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2 +M對于O "抵銷，其中M是水道內(nèi)壁和附近的一個組件間所允許的最短距離。Oi的增長有效的減少了水道Ci的長度對于直線Li來說。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相對于線段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有交匯點，那么，原來的水道Ci將不會與模具組件相交。（a）水道Ci和模具的（b）模具組件和Ci相對Li的偏移三個組件（c）模具組件和Ci相對Pi點的補償（d）Ci的自由度（e）Oi相對Pi的減少量（f）Ci的自由點Fri 圖4在一個通道CI自由區(qū)FRi施工的主要步驟水道是通過鉆孔從對模具的表面插入的，任何如Oi的障礙以及鉆孔深度將會影響水道的構(gòu)建。鉆孔深度及Oi的補償O"沿鉆孔的方向延伸，直到鉆到模具對應(yīng)的另一面生成水道為止。Oi相對 Pi沿直線Li的減少至Li的終點。如圖4（c）所示，如果點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所有可能的變化當僅考慮插入的模具幾何形狀時。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空間在一個冷卻系統(tǒng)的C空間中確定自由區(qū)域FRF，每個冷卻水道的自由區(qū)域必須以一個適當?shù)姆绞健敖徊妗?，以使障礙的效果能恰當?shù)耐ㄟ^FRF描繪對于所有水道來說。然而在兩個不同水道之間的自由區(qū)域的標準布爾交叉口無法執(zhí)行，因為他們的C空間在一般跨距于不同的軸線。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，從一個渠道和另一個渠道的C空間中推算一個地區(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。標記法用于描述高維空間 S n是指一個通過坐標定義的n維空間 = {X 1, X 2, . . . , X n}. Sn是指一個通過坐標定義的m維空間 = {X , X , . . . , X }. Pn 是指在Sn 的一個點 p n = (x 1, x 2, . . . , x n) Rn屬于區(qū)間S n(R n S n) 標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，它是獨立水道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，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明了突出的例子 (i) (ii) ；而(iii) ，且對(i)Pn 和Pm的坐標是一樣的如果Sn和Sm在同一區(qū)間時。對(ii)和(iii)Pn在區(qū)間Rm中。因為Pm在Rm中，當點位于Sn和Sm中時Pn等于Pm。而對另一坐標Pm其可以是任意值；特別對(ii)和(iii)，假設(shè)水道Cn和Cm，因為它們相近所以必須連接。這樣它們的C空間Sn、Sm有相同的坐標值。假設(shè)那是一個結(jié)論？對應(yīng)到在S n中一個點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Pm與P n共用相同坐標在共同的軸線。這意味著Pm和PN可以是任何點在區(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中每一點的簡化。圖5（ b ）說明了相應(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 )是一個點=(x,x,……,x),因為X = Xj, x = xj 因為i [1,m]。為了在隨后的討論中簡化符號，這一投影是被視為單獨點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P為冷卻系統(tǒng)的每個自由度給定了一個值，使水道與其他模具組件在幾何空間是不會發(fā)生任何干涉。另一方面， P相對每個點s i的投影是，在Ci的每個自由區(qū)域FR中。因此，F(xiàn)R定義如下。定義2 （一個冷卻系統(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單一的點，因為跨距s i始終是s n一個子軸線. Ci的每一個自由區(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，都已在附錄中標出。 2.4.C -空間的表示和計算為了表示自由區(qū)域和便于在一個高維空間的區(qū)域布爾交叉口之間的計算，我們可以利用類似 [ 21,24 ]中的一種細胞枚舉法。基本思路是用一高維立方體在中逐漸靠近一高維區(qū)間。每個立方體是通過對每個軸指定間隔來確定的。兩個區(qū)間的交匯點是通過兩個立方塊交匯點所取得的。兩個高維立方體的交叉點只不過是在每個軸的立方體之間間隔的普通交叉點。假設(shè)每個FR是近似由m個三維立方體組成，投影PROJ S（FR）便可近似由維立方體組成。使用定理1對的構(gòu)建，需要在n-三維立方體中交叉，是用一個n-三維立方體只中的最大值表示。雖然用來代表交叉點中間結(jié)果的立方體的數(shù)量和可通過特殊技術(shù)減少，可以預(yù)料到記憶和計算的要求仍然是這種方法的主要問題。在下一節(jié)中將介紹一種更先進的方法。（二）在配置空間Si中每個水道的自由區(qū)域。（一）一個擁有四個水道和四個自由度的簡單冷卻系統(tǒng) 3.C空間構(gòu)建的一種有效率技術(shù) 對的表示和構(gòu)建時為了避免高的內(nèi)存和計算的要求，我們選擇不表示和不計算。相反，我們專注于對每一獨立水道的C -空間計算過程是否有效的技術(shù)。首先，我們看顯示在圖6的簡化設(shè)計例子。假設(shè)在這個例子中模具沿z方向插入時在FR中不存在變異，那么冷卻系統(tǒng)有四個如圖6（ a ）所示的自由度。每個水道的Si是兩維和假設(shè)的FR如圖6（ b ）所示。為水道考慮一個簡單的設(shè)計方法。首先，點可以從FR中選擇，以使不會和任何障礙發(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)計的所有可行點，是與 “交叉”。這個“交叉點”結(jié)果如圖6（ c ）所示，這是通過移動區(qū)間x 2 < 6得到的，因為該自由區(qū)域，× 2[ 6 ， 10 ] ?，F(xiàn)在，如圖6（ c ）所以示給定一個與任何障礙不發(fā)生干涉的水道，并在其自由區(qū)間的任何一點的選定，始終為C 2存在著這樣一種設(shè)計：例如，它可以連接到（他們都有一個共同的× 2值）并和任何障礙不發(fā)生干涉。然而，這個簡單方法的一個主要問題是在為C 1和C 2進行有效的設(shè)計時并不保證冷卻系統(tǒng)其他水道存在有效的設(shè)計。例如，如果一個點選定如圖6（d）所示，則× 2 ∈[ 8 ,10 ] ,那么由，× 3 ∈ [6 ，8 ] ，在并沒有有效點和在這個區(qū)間。上述例證表明，在為水道設(shè)計時，只考慮與相鄰并有一個共同軸的的自由區(qū)域和是不恰當?shù)摹Ｊ聦嵣?，其他所有的都必須加以考慮，盡管他們的C 空間并沒有共同軸和（且他們也不和C 1相鄰），因為組成冷卻系統(tǒng)的冷卻水道是相接的。一個自由度的選擇會影響冷卻系統(tǒng)另一自由度的選擇。為每一個獨立水道的C空間發(fā)展一個設(shè)計的過程，主要關(guān)注的是：在一個水道C的空間選擇一個點后，必須始終存在和所有其他s i相應(yīng)的點，以使所有的水道可以連接到一起形成一個有效的冷卻系統(tǒng)。為解決這一問題，到每個量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)計的一個有效點P1使C4成為無效的設(shè)計。圖6 定義3 。定義為到投影 = PRO () 顯然，對在選定的任何點，始終存在著相應(yīng)的點在中，因為和都是點在的投影，在中選中的任何點，很明顯總是有一些相應(yīng)的設(shè)計對應(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，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，這個方法通過一個工作在空間三維或更少維數(shù)的序列運行來建構(gòu)。該方法正式介紹之前，在圖6所舉的例子再次被使用來說明這種方法的基本概念。為了開始一個設(shè)計過程，在的點P 1 =（× 1，× 2 ）首先被選擇如圖7所示。因為有一點在中，必須有一個值，以使我們可以找到=（× 2 ，× 3 ）在。又有一個坐標在，坐標必須有一個值，以使我們可以找到=（× 3 ，× 4 ）在。此外，因為在有和，=（，）必須在。圖7顯示了為水道構(gòu)建一個有效設(shè)計的點、、和的順序。上述例子顯示，為了在代表所有的有效設(shè)計的中確定有效的區(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。為達到這一目的，組合的運作正式被界定。定義4 （組成）對于在一個冷卻系統(tǒng)里的兩個相鄰水道和，他們從到的自由區(qū)域的組合，標注為，而他們從到自由區(qū)域的組合，標注為，定義如下： (b)FRi每個通道的自由地區(qū)Si的配置空間圖6冷卻系統(tǒng)設(shè)計的一個簡化的例子對于冷卻系統(tǒng)一個水道{Ci}序列的構(gòu)成，從到自由區(qū)域的組成，標注為，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，對于CR中的任意一點，可以保證冷卻系統(tǒng)的一個有效設(shè)計可以被構(gòu)造。通過組合序列的運用，一個有效的設(shè)計可以通過在每個中選擇點獲得。在其他所有水道的自由區(qū)域已經(jīng)組合到中時。不過，我們也想確保沒有將有效的設(shè)計從自由區(qū)域中排除，當組合序列被應(yīng)用以后。否則，有些可能提供更佳的冷卻性能的有效設(shè)計將不能用這個方法得到。以C的設(shè)計為例，圖8（c）的CR不僅僅代表著C一部份有效設(shè)計，而且代表著C所有的有效設(shè)計，這對C來說尤為重要。為了解決這一問題，我們提出以下定理：應(yīng)用水道{C}的一個序列{C},i[1,]到冷卻系統(tǒng)。定理2 定理2說明代表水道C所有有效的設(shè)計PR，可以通過和之間的一個布爾交點得到。這定理的一個重要特點是PR可以在三維立體空間中計算得到，因和都在S中，所以交點在S中。此外和也可以通過在中的區(qū)間相交得到。這樣，PR可以通過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)說明再次被使用，即是說如果每個通過M個三維立方體近似得到，那么和PR也可以用M個三維立方體表示。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區(qū)域的需要，并可以避免高容量和繁瑣計算的要求如在定理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 因為p 和有相同的坐標在和用同樣的方法，我們可以確定一點以使和具有相同的坐標在和。使用這種方法，我們也可以確定一系列點，k[1,i -1],以使，那么和具有相同的坐標在軸線和。（ii）（b）由PROJ()構(gòu)建用類似的方法，我們可以確定另一系列點，k[i+1,],以使，那么和具有相同的坐標在軸線和。由（i）及（ii）知，我們確定了一系列的點，k[1,],以使，在連續(xù)的任何兩個相鄰的點具有相同的坐標在他們的共同軸線。對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，k[1,]將為的每個軸提供唯一的坐標。令為由坐標構(gòu)建的點。很明顯：（c）由PROJ()構(gòu)建用類似的方法，可以得到：初始設(shè)計給定一個為冷卻系統(tǒng)指定一系列水道和他們理想幾何尺寸的初步設(shè)計，第一步是為每個水道建構(gòu)一個。然后，每個水道的可以通過應(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)計的一個方法是，是要從中選出一套坐標。為了簡化解釋，假設(shè)每個水道詞擁有自由度和，而和相鄰的水道有著相同的坐標。為了生成一個設(shè)計，在的點（，）必須被選擇。然后，點被選擇為了讓（，）在中。此選擇 4．候選設(shè)計產(chǎn)生由于冷卻系統(tǒng)初始設(shè)計對水道系列和它們的理想幾何結(jié)構(gòu)進行了具體化，第一步要做的是為每個水道建立FRi，然后通過將復(fù)合應(yīng)用應(yīng)用到定理2中得到每個水道的PRi。一個產(chǎn)生冷卻系統(tǒng)候選設(shè)計的方法是從如后PRi系列中選出坐標系。為簡化闡述，假設(shè)每個水道C的自由度為和，被鄰近水道共用。為得到一個設(shè)計，選擇了PR1中的一個點（X1，X2），然后，選擇一個X3使（X3，X2）在PR2內(nèi)。這個選擇過程在下一個水道PR坐標中重復(fù)，直到確定所有的自由度時停止。此方法的一個重要的特點是在一個步進中無論坐標值如何選取，后續(xù)步驟中總存在一個下一坐標可選有效值。 5．應(yīng)用源運算法則的自動化設(shè)計過程為測試C-空間方法在支持自動化布局設(shè)計過程時的可行性，在C-空間建立項目中插入與應(yīng)用了一個簡單源運算法則（GA）[25]。在實施GA時候用到了一個簡單的染色體結(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。為得帶一個形狀設(shè)計，用到了前面部分提到的方法和應(yīng)用g作為一個百分比值來選擇坐標。例如，中坐標的有效值的在區(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)）一個單點交叉操作，一個轉(zhuǎn)化操作和轉(zhuǎn)跡線輪選擇方法[26]被用于GA過程中。之前研究中提到的模糊記值方法[13,14]對相對于機構(gòu)的候選設(shè)計的適合性進行快速評定。必須注意的是在在GA過程開始之前，建立起每個水道的，經(jīng)過一次建立得到，因此不會影響GA演變過程的計算時間。下一部分給出了一些由GA過程得到的布局設(shè)計實例。 6.實例研究圖9（a）顯示出了實例部分的2個觀察結(jié)果。圖9（b）顯示了當只考慮系統(tǒng)冷卻效果時，具體給出每個冷卻水道的理想位置的冷卻系統(tǒng)的初始設(shè)計。（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。用提出的方法進行布局設(shè)計，自動化，就建立起了每個水道的。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。在所有的計算完成之后，GA過程開始調(diào)用，圖9（j）顯示了演變過程中得到的初始設(shè)計最大適合值。最大適合值在產(chǎn)生值接近600時開始收斂。如圖9（c）所示，冷卻系統(tǒng)由15個自由度組成，他們的值在表1中列出。叫“初始設(shè)計”的行顯示初始設(shè)計的值。下一行顯示設(shè)計1的值，它是GA過程在1000生產(chǎn)后得到最好的設(shè)計。如表中明顯之處，涉及1通過減小1.21mm得到。圖9（d）顯示設(shè)計1，這個調(diào)整對應(yīng)于沿著Z方向減小以消除和之間的干涉。這個調(diào)整對水道和到也適用。表1也顯示設(shè)計1中所有其它的值都保持在規(guī)定初始至0.2mm誤差以內(nèi)。為更好的表征C-空間方法，模成分沿著Y方向移動同相截，如圖9（e）所示。這個新障礙增加了自由區(qū)域的約束以至于方向體移動性受到很大限制。這個效應(yīng)在更新中顯示出來，如圖9（i）所示，其中只有的上部分在圖9（h）中顯示出來。以所有水道新的再次調(diào)用GA過程以獲得設(shè)計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1中獲得的要小。這很合理，因為約束的增加，偏移量與真實值的差距很大。又GA過程獲得的值在表1的最后一行中顯示出來。如表中所示，調(diào)整5mm以清除同的干涉。這同沿Z方向移動水道到相對應(yīng)?，F(xiàn)在和截面不能通過調(diào)整使其光亮。而調(diào)整和，相應(yīng)地將沿-Y方向移動2.94mm，沿-X方向移動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2顯示，當一個水道的約束數(shù)（如）變化時，提出C-空間方法很好的將這個效應(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)計組得到相應(yīng)的調(diào)整。 C-模型冷卻分析用于分析設(shè)計得到的布局圖。從圖10（a）到（d）可見，兩個設(shè)計中，冷卻時間為20s時，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意的設(shè)計布局。從圖10（c）和（d）觀察得到，同設(shè)計2比較，涉及1中工件大部分沒有產(chǎn)生變色。這表明在設(shè)計1中很多工件的溫度偏差在以內(nèi)。這是因為在設(shè)計2中，隨著空腔中的水道向模壓移動了5mm，冷卻效果變得不均勻，這表示當施加很多約束時，保持初始理想冷卻效果很困難。它也解釋了為什么設(shè)計2的最大適切性稍微小于設(shè)計1的最大適切性。（a）示例零件（b）冷卻系統(tǒng)的初始設(shè)計（a）冷卻系統(tǒng)的15個自由度（b）設(shè)計1 （b）移動和相交（b）設(shè)計2 圖9 分層設(shè)計表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，一個單元列舉方案被用于簡化這個方法的執(zhí)行，在目前的執(zhí)行中，C-空間一維分辨率為0.15mm。對冷卻系統(tǒng)設(shè)這個分辨率是足夠的，因為對一個好的調(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)項目。實際上，基于理論2的方法，所有C-空間計算和存儲都在3維空間內(nèi)完成，因此標準校核模型技巧可以應(yīng)用。該研究的一個主要貢獻是發(fā)展了一個特別的支持布局設(shè)計的C-空間方法。應(yīng)用這個C-空間方法，所有的可行布局設(shè)計很好的被顯示出來。同時我們得出了該方法不僅可以用于冷卻系統(tǒng)設(shè)計的優(yōu)化設(shè)計支持，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)計的局限，如前面的方法[13，14]。這個C-空間方法能夠獨立作為一個系統(tǒng)去支持互動布局設(shè)計。它使設(shè)計者在不用檢查冷卻系統(tǒng)截面和其它模型插件能夠開發(fā)出設(shè)計方法。該研究主要目的集中在冷卻系統(tǒng)設(shè)計的幾何形狀構(gòu)成方面。在設(shè)計冷卻系統(tǒng)時，其它參數(shù)如冷流率，冷卻時間，包裝時間，擠出時間都需要被考慮進來。一個可行的方法就是將這些所有參數(shù)進行考慮插入配備更復(fù)雜的GA的C-空間方法，如[8]報道所示。需要對該方法進一步研究，其他研究方向包括C-空間方法處理冷卻系統(tǒng)拓撲變化和具體設(shè)計約束，如初始設(shè)計選擇水道之間的變化幾何形狀和拓撲約束扽。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（項目No.7001775）的大力支持。（a）設(shè)計1的模具溫度（b）設(shè)計2的模具溫度（c）設(shè)計1零件的不同溫度（d）設(shè)計2零件的不同溫度圖10。用CAE模具冷卻分析系統(tǒng)比較這兩個布圖設(shè)計引理2 在中給定兩個區(qū)間和。如果，那么引理3 在中給定，那么引理4 在中給定任意兩個和。如果，則它們對的投影滿足：引理5 給定兩個區(qū)間和滿足。則在中的區(qū)間滿足：引理6 給定三個區(qū)間、和滿足和。則在中的區(qū)間滿足：引理7 引理8 給定兩個區(qū)間和滿足，其中點在中，點在中，如果那么：定理1 參考文獻： 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

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