外文翻译--用配置空间的方法对注塑模冷却系统进行设计 英文版

Computer-Aided Design 40 (2008)spaceC.L.productimoulded 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 feasibilityof building the cooling system inside the mould insert without interfering with the other mould components. This paper reports a configurationspace (C-space) method to address this important issue. While a high-dimensional C-space is generally required to deal with a complex systemsuch as a cooling system, the special characteristics of cooling system design are exploited in the present study, and special techniques that allowC-space computation and storage in three-dimensional or lower dimension are developed. This new method is an improvement on the heuristicmethod developed previously by the authors, because the C-space representation enables an automatic layout design system to conduct a moresystematic search among all of the feasible designs. A simple genetic algorithm is implemented and integrated with the C-space representation toautomatically generate candidate layout designs. Design examples generated by the genetic algorithm are given to demonstrate the feasibility ofthe method.c 2007 Elsevier Ltd. All rights reserved.Keywords: Cooling system design； Plastic injection mould； Configuration space method1. IntroductionThe cooling system of an injection mould is very importantto the productivity of the injection moulding process andthe quality of the moulded part. Extensive research has beenconducted into the analysis of cooling systems 1,2, andcommercial CAE systems such as MOLDFLOW 3 andMoldex3D 4 are widely used in the industry. Researchinto techniques to optimize a given cooling system has alsobeen reported 58. Recently, methods to build better coolingsystems by using new forms of fabrication technology havebeen reported. Xu et al. 9 reported the design and fabricationof conformal cooling channels that maintain a constant distancefrom the mould impression. Sun et al. 10,11 used CNCDespitethevariousresearcheffortsthathavefocusedmainlyon the preliminary design phase of the cooling system designprocess in which the major concern is the performance ofthe cooling function of the system, support for the layoutdesign phase in which the feasibility and manufacturability ofthe cooling system design are addressed has not been welldeveloped. A major concern in the layout design phase is thefeasibility of building the cooling system inside the mouldinsert without interfering with the other mould components.Consider the example shown in Fig. 1. It can be seen thatmany different components of the various subsystems of theinjection mould, such as ejector pins, slides, sub-inserts, andso forth, have to be packed into the mould insert. Finding thebest location for each channel of the cooling circuit to optimizePlastic injection mould coolingconfigurationC.G. Li,Department of Manufacturing Engineering and EngineeringReceived 3 May 2007； acceptedAbstractThe cooling system of an injection mould is very important to themilling to produce U-shaped milled grooves for coolingchannels and Yu 12 proposed a scaffolding structure for thedesign of conformal cooling. Corresponding author.E-mail address: .hk (C.L. Li).0010-4485/$ - see front matter c 2007 Elsevier Ltd. All rights reserved.doi:10.1016//locate/cadsystem design by themethodLiManagement, City University of Hong Kong, Hong Kong18 November 2007vity of the injection moulding process and the quality of thethe cooling performance of the cooling system and to avoidinterference with the other components is not a simple task.Another issue that further complicates the layout designproblem is that the individual cooling channels need to beconnected to form a path that connects between the inlet andthe outlet. Therefore, changing the location of a channel may335Fig.1. Thecoolingsystemcomponents.require changing theexample shown into optimize the coolingin Fig. 2(a). Assumeother mould componentsmould componentAs C1 cannot be mointerference with otherC2 is moved and Cconnectivity, as shoC3 is found to interferemould components,is very tedious.that supports thethis new technique,used to provide alayout designs. Thean efficient methodthe layout designto generate layoutsystem developedw C-space methodto conduct a morelayout designs.is the space thatsystem is treatedthe configurationfree region. Pointsof thethe componentscorrespond toof the systeminitially formalizedplanning problemsshortenedand further modification is needed, which results in the finaldesign shown in Fig. 2(c). Given that a typical injection mouldmay have more than ten cooling channels, with each channel(a) Interference occurs between cooling channel C1and mould component O1 at the ideal location ofC1.(c) C3 is moved and C2 isdesign.Fig. 2. An example showing the tediousnessand a survey in this area of research has been reported byWise and Bowyer 16. The C-space method has also beenused to solve problems in qualitative reasoning (e.g., 17,18)(b) Channel C1 is shortened, C2 is moved, and C3 iselongated.to give the finalC.G. Li, C.L. Li / Computer-Aided Design 40 (2008) 334349insideamouldinsertpackedwithmanyothermouldother channels as well. Consider theFig. 2. The ideal location of each channelperformance of the system is shownthat when the cooling system and theare built into the mould insert, aO1 is found to interfere with channel C1.ved to a nearby location due to the possiblecomponents, it is shortened. As a result,3 is elongated accordingly to maintain thewn in Fig. 2(b). Owing to its new length,with another mould component, O2,potentially interfering with a few otherfinding an optimal layout design manuallyThis paper reports a new techniqueautomation of the layout design process. Ina configuration space (C-space) method isconcise representation of all of the feasibleC-space representation is constructed bythat exploits the special characteristics ofproblem. Instead of using heuristic rulesdesigns, as in the automatic layout designpreviously by the authors 13,14, this neenables an automatic layout design systemsystematic search among all of the feasible2. The configuration space methodIn general, the C-space of a systemresults when each degree of freedom of thatas a dimension of the space. Regions inspace are labeled as blocked region orin the free regions correspond to valid configurationssystem where there is no interference betweenof the system. Points in the blocked regionsinvalid configurations where the componentsinterfere with one another. C-space wasby Lozano-Perez 15 to solve robot pathof the layout design process.336and(e.g.,automatic232.1.theyc3se(e)a cooling system. Fig. 3 gives an example. The preliminarydesign of this cooling system consists of four cooling channels.To generate a layout design from the preliminary design, thecenters and lengths of the channels are adjusted. As shown inFig. 3, the center of channel C1 can be moved along the X1and X2 directions, and its length can be adjusted along the X3direction. Similarly, the length of C2 can be adjusted along theX4 direction, while its center adjustment is described by X1and X3 and thus must be the same as the adjustment of C1 tomaintain the connectivity. By applying similar arguments to theother channels, it can be seen that the cooling system has 5(a) Channel Ci and three mouldcomponents inside the mould insert.(b) Offsets of the mouldCi represented by line(d) The initial free region of Ci.Fig. 4. The major steps in the constructionconsidered. To account for the diameter D, Oi is first offsetby D/2 + M to give Oprimei, where M is the minimum allowabledistance 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 achannel Ci and three mould components, O1, O2, and O3, thatmay interfere with Ci. Fig. 4(b) shows the offsets Oprime1, Oprime2,and Oprime3 of the mould components, and the reduction of Ci toa line segment Li that is coincident with the axis of Ci. Ifthere is no intersection between Li and the offsets of the mouldcomponents, then the original channel Ci will not intersect withcomponents andgment Li.(c) Sweeping the offsets of the mouldcomponents 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) 334349Fig. 3. An example showing the degrees of freedom of a cooling system.the analysis and design automation of kinematic devices1921).TheauthorinvestigatedaC-spacemethodinthedesign synthesis of multiple-state mechanisms 22, in previous research.C-space of a cooling systemA high-dimensional C-space can be used to represent all offeasible layout designs of a given preliminary design ofdegreesoffreedom,andtheyaredenotedas Xi,i = 1,2,.,5.In principle, the C-space is a five-dimensional space and anpoint in the free region of this space gives a set of coordinatevalues on the Xi axes that can be used to define the geometry ofthe channels without causing interference with the other mouldcomponents.Todeterminethefreeregioninahigh-dimensionalC-spaceofacoolingsystem,thefirststepistoconstructthefreeregions in the C-spaces of the individual channels.2.2. C-space construction of individual cooling channelsWhen an individual channel Ci is considered alone, it hasthree degrees of freedom, say X1 and X2 for its center locationand X3 for its length. As the ideal center location and lengthhave already been specified in the preliminary design, it isreasonable to assume a fixed maximum allowable variationfor X1, X2, and X3. The initial free region in the C-spaceof channel Ci is thus a three-dimensional cube Bi with thedimensionsc c c.To avoid any possible interference with a mould componentOi when channel Ci is built into the mould insert by drilling,a drilling diameter D and drilling depth along X have to beof the free region FRi of a channel Ci.C.G. Li, C.L. Li / Computer-Aidedthe mould components. This growing or offset of an obstacle isa standard technique in the C-space method 15.A channel is formed by drilling from a face of the mouldinsert, and any obstacle Oi within the drilling depth will affectthe construction of the channel. To account for the drillingdepth, the offset Oprimei of Oi is swept along the drilling directionuntil the opposite face of the mould insert is reached to generateOprimeprimei . This sweeping of Oprimei in effect reduces line Li to a point Pilocated at the end of Li. As shown in Fig. 4(c), if the point Piis 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 centerat Pi as shown in Fig. 4(d). Bi then intersects with the mouldinsert to obtain Bprimei. Bprimei represents all of the possible variationsof Ci when only the geometric shape of the mould insert isconsidered. Then, FRi is obtained by subtracting from Bprimei theOprimeprimei of all of the obstacles. Fig. 4(e) and (f) show the subtractionand the resulting FRi of the example.2.3. Basic approach to the construction of the C-space ofcooling systemTo determine the free region FRF in the C-space of acooling system, the free regions of each cooling channel haveto be “intersected” in a proper manner so that the effect ofthe obstacles to all of the channels are properly representedby FRF. However, the standard Boolean intersection betweenthe free regions of two different channels cannot be performedbecause their C-spaces are in general spanned by different setsof axes. Referring to the example in Fig. 3, the C-spaces ofC1 and C2 are spanned by X1, X2, X3 and X1, X3, X4,respectively. To facilitate the intersection between free regionsin different C-spaces, the projection of a region from the C-space of one channel to that of another channel is needed. Thefollowing notations are first introduced and will be used inthe subsequent discussions on projections and the rest of thepaper.Notations used in describing high-dimensional spacesSn denotes an n-dimensional space spanned by the set of axesXn = X1, X2,., Xn.Sm denotes an m-dimensional space spanned by the set of axesXm = Xprime1, Xprime2,., Xprimem.pn denotes a point in Sn and pn = (x1,x2,.,xn), where xidenotes 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 toSm.PROJSm(Rn) denotes the projection of a region Rn from Sn toSm.Notations used in describing a cooling systemnC 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) 334349 337FRi denotes the free region in Si. That is, it is the free region ofan 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 regionof the cooling system.Consider the projection of a point pn in Sn to a point pm inSm. Fig. 5(a) illustrates examples of projection using spaces ofone dimension to three dimensions. Projections are illustratedforthreecases:(i) Xm Xn；(ii) Xm Xn；and(iii) Xm negationslash Xn,Xn negationslash Xm, and Xn Xm negationslash= . For (i), each coordinate ofpm is equal to a corresponding coordinate of pn that is on thesame 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 thatof 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, inparticular, for cases (ii) and (iii), is as follows. Consider twoadjacent channels Cn and Cm. As they are adjacent, they mustbe connected and thus their C-spacesSn and Sm share somecommon axes. Assume that a configuration that correspondsto a point pn in Sn has been selected for Cn. To maintainthe connectivity, the configuration for Cm must be selectedsuch that the corresponding point pm in Sm shares the samecoordinates with pn on their common axes. This implies thatpm can be any point within the projection of pn on Sm, wherethe method of projection is defined above. The projections of aregion Rn in Sn to Sm are simply the projections of every pointin Rn to Sm. Fig. 5(b) illustrates the region projections. Theformal 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 thesingle 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)isa region Rm = pm|PROJSI (pm) = PROJSI (pn), whereSI 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 avalue for each degree of freedom of the cooling system so thatthe geometry of the channels is free from interference with theother mould components. In other words, the projection of pFto each Si is in the free region FRi of each Ci. Thus, FRF isdefined as follows.Definition 2 (Free Region in the C-space of a Cooling System).FRF = pF|PROJSi (pF) FRi,i 1,nC-AidedNote that according toto Si always contains onlythat span Si is always a subsetThe construction of thealready been explained inthe following theorem is useful.Theorem 1.FRF =nCintersectiondisplayi=1PROJSF(FRi).Intuitively, this theorem saysfirst projected to the C-spacecan then be obtained by performingamong the projections. Theused in the proof are givenof the C-spaceF and to facilitate thebetween the regionscan use a kind of cellused in 21,24. Theregion RF inEach box is defined bySF. The intersection ofof the two sets ofhigh-dimensional boxesintervals of each of theby m three-OJSF(FRi) can then beboxes. The constructionFig. 5. The projections of points and regions in Sn to Sm.Definition 1.1, the projection of pFa single point because the set of axesof the axes that span Sn.free region FRi of each Ci hasSection 2.2. To find FRF from FRi,that to find FRF, all of the FRi areof the cooling system SF. FRFthe Boolean intersectionsproof of Theorem 1 and the lemmas2.4. Representation and computationTo represent the free region FRcomputation of the Boolean intersectionsin a high-dimensional space, weenumeration method similar to the onebasic idea is to approximate a high-dimensionalSF by a set of high-dimensional boxes.specifying an interval on each axis oftwo regions is achieved by the intersectionboxes. The intersection between twois simply the intersection between theboxes in each axis.Assuming that each FRi is approximateddimensional boxes, the projection PRapproximated by mnF-dimensional338 C.G. Li, C.L. Li / Computerin the Appendix.Design 40 (2008) 334349of FRF that uses Theorem 1 then requires mnC intersectionsbetween nF-dimensionalmaximum of mnCnFof boxes used to representintersections and FRis anticipated that theare still major problemsimproved method is3. An efficient techniqueTo avoid the highfor the representation. Instead, weprocess toexample shown inis assumed in thisalong the Z directionhasfourdegreeseach channel Ci areshown 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 numberthe intermediate results of theF can be reduced by special techniques, itmemory and computational requirementsof this method. In the next section, andeveloped.for C-space constructionto represent and not to compute FRF explicitlyfocus on a technique that enables the computationalwork on the C-spaces of each individual channel.First, consider the simplified designFig. 6. For the purpose of illustration, itexample that there is no variation in FRiofthemouldinsertandthusthecoolingsystemof freedom as shown in Fig. 6(a). The Si oftwo dimensional and the assumed FRi areConsider a simple method for designingC.G. Li, C.L. Li / Computer-Aidedmemory and computational requirementsand construction of FRF, we choose notDesign 40 (2008) 334349 339point p1 can be selected from within FR1 so that C1 is freefrom interference with any obstacle. However, S1 is spanned-Aidedcontinuedeven though their C-spacesof C1, (i.e., they areas well, because thesystem are connected.have an effect in thecooling system.To develop a designof each individual channels,selection of a pointalways exist a correspondingthat all of the channelssystem. To address thisSi is needed.Definition 3. PRi isPRi = PROJSi (FRFObviously, for analways a correspondiFR2. Again, as p2x3 must have a valueFR3. Also, asmust also be insidep1, p2, p3, and p4C1.determine the validdesigns for C1, theThe effect of FR4valid region in FR3,finally in S1. Theall of the effects ofis formallychannels Ci andof their free regionsdo not have an axis common to thatnot adjacent to C1), have to be consideredcooling channels that make up the coolingA choice in one degree of freedom willchoice of another degree of freedom of theprocess that works on the C-spacesa major concern is that after thein the C-space of one channel, there mustpoint in all of the other Si suchcan be connected to form a valid coolingconcern, the projection of FRF to eachdefined as the projection of FRF to Si.)which we can find a p2 = (x2,x3) withinhas a coordinate x3 in X3, the coordinatefor which we can find a p3 = (x3,x4) withinFR4 has axes X3 and X4, p4 = (x3,x4)FR4. Fig. 7 shows the sequence of pointsthat constitutes a valid design for channelThe above illustration reveals that toregion in S1 that represents all of the validfree region FR4 should be considered first.should be “reflected” in S3 to determine thewhich should then be “reflected” in S2, andresulting valid region in S1 then includesFR4, FR3, FR2, and FR1. A composition operationdefined for this purpose.Definition 4 (Composition). For two adjacentCi+1 in a cooling system, the composition340 C.G. Li, C.L. Li / Computer(c) Free regions in S1 after “intersection” withFR2.Fig. 6. (by X1 and X2, and X2 is shared by S2. Hence, the constraintsimposed by those obstacles in S2 must also be considered. Inan attempt to find all of the feasible points for designing C1,FR1 is “intersected” with FR2. The result of this “intersection”is shown in Fig. 6(c), which is obtained by removing the regionin FR1 where x2 j, CRj,iIf i j, CRj,iIf i = j, CRj,iAs an example,that leads to theconstruct CR3,4, whichFR3, as shown inCR2,4 = PROFinally, CR1,4 is gishown in Fig. 8(c).CR1,4 takes into accountthe channels thatany point in CR1,4cooling system canBy applying thebe obtained by selectinginto Si.no valid designthe compositiondesigns that mayobtained by thisit is importantpart of the validvalid designs forwing theoremof a sequence ofall of theby a Booleanimportant featureby computationsCRi,1 and CRi,nCperformed in Si.the intersectionFig. 7. A sequence of points in FRi that gives a valid design of the cooling system.i+1(FRi) FRi+1.i (FRi+1) FRi.that consists of a sequence of channelsofthefreeregionsfromCi toCj,denotedbelow.= PROJSj (CRj+1,i) FRj.= PROJSj (CRj1,i) FRj.= FRi.Fig. 8 shows the sequence of compositionsconstruction of CR1,4. The first step is tois given by CR3,4 = PROJS3(FR4) Fig. 8(a). Then, CR2,4 is constructed byJS2(CR3,4) FR2, as shown in Fig. 8(b).ven by CR1,4 = PROJS1(CR2,4) FR1, asIt is obvious from Fig. 8(c) that the resultingthe effects of the free regions of all ofmake up the cooling system. Therefore, for, it is guaranteed that a valid design for thebe constructed.of all of the other channels have been “composited”However, we would also like to ensure thatis being excluded from the free region afteroperations are applied. Otherwise, some validgive better cooling performance can never bemethod. Taking the design of C1 as an example,that CR1,4 in Fig. 8(c) not only represent adesign for C1, but also represent all of theC1. To address this issue, we introduce the follothat applies to a cooling system that consistschannels Ci, i 1,nc.Theorem 2.PRi = CRi,1 CRi,nC.Theorem 2 states that PRi, which representsvalid designs for channel Ci, can be obtainedintersection between CRi,1 and CRi,nC. Anof this theorem is that PRi can be obtainedin three-dimensional spaces, because bothare regions in Si and thus the intersection isMoreover, CRi,1 and CRi,nC are obtained byC.G. Li, C.L. Li / Computer-Aidedcomposition operations, a valid design canpoints in each Si after the free regionsDesign 40 (2008) 334349 341of regions in Sj. That is, PRi is obtained by a sequenceof operations in three-dimensional spaces. If the assumptionstated in Section 2.4approximated by m three-dimensionaland PRi can also bea total of ncm three-dimensionalall of the PRi. It canbetween three-dimensionalthe PRi. Therefore, thestore regions in a high-dimensionalmemory and computationalin Theorem 1.The following givestwo parts: the proof ofof CRi,1 CRi,nC stated in the Appendix3.1. Proof of Theorem(1) To prove: CRi,1 pi CRi,1 CRi,nC pi CRi,1, pi (i) Inducing from pi pi CRi,14)Si (pi1)in the common axes)a point pi2 same coordinates indetermine a series ofk, and pk and pk+1axes of Sk and Sk+1.(a) Constructing CR3,4 by PROJS3(FR4) FR3.Fig. 8. The sequence of operations by which CR1,4 is constructed.is used again, that is, if each FRi isboxes, then both CRi,jrepresented by m 3D boxes. Therefore,boxes is needed to representbe shown that O(ncm2) intersectionsboxes are needed to generate all ofuse of Theorem 2 prevents the need tospace, and avoids the highrequirements of the method giventhe proof of Theorem 2. It consists ofCRi,1 CRi,nC PRi and the proofPRi. The lemmas used in the proof are.2CRi,nC PRiCRi,nC.CRi,1CRi,1 = PROJSi (CRi1,1) FRi (By Definition pi PROJSi (CRi1,1)pi FRi pi PROJSi (CRi1,1) pi1 CRi1,1 such that pi PROJ(By Definition 1.4) pi PROJSi (pi1) pi and pi1 have the same coordinatesof Si and Si1. (By Definitions 1.11.3 pi1 CRi1,1CRi1,1 = PROJSi1(CRi2,1) FRi1 pi1 FRi1.Using the same method, we can determineFRi2 such that pi1 and pi2 have thethe common axes Si1 and Si2.Repeatedly using this method, we canpoints pk,k 1,i 1, such that pk FRhave the same coordinates in the common(ii) Inducing from pi CRi,nC342 C.G. Li, C.L. Li / Computer-Aided Design 40 (2008) 334349 pi CRi,nC.Using a similar method,points pk,k i +pk1 have the sameSk1.By (i) and (ii), wesuch that pk FRhave the same coordinatesFor a cooling systemchannels Ci, thechannelsCi andCiof the physical connectionif there is a commonand Cj, Xc mustchannels betweenpk,k 1,nC constructedunique coordinateconstructed from thispk = PROJSk(pF pk FRk,k 1 PROJSk(pF) pF PROJSF(i.e. pF intersectiondisplaynCk=1 PRk)parenrightBigg(FRk)parenrightBigg1)3).1,1) FRiparenrightbigF(FRi)(By Lemma 6) PROJSF(FRi)SF(FRi1)(b) Constructing CR2,4 by PROJS2(CR3,4) FR2.Fig. 8. (continued)we can determine another series of1,nC, such that pk FRk, and pk andcoordinates in the common axes of Sk andobtain a series of points pk,k 1,nC,k, and any two adjacent points in this seriesin their common axes.that consists of a sequence of coolingC-spaces Si and Si+1 of two adjacent+1 alwayssharesomecommonaxesbecausebetween the channels. Furthermore,axis Xc in the C-spaces of channels Cialso appear in all of the C-spaces of theCi and Cj. Therefore, the series of pointsby the above method will give afor each axis of SF. Let pF be the pointset of coordinates. It is obvious that),k 1,nC,nCFRk,k 1,nCFRk),k 1,nC (By Lemma 8) PROJSi (pF) PROJSiparenleftBiggnCintersectiondisplayk=1PROJSF(FR(By Lemma 1) pi PROJSiparenleftBiggnCintersectiondisplayk=1PROJSF(FRk)parenrightBigg CSi,1 CSi,nC PROJSiparenleftBiggnCintersectiondisplayk=1PROJSF CSi,1 CSi,nC PROJSi (FRF) (By Theorem CSi,1 CSi,nC PRi (By Definition(2) To prove: CRi,1 CRi,nC PRiPROJSF(CRi,1) = PROJSF parenleftbigPROJSi (CRi= PROJSF(PROJSi (CRi1,1) PROJS(By Lemma 4) PROJSF(CRi1,1) PROJSF(FRi)= PROJSF(PROJSi1(CRi2,1) FRi1= PROJSF(PROJSi1(CRi2,1) PROJ PROJSF(FRi) (By Lemma 4)C.G. Li, C.L. Li / Computer-AidedOJSF(FRk)Design 40 (2008) 334349 343 PROJSF(CRi2,1) PROJSF(FRi1) PROJSF(FRi)-Aided(By Lemma 6). PROJSF(FR1) =iintersectiondisplayk=1PROJSF(FRUsing a similar method,PROJSF(CRi,nC) PROJSF(CRi,1) PROJSF(CRi,1 C PROJSF(CRi,nC PROJSF(CRi,1 PROJSi (PROJSF PROJSiparenleftBiggnCintersectiondisplayk=1PRF(FRk)parenrightBiggTheorem 1)3).systemthatspecifies, the first stepThen, the PRi foroperationa candidateset of coordinatesthe explanation,of freedom Xi andchannel Ci+1. To(c) Constructing CR1,4 by PROJS1(CR2,4) FR1.Fig. 8. (continued)PROJSF(FR2) PROJSF(FRi)k).we can obtain:nCintersectiondisplayk=iPROJSF(FRk)PROJSF(CRi,nC) nCintersectiondisplayk=1PROJSF(FRk)Ri,nC) = PROJSF(CRi,1) (By Lemma 4)CRi,nC) nCintersectiondisplayk=1PROJSF(FRk)(CRi,1 CRi,nC)parenrightBigg CRi,1 CRi,nC PROJSiparenleftBiggnCintersectiondisplayk=1PROJS(By Lemma 5) CRi,1 CRi,nC PROJSi (FRF) (By CRi,1 CRi,nC PRi (By DefinitionBy (1) and (2): PRi = CRi,1 CRi,n.4. Generation of candidate designsGivenapreliminarydesignofacoolinga sequence of channels and their ideal geometryis to construct an FRi for each channel.each channel is obtained by applying the compositionas specified in Theorem 2. One way to generatedesign for a cooling system is to select afrom the set of PRi as follows. To simplifyassume that each channel Ci has degreesXi+1, and Xi+1 is shared by the adjacent344 C.G. Li, C.L. Li / ComputerOJSF(FRk) (By Lemma 2)Design 40 (2008) 334349generate a design, a point (x1,x2) in PR1 is chosen. Then, anx3 is chosen such that (x2,x3) is within PR2. This selectionC.G. Li, C.L. Li / Computer-Aidedprocess is then repeated for the next coordinate in the PR ofthe next channel until the coordinates of all of the degrees offreedom are determined. An important feature of the methodis that whatever value is selected for a coordinate in one step,there always exist valid values that can be selected for the nextcoordinate in a subsequent step.5. The automation of the design process using a geneticalgorithmTo test the feasibility of the C-space method in supportingthe automation of the layout design process, a simple geneticalgorithm (GA) 25 is implemented and integrated with the C-space construction program. A simple chromosome structure isused in the implementation of the GA. It consists of a stringof nF real values g1g2 .gnF, for which each gi has a realvalue between 0 and 1, and nF is the number of the degrees offreedom of the cooling system. To generate a design from thechromosome, the approach described in the previous sectionis used, with the gi used as a percentage value to select acoordinate. For example, if the valid values for a coordinate xiin PRi exist in the intervals x1i ,x2i and x3i ,x4i , where x1i x2i x3i ,x4i , then the selected value for xi will be x1i +gi(x2i x1i )+(x4i x3i ) if gi (x2i x1i )/(x2i x1i +x4i x3i ) (i.e., xilies in the first interval). Otherwise, xi will be set to x3i +(gi 1)(x2i x1i ) + gi(x4i x3i ) (i.e., xi lies in the second interval).A standard one-point crossover operation, a mutationoperation,andtheroulettewheelselectionmethod26areusedin the GA process. The fuzzy evaluation method developed inour previous research 13,14 is used to perform fast evaluationof the fitness of the candidate design that corresponds to achromosome. Note also that before the GA process starts,the PRi for each channel is constructed. The constructionof the PRi is done only once and thus it will not affectthe computational time for the evolution process of the GA.Examples of the layout designs generated by the GA processare given in the next section.6. Case studyTwo views of an example part are shown in Fig. 9(a).Fig.9(b)illustratesthepreliminarydesignofthecoolingsystemthat specifies the ideal location of each cooling channel whenonly the cooling performance of the system is considered (forthe purpose of illustration, only the cooling system in thecavity half is shown). In the ideal location, interference occursbetween channel C5 and a mould component O1. Using theproposed method to automate the layout design, the FRi andthen the PRi of each channel are constructed. As an example,Fig. 9(g) and (h) show FR4 and PR4 for channel C4. It isnotedthat PR4 isobtainedfrom FR4 bycompositionwithotherFRi, and thus PR4 is a subset of FR4, as is evident from thefigures. After all of the PRi are computed, the GA process isinvoked, and the maximum fitness value among the candidatedesigns generated in each generation during the evolutionaryprocess is shown in Fig. 9(j). The maximum fitness value startsto converge after approximately 600 generations. As shown inDesign 40 (2008) 334349 345Fig. 9(c), the cooling system consists of 15 degrees of freedomXi and their values are listed in Table 1. In the table, therow labeled “Preliminary design” shows the xi values for thepreliminary design. The next row lists the values for Design 1,whichisthebestdesigngeneratedbytheGAprocessafter1000generations. As highlighted in the table, Design 1 is obtainedby reducing x6 by 1.21 mm. Fig. 9(d), which shows Design 1,this adjustment corresponds to the lowering of C5 along the Zdirection to clear the interference between C5 and O1. Due tothe connectivity among the channels, the same adjustment alsoapplies to channels C4 and C6 to C13. Table 1 also shows thatall of the other xi in Design 1 are maintained to within 0.2 mmfrom values specified in the preliminary design.TodemonstratefurtherthecapabilityoftheC-spacemethod,the mould component O2 is moved along the Y direction sothat it interferes with C13, as shown in Fig. 9(e). This newobstacle imposes a new constraint in the free region of C13 sothat the feasible adjustment along X6 is further limited. Thiseffect is demonstrated in the updated PR4 shown in Fig. 9(i) inwhich only the upper portion of the PR 4 shown in Fig. 9(h)is retained. The GA process is invoked again with the newPRi of all of the channels to generate Design 2. The fitnessvalue is shown in Fig. 9(k). Note that the best fitness valueattained is lower than that of Design 1. This is justified becausewith the imposition of more constraints, larger deviation fromthe ideal design is expected. The values of the xi obtainedfrom the GA process are shown in the last row of Table 1. Ashighlighted in the table, x6 is adjusted for 5 mm to clear theinterference with O2. This corresponds to moving channels C4to C13 along the Z direction. Now, the interference between C5and O1 can no longer be cleared by adjusting x6. Instead, x4and x5 are adjusted, which corresponds to moving C5 2.94 mmalong the Y direction, and moving C4 6.22 mm along theX direction, as shown in Fig. 9(f). Channels C2 and C3 arealsoadjustedaccordinglytomaintaintheconnectivity.Design2demonstrates that when the constraint in one channel (e.g., C13)is changed, the proposed C-space method properly propagatesthis effect to the other channels (e.g., C4 and C5) so that theset of all of the feasible designs for these channels is adjustedaccordingly.Cooling analysis with C-Mold has been used to analyzethe layout designs generated. It can be seen from Fig. 10(a)through (d) that the maximum mould-wall temperature is about46 C with a cooling time of 20 s for both designs, and thattheir maximum temperature differences are less than 8 C,which indicate that the proposed method is able to generatesatisfactory layout designs for both cases. It is also observedfrom Fig. 10(c) and (d) that a much larger portion of the part inDesign 1 is not colored when Design 1 is compared to Design2. This indicates that the temperature difference in most of thepart in Design 1 is less than 5.5 C. This is because in Design2 as the channels in the cavity half are moved by 5 mm towardsthe mould impression, the cooling effect becomes less uniform,which demonstrates that when more constraints are imposed,maintaining the ideal cooling effect of the preliminary designbecomes more difficult. It also explains why the maximumfitnessvalueforDesign2isslightlylowerthanthatofDesign1.346 C.G. Li, C.L. Li / Computer-Aided Design 40 (2008) 334349(a) An example part. (b) Preliminary design of the cooling system.(c) The 15 degrees of freedom of the cooling system. (d) Design 1.(e) O2 moved to interfere with C13. (f) Design 2.Fig. 9. The layout design examples generated by the proposed method.Table 1Degrees of freedom of the cooling systemDegree offreedomX1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13 X14 X15Preliminarydesign32.5 169 55 212.5 139.5 29 32.5 110 212.5 80.5 32.5 212.5 51 21.5 32.5Design 1 32.41 168.91 54.95 212.34 139.49 27.79 32.36 109.97 212.51 80.60 32.30 212.52 51.10 21.56 32.52Design 2 32.51 169.04 55.13 206.28 136.56 34.00 32.51 109.84 212.49 80.45 32.50 212.52 50.93 21.66 32.64R7. DiscussionIn theenumerationIn the currentrepresentationbe adequatefine adjustment,performanceand theoremsparticular schemebased on Theoremdone in threetechniques canA majorof a specificprocess. Usingdesigns arethe C-spacein which notcooling performancemanufacturable.relying onas in our prealso be usedlayoutdesign.betweenof thetime,designingall thesemethodin 8.and otherC-spacesystemandin thebyy of HongSn. If(j) Fitness value against the generation for Design 1. (k) Fitness value against the generation for Design 2.Fig. 9. (continued)and conclusionimplementation of the C-space method, a cellscheme is used to simplify the implementation.implementation, the resolution in the C-spaceis 0.15 mm in each dimension, which shouldfor the cooling system design because for a verysay 0.01 mm, the change in the coolingmay not be noticeable. However, the methodologydeveloped in this research are not limited to afor representation. In fact, for the method2, all C-space computation and storage aredimension, and thus standard geometric modelingbe used.contribution of this research is the developmentC-space method that supports the layout designthe C-space method, all of the feasible layoutproperly represented. We have demonstrated thatmethod can be used to support design generationonly designs that are optimal in terms ofare generated, but also, the designs areThis new method overcomes the limitation ofspecific heuristics to generate the layout design,vious method 13,14. This C-space method caninteractively without having to check for interferencethe cooling system and the other mould insert components.The focus of this research is on the geometric aspectcooling system design. It is understood that other parameters,such as the coolant flow rate, cooling time, packingejectiontime,etc.needtobeconsideredaswellwhena cooling system. One possible approach to takeparameters into account is to integrate the C-spacewith a more sophisticated GA such as the one reportedFurther investigation on this approach is neededfurther research directions include the extension of themethod to deal with topology changes in the coolingand specific design constraints, such as various geometrytopology constraints specified between selected channelspreliminary design.AcknowledgementThe work described in this paper was fully supporteda Strategic Research Grant from the City UniversitKong (Project No. 7001775).AppendixLemma 1. Given a region Rn and a point pn in spacepn Rn, thenC.G. Li, C.L. Li / Computer-Aided(g) FR4. (h) Pas a stand-alone system to support interactiveItallowsadesignertoexploredesignalternativesDesign 40 (2008) 334349 3474. (i)UpdatedPR4.PROJSm(pn) PROJSm(Rn).-Aided Design 40 (2008) 334349Lemma 2. GivenRn, thenPROJSm(Rprimen) Lemma 3. GivenPROJSmparenleftBigg Lintersectiondisplayk=1Lemma 4. GivenXn, then theirPROJSn(Rm) Lemma 5. GivenXn. A region RPROJSm(PROJaxes setsSl satisfies theaxes setsRm in Sm. IfFig. 10. A comparison of the two layout designs using CAE mould cooling analysis.two regions Rn and Rprimen in space Sn. If Rprimen PROJSm(Rn).L regions Rkn,k 1, L in space Sn, thenRknparenrightBiggLintersectiondisplayk=1PROJSm(Rkn).any two regions Rm and Rprimem in Sm. If Xm projections to Sn satisfy the following:PROJSn(Rprimem) = PROJSn(Rm Rprimem).two spaces Sn and Sm having axes sets Xm m in Sm satisfies the following:Lemma 6. Given three spaces Sn, Sm and Sl havingXm Xn and Xl Xn. A region Rl in spacefollowing:PROJSn(Rl) PROJSn(PROJSm(Rl).Lemma 7. PRi FRi.Lemma 8. Given two spaces Sn and Sm havingXm Xn, a point pn in space