外文翻译--用配置空间的方法对注塑模冷却系统进行设计【中英文文献译文】
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Computer-Aided Design 40 (2008) 334349/locate/cadPlastic injection mould cooling system design by theconfiguration space methodC.G. Li, C.L. LiDepartment of Manufacturing Engineering and Engineering Management, City University of Hong Kong, Hong KongReceived 3 May 2007; accepted 18 November 2007AbstractThe cooling system of an injection mould is very important to the productivity of the injection moulding process and the quality of themoulded 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 CNCmilling 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).Despite the various research efforts that have focused mainlyon 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 optimizethe 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 may0010-4485/$ - see front matter c ? 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.cad.2007.11.010C.G. Li, C.L. Li / Computer-Aided Design 40 (2008) 334349335Fig.1. Thecoolingsysteminsideamouldinsertpackedwithmanyothermouldcomponents.require changing the other channels as well. Consider theexample shown in Fig. 2. The ideal location of each channelto optimize the cooling performance of the system is shownin Fig. 2(a). Assume that when the cooling system and theother mould components are built into the mould insert, amould component O1is found to interfere with channel C1.As C1cannot be moved to a nearby location due to the possibleinterference with other components, it is shortened. As a result,C2is moved and C3is elongated accordingly to maintain theconnectivity, as shown in Fig. 2(b). Owing to its new length,C3is found to interfere with another mould component, O2,and 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 channelpotentially interfering with a few other mould components,finding an optimal layout design manually is very tedious.This paper reports a new technique that supports theautomation of the layout design process. In this new technique,a configuration space (C-space) method is used to provide aconcise representation of all of the feasible layout designs. TheC-space representation is constructed by an efficient methodthat exploits the special characteristics of the layout designproblem. Instead of using heuristic rules to generate layoutdesigns, as in the automatic layout design system developedpreviously by the authors 13,14, this new C-space methodenables an automatic layout design system to conduct a moresystematic search among all of the feasible layout designs.2. The configuration space methodIn general, the C-space of a system is the space thatresults when each degree of freedom of that system is treatedas a dimension of the space. Regions in the configurationspace are labeled as blocked region or free region. Pointsin the free regions correspond to valid configurations of thesystem where there is no interference between the componentsof the system. Points in the blocked regions correspond toinvalid configurations where the components of the systeminterfere with one another. C-space was initially formalizedby Lozano-Perez 15 to solve robot path planning problemsand 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)(a) Interference occurs between cooling channel C1and mould component O1at the ideal location ofC1.(b) Channel C1is shortened, C2is moved, and C3iselongated.(c) C3is moved and C2is shortened to give the finaldesign.Fig. 2. An example showing the tediousness of the layout design process.336C.G. Li, C.L. Li / Computer-Aided Design 40 (2008) 334349Fig. 3. An example showing the degrees of freedom of a cooling system.and the analysis and design automation of kinematic devices(e.g., 1921). The author investigated a C-space method in theautomatic design synthesis of multiple-state mechanisms 22,23 in previous research.2.1. C-space of a cooling systemA high-dimensional C-space can be used to represent all ofthe feasible layout designs of a given preliminary design ofa 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 C1can be moved along the X1and X2directions, and its length can be adjusted along the X3direction. Similarly, the length of C2can be adjusted along theX4direction, while its center adjustment is described by X1and X3and thus must be the same as the adjustment of C1tomaintain the connectivity. By applying similar arguments to theother channels, it can be seen that the cooling system has 5degreesoffreedom,andtheyaredenotedas Xi,i = 1,2,.,5.In principle, the C-space is a five-dimensional space and anypoint in the free region of this space gives a set of coordinatevalues on the Xiaxes 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 Ciis considered alone, it hasthree degrees of freedom, say X1and X2for its center locationand X3for its length. As the ideal center location and lengthhave already been specified in the preliminary design, it isreasonable to assume a fixed maximum allowable variation cfor X1, X2, and X3. The initial free region in the C-spaceof channel Ciis thus a three-dimensional cube Biwith thedimensions c c c.To avoid any possible interference with a mould componentOiwhen channel Ciis built into the mould insert by drilling,a drilling diameter D and drilling depth along X3have to beconsidered. To account for the diameter D, Oiis first offsetby D/2 + M to give O0i, where M is the minimum allowabledistance between the channel wall and the face of a component.This growing of Oiin effect reduces channel Cito a line Li.Consider the example illustrated in Fig. 4. Fig. 4(a) shows achannel Ciand three mould components, O1, O2,and O3,thatmay interfere with Ci. Fig. 4(b) shows the offsets O01, O02,and O03of the mould components, and the reduction of Citoa line segment Lithat is coincident with the axis of Ci. Ifthere is no intersection between Liand the offsets of the mouldcomponents, then the original channel Ciwill not intersect with(a) Channel Ciand three mouldcomponents inside the mould insert.(b) Offsets of the mould components andCirepresented by line segment Li.(c) Sweeping the offsets of the mouldcomponents and Cirepresented by point Pi.(d) The initial free region of Ci.(e) Subtracting O00ifrom B0i.(f) The free region FRiof Ci.Fig. 4. The major steps in the construction of the free region FRiof a channel Ci.C.G. Li, C.L. Li / Computer-Aided Design 40 (2008) 334349337the 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 Oiwithin the drilling depth will affectthe construction of the channel. To account for the drillingdepth, the offset O0iof Oiis swept along the drilling directionuntil the opposite face of the mould insert is reached to generateO00i. This sweeping of O0iin effect reduces line Lito a point Pilocated at the end of Li. As shown in Fig. 4(c), if the point Piis outside O00i, the drilling along Lito produce Ciis feasible.The free region FRiof channel Ciis obtained as follows.First, the initial free region Biis constructed with its centerat Pias shown in Fig. 4(d). Bithen intersects with the mouldinsert to obtain B0i. B0irepresents all of the possible variationsof Ciwhen only the geometric shape of the mould insert isconsidered. Then, FRiis obtained by subtracting from B0itheO00iof all of the obstacles. Fig. 4(e) and (f) show the subtractionand the resulting FRiof the example.2.3. Basic approach to the construction of the C-space ofcooling systemTo determine the free region FRFin 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 ofC1and C2are 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 spacesSndenotes an n-dimensional space spanned by the set of axesXn= X1, X2,., Xn.Smdenotes an m-dimensional space spanned by the set of axesXm= X01, X02,., X0m.pndenotes a point in Snand pn= (x1,x2,.,xn), where xidenotes a coordinate on the ith axis Xi.Rndenotes a region in Sn(Rn Sn). Rnis a set of points in Sn.PROJSm(pn) denotes the projection of a point pnfrom SntoSm.PROJSm(Rn) denotes the projection of a region Rnfrom SntoSm.Notations used in describing a cooling systemnCdenotes the number of channels in the cooling system.nFdenotes the total degrees of freedom of the cooling system.Cidenotes the ith channel of the cooling system.Sidenotes the C-space of Ci.FRidenotes the free region in Si. That is, it is the free region ofan individual channel Ci.SFdenotes the C-space of the cooling system.FRFdenotes the free region in SF. That is, it is the free regionof the cooling system.Consider the projection of a point pnin Snto a point pminSm. Fig. 5(a) illustrates examples of projection using spaces ofone dimension to three dimensions. Projections are illustratedfor three cases: (i)XmXn; (ii)XmXn; and (iii)Xm6Xn,Xn6Xm, andXnXm6= . For (i), each coordinate ofpmis equal to a corresponding coordinate of pnthat is on thesame axis. For (ii) and (iii), the projection of pnis a region Rm.For each point pmin Rm, a coordinate of pmis equal to thatof pnif that coordinate is on a common axis of Snand 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 Cnand Cm. As they are adjacent, they mustbe connected and thus their C-spacesSnand Smshare somecommon axes. Assume that a configuration that correspondsto a point pnin Snhas been selected for Cn. To maintainthe connectivity, the configuration for Cmmust be selectedsuch that the corresponding point pmin Smshares the samecoordinates with pnon their common axes. This implies thatpmcan be any point within the projection of pnon Sm, wherethe method of projection is defined above. The projections of aregion Rnin Snto Smare simply the projections of every pointin Rnto Sm. Fig. 5(b) illustrates the region projections. Theformal definition of projection is given below.Definition 1 (Projection).1.1. IfXmXn, PROJSm(pn) is a pointpm=(x01,x02,.,x0m), where for X0i= Xj, x0i= xjfor 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. IfXmXn, PROJSm(pn) is a regionRm=pm|PROJSn(pm) = pn.1.3. IfXm6Xn,Xn6Xm, andXnXm6= , PROJSm(pn)isa region Rm= pm|PROJSI(pm) = PROJSI(pn), whereSIis the space spanned byXnXm. IfXnXm= ,PROJSm(pn) is defined as Sm.1.4. PROJSm(Rn) is defined as the region Rm= pm|pmPROJSm(pn), pn Rn.As discussed in Section 2.1, any point pFin FRFgives 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 Siis in the free region FRiof each Ci. Thus, FRFisdefined as follows.Definition 2 (Free Region in the C-space of a Cooling System).FRF= pF|PROJSi(pF) FRi,i 1,nC338C.G. Li, C.L. Li / Computer-Aided Design 40 (2008) 334349Fig. 5. The projections of points and regions in Snto Sm.Note that according to Definition 1.1, the projection of pFto Sialways contains only a single point because the set of axesthat span Siis always a subset of the axes that span Sn.The construction of the free region FRiof each Cihasalready been explained in Section 2.2. To find FRFfrom FRi,the following theorem is useful.Theorem 1.FRF=nCi=1PROJSF(FRi).Intuitively, this theorem says that to find FRF, all of the FRiarefirst projected to the C-space of the cooling system SF. FRFcan then be obtained by performing the Boolean intersectionsamong the projections. The proof of Theorem 1 and the lemmasused in the proof are given in the Appendix.2.4. Representation and computation of the C-spaceTo represent the free region FRFand to facilitate thecomputation of the Boolean intersections between the regionsin a high-dimensional space, we can use a kind of cellenumeration method similar to the one used in 21,24. Thebasic idea is to approximate a high-dimensional region RFinSFby a set of high-dimensional boxes. Each box is defined byspecifying an interval on each axis of SF. The intersection oftwo regions is achieved by the intersection of the two sets ofboxes. The intersection between two high-dimensional boxesis simply the intersection between the intervals of each of theboxes in each axis.Assuming that each FRiis approximated by m three-dimensional boxes, the projection PROJSF(FRi) can then beapproximated by mnF-dimensional boxes. The constructionof FRFthat uses Theorem 1 then requires mnCintersectionsC.G. Li, C.L. Li / Computer-Aided Design 40 (2008) 334349339(a) A simple cooling system with four channels and four degrees of freedom.(b) The free region FRiof each channel in its configuration space Si.Fig. 6. A simplified example of a cooling system design.between nF-dimensional boxes, and FRFis represented by amaximum of mnCnF-dimensional boxes. Although the numberof boxes used to represent the intermediate results of theintersections and FRFcan be reduced by special techniques, itis anticipated that the memory and computational requirementsare still major problems of this method. In the next section, animproved method is developed.3. An efficient technique for C-space constructionTo avoid the high memory and computational requirementsfor the representation and construction of FRF, we choose notto represent and not to compute FRFexplicitly. Instead, wefocus on a technique that enables the computational process towork on the C-spaces of each individual channel.First, consider the simplified design example shown inFig. 6. For the purpose of illustration, it is assumed in thisexample that there is no variation in FRialong the Z directionof the mould insert and thus the cooling system has four degreesof freedom as shown in Fig. 6(a). The Siof each channel Ciaretwo dimensional and the assumed FRiare shown in Fig. 6(b).Consider a simple method for designing channel C1. First, apoint p1can be selected from within FR1so that C1is freefrom interference with any obstacle. However, S1is spanned340C.G. Li, C.L. Li / Computer-Aided Design 40 (2008) 334349(c) Free regions in S1after “intersection” withFR2.(d) A valid point p1for the designs of C1and C2results in an invalid design for C4.Fig. 6. (continued)by X1and X2, and X2is shared by S2. Hence, the constraintsimposed by those obstacles in S2must also be considered. Inan attempt to find all of the feasible points for designing C1,FR1is “intersected” with FR2. The result of this “intersection”is shown in Fig. 6(c), which is obtained by removing the regionin FR1where x2 j,CRj,i= PROJSj(CRj+1,i) FRj.If i j,CRj,i= PROJSj(CRj1,i) FRj.If i = j,CRj,i= FRi.As an example, Fig. 8 shows the sequence of compositionsthat leads to the construction of CR1,4. The first step is toconstruct CR3,4, which is given by CR3,4= PROJS3(FR4) FR3, as shown in Fig. 8(a). Then, CR2,4is constructed byCR2,4=PROJS2(CR3,4) FR2, as shown in Fig. 8(b).Finally, CR1,4is given by CR1,4= PROJS1(CR2,4) FR1, asshown in Fig. 8(c). It is obvious from Fig. 8(c) that the resultingCR1,4takes into account the effects of the free regions of all ofthe channels that make up the cooling system. Therefore, forany point in CR1,4, it is guaranteed that a valid design for thecooling system can be constructed.By applying the composition operations, a valid design canbe obtained by selecting points in each Siafter the free regionsof all of the other channels have been “composited” into Si.However, we would also like to ensure that no valid designis being excluded from the free region after the compositionoperations are applied. Otherwise, some valid designs that maygive better cooling performance can never be obtained by thismethod. Taking the design of C1as an example, it is importantthat CR1,4in Fig. 8(c) not only represent a part of the validdesign for C1, but also represent all of the valid designs forC1. To address this issue, we introduce the following theoremthat applies to a cooling system that consists of a sequence ofchannels Ci, i 1,nc.Theorem 2.PRi= CRi,1 CRi,nC.Theorem 2 states that PRi, which represents all of thevalid designs for channel Ci, can be obtained by a Booleanintersection between CRi,1and CRi,nC. An important featureof this theorem is that PRican be obtained by computationsin three-dimensional spaces, because both CRi,1and CRi,nCare regions in Siand thus the intersection is performed in Si.Moreover, CRi,1and CRi,nCare obtained by the intersectionof regions in Sj. That is, PRiis obtained by a sequenceof operations in three-dimensional spaces. If the assumption342C.G. Li, C.L. Li / Computer-Aided Design 40 (2008) 334349(a) Constructing CR3,4by PROJS3(FR4) FR3.Fig. 8. The sequence of operations by which CR1,4is constructed.stated in Section 2.4 is used again, that is, if each FRiisapproximated by m three-dimensional boxes, then both CRi,jand PRican also be represented by m 3D boxes. Therefore,a total of ncm three-dimensional boxes is needed to representall of the PRi. It can be shown that O(ncm2) intersectionsbetween three-dimensional boxes are needed to generate all ofthe PRi. Therefore, the use of Theorem 2 prevents the need tostore regions in a high-dimensional space, and avoids the highmemory and computational requirements of the method givenin Theorem 1.The following gives the proof of Theorem 2. It consists oftwo parts: the proof of CRi,1 CRi,nC PRiand the proofof CRi,1 CRi,nC PRi. The lemmas used in the proof arestated in the Appendix.3.1. Proof of Theorem 2(1) To prove: CRi,1 CRi,nC PRipi CRi,1 CRi,nC pi CRi,1,pi CRi,nC.(i) Inducing from pi CRi,1pi CRi,1CRi,1= PROJSi(CRi1,1) FRi(By Definition 4) pi PROJSi(CRi1,1)pi FRipi PROJSi(CRi1,1) pi1 CRi1,1such that pi PROJSi(pi1)(By Definition 1.4)pi PROJSi(pi1) piand pi1have the same coordinates in the common axesof Siand Si1.(By Definitions 1.11.3)pi1 CRi1,1CRi1,1= PROJSi1(CRi2,1) FRi1 pi1 FRi1.Using the same method, we can determine a point pi2FRi2such that pi1and pi2have the same coordinates inthe common axes Si1and Si2.Repeatedly using this method, we can determine a series ofpoints pk,k 1,i 1, such that pk FRk, and pkand pk+1have the same coordinates in the common axes of Skand Sk+1.(ii) Inducing from pi CRi,nCpi CRi,nC.C.G. Li, C.L. Li / Computer-Aided Design 40 (2008) 334349343(b) Constructing CR2,4by PROJS2(CR3,4) FR2.Fig. 8. (continued)Using a similar method, we can determine another series ofpoints pk,k i + 1,nC, such that pk FRk, and pkandpk1have the same coordinates in the common axes of SkandSk1.By (i) and (ii), we obtain a series of points pk,k 1,nC,such that pk FRk, and any two adjacent points in this serieshave the same coordinates in their common axes.For a cooling system that consists of a sequence of coolingchannels Ci, the C-spaces Siand Si+1of two adjacentchannels Ciand Ci+1always share some common axes becauseof the physical connection between the channels. Furthermore,if there is a common axis Xcin the C-spaces of channels Ciand Cj, Xcmust also appear in all of the C-spaces of thechannels between Ciand Cj. Therefore, the series of pointspk,k 1,nC constructed by the above method will give aunique coordinate for each axis of SF. Let pFbe the pointconstructed from this set of coordinates. It is obvious thatpk = PROJSk(pF),k 1,nCpk FRk,k 1,nC PROJSk(pF) FRk,k 1,nC pF PROJSF(FRk),k 1,nC(By Lemma 8)i.e. pFnCk=1PROJSF(FRk) PROJSi(pF) PROJSi nCk=1PROJSF(FRk)!(By Lemma 1) pi PROJSi nCk=1PROJSF(FRk)! CSi,1 CSi,nC PROJSi nCk=1PROJSF(FRk)! CSi,1 CSi,nC PROJSi(FRF)(By Theorem 1) CSi,1 CSi,nC PRi(By Definition 3).(2) To prove: CRi,1 CRi,nC PRiPROJSF(CRi,1) = PROJSF?PROJSi(CRi1,1) FRi?= PROJSF(PROJSi(CRi1,1) PROJSF(FRi)(By Lemma 4) PROJSF(CRi1,1) PROJSF(FRi)(By Lemma 6)= PROJSF(PROJSi1(CRi2,1) FRi1) PROJSF(FRi)= PROJSF(PROJSi1(CRi2,1) PROJSF(FRi1) PROJSF(FRi)(By Lemma 4) PROJSF(CRi2,1) PROJSF(FRi1) PROJSF(FRi)344C.G. Li, C.L. Li / Computer-Aided Design 40 (2008) 334349(c) Constructing CR1,4by PROJS1(CR2,4) FR1.Fig. 8. (continued)(By Lemma 6). PROJSF(FR1) PROJSF(FR2) PROJSF(FRi)=ik=1PROJSF(FRk).Using a similar method, we can obtain:PROJSF(CRi,nC) nCk=iPROJSF(FRk)PROJSF(CRi,1) PROJSF(CRi,nC) nCk=1PROJSF(FRk)PROJSF(CRi,1 CRi,nC) = PROJSF(CRi,1) PROJSF(CRi,nC)(By Lemma 4) PROJSF(CRi,1 CRi,nC) nCk=1PROJSF(FRk) PROJSi(PROJSF(CRi,1 CRi,nC) PROJSi nCk=1PROJSF(FRk)!(By Lemma 2) CRi,1 CRi,nC PROJSi nCk=1PROJSF(FRk)!(By Lemma 5) CRi,1 CRi,nC PROJSi(FRF)(By Theorem 1) CRi,1 CRi,nC PRi(By Definition 3).By (1) and (2):PRi= CRi,1 CRi,n.4. Generation of candidate designsGiven a preliminary design of a cooling system that specifiesa sequence of channels and their ideal geometry, the first stepis to construct an FRifor each channel. Then, the PRiforeach channel is obtained by applying the composition operationas specified in Theorem 2. One way to generate a candidatedesign for a cooling system is to select a set of coordinatesfrom the set of PRias follows. To simplify the explanation,assume that each channel Cihas degrees of freedom XiandXi+1, and Xi+1is shared by the adjacent channel Ci+1. Togenerate a design, a point (x1,x2) in PR1is chosen. Then, anx3is chosen such that (x2,x3) is within PR2. This selectionC.G. Li, C.L. Li / Computer-Aided Design 40 (2008) 334349345process 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 nFreal values g1g2.gnF, for which each gihas a realvalue between 0 and 1, and nFis 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 giused as a percentage value to select acoordinate. For example, if the valid values for a coordinate xiin PRiexist in the intervals x1i,x2i and x3i,x4i, where x1ix2i x3i,x4i, then the selected value for xiwill be x1i+gi(x2ix1i)+(x4ix3i) if gi (x2ix1i)/(x2ix1i+x4ix3i) (i.e., xilies in the first interval). Otherwise, xiwill be set to x3i+ (gi1)(x2i x1i) + gi(x4i x3i) (i.e., xilies 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 PRifor each channel is constructed. The constructionof the PRiis 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 C5and a mould component O1. Using theproposed method to automate the layout design, the FRiandthen the PRiof each channel are constructed. As an example,Fig. 9(g) and (h) show FR4and PR4for channel C4. It isnoted that PR4is obtained from FR4by composition with otherFRi, and thus PR4is a subset of FR4, as is evident from thefigures. After all of the PRiare 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 inFig. 9(c), the cooling system consists of 15 degrees of freedomXiand their values are listed in Table 1. In the table, therow labeled “Preliminary design” shows the xivalues for thepreliminary design. The next row lists the values for Design 1,which is the best design generated by the GA process after 1000generations. As highlighted in the table, Design 1 is obtainedby reducing x6by 1.21 mm. Fig. 9(d), which shows Design 1,this adjustment corresponds to the lowering of C5along the Zdirection to clear the interference between C5and O1. Due tothe connectivity among the channels, the same adjustment alsoapplies to channels C4and C6to C13. Table 1 also shows thatall of the other xiin Design 1 are maintained to within 0.2 mmfrom values specified in the preliminary design.To demonstrate further the capability of the C-space method,the mould component O2is 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 C13sothat the feasible adjustment along X6is further limited. Thiseffect is demonstrated in the updated PR4shown in Fig. 9(i) inwhich only the upper portion of the PR4shown in Fig. 9(h)is retained. The GA process is invoked again with the newPRiof 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 xiobtainedfrom the GA process are shown in the last row of Table 1. Ashighlighted in the table, x6is adjusted for 5 mm to clear theinterference with O2. This corresponds to moving channels C4to C13along the Z direction. Now, the interference between C5and O1can no longer be cleared by adjusting x6. Instead, x4and x5are adjusted, which corresponds to moving C52.94 mmalong the Y direction, and moving C46.22 mm along theX direction, as shown in Fig. 9(f). Channels C2and C3arealso adjustedaccordingly to maintain theconnectivity.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., C4and 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 about46C with a cooling time of 20 s for both designs, and thattheir maximum temperature differences are less than 8C,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 5.5C. 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 maximumfitness value for Design 2 is slightly lower than that of Design 1.346C.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) O2moved 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 offreedomX1X2X3X4X5X6X7X8X9X10X11X12X13X14X15Preliminarydesign32.516955212.5139.52932.5110212.580.532.5212.55121.532.5Design 132.41168.9154.95212.34139.4927.7932.36109.97212.5180.6032.30212.5251.1021.5632.52Design 232.51169.0455.13206.28136.5634.0032.51109.84212.4980.4532.50212.5250.9321.6632.64C.G. Li, C.L. Li / Computer-Aided Design 40 (2008) 334349347(g) FR4.(h) PR4.(i)UpdatedPR4.(j) Fitness value against the generation for Design 1.(k) Fitness value against the generation for Design 2.Fig. 9. (continued)7. Discussion and conclusionIn the implementation of the C-space method, a cellenumeration scheme is used to simplify the implementation.In the current implementation, the resolution in the C-spacerepresentation is 0.15 mm in each dimension, which shouldbe adequate for the cooling system design because for a veryfine adjustment, say 0.01 mm, the change in the coolingperformance may not be noticeable. However, the methodologyand theorems developed in this research are not limited to aparticular scheme for representation. In fact, for the methodbased on Theorem 2, all C-space computation and storage aredone in three dimension, and thus standard geometric modelingtechniques can be used.A major contribution of this research is the developmentof a specific C-space method that supports the layout designprocess. Using the C-space method, all of the feasible layoutdesigns are properly represented. We have demonstrated thatthe C-space method can be used to support design generationin which not only designs that are optimal in terms ofcooling performance are generated, but also, the designs aremanufacturable. This new method overcomes the limitation ofrelying on specific heuristics to generate the layout design,as in our previous method 13,14. This C-space method canalso be used as a stand-alone system to support interactivelayout design. It allows a designer to explore design alternativesinteractively without having to check for interference betweenthe cooling system and the other mould insert components.The focus of this research is on the geometric aspect of thecooling system design. It is understood that other parameters,such as the coolant flow rate, cooling time, packing time,ejection time, etc. need to be considered as well when designinga cooling system. One possible approach to take all theseparameters into account is to integrate the C-space methodwith a more sophisticated GA such as the one reported in 8.Further investigation on this approach is needed and otherfurther research directions include the extension of the C-spacemethod to deal with topology changes in the cooling systemand specific design constraints, such as various geometry andtopology constraints specified between selected channels in thepreliminary design.AcknowledgementThe work described in this paper was fully supported bya Strategic Research Grant from the City University of HongKong (Project No. 7001775).AppendixLemma 1. Given a region Rnand a point pnin space Sn. Ifpn Rn, thenPROJSm(pn) PROJSm(Rn).348C.G. Li, C.L. Li / Computer-Aided Design 40 (2008) 334349Fig. 10. A comparison of the two layout designs using CAE mould cooling analysis.Lemma 2. Given two regions Rnand R0nin space Sn. If R0nRn, thenPROJSm(R0n) PROJSm(Rn).Lemma 3. Given L regions Rkn,k 1, L in space Sn, thenPROJSm Lk=1Rkn!Lk=1PROJSm(Rkn).Lemma 4. Given any two regions Rmand R0min Sm. IfXmXn, then their projections to Snsatisfy the following:PROJSn(Rm) PROJSn(R0m) = PROJSn(Rm R0m).Lemma 5. Given two spaces Snand Smhaving axes setsXmXn. A region Rmin Smsatisfies the following:PROJSm(PROJSn(Rm) = Rm.Lemma 6. Given three spaces Sn, Smand Slhaving axes setsXmXnandXlXn. A region Rlin space Slsatisfies thefollowing:PROJSn(Rl) PROJSn(PROJSm(Rl).Lemma 7. PRi FRi.Lemma 8. Given two spaces Snand Smhaving axes setsXmXn, a point pn
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