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用配置空間的方法對(duì)注塑模冷卻系統(tǒng)進(jìn)行設(shè)計(jì)
c.g.李， c.l.李*
香港城市大學(xué)制造工程及工程管理部，香港
2007年5月3日收到; 2007年11月18日接納
摘要
注塑模的冷卻系統(tǒng)對(duì)注射模具的成型過(guò)程和塑料零件質(zhì)量影響是非常重要的。盡管已有各種針對(duì)冷卻系統(tǒng)的分析、優(yōu)化和制作的研究，但冷卻系統(tǒng)的布局設(shè)計(jì)方面并沒(méi)有得到很好的發(fā)展。在規(guī)劃設(shè)計(jì)階段，我們主要關(guān)注的是冷卻系統(tǒng)的可行性和其他模具組件插入是否發(fā)生干預(yù)。本文介紹了利用配置空間（C空間）的方法來(lái)解決這一重要問(wèn)題。然而高維配置空間方法一般需要處理一個(gè)如冷卻系統(tǒng)般復(fù)雜的系統(tǒng)，冷卻系統(tǒng)的特殊特點(diǎn)設(shè)計(jì)目前正在探索研究中，利用C空間在三維空間或更低維空間計(jì)算和存儲(chǔ)的特別技術(shù)也在發(fā)展中。這種新方法是由作者對(duì)以前啟發(fā)式方法的改善，因?yàn)镃空間的代表性能使自動(dòng)布局設(shè)計(jì)系統(tǒng)在所有可行的設(shè)計(jì)中進(jìn)行更系統(tǒng)的搜索。自動(dòng)生成候選布局設(shè)計(jì)的一個(gè)簡(jiǎn)單的遺傳算法是C空間代表性的實(shí)施和綜合。遺傳算法所產(chǎn)生的設(shè)計(jì)實(shí)例，給這
種方法提供了可行性證明。 c 2007 Elsevier公司有限公司，保留所有權(quán)利。
關(guān)鍵詞: 冷卻系統(tǒng)設(shè)計(jì)；注塑模具；配置空間的方法
1.導(dǎo)言
注塑模的冷卻系統(tǒng)對(duì)注射模具的成型過(guò)程和塑料零件質(zhì)量影響是非常重要的。大量涉及對(duì)冷卻系統(tǒng)分析[ 1,2 ] ，及商業(yè)CAE系統(tǒng)，如Moldflow [ 3 ]和moldex3d [ 4 ] 的研究被廣泛應(yīng)用于工業(yè)。以?xún)?yōu)化某一特定的冷卻系統(tǒng)的研究技術(shù)亦已報(bào)道[ 5-8 ] 。最近，通過(guò)使用新形式的制造技術(shù)以建立更好的冷卻系統(tǒng)的研究已被報(bào)告。徐等人[ 9 ]報(bào)道了他們的模具意念：保持一定距離的冷卻水道的設(shè)計(jì)和制作。孫等人[ 10,11 ]用數(shù)控銑床銑削生產(chǎn)U形槽冷卻渠道和俞[ 12 ]提出了一個(gè)棚架形冷卻結(jié)構(gòu)的設(shè)計(jì)。
盡管各種研究的重點(diǎn)主要集中在冷卻系統(tǒng)的初步設(shè)計(jì)過(guò)程中冷卻系統(tǒng)的功能實(shí)現(xiàn)問(wèn)題，布局設(shè)計(jì)階段過(guò)程中沒(méi)有得到很好發(fā)展的冷卻系統(tǒng)的可行性和可制造性設(shè)計(jì)問(wèn)題。關(guān)注的重點(diǎn)主要是：在初步設(shè)計(jì)階段冷卻系統(tǒng)的可行性且與其他的模具部件是否干預(yù)。如圖1所示 。從中可以看到注塑模的各子系統(tǒng)許多不同的組成部分，如噴射器的管腳，滑塊等等，都必須裝入模具中。為每個(gè)回路冷卻水道尋找最佳位置以?xún)?yōu)化冷卻性能并避免與其他組件干擾不是一項(xiàng)簡(jiǎn)單的任務(wù)。另一個(gè)讓規(guī)劃布局設(shè)計(jì)更復(fù)雜的問(wèn)題是，單獨(dú)的冷卻水道需要和出水道和進(jìn)水道連接而形成一條環(huán)形水道。因此，改變一條水道的位置，其他水道可能也需要改變。 在圖 2所示 。優(yōu)化冷卻系統(tǒng)的每個(gè)水道的理想位置都如圖2（a）所示 。假設(shè)當(dāng)冷卻系統(tǒng)及其他模具組件都裝入模具內(nèi)部時(shí)，模具組件O1和水道C1是干擾的。因?yàn)镃1與其他組件可能的干擾而無(wú)法移到附近的一個(gè)位置，它必須被縮短長(zhǎng)度。因此， 通過(guò)移動(dòng)C2和延長(zhǎng)C3使他們保持連接，如圖2（b）所示。基于其新的長(zhǎng)度， C3又與其他模具組件O2發(fā)生干擾，進(jìn)一步修改是必要的，最后的設(shè)計(jì)結(jié)果如圖2（c）所示 。鑒于一個(gè)典型的注塑模具可能有10條以上的冷卻水道，每個(gè)水道與其他模具組件都可能存在著潛在的干擾，手工找出一個(gè)優(yōu)化布置設(shè)計(jì)是非常繁瑣的。 本文介紹了一種在設(shè)計(jì)過(guò)程中支持自動(dòng)布局的新技術(shù)。對(duì)于這種新技術(shù)，配置空間（C空間）的方法是用來(lái)在所有可行的設(shè)計(jì)中提供一個(gè)簡(jiǎn)潔的有代表性的布圖設(shè)計(jì)。C空間的代表性是通過(guò)利用解決布局設(shè)計(jì)問(wèn)題這個(gè)特殊特點(diǎn)的有效方法構(gòu)建的，而不是采用啟發(fā)式規(guī)則來(lái)生成的布局設(shè)計(jì)，這就就好比以前作者開(kāi)發(fā)的自動(dòng)布局設(shè)計(jì)系統(tǒng) [ 13,14 ] ，這個(gè)新的C空間方法能使自動(dòng)布局設(shè)計(jì)系統(tǒng)在所有可行的布圖設(shè)計(jì)中進(jìn)行更系統(tǒng)的搜索。
2.配置空間的方法
一般來(lái)說(shuō)， 一個(gè)系統(tǒng)的C空間是當(dāng)該系統(tǒng)的每個(gè)自由度被視為一個(gè)層面的結(jié)果而導(dǎo)致的空間。配置空間中的區(qū)域被標(biāo)記為堵塞區(qū)域或自由區(qū)域。在自由地區(qū)的點(diǎn)對(duì)應(yīng)于組件間沒(méi)有相互干擾的系統(tǒng)的有效配置。在被堵塞區(qū)域的點(diǎn)對(duì)應(yīng)于組件間相互干擾的系統(tǒng)的無(wú)效配置。 C空間最初被洛薩諾-佩雷斯定形 [ 15 ]以解決機(jī)器人路徑規(guī)劃的問(wèn)題和關(guān)于這方面的研究一項(xiàng)調(diào)查已被明智和鮑耶 [ 16 ] 報(bào)道 。C空間的方法也被用來(lái)解決定性推理方面的問(wèn)題（例如，[ 17,18 ] ）和運(yùn)動(dòng)裝置的自動(dòng)化分析與設(shè)計(jì)（例如， [ 19-21 ] ） 。作者在由多個(gè)國(guó)家組成的自動(dòng)設(shè)計(jì)機(jī)構(gòu)做研究時(shí)[ 22 ， 23日]研究了一種C空間的方法。
（a） 冷卻水道C1 和模具組件 （b）c1截短，c2移動(dòng)，c3延長(zhǎng)
O1干擾發(fā)生在理想的位置
（c）c3移動(dòng)，c2截短從而效果最佳
圖3冷卻系統(tǒng)的自由度
2.1一個(gè)冷卻系統(tǒng)的C空間
一個(gè)高維C空間可以用來(lái)表示給定的某一冷卻系統(tǒng)的初步設(shè)計(jì)中所有可行的布圖設(shè)計(jì)。圖3給出了一個(gè)例子。冷卻系統(tǒng)的初步設(shè)計(jì)由4冷卻水道組成。從初步設(shè)計(jì)中生成一個(gè)布局設(shè)計(jì)，渠道的中心和長(zhǎng)度需要被調(diào)整。正如圖3所示，該水道c 1的中心可沿著X1 和X 2方向移動(dòng)，其長(zhǎng)度可以沿X 3 方向調(diào)整。同樣地，C2長(zhǎng)度的可以沿X 4方向調(diào)整，而其中心可以按X1 和X 3所描述的調(diào)整 ，因此必須與調(diào)整C 1保持連接性的情況相同。通過(guò)運(yùn)用類(lèi)似的觀點(diǎn)對(duì)其他水道，可以看出，冷卻系統(tǒng)有5個(gè)自由度，它們都是標(biāo)注為Xi，i= 1 ， 2 ，…… 5 。原則上， C空間是一個(gè)五維空間而這個(gè)空間的自由區(qū)域中的任何一點(diǎn)都給定了一個(gè)對(duì)應(yīng)的坐標(biāo)值在X i軸上，可以用來(lái)界定渠道的幾何位置且沒(méi)有與其他模具組件造成干擾。在一個(gè)冷卻系統(tǒng)的高維C空間中確定一個(gè)自由區(qū)域，第一步是在獨(dú)立水道的C空間中構(gòu)建自由區(qū)域。
2.2 獨(dú)立水道的c空間構(gòu)造
當(dāng)一個(gè)獨(dú)立的水道c1被確定為單獨(dú)時(shí)，它有三個(gè)自由度，則X 1和X 2為其中心位置而X 3是它的長(zhǎng)度。因?yàn)槔硐氲闹行奈恢煤烷L(zhǎng)度已經(jīng)在初步設(shè)計(jì)中指明，因此假定一個(gè)固定的允許最大變化量δ C為X1 ，X2 ，X3是合理的。c1水道的C空間中最初確定的自由區(qū)域，是一個(gè)尺寸為δc×δc×δC的三維立方體。為避免與模具組件oi發(fā)生任何可能的干擾當(dāng)水道通過(guò)鉆孔插入模具內(nèi)部時(shí)，鉆頭直徑D和沿X3的鉆孔深度必須考慮。假設(shè)直徑D ，Oi開(kāi)始時(shí)用D/2 +M對(duì)于O "抵銷(xiāo)，其中M是水道內(nèi)壁和附近的一個(gè)組件間所允許的最短距離。Oi的增長(zhǎng)有效的減少了水道Ci的長(zhǎng)度對(duì)于直線(xiàn)Li來(lái)說(shuō) 。以圖4為例子 。圖4（a）表明了水道Ci和三模具組件O1、O2、O3可能會(huì)與Ci發(fā)生干擾。圖4（b）顯示了模具組件O' , O ', O '和O "的偏移及 Ci相對(duì)于線(xiàn)段Li的減少量與Ci的x值相符情況。如果Li和模具其他組成部分沒(méi)有交匯點(diǎn)，那么，原來(lái)的水道Ci將不會(huì)與模具組件相交。
（a） 水道Ci和模具的 （b）模具組件和Ci相對(duì)Li的偏移
三個(gè)組件
（c） 模具組件和Ci相對(duì)Pi點(diǎn)的補(bǔ)償 （d）Ci的自由度
（e）Oi相對(duì)Pi的減少量 （f）Ci的自由點(diǎn)Fri
圖4在一個(gè)通道CI自由區(qū)FRi施工的主要步驟
水道是通過(guò)鉆孔從對(duì)模具的表面插入的，任何如Oi的障礙以及鉆孔深度將會(huì)影響水道的構(gòu)建。鉆孔深度及Oi的補(bǔ)償O"沿鉆孔的方向延伸，直到鉆到模具對(duì)應(yīng)的另一面生成水道為止。Oi相對(duì) Pi沿直線(xiàn)Li的減少至Li的終點(diǎn)。如圖4（c）所示，如果點(diǎn)Pi位于Oi"之外 ，沿Li鉆孔產(chǎn)生水道Ci是可行的。
水道Ci的自由區(qū)域Fri用如下方法取得。首先，初始自由地區(qū)Bi是用如圖4（d）所示的Pi點(diǎn)作為中心構(gòu)建的。然后插入與模具交叉取得B 0 。 B 0代表Ci所有可能的變化當(dāng)僅考慮插入的模具幾何形狀時(shí)。然后Fri是從所有障礙的Oi中減去Bi獲得。圖4（e）和（f）顯示了這種減法以及這種例子的結(jié)果FRi。
2.3 基本接近法構(gòu)建冷卻系統(tǒng)的C空間
在一個(gè)冷卻系統(tǒng)的C空間中確定自由區(qū)域FRF，每個(gè)冷卻水道的自由區(qū)域必須以一個(gè)適當(dāng)?shù)姆绞健敖徊妗保允拐系K的效果能恰當(dāng)?shù)耐ㄟ^(guò)FRF描繪對(duì)于所有水道來(lái)說(shuō)。然而在兩個(gè)不同水道之間的自由區(qū)域的標(biāo)準(zhǔn)布爾交叉口無(wú)法執(zhí)行，因?yàn)樗麄兊腃空間在一般跨距于不同的軸線(xiàn)。以圖3為例子 ，C1和C2的C空間分別為（ X1 ， X2 ，X 3 ）和（X 1 ，X3 ，X4 ）。為了更方便在不同的C 空間中的自由區(qū)域之間確定交叉口，從一個(gè)渠道和另一個(gè)渠道的C空間中推算一個(gè)地區(qū)是必要的。以下批注首先介紹了并將用于隨后的討論和其余的文件。
標(biāo)記法用于描述高維空間
S n是指一個(gè)通過(guò)坐標(biāo)定義的n維空間
= {X 1, X 2, . . . , X n}.
Sn是指一個(gè)通過(guò)坐標(biāo)定義的m維空間
= {X , X , . . . , X }.
Pn 是指在Sn 的一個(gè)點(diǎn) p n = (x 1, x 2, . . . , x n)
Rn屬于區(qū)間S n(R n S n)
標(biāo)記法用于描述冷卻系統(tǒng)
n c指在冷卻系統(tǒng)中水道的數(shù)目。
n f指冷卻系統(tǒng)總的自由度。
ci指冷卻系統(tǒng)第i個(gè)水道。
s i指Ci的C空間。
FRi是指在Si中的自由地區(qū)。也就是說(shuō)，它是獨(dú)立水道Ci的自由區(qū)域。
SF指冷卻系統(tǒng)的C空間。
FRF是指SF中的自由區(qū)域。也就是說(shuō)，它是冷卻系統(tǒng)的自由區(qū)域。
假設(shè)Pn在Sn中，Pm在Sm中，圖5（a）用一唯和三唯的的空間點(diǎn)明了突出的例子
(i) (ii) ；而(iii) ， 且
對(duì)(i)Pn 和Pm的坐標(biāo)是一樣的如果Sn和Sm在同一區(qū)間時(shí)。對(duì)(ii)和(iii)Pn在區(qū)間Rm中。因?yàn)镻m在Rm中，當(dāng)點(diǎn)位于Sn和Sm中時(shí)Pn等于Pm。而對(duì)另一坐標(biāo)Pm其可以是任意值；特別對(duì)(ii)和(iii)，假設(shè)水道Cn和Cm，因?yàn)樗鼈兿嘟员仨氝B接。這樣它們的C空間Sn、Sm有相同的坐標(biāo)值。假設(shè)那是一個(gè)結(jié)論？對(duì)應(yīng)到在S n中一個(gè)點(diǎn)P n已選定為Cn。保持連通性，結(jié)論呢？ Cm必須被選擇在以使Sm中的相應(yīng)點(diǎn)Pm與P n共用相同坐標(biāo)在共同的軸線(xiàn)。這意味著Pm和PN可以是任何點(diǎn)在區(qū)間Sm中，該方法已經(jīng)在前面予以定義。在區(qū)域Sn和Sm中的一區(qū)域Rn是Rn和Sm中每一點(diǎn)的簡(jiǎn)化。圖5（ b ）說(shuō)明了相應(yīng)的區(qū)域。投影的正式定義如下面所示。
定義1 （投影）
1.1.如果Xˉ m? Xˉ n, PROJ Sm ( pn )是一個(gè)點(diǎn)=(x,x,……,x),因?yàn)閄 = Xj, x = xj
因?yàn)閕 [1,m]。為了在隨后的討論中簡(jiǎn)化符號(hào)，這一投影是被視為單獨(dú)點(diǎn)Pm的區(qū)間。也即是PROJ Sm ( pn )={Pm}。
. 1.2.如果Xˉ m? Xˉ n，PROJ Sm ( pn )是一個(gè)區(qū)間{ pm |PROJ Sn ( pm ) = { pn }}.
1.3.如果Xˉ m Xˉn , Xˉ n Xˉm ,并且 Xˉ n∩ Xˉ m , PROJ Sm( pn )是一個(gè)區(qū)間Rm = { pm|PROJ SI( pm ) = PROJ SI( pn )},其中Si位于區(qū)間Xˉ n∩ Xˉ m ，如果 n∩ Xˉ m =，PROJ Sm( pn )則定義為Sm。
1.4.ROJSm(Rn) 定義在區(qū)間Rm={Pm|PmPROJ(Pn),PnRn}.
正如在2.1節(jié)所討論的，在FR中的任意點(diǎn)P為冷卻系統(tǒng)的每個(gè)自由度給定了一個(gè)值，使水道與其他模具組件在幾何空間是不會(huì)發(fā)生任何干涉。另一方面， P相對(duì)每個(gè)點(diǎn)s i的投影是，在Ci的每個(gè)自由區(qū)域FR中。因此，F(xiàn)R定義如下。
定義2 （一個(gè)冷卻系統(tǒng)C空間的自由區(qū)域）
FRF = { pF | P R O JSi ( pF ) FRi , i ∈ [1, nC ]}
圖 5 點(diǎn)和區(qū)間在Sn至Sm區(qū)間中的投影。
根據(jù)定義1.1知道， 從到的區(qū)間投影始終只包含一個(gè)單一的點(diǎn)，因?yàn)榭缇鄐 i始終是s n一個(gè)子軸線(xiàn). Ci的每一個(gè)自由區(qū)域FR的構(gòu)造，已經(jīng)在第2.2節(jié)中解釋。從FR中找出FRF，下面的定理是很有用的。
定理1 .
這定理很直觀表明為找出，所有的FR首先投影到冷卻系統(tǒng)的C空間. 可以從投影的布爾交叉口得到。定理1的證明和所用的引理，都已在附錄中標(biāo)出。
2.4.C -空間的表示和計(jì)算
為了表示自由區(qū)域和便于在一個(gè)高維空間的區(qū)域布爾交叉口之間的計(jì)算，我們可以利用類(lèi)似 [ 21,24 ]中的一種細(xì)胞枚舉法?；舅悸肥怯靡桓呔S立方體在中逐漸靠近一高維區(qū)間。每個(gè)立方體是通過(guò)對(duì)每個(gè)軸指定間隔來(lái)確定的。兩個(gè)區(qū)間的交匯點(diǎn)是通過(guò)兩個(gè)立方塊交匯點(diǎn)所取得的。兩個(gè)高維立方體的交叉點(diǎn)只不過(guò)是在每個(gè)軸的立方體之間間隔的普通交叉點(diǎn)。
假設(shè)每個(gè)FR是近似由m個(gè)三維立方體組成，投影PROJ S（FR）便可近似由維立方體組成。使用定理1對(duì)的構(gòu)建，需要在n-三維立方體中交叉，是用一個(gè)n-三維立方體只中的最大值表示。雖然用來(lái)代表交叉點(diǎn)中間結(jié)果的立方體的數(shù)量和 可通過(guò)特殊技術(shù)減少，可以預(yù)料到記憶和計(jì)算的要求仍然是這種方法的主要問(wèn)題。在下一節(jié)中將介紹一種更先進(jìn)的方法。
（二）在配置空間Si中每個(gè)水道的自由區(qū)域。
（一）一個(gè)擁有四個(gè)水道和四個(gè)自由度的簡(jiǎn)單冷卻系統(tǒng)
3.C空間構(gòu)建的一種有效率技術(shù)
對(duì)的表示和構(gòu)建時(shí)為了避免高的內(nèi)存和計(jì)算的要求，我們選擇不表示和不計(jì)算。相反，我們專(zhuān)注于對(duì)每一獨(dú)立水道的C -空間計(jì)算過(guò)程是否有效的技術(shù)。首先，我們看顯示在圖6的簡(jiǎn)化設(shè)計(jì)例子 。假設(shè)在這個(gè)例子中模具沿z方向插入時(shí)在FR中不存在變異，那么冷卻系統(tǒng)有四個(gè)如圖6（ a ）所示的自由度。每個(gè)水道的Si是兩維和假設(shè)的FR如圖6（ b ）所示。為水道考慮一個(gè)簡(jiǎn)單的設(shè)計(jì)方法。首先，點(diǎn)可以從FR中選擇，以使不會(huì)和任何障礙發(fā)生干涉。然而，由X 1和X 2確定 ，而X2在S 2中 。因此那些在S 2中的障礙所施加的約束，還必須考慮。為了找出設(shè)計(jì)的所有可行點(diǎn)，是與 “交叉”。這個(gè)“交叉點(diǎn)”結(jié)果如圖6（ c ）所示，這是通過(guò)移動(dòng)區(qū)間x 2 < 6得到的 ，因?yàn)樵撟杂蓞^(qū)域，× 2[ 6 ， 10 ] ?，F(xiàn)在，如圖6（ c ）所以示給定一個(gè)與任何障礙不發(fā)生干涉的水道，并在其自由區(qū)間的任何一點(diǎn)的選定，始終為C 2存在著這樣一種設(shè)計(jì)：例如，它可以連接到（他們都有一個(gè)共同的× 2值）并和任何障礙不發(fā)生干涉。然而，這個(gè)簡(jiǎn)單方法的一個(gè)主要問(wèn)題是在為C 1和C 2進(jìn)行有效的設(shè)計(jì)時(shí)并不保證冷卻系統(tǒng)其他水道存在有效的設(shè)計(jì)。例如，如果一個(gè)點(diǎn)選定如圖6（d）所示，則× 2 ∈[ 8 ,10 ] ,那么由，× 3 ∈ [6 ，8 ] ，在并沒(méi)有有效點(diǎn)和在這個(gè)區(qū)間。
上述例證表明，在為水道設(shè)計(jì)時(shí)，只考慮與相鄰并有一個(gè)共同軸的的自由區(qū)域和是不恰當(dāng)?shù)?。事?shí)上，其他所有的都必須加以考慮，盡管他們的C 空間并沒(méi)有共同軸和（且他們也不和C 1相鄰 ），因?yàn)榻M成冷卻系統(tǒng)的冷卻水道是相接的。一個(gè)自由度的選擇會(huì)影響冷卻系統(tǒng)另一自由度的選擇。
為每一個(gè)獨(dú)立水道的C空間發(fā)展一個(gè)設(shè)計(jì)的過(guò)程，主要關(guān)注的是：在一個(gè)水道C的空間選擇一個(gè)點(diǎn)后，必須始終存在和所有其他s i相應(yīng)的點(diǎn)，以使所有的水道可以連接到一起形成一個(gè)有效的冷卻系統(tǒng)。為解決這一問(wèn)題，到每個(gè)量s i的投影是必要的。
（c）在與相交以后的自由區(qū)間 （d）為C1和C2設(shè)計(jì)的一個(gè)有效點(diǎn)P1使C4成為無(wú)效的設(shè)計(jì)。
圖6
定義3 。定義為到投影
= PRO ()
顯然，對(duì)在選定的任何點(diǎn)，始終存在著相應(yīng)的點(diǎn)在中 ，因?yàn)楹投际屈c(diǎn)在的投影，在中選中的任何點(diǎn)，很明顯總是有一些相應(yīng)的設(shè)計(jì)對(duì)應(yīng)其他所有的渠道以使這些水道可以連接在一起形成一個(gè)有效的冷卻系統(tǒng)。因此，為了保證冷卻系統(tǒng)能有效的設(shè)計(jì)，的構(gòu)建是很重要的。根據(jù)定理3，為到投影。然而，如在第2.4節(jié)所討論的，我們并不想構(gòu)建基于大容量空間和繁瑣計(jì)算要求。另一種可供選擇的更有效的方法是直接構(gòu)建。而不是作用在高維空間，這個(gè)方法通過(guò)一個(gè)工作在空間三維或更少維數(shù)的序列運(yùn)行來(lái)建構(gòu)。
該方法正式介紹之前，在圖6所舉的例子再次被使用來(lái)說(shuō)明這種方法的基本概念。為了開(kāi)始一個(gè)設(shè)計(jì)過(guò)程，在的點(diǎn)P 1 =（× 1，× 2 ）首先被選擇如圖7所示 。因?yàn)?有一點(diǎn)在中 ，必須有一個(gè)值，以使我們可以找到=（× 2 ，× 3 ）在。又有一個(gè)坐標(biāo)在，坐標(biāo)必須有一個(gè)值，以使我們可以找到=（× 3 ，× 4 ）在 。此外，因?yàn)樵谟泻?，=（，）必須在。圖7顯示了為水道構(gòu)建一個(gè)有效設(shè)計(jì)的點(diǎn)、、和的順序。
上述例子顯示，為了在代表所有的有效設(shè)計(jì)的中確定有效的區(qū)間，自由區(qū)域應(yīng)首先考慮。的影響應(yīng)該可以 “促使”以確定有效的區(qū)間在中，然后是，最后是。在的有效區(qū)域產(chǎn)生的結(jié)果包括、、、的所有影響。為達(dá)到這一目的，組合的運(yùn)作正式被界定。
定義4 （組成）
對(duì)于在一個(gè)冷卻系統(tǒng)里的兩個(gè)相鄰水道和，他們從到的自由區(qū)域的組合，標(biāo)注為，而他們從到自由區(qū)域的組合，標(biāo)注為，定義如下：
(b)FRi每個(gè)通道的自由地區(qū)Si的配置空間
圖6冷卻系統(tǒng)設(shè)計(jì)的一個(gè)簡(jiǎn)化的例子
對(duì)于冷卻系統(tǒng)一個(gè)水道{Ci}序列的構(gòu)成， 從到自由區(qū)域的組成，標(biāo)注為，定義如下文。
如果
如果
如果
圖8顯示了促使 構(gòu)建的組合序列。第一步是要構(gòu)建，就像圖8（a）所示這已被給定在=PROJ(FR)FR， 。然后如在圖8（b）所示CR的構(gòu)建由公式CR=PROJ()FR得。最后，CR，由CR=PROJ( CR)FR。如圖8（c）所示。從圖8（c）很明顯的得出，CR對(duì)組成冷卻系統(tǒng)的所有水道的自由區(qū)域存在著影響。因此，對(duì)于CR中的任意一點(diǎn)，可以保證冷卻系統(tǒng)的一個(gè)有效設(shè)計(jì)可以被構(gòu)造。
通過(guò)組合序列的運(yùn)用，一個(gè)有效的設(shè)計(jì)可以通過(guò)在每個(gè)中選擇點(diǎn)獲得。在其他所有水道的自由區(qū)域已經(jīng)組合到中時(shí)。不過(guò)，我們也想確保沒(méi)有將有效的設(shè)計(jì)從自由區(qū)域中排除，當(dāng)組合序列被應(yīng)用以后。否則，有些可能提供更佳的冷卻性能的有效設(shè)計(jì)將不能用這個(gè)方法得到。以C的設(shè)計(jì)為例，圖8（c）的CR不僅僅代表著C一部份有效設(shè)計(jì)，而且代表著C所有的有效設(shè)計(jì)，這對(duì)C來(lái)說(shuō)尤為重要。為了解決這一問(wèn)題，我們提出以下定理：應(yīng)用水道{C}的一個(gè)序列{C},i[1,]到冷卻系統(tǒng)。
定理2
定理2說(shuō)明代表水道C所有有效的設(shè)計(jì)PR，可以通過(guò)和之間的一個(gè)布爾交點(diǎn)得到。這定理的一個(gè)重要特點(diǎn)是PR可以在三維立體空間中計(jì)算得到，因和都在S中，所以交點(diǎn)在S中。此外和也可以通過(guò)在中的區(qū)間相交得到。這樣，PR可以通過(guò)在三維立體空間的序列得到。如果在第2.4節(jié)中的假設(shè)說(shuō)明再次被使用，即是說(shuō)如果每個(gè)通過(guò)M個(gè)三維立方體近似得到，那么和PR也可以用M個(gè)三維立方體表示。所以，nm所有的三維立方體需要代表所有的PR。因此可以證明三維立方體之間的交點(diǎn)O需要產(chǎn)生所有的PR。因此，使用定理2可以防止在高維空間存儲(chǔ)區(qū)域的需要，并可以避免高容量和繁瑣計(jì)算的要求如在定理1所證明的。
圖8 CR構(gòu)建所用的序列
以下給出了定理2的證明 。它由兩部分組成：
該引理中所使用的證明如附錄所示。
3.1定理2證明
（1） 為了證明：
（i） 由p
因?yàn)閜
和 有相同的坐標(biāo)在和
用同樣的方法，我們可以確定一點(diǎn)以使和具有相同的坐標(biāo)在和。 使用這種方法，我們也可以確定一系列點(diǎn)，k[1,i -1],以使，那么和
具有相同的坐標(biāo)在軸線(xiàn)和。
（ii）
（b）由PROJ()構(gòu)建
用類(lèi)似的方法，我們可以確定另一系列點(diǎn)，k[i+1,],以使，那么和
具有相同的坐標(biāo)在軸線(xiàn)和。
由（i）及（ii）知，我們確定了一系列的點(diǎn)，k[1,],以使，在連續(xù)的任何兩個(gè)相鄰的點(diǎn)具有相同的坐標(biāo)在他們的共同軸線(xiàn)。
對(duì)于由一系列冷卻水道{}構(gòu)成的冷卻系統(tǒng)，在兩相鄰水道和的C空間和總是存在著一些共同的軸線(xiàn)由于它們之間的空間聯(lián)系。此外，如果在和的C空間有一個(gè)公共軸，也必須存在于和間所有水道的C 空間。所以，由上述方法構(gòu)建的一系列點(diǎn)，k[1,]將為的每個(gè)軸提供唯一的坐標(biāo)。令為由坐標(biāo)構(gòu)建的點(diǎn)。很明顯：
（c）由PROJ()構(gòu)建
用類(lèi)似的方法，可以得到：
初始設(shè)計(jì)
給定一個(gè)為冷卻系統(tǒng)指定一系列水道和他們理想幾何尺寸的初步設(shè)計(jì)，第一步是為每個(gè)水道建構(gòu)一個(gè)。然后，每個(gè)水道的可以通過(guò)應(yīng)用定理2的組合操作得到。為冷卻系統(tǒng)產(chǎn)生初始設(shè)計(jì)的一個(gè)方法是，是要從中選出一套坐標(biāo)。為了簡(jiǎn)化解釋?zhuān)僭O(shè)每個(gè)水道詞擁有自由度和，而和相鄰的水道有著相同的坐標(biāo)。為了生成一個(gè)設(shè)計(jì)，在的點(diǎn)（，）必須被選擇。然后，點(diǎn)被選擇為了讓?zhuān)ǎ┰谥?。此選擇
4．候選設(shè)計(jì)產(chǎn)生
由于冷卻系統(tǒng)初始設(shè)計(jì)對(duì)水道系列和它們的理想幾何結(jié)構(gòu)進(jìn)行了具體化，第一步要做的是為每個(gè)水道建立FRi，然后通過(guò)將復(fù)合應(yīng)用應(yīng)用到定理2中得到每個(gè)水道的PRi。一個(gè)產(chǎn)生冷卻系統(tǒng)候選設(shè)計(jì)的方法是從如后PRi系列中選出坐標(biāo)系。為簡(jiǎn)化闡述，假設(shè)每個(gè)水道C的自由度為和，被鄰近水道共用。為得到一個(gè)設(shè)計(jì)，選擇了PR1中的一個(gè)點(diǎn)（X1，X2），然后，選擇一個(gè)X3使（X3，X2）在PR2內(nèi)。這個(gè)選擇過(guò)程在下一個(gè)水道PR坐標(biāo)中重復(fù)，直到確定所有的自由度時(shí)停止。此方法的一個(gè)重要的特點(diǎn)是在一個(gè)步進(jìn)中無(wú)論坐標(biāo)值如何選取，后續(xù)步驟中總存在一個(gè)下一坐標(biāo)可選有效值。
5．應(yīng)用源運(yùn)算法則的自動(dòng)化設(shè)計(jì)過(guò)程
為測(cè)試C-空間方法在支持自動(dòng)化布局設(shè)計(jì)過(guò)程時(shí)的可行性，在C-空間建立項(xiàng)目中插入與應(yīng)用了一個(gè)簡(jiǎn)單源運(yùn)算法則（GA）[25]。在實(shí)施GA時(shí)候用到了一個(gè)簡(jiǎn)單的染色體結(jié)構(gòu)，它由一系列nF真值[g1g2…gnF]組成，其中g(shù)i的真值在0～1之間，nF冷卻系統(tǒng)的自由度。為得帶一個(gè)形狀設(shè)計(jì)，用到了前面部分提到的方法和應(yīng)用g作為一個(gè)百分比值來(lái)選擇坐標(biāo)。例如，中坐標(biāo)的有效值的在區(qū)間和，其中，就得的選取值為，（也就是在第一區(qū)間）否則就設(shè)置為（也就是在第二區(qū)間內(nèi)）一個(gè)單點(diǎn)交叉操作，一個(gè)轉(zhuǎn)化操作和轉(zhuǎn)跡線(xiàn)輪選擇方法[26]被用于GA過(guò)程中。之前研究中提到的模糊記值方法[13,14]對(duì)相對(duì)于機(jī)構(gòu)的候選設(shè)計(jì)的適合性進(jìn)行快速評(píng)定。必須注意的是在在GA過(guò)程開(kāi)始之前，建立起每個(gè)水道的，經(jīng)過(guò)一次建立得到，因此不會(huì)影響GA演變過(guò)程的計(jì)算時(shí)間。下一部分給出了一些由GA過(guò)程得到的布局設(shè)計(jì)實(shí)例。
6.實(shí)例研究
圖9（a）顯示出了實(shí)例部分的2個(gè)觀察結(jié)果。圖9（b）顯示了當(dāng)只考慮系統(tǒng)冷卻效果時(shí)，具體給出每個(gè)冷卻水道的理想位置的冷卻系統(tǒng)的初始設(shè)計(jì)。（為了便于表征，只給出了行腔部分冷卻系統(tǒng)的圖示）。在理想位置上，水稻C5和模具組成發(fā)生干涉現(xiàn)象。用提出的方法進(jìn)行布局設(shè)計(jì)，自動(dòng)化，就建立起了每個(gè)水道的。例如，圖9（g）和（h）顯示了水道的和。值得注意的是是通過(guò)將和其他復(fù)合得到，因此是亞設(shè)置，如數(shù)據(jù)明顯指出。在所有的計(jì)算完成之后，GA過(guò)程開(kāi)始調(diào)用，圖9（j）顯示了演變過(guò)程中得到的初始設(shè)計(jì)最大適合值。最大適合值在產(chǎn)生值接近600時(shí)開(kāi)始收斂。如圖9（c）所示，冷卻系統(tǒng)由15個(gè)自由度組成，他們的值在表1中列出。叫“初始設(shè)計(jì)”的行顯示初始設(shè)計(jì)的值。下一行顯示設(shè)計(jì)1的值，它是GA過(guò)程在1000生產(chǎn)后得到最好的設(shè)計(jì)。如表中明顯之處，涉及1通過(guò)減小1.21mm得到。圖9（d）顯示設(shè)計(jì)1，這個(gè)調(diào)整對(duì)應(yīng)于沿著Z方向減小以消除和之間的干涉。這個(gè)調(diào)整對(duì)水道和到也適用。表1也顯示設(shè)計(jì)1中所有其它的值都保持在規(guī)定初始至0.2mm誤差以?xún)?nèi)。
為更好的表征C-空間方法，模成分沿著Y方向移動(dòng)同相截，如圖9（e）所示。這個(gè)新障礙增加了自由區(qū)域的約束以至于方向體移動(dòng)性受到很大限制。這個(gè)效應(yīng)在更新中顯示出來(lái)，如圖9（i）所示，其中只有的上部分在圖9（h）中顯示出來(lái)。以所有水道新的再次調(diào)用GA過(guò)程以獲得設(shè)計(jì)2。適合值在圖9（k）中顯示。值得注意的是最佳適合值比設(shè)計(jì)1中獲得的要小。這很合理，因?yàn)榧s束的增加，偏移量與真實(shí)值的差距很大。又GA過(guò)程獲得的值在表1的最后一行中顯示出來(lái)。如表中所示，調(diào)整5mm以清除同的干涉。這同沿Z方向移動(dòng)水道到相對(duì)應(yīng)?，F(xiàn)在和截面不能通過(guò)調(diào)整使其光亮。而調(diào)整和，相應(yīng)地將沿-Y方向移動(dòng)2.94mm，沿-X方向移動(dòng)6.22mm如圖9（e）所示。為保持連結(jié)性，和也作相應(yīng)的調(diào)整。設(shè)計(jì)2顯示，當(dāng)一個(gè)水道的約束數(shù)（如）變化時(shí)，提出C-空間方法很好的將這個(gè)效應(yīng)傳播到其它水道（如和）中去，以至于所有這些水道的可行設(shè)計(jì)組得到相應(yīng)的調(diào)整。
C-模型冷卻分析用于分析設(shè)計(jì)得到的布局圖。從圖10（a）到（d）可見(jiàn)，兩個(gè)設(shè)計(jì)中，冷卻時(shí)間為20s時(shí)，最高模-壁溫度在以上。它們的最大溫度偏差小于，這表明兩種情形下，提出的方法能夠得到滿(mǎn)意的設(shè)計(jì)布局。從圖10（c）和（d）觀察得到，同設(shè)計(jì)2比較，涉及1中工件大部分沒(méi)有產(chǎn)生變色。這表明在設(shè)計(jì)1中很多工件的溫度偏差在以?xún)?nèi)。這是因?yàn)樵谠O(shè)計(jì)2中，隨著空腔中的水道向模壓移動(dòng)了5mm，冷卻效果變得不均勻，這表示當(dāng)施加很多約束時(shí)，保持初始理想冷卻效果很困難。它也解釋了為什么設(shè)計(jì)2的最大適切性稍微小于設(shè)計(jì)1的最大適切性。
（a）示例零件
（b）冷卻系統(tǒng)的初始設(shè)計(jì)
（a）冷卻系統(tǒng)的15個(gè)自由度
（b）設(shè)計(jì)1
（b）移動(dòng)和相交
（b） 設(shè)計(jì)2
圖9 分層設(shè)計(jì)
表1.冷卻系統(tǒng)的自由度
7．討論與結(jié)論
在執(zhí)行C-空間方法中，一個(gè)單元列舉方案被用于簡(jiǎn)化這個(gè)方法的執(zhí)行，在目前的執(zhí)行中，C-空間一維分辨率為0.15mm。對(duì)冷卻系統(tǒng)設(shè)這個(gè)分辨率是足夠的，因?yàn)閷?duì)一個(gè)好的調(diào)整，如0.01mm，冷卻系統(tǒng)的功能變化是很難發(fā)現(xiàn)的，然而，該研究中所發(fā)展的理論與方法并不局限于相應(yīng)的表現(xiàn)項(xiàng)目。實(shí)際上，基于理論2的方法，所有C-空間計(jì)算和存儲(chǔ)都在3維空間內(nèi)完成，因此標(biāo)準(zhǔn)校核模型技巧可以應(yīng)用。
該研究的一個(gè)主要貢獻(xiàn)是發(fā)展了一個(gè)特別的支持布局設(shè)計(jì)的C-空間方法。應(yīng)用這個(gè)C-空間方法，所有的可行布局設(shè)計(jì)很好的被顯示出來(lái)。同時(shí)我們得出了該方法不僅可以用于冷卻系統(tǒng)設(shè)計(jì)的優(yōu)化設(shè)計(jì)支持，還可以用于生產(chǎn)制造。該方法克服特殊啟發(fā)產(chǎn)生布局設(shè)計(jì)的局限，如前面的方法[13，14]。這個(gè)C-空間方法能夠獨(dú)立作為一個(gè)系統(tǒng)去支持互動(dòng)布局設(shè)計(jì)。它使設(shè)計(jì)者在不用檢查冷卻系統(tǒng)截面和其它模型插件能夠開(kāi)發(fā)出設(shè)計(jì)方法。
該研究主要目的集中在冷卻系統(tǒng)設(shè)計(jì)的幾何形狀構(gòu)成方面。在設(shè)計(jì)冷卻系統(tǒng)時(shí)，其它參數(shù)如冷流率，冷卻時(shí)間，包裝時(shí)間，擠出時(shí)間都需要被考慮進(jìn)來(lái)。一個(gè)可行的方法就是將這些所有參數(shù)進(jìn)行考慮插入配備更復(fù)雜的GA的C-空間方法，如[8]報(bào)道所示。需要對(duì)該方法進(jìn)一步研究，其他研究方向包括C-空間方法處理冷卻系統(tǒng)拓?fù)渥兓途唧w設(shè)計(jì)約束，如初始設(shè)計(jì)選擇水道之間的變化幾何形狀和拓?fù)浼s束扽。
鳴謝
該文章中所完成的工作得到香港城市大學(xué)戰(zhàn)略研究部（項(xiàng)目No.7001775）的大力支持。
（a）設(shè)計(jì)1的模具溫度 （b）設(shè)計(jì)2的模具溫度
（c）設(shè)計(jì)1零件的不同溫度 （d）設(shè)計(jì)2零件的不同溫度
圖10。用CAE模具冷卻分析系統(tǒng)比較這兩個(gè)布圖設(shè)計(jì)
引理2
在中給定兩個(gè)區(qū)間和。如果，那么
引理3
在中給定，那么
引理4
在中給定任意兩個(gè)和。如果，則它們對(duì)的投影滿(mǎn)足：
引理5
給定兩個(gè)區(qū)間和滿(mǎn)足。則在中的區(qū)間滿(mǎn)足：
引理6
給定三個(gè)區(qū)間、和滿(mǎn)足和。則在中的區(qū)間滿(mǎn)足：
引理7
引理8
給定兩個(gè)區(qū)間和滿(mǎn)足，其中點(diǎn)在中，點(diǎn)在中，如果
那么：
定理1
參考文獻(xiàn)：
Computer Aided Design 40 2008 space C L producti moulded part Despite the various research efforts that have been directed towards the analysis optimization and fabrication of cooling systems support for the layout design of the cooling system has not been well developed In the layout design phase a major concern is the feasibility of building the cooling system inside the mould insert without interfering with the other mould components This paper reports a configuration space C space method to address this important issue While a high dimensional C space is generally required to deal with a complex system such as a cooling system the special characteristics of cooling system design are exploited in the present study and special techniques that allow C space computation and storage in three dimensional or lower dimension are developed This new method is an improvement on the heuristic method developed previously by the authors because the C space representation enables an automatic layout design system to conduct a more systematic search among all of the feasible designs A simple genetic algorithm is implemented and integrated with the C space representation to automatically generate candidate layout designs Design examples generated by the genetic algorithm are given to demonstrate the feasibility of the method c 2007 Elsevier Ltd All rights reserved Keywords Cooling system design Plastic injection mould Configuration space method 1 Introduction The cooling system of an injection mould is very important to the productivity of the injection moulding process and the quality of the moulded part Extensive research has been conducted into the analysis of cooling systems 1 2 and commercial CAE systems such as MOLDFLOW 3 and Moldex3D 4 are widely used in the industry Research into techniques to optimize a given cooling system has also been reported 5 8 Recently methods to build better cooling systems by using new forms of fabrication technology have been reported Xu et al 9 reported the design and fabrication of conformal cooling channels that maintain a constant distance from the mould impression Sun et al 10 11 used CNC Despitethevariousresearcheffortsthathavefocusedmainly on the preliminary design phase of the cooling system design process in which the major concern is the performance of the cooling function of the system support for the layout design phase in which the feasibility and manufacturability of the cooling system design are addressed has not been well developed A major concern in the layout design phase is the feasibility of building the cooling system inside the mould insert without interfering with the other mould components Consider the example shown in Fig 1 It can be seen that many different components of the various subsystems of the injection mould such as ejector pins slides sub inserts and so forth have to be packed into the mould insert Finding the best location for each channel of the cooling circuit to optimize Plastic injection mould cooling configuration C G Li Department of Manufacturing Engineering and Engineering Received 3 May 2007 accepted Abstract The cooling system of an injection mould is very important to the milling to produce U shaped milled grooves for cooling channels and Yu 12 proposed a scaffolding structure for the design of conformal cooling Corresponding author E mail address meclli cityu edu hk C L Li 0010 4485 see front matter c 2007 Elsevier Ltd All rights reserved doi 10 1016 j cad 2007 11 010 334 349 system design by the method Li Management City University of Hong Kong Hong Kong 18 November 2007 vity of the injection moulding process and the quality of the the cooling performance of the cooling system and to avoid interference with the other components is not a simple task Another issue that further complicates the layout design problem is that the individual cooling channels need to be connected to form a path that connects between the inlet and the outlet Therefore changing the location of a channel may 335 Fig 1 Thecoolingsystem components require changing the example shown in to optimize the cooling in Fig 2 a Assume other mould components mould component As C1 cannot be mo interference with other C2 is moved and C connectivity as sho C3 is found to interfere mould components is very tedious that supports the this new technique used to provide a layout designs The an efficient method the layout design to generate layout system developed w C space method to conduct a more layout designs is the space that system is treated the configuration free region Points of the the components correspond to of the system initially formalized planning problems shortened and further modification is needed which results in the final design shown in Fig 2 c Given that a typical injection mould may have more than ten cooling channels with each channel a Interference occurs between cooling channel C1 and mould component O1 at the ideal location of C1 c C3 is moved and C2 is design Fig 2 An example showing the tediousness and a survey in this area of research has been reported by Wise and Bowyer 16 The C space method has also been used to solve problems in qualitative reasoning e g 17 18 b Channel C1 is shortened C2 is moved and C3 is elongated to give the final C G Li C L Li Computer Aided Design 40 2008 334 349 insideamouldinsertpackedwithmanyothermould other channels as well Consider the Fig 2 The ideal location of each channel performance of the system is shown that when the cooling system and the are built into the mould insert a O1 is found to interfere with channel C1 ved to a nearby location due to the possible components it is shortened As a result 3 is elongated accordingly to maintain the wn in Fig 2 b Owing to its new length with another mould component O2 potentially interfering with a few other finding an optimal layout design manually This paper reports a new technique automation of the layout design process In a configuration space C space method is concise representation of all of the feasible C space representation is constructed by that exploits the special characteristics of problem Instead of using heuristic rules designs as in the automatic layout design previously by the authors 13 14 this ne enables an automatic layout design system systematic search among all of the feasible 2 The configuration space method In general the C space of a system results when each degree of freedom of that as a dimension of the space Regions in space are labeled as blocked region or in the free regions correspond to valid configurations system where there is no interference between of the system Points in the blocked regions invalid configurations where the components interfere with one another C space was by Lozano Perez 15 to solve robot path of the layout design process 336 and e g automatic 23 2 1 the y c 3 se e a cooling system Fig 3 gives an example The preliminary design of this cooling system consists of four cooling channels To generate a layout design from the preliminary design the centers and lengths of the channels are adjusted As shown in Fig 3 the center of channel C1 can be moved along the X1 and X2 directions and its length can be adjusted along the X3 direction Similarly the length of C2 can be adjusted along the X4 direction while its center adjustment is described by X1 and X3 and thus must be the same as the adjustment of C1 to maintain the connectivity By applying similar arguments to the other channels it can be seen that the cooling system has 5 a Channel Ci and three mould components inside the mould insert b Offsets of the mould Ci represented by line d The initial free region of Ci Fig 4 The major steps in the construction considered To account for the diameter D Oi is first offset by D 2 M to give Oprimei where M is the minimum allowable distance between the channel wall and the face of a component This growing of Oi in effect reduces channel Ci to a line Li Consider the example illustrated in Fig 4 Fig 4 a shows a channel Ci and three mould components O1 O2 and O3 that may interfere with Ci Fig 4 b shows the offsets Oprime1 Oprime2 and Oprime3 of the mould components and the reduction of Ci to a line segment Li that is coincident with the axis of Ci If there is no intersection between Li and the offsets of the mould components then the original channel Ci will not intersect with components and gment Li c Sweeping the offsets of the mould components and Ci represented by point Pi Subtracting Oprimeprimei from Bprimei f The free region FRi of Ci C G Li C L Li Computer Aided Design 40 2008 334 349 Fig 3 An example showing the degrees of freedom of a cooling system the analysis and design automation of kinematic devices 19 21 TheauthorinvestigatedaC spacemethodinthe design synthesis of multiple state mechanisms 22 in previous research C space of a cooling system A high dimensional C space can be used to represent all of feasible layout designs of a given preliminary design of degreesoffreedom andtheyaredenotedas Xi i 1 2 5 In principle the C space is a five dimensional space and an point in the free region of this space gives a set of coordinate values on the Xi axes that can be used to define the geometry of the channels without causing interference with the other mould components Todeterminethefreeregioninahigh dimensional C spaceofacoolingsystem thefirststepistoconstructthefree regions in the C spaces of the individual channels 2 2 C space construction of individual cooling channels When an individual channel Ci is considered alone it has three degrees of freedom say X1 and X2 for its center location and X3 for its length As the ideal center location and length have already been specified in the preliminary design it is reasonable to assume a fixed maximum allowable variation for X1 X2 and X3 The initial free region in the C space of channel Ci is thus a three dimensional cube Bi with the dimensions c c c To avoid any possible interference with a mould component Oi when channel Ci is built into the mould insert by drilling a drilling diameter D and drilling depth along X have to be of the free region FRi of a channel Ci C G Li C L Li Computer Aided the mould components This growing or offset of an obstacle is a standard technique in the C space method 15 A channel is formed by drilling from a face of the mould insert and any obstacle Oi within the drilling depth will affect the construction of the channel To account for the drilling depth the offset Oprimei of Oi is swept along the drilling direction until the opposite face of the mould insert is reached to generate Oprimeprimei This sweeping of Oprimei in effect reduces line Li to a point Pi located at the end of Li As shown in Fig 4 c if the point Pi is outside Oprimeprimei the drilling along Li to produce Ci is feasible The free region FRi of channel Ci is obtained as follows First the initial free region Bi is constructed with its center at Pi as shown in Fig 4 d Bi then intersects with the mould insert to obtain Bprimei Bprimei represents all of the possible variations of Ci when only the geometric shape of the mould insert is considered Then FRi is obtained by subtracting from Bprimei the Oprimeprimei of all of the obstacles Fig 4 e and f show the subtraction and the resulting FRi of the example 2 3 Basic approach to the construction of the C space of cooling system To determine the free region FRF in the C space of a cooling system the free regions of each cooling channel have to be intersected in a proper manner so that the effect of the obstacles to all of the channels are properly represented by FRF However the standard Boolean intersection between the free regions of two different channels cannot be performed because their C spaces are in general spanned by different sets of axes Referring to the example in Fig 3 the C spaces of C1 and C2 are spanned by X1 X2 X3 and X1 X3 X4 respectively To facilitate the intersection between free regions in different C spaces the projection of a region from the C space of one channel to that of another channel is needed The following notations are first introduced and will be used in the subsequent discussions on projections and the rest of the paper Notations used in describing high dimensional spaces Sn denotes an n dimensional space spanned by the set of axes Xn X1 X2 Xn Sm denotes an m dimensional space spanned by the set of axes Xm Xprime1 Xprime2 Xprimem pn denotes a point in Sn and pn x1 x2 xn where xi denotes a coordinate on the ith axis Xi Rn denotes a region in Sn Rn Sn Rn is a set of points in Sn PROJSm pn denotes the projection of a point pn from Sn to Sm PROJSm Rn denotes the projection of a region Rn from Sn to Sm Notations used in describing a cooling system nC denotes the number of channels in the cooling system nF denotes the total degrees of freedom of the cooling system Ci denotes the ith channel of the cooling system Si denotes the C space of Ci Design 40 2008 334 349 337 FRi denotes the free region in Si That is it is the free region of an individual channel Ci SF denotes the C space of the cooling system FRF denotes the free region in SF That is it is the free region of the cooling system Consider the projection of a point pn in Sn to a point pm in Sm Fig 5 a illustrates examples of projection using spaces of one dimension to three dimensions Projections are illustrated forthreecases i Xm Xn ii Xm Xn and iii Xm negationslash Xn Xn negationslash Xm and Xn Xm negationslash For i each coordinate of pm is equal to a corresponding coordinate of pn that is on the same axis For ii and iii the projection of pn is a region Rm For each point pm in Rm a coordinate of pm is equal to that of pn if that coordinate is on a common axis of Sn and Sm For the other coordinates of pm any value can be assigned The reason for this specific definition of the projections in particular for cases ii and iii is as follows Consider two adjacent channels Cn and Cm As they are adjacent they must be connected and thus their C spacesSn and Sm share some common axes Assume that a configuration that corresponds to a point pn in Sn has been selected for Cn To maintain the connectivity the configuration for Cm must be selected such that the corresponding point pm in Sm shares the same coordinates with pn on their common axes This implies that pm can be any point within the projection of pn on Sm where the method of projection is defined above The projections of a region Rn in Sn to Sm are simply the projections of every point in Rn to Sm Fig 5 b illustrates the region projections The formal definition of projection is given below Definition 1 Projection 1 1 If Xm Xn PROJSm pn is a point pm xprime1 xprime2 xprimem where for Xprimei X j xprimei xj for all i 1 m To simplify the notations in subsequent discussion this projection is regarded as a region that consists of the single point pm That is PROJSm pn pm 1 2 If Xm Xn PROJSm pn is a region Rm pm PROJSn pm pn 1 3 If Xm negationslash Xn Xn negationslash Xm and Xn Xm negationslash PROJSm pn is a region Rm pm PROJSI pm PROJSI pn where SI is the space spanned by Xn Xm If Xn Xm PROJSm pn is defined as Sm 1 4 PROJSm Rn is defined as the region Rm pm pm PROJSm pn pn Rn As discussed in Section 2 1 any point pF in FRF gives a value for each degree of freedom of the cooling system so that the geometry of the channels is free from interference with the other mould components In other words the projection of pF to each Si is in the free region FRi of each Ci Thus FRF is defined as follows Definition 2 Free Region in the C space of a Cooling System FRF pF PROJSi pF FRi i 1 nC Aided Note that according to to Si always contains only that span Si is always a subset The construction of the already been explained in the following theorem is useful Theorem 1 FRF nCintersectiondisplay i 1 PROJSF FRi Intuitively this theorem says first projected to the C space can then be obtained by performing among the projections The used in the proof are given of the C space F and to facilitate the between the regions can use a kind of cell used in 21 24 The region RF in Each box is defined by SF The intersection of of the two sets of high dimensional boxes intervals of each of the by m three OJSF FRi can then be boxes The construction Fig 5 The projections of points and regions in Sn to Sm Definition 1 1 the projection of pF a single point because the set of axes of the axes that span Sn free region FRi of each Ci has Section 2 2 To find FRF from FRi that to find FRF all of the FRi are of the cooling system SF FRF the Boolean intersections proof of Theorem 1 and the lemmas 2 4 Representation and computation To represent the free region FR computation of the Boolean intersections in a high dimensional space we enumeration method similar to the one basic idea is to approximate a high dimensional SF by a set of high dimensional boxes specifying an interval on each axis of two regions is achieved by the intersection boxes The intersection between two is simply the intersection between the boxes in each axis Assuming that each FRi is approximated dimensional boxes the projection PR approximated by mnF dimensional 338 C G Li C L Li Computer in the Appendix Design 40 2008 334 349 of FRF that uses Theorem 1 then requires mnC intersections between nF dimensional maximum of mnCnF of boxes used to represent intersections and FR is anticipated that the are still major problems improved method is 3 An efficient technique To avoid the high for the representation Instead we process to example shown in is assumed in this along the Z direction hasfourdegrees each channel Ci are shown in Fig 6 b channel C1 First a a A simple cooling system with four channels and four degrees of freedom b The free region FRi of each channel in its configuration space Si Fig 6 A simplified example of a cooling system design boxes and FRF is represented by a dimensional boxes Although the number the intermediate results of the F can be reduced by special techniques it memory and computational requirements of this method In the next section an developed for C space construction to represent and not to compute FRF explicitly focus on a technique that enables the computational work on the C spaces of each individual channel First consider the simplified design Fig 6 For the purpose of illustration it example that there is no variation in FRi ofthemouldinsertandthusthecoolingsystem of freedom as shown in Fig 6 a The Si of two dimensional and the assumed FRi are Consider a simple method for designing C G Li C L Li Computer Aided memory and computational requirements and construction of FRF we choose not Design 40 2008 334 349 339 point p1 can be selected from within FR1 so that C1 is free from interference with any obstacle However S1 is spanned Aided continued even though their C spaces of C1 i e they are as well because the system are connected have an effect in the cooling system To develop a design of each individual channels selection of a point always exist a corresponding that all of the channels system To address this Si is needed Definition 3 PRi is PRi PROJSi FRF Obviously for an always a correspondi FR2 Again as p2 x3 must have a value FR3 Also as must also be inside p1 p2 p3 and p4 C1 determine the valid designs for C1 the The effect of FR4 valid region in FR3 finally in S1 The all of the effects of is formally channels Ci and of their free regions do not have an axis common to that not adjacent to C1 have to be considered cooling channels that make up the cooling A choice in one degree of freedom will choice of another degree of freedom of the process that works on the C spaces a major concern is that after the in the C space of one channel there must point in all of the other Si such can be connected to form a valid cooling concern the projection of FRF to each defined as the projection of FRF to Si which we can find a p2 x2 x3 within has a coordinate x3 in X3 the coordinate for which