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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: meclli.hk (C.L. Li).0010-4485/$ - see front matter c 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.cad.2007.11.010334349/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 de
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