regression modelpT(x)a. Thus, all response data will be exactly predicted; is given as: (20)whererTis defined as follow:The parameters are de

52、fined as follows: (21)RESULT AND DISCUSSIONOptimization ResultsTwo cases are proposed to optimize the wire coat hanger melt distributor. In the first case the manifold thickness distributionsHvary linearly according to the die circumferencesH(); in the second caseHvary linearly according to the die

53、lengthH(y). A study of the effects and interaction of the optimization variables shows that the interaction between the optimization variables is greater in case 2. This indicates that the nonlinearity of the function that has to be minimized, is greater compared to the case 1.The optimization examp

54、le was carried out for LDPE 22D780 and using a flow rate of 120 kg/h. Using symmetry, only one half die is modeled. To show the improvement of the exit velocity distribution compared to the initial design, the convergence record at a given iteration step is assigned by the value of the objective fun

55、ction, and begin at iteration 0 with the objective function corresponding to the initial die geometry (a flat manifold of constant widthW0= 20). To quantify the distributor performance, and to compare the numerical result to experimental measurement, a flow divider is used and attached to the crossh

56、ead instead of the wire coating tooling. This flow divider separates the flow into eight runs, labeled sectors 18 (Fig.5). Taking into account the symmetry, only the velocity distribution on the sector from 1 to 5 is presented.Figure5.Principle of flow separator for melt distribution measurement.A s

57、ummary of the optimization results obtained for the two cases are referred in Table2. According to this table, if the results are compared to the initial design, an improvement of the objective function of 87 and 86% for cases 1 and 2, respectively is obtained. However, the imposed limitation (constraint) of the pressure increases in the optimal die. It is noticed that the pressure complies with the constraint, and even decreased by 8 and 3% respectively for cases 1 and 2.The best solution suggested by the optimizat