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COMPUTERAIDEDDESIGN402008SPACECLPRODUCTIMOULDEDPARTDESPITETHEVARIOUSRESEARCHEFFORTSTHATHAVEBEENDIRECTEDTOWARDSTHEANALYSIS,OPTIMIZATION,ANDFABRICATIONOFCOOLINGSYSTEMS,SUPPORTFORTHELAYOUTDESIGNOFTHECOOLINGSYSTEMHASNOTBEENWELLDEVELOPEDINTHELAYOUTDESIGNPHASE,AMAJORCONCERNISTHEFEASIBILITYOFBUILDINGTHECOOLINGSYSTEMINSIDETHEMOULDINSERTWITHOUTINTERFERINGWITHTHEOTHERMOULDCOMPONENTSTHISPAPERREPORTSACONFIGURATIONSPACECSPACEMETHODTOADDRESSTHISIMPORTANTISSUEWHILEAHIGHDIMENSIONALCSPACEISGENERALLYREQUIREDTODEALWITHACOMPLEXSYSTEMSUCHASACOOLINGSYSTEM,THESPECIALCHARACTERISTICSOFCOOLINGSYSTEMDESIGNAREEXPLOITEDINTHEPRESENTSTUDY,ANDSPECIALTECHNIQUESTHATALLOWCSPACECOMPUTATIONANDSTORAGEINTHREEDIMENSIONALORLOWERDIMENSIONAREDEVELOPEDTHISNEWMETHODISANIMPROVEMENTONTHEHEURISTICMETHODDEVELOPEDPREVIOUSLYBYTHEAUTHORS,BECAUSETHECSPACEREPRESENTATIONENABLESANAUTOMATICLAYOUTDESIGNSYSTEMTOCONDUCTAMORESYSTEMATICSEARCHAMONGALLOFTHEFEASIBLEDESIGNSASIMPLEGENETICALGORITHMISIMPLEMENTEDANDINTEGRATEDWITHTHECSPACEREPRESENTATIONTOAUTOMATICALLYGENERATECANDIDATELAYOUTDESIGNSDESIGNEXAMPLESGENERATEDBYTHEGENETICALGORITHMAREGIVENTODEMONSTRATETHEFEASIBILITYOFTHEMETHODC2007ELSEVIERLTDALLRIGHTSRESERVEDKEYWORDSCOOLINGSYSTEMDESIGNPLASTICINJECTIONMOULDCONFIGURATIONSPACEMETHOD1INTRODUCTIONTHECOOLINGSYSTEMOFANINJECTIONMOULDISVERYIMPORTANTTOTHEPRODUCTIVITYOFTHEINJECTIONMOULDINGPROCESSANDTHEQUALITYOFTHEMOULDEDPARTEXTENSIVERESEARCHHASBEENCONDUCTEDINTOTHEANALYSISOFCOOLINGSYSTEMS1,2,ANDCOMMERCIALCAESYSTEMSSUCHASMOLDFLOW3ANDMOLDEX3D4AREWIDELYUSEDINTHEINDUSTRYRESEARCHINTOTECHNIQUESTOOPTIMIZEAGIVENCOOLINGSYSTEMHASALSOBEENREPORTED58RECENTLY,METHODSTOBUILDBETTERCOOLINGSYSTEMSBYUSINGNEWFORMSOFFABRICATIONTECHNOLOGYHAVEBEENREPORTEDXUETAL9REPORTEDTHEDESIGNANDFABRICATIONOFCONFORMALCOOLINGCHANNELSTHATMAINTAINACONSTANTDISTANCEFROMTHEMOULDIMPRESSIONSUNETAL10,11USEDCNCDESPITETHEVARIOUSRESEARCHEFFORTSTHATHAVEFOCUSEDMAINLYONTHEPRELIMINARYDESIGNPHASEOFTHECOOLINGSYSTEMDESIGNPROCESSINWHICHTHEMAJORCONCERNISTHEPERFORMANCEOFTHECOOLINGFUNCTIONOFTHESYSTEM,SUPPORTFORTHELAYOUTDESIGNPHASEINWHICHTHEFEASIBILITYANDMANUFACTURABILITYOFTHECOOLINGSYSTEMDESIGNAREADDRESSEDHASNOTBEENWELLDEVELOPEDAMAJORCONCERNINTHELAYOUTDESIGNPHASEISTHEFEASIBILITYOFBUILDINGTHECOOLINGSYSTEMINSIDETHEMOULDINSERTWITHOUTINTERFERINGWITHTHEOTHERMOULDCOMPONENTSCONSIDERTHEEXAMPLESHOWNINFIG1ITCANBESEENTHATMANYDIFFERENTCOMPONENTSOFTHEVARIOUSSUBSYSTEMSOFTHEINJECTIONMOULD,SUCHASEJECTORPINS,SLIDES,SUBINSERTS,ANDSOFORTH,HAVETOBEPACKEDINTOTHEMOULDINSERTFINDINGTHEBESTLOCATIONFOREACHCHANNELOFTHECOOLINGCIRCUITTOOPTIMIZEPLASTICINJECTIONMOULDCOOLINGCONFIGURATIONCGLI,DEPARTMENTOFMANUFACTURINGENGINEERINGANDENGINEERINGRECEIVED3MAY2007ACCEPTEDABSTRACTTHECOOLINGSYSTEMOFANINJECTIONMOULDISVERYIMPORTANTTOTHEMILLINGTOPRODUCEUSHAPEDMILLEDGROOVESFORCOOLINGCHANNELSANDYU12PROPOSEDASCAFFOLDINGSTRUCTUREFORTHEDESIGNOFCONFORMALCOOLINGCORRESPONDINGAUTHOREMAILADDRESSMECLLICITYUEDUHKCLLI00104485/SEEFRONTMATTERC2007ELSEVIERLTDALLRIGHTSRESERVEDDOI101016/JCAD200711010334349WWWELSEVIERCOM/LOCATE/CADSYSTEMDESIGNBYTHEMETHODLIMANAGEMENT,CITYUNIVERSITYOFHONGKONG,HONGKONG18NOVEMBER2007VITYOFTHEINJECTIONMOULDINGPROCESSANDTHEQUALITYOFTHETHECOOLINGPERFORMANCEOFTHECOOLINGSYSTEMANDTOAVOIDINTERFERENCEWITHTHEOTHERCOMPONENTSISNOTASIMPLETASKANOTHERISSUETHATFURTHERCOMPLICATESTHELAYOUTDESIGNPROBLEMISTHATTHEINDIVIDUALCOOLINGCHANNELSNEEDTOBECONNECTEDTOFORMAPATHTHATCONNECTSBETWEENTHEINLETANDTHEOUTLETTHEREFORE,CHANGINGTHELOCATIONOFACHANNELMAY335FIG1THECOOLINGSYSTEMCOMPONENTSREQUIRECHANGINGTHEEXAMPLESHOWNINTOOPTIMIZETHECOOLINGINFIG2AASSUMEOTHERMOULDCOMPONENTSMOULDCOMPONENTASC1CANNOTBEMOINTERFERENCEWITHOTHERC2ISMOVEDANDCCONNECTIVITY,ASSHOC3ISFOUNDTOINTERFEREMOULDCOMPONENTS,ISVERYTEDIOUSTHATSUPPORTSTHETHISNEWTECHNIQUE,USEDTOPROVIDEALAYOUTDESIGNSTHEANEFFICIENTMETHODTHELAYOUTDESIGNTOGENERATELAYOUTSYSTEMDEVELOPEDWCSPACEMETHODTOCONDUCTAMORELAYOUTDESIGNSISTHESPACETHATSYSTEMISTREATEDTHECONFIGURATIONFREEREGIONPOINTSOFTHETHECOMPONENTSCORRESPONDTOOFTHESYSTEMINITIALLYFORMALIZEDPLANNINGPROBLEMSSHORTENEDANDFURTHERMODIFICATIONISNEEDED,WHICHRESULTSINTHEFINALDESIGNSHOWNINFIG2CGIVENTHATATYPICALINJECTIONMOULDMAYHAVEMORETHANTENCOOLINGCHANNELS,WITHEACHCHANNELAINTERFERENCEOCCURSBETWEENCOOLINGCHANNELC1ANDMOULDCOMPONENTO1ATTHEIDEALLOCATIONOFC1CC3ISMOVEDANDC2ISDESIGNFIG2ANEXAMPLESHOWINGTHETEDIOUSNESSANDASURVEYINTHISAREAOFRESEARCHHASBEENREPORTEDBYWISEANDBOWYER16THECSPACEMETHODHASALSOBEENUSEDTOSOLVEPROBLEMSINQUALITATIVEREASONINGEG,17,18BCHANNELC1ISSHORTENED,C2ISMOVED,ANDC3ISELONGATEDTOGIVETHEFINALCGLI,CLLI/COMPUTERAIDEDDESIGN402008334349INSIDEAMOULDINSERTPACKEDWITHMANYOTHERMOULDOTHERCHANNELSASWELLCONSIDERTHEFIG2THEIDEALLOCATIONOFEACHCHANNELPERFORMANCEOFTHESYSTEMISSHOWNTHATWHENTHECOOLINGSYSTEMANDTHEAREBUILTINTOTHEMOULDINSERT,AO1ISFOUNDTOINTERFEREWITHCHANNELC1VEDTOANEARBYLOCATIONDUETOTHEPOSSIBLECOMPONENTS,ITISSHORTENEDASARESULT,3ISELONGATEDACCORDINGLYTOMAINTAINTHEWNINFIG2BOWINGTOITSNEWLENGTH,WITHANOTHERMOULDCOMPONENT,O2,POTENTIALLYINTERFERINGWITHAFEWOTHERFINDINGANOPTIMALLAYOUTDESIGNMANUALLYTHISPAPERREPORTSANEWTECHNIQUEAUTOMATIONOFTHELAYOUTDESIGNPROCESSINACONFIGURATIONSPACECSPACEMETHODISCONCISEREPRESENTATIONOFALLOFTHEFEASIBLECSPACEREPRESENTATIONISCONSTRUCTEDBYTHATEXPLOITSTHESPECIALCHARACTERISTICSOFPROBLEMINSTEADOFUSINGHEURISTICRULESDESIGNS,ASINTHEAUTOMATICLAYOUTDESIGNPREVIOUSLYBYTHEAUTHORS13,14,THISNEENABLESANAUTOMATICLAYOUTDESIGNSYSTEMSYSTEMATICSEARCHAMONGALLOFTHEFEASIBLE2THECONFIGURATIONSPACEMETHODINGENERAL,THECSPACEOFASYSTEMRESULTSWHENEACHDEGREEOFFREEDOMOFTHATASADIMENSIONOFTHESPACEREGIONSINSPACEARELABELEDASBLOCKEDREGIONORINTHEFREEREGIONSCORRESPONDTOVALIDCONFIGURATIONSSYSTEMWHERETHEREISNOINTERFERENCEBETWEENOFTHESYSTEMPOINTSINTHEBLOCKEDREGIONSINVALIDCONFIGURATIONSWHERETHECOMPONENTSINTERFEREWITHONEANOTHERCSPACEWASBYLOZANOPEREZ15TOSOLVEROBOTPATHOFTHELAYOUTDESIGNPROCESS336ANDEG,AUTOMATIC2321THEYC3SEEACOOLINGSYSTEMFIG3GIVESANEXAMPLETHEPRELIMINARYDESIGNOFTHISCOOLINGSYSTEMCONSISTSOFFOURCOOLINGCHANNELSTOGENERATEALAYOUTDESIGNFROMTHEPRELIMINARYDESIGN,THECENTERSANDLENGTHSOFTHECHANNELSAREADJUSTEDASSHOWNINFIG3,THECENTEROFCHANNELC1CANBEMOVEDALONGTHEX1ANDX2DIRECTIONS,ANDITSLENGTHCANBEADJUSTEDALONGTHEX3DIRECTIONSIMILARLY,THELENGTHOFC2CANBEADJUSTEDALONGTHEX4DIRECTION,WHILEITSCENTERADJUSTMENTISDESCRIBEDBYX1ANDX3ANDTHUSMUSTBETHESAMEASTHEADJUSTMENTOFC1TOMAINTAINTHECONNECTIVITYBYAPPLYINGSIMILARARGUMENTSTOTHEOTHERCHANNELS,ITCANBESEENTHATTHECOOLINGSYSTEMHAS5ACHANNELCIANDTHREEMOULDCOMPONENTSINSIDETHEMOULDINSERTBOFFSETSOFTHEMOULDCIREPRESENTEDBYLINEDTHEINITIALFREEREGIONOFCIFIG4THEMAJORSTEPSINTHECONSTRUCTIONCONSIDEREDTOACCOUNTFORTHEDIAMETERD,OIISFIRSTOFFSETBYD/2MTOGIVEOPRIMEI,WHEREMISTHEMINIMUMALLOWABLEDISTANCEBETWEENTHECHANNELWALLANDTHEFACEOFACOMPONENTTHISGROWINGOFOIINEFFECTREDUCESCHANNELCITOALINELICONSIDERTHEEXAMPLEILLUSTRATEDINFIG4FIG4ASHOWSACHANNELCIANDTHREEMOULDCOMPONENTS,O1,O2,ANDO3,THATMAYINTERFEREWITHCIFIG4BSHOWSTHEOFFSETSOPRIME1,OPRIME2,ANDOPRIME3OFTHEMOULDCOMPONENTS,ANDTHEREDUCTIONOFCITOALINESEGMENTLITHATISCOINCIDENTWITHTHEAXISOFCIIFTHEREISNOINTERSECTIONBETWEENLIANDTHEOFFSETSOFTHEMOULDCOMPONENTS,THENTHEORIGINALCHANNELCIWILLNOTINTERSECTWITHCOMPONENTSANDGMENTLICSWEEPINGTHEOFFSETSOFTHEMOULDCOMPONENTSANDCIREPRESENTEDBYPOINTPISUBTRACTINGOPRIMEPRIMEIFROMBPRIMEIFTHEFREEREGIONFRIOFCICGLI,CLLI/COMPUTERAIDEDDESIGN402008334349FIG3ANEXAMPLESHOWINGTHEDEGREESOFFREEDOMOFACOOLINGSYSTEMTHEANALYSISANDDESIGNAUTOMATIONOFKINEMATICDEVICES1921THEAUTHORINVESTIGATEDACSPACEMETHODINTHEDESIGNSYNTHESISOFMULTIPLESTATEMECHANISMS22,INPREVIOUSRESEARCHCSPACEOFACOOLINGSYSTEMAHIGHDIMENSIONALCSPACECANBEUSEDTOREPRESENTALLOFFEASIBLELAYOUTDESIGNSOFAGIVENPRELIMINARYDESIGNOFDEGREESOFFREEDOM,ANDTHEYAREDENOTEDASXI,I1,2,5INPRINCIPLE,THECSPACEISAFIVEDIMENSIONALSPACEANDANPOINTINTHEFREEREGIONOFTHISSPACEGIVESASETOFCOORDINATEVALUESONTHEXIAXESTHATCANBEUSEDTODEFINETHEGEOMETRYOFTHECHANNELSWITHOUTCAUSINGINTERFERENCEWITHTHEOTHERMOULDCOMPONENTSTODETERMINETHEFREEREGIONINAHIGHDIMENSIONALCSPACEOFACOOLINGSYSTEM,THEFIRSTSTEPISTOCONSTRUCTTHEFREEREGIONSINTHECSPACESOFTHEINDIVIDUALCHANNELS22CSPACECONSTRUCTIONOFINDIVIDUALCOOLINGCHANNELSWHENANINDIVIDUALCHANNELCIISCONSIDEREDALONE,ITHASTHREEDEGREESOFFREEDOM,SAYX1ANDX2FORITSCENTERLOCATIONANDX3FORITSLENGTHASTHEIDEALCENTERLOCATIONANDLENGTHHAVEALREADYBEENSPECIFIEDINTHEPRELIMINARYDESIGN,ITISREASONABLETOASSUMEAFIXEDMAXIMUMALLOWABLEVARIATIONFORX1,X2,ANDX3THEINITIALFREEREGIONINTHECSPACEOFCHANNELCIISTHUSATHREEDIMENSIONALCUBEBIWITHTHEDIMENSIONSCCCTOAVOIDANYPOSSIBLEINTERFERENCEWITHAMOULDCOMPONENTOIWHENCHANNELCIISBUILTINTOTHEMOULDINSERTBYDRILLING,ADRILLINGDIAMETERDANDDRILLINGDEPTHALONGXHAVETOBEOFTHEFREEREGIONFRIOFACHANNELCICGLI,CLLI/COMPUTERAIDEDTHEMOULDCOMPONENTSTHISGROWINGOROFFSETOFANOBSTACLEISASTANDARDTECHNIQUEINTHECSPACEMETHOD15ACHANNELISFORMEDBYDRILLINGFROMAFACEOFTHEMOULDINSERT,ANDANYOBSTACLEOIWITHINTHEDRILLINGDEPTHWILLAFFECTTHECONSTRUCTIONOFTHECHANNELTOACCOUNTFORTHEDRILLINGDEPTH,THEOFFSETOPRIMEIOFOIISSWEPTALONGTHEDRILLINGDIRECTIONUNTILTHEOPPOSITEFACEOFTHEMOULDINSERTISREACHEDTOGENERATEOPRIMEPRIMEITHISSWEEPINGOFOPRIMEIINEFFECTREDUCESLINELITOAPOINTPILOCATEDATTHEENDOFLIASSHOWNINFIG4C,IFTHEPOINTPIISOUTSIDEOPRIMEPRIMEI,THEDRILLINGALONGLITOPRODUCECIISFEASIBLETHEFREEREGIONFRIOFCHANNELCIISOBTAINEDASFOLLOWSFIRST,THEINITIALFREEREGIONBIISCONSTRUCTEDWITHITSCENTERATPIASSHOWNINFIG4DBITHENINTERSECTSWITHTHEMOULDINSERTTOOBTAINBPRIMEIBPRIMEIREPRESENTSALLOFTHEPOSSIBLEVARIATIONSOFCIWHENONLYTHEGEOMETRICSHAPEOFTHEMOULDINSERTISCONSIDEREDTHEN,FRIISOBTAINEDBYSUBTRACTINGFROMBPRIMEITHEOPRIMEPRIMEIOFALLOFTHEOBSTACLESFIG4EANDFSHOWTHESUBTRACTIONANDTHERESULTINGFRIOFTHEEXAMPLE23BASICAPPROACHTOTHECONSTRUCTIONOFTHECSPACEOFCOOLINGSYSTEMTODETERMINETHEFREEREGIONFRFINTHECSPACEOFACOOLINGSYSTEM,THEFREEREGIONSOFEACHCOOLINGCHANNELHAVETOBE“INTERSECTED”INAPROPERMANNERSOTHATTHEEFFECTOFTHEOBSTACLESTOALLOFTHECHANNELSAREPROPERLYREPRESENTEDBYFRFHOWEVER,THESTANDARDBOOLEANINTERSECTIONBETWEENTHEFREEREGIONSOFTWODIFFERENTCHANNELSCANNOTBEPERFORMEDBECAUSETHEIRCSPACESAREINGENERALSPANNEDBYDIFFERENTSETSOFAXESREFERRINGTOTHEEXAMPLEINFIG3,THECSPACESOFC1ANDC2ARESPANNEDBYX1,X2,X3ANDX1,X3,X4,RESPECTIVELYTOFACILITATETHEINTERSECTIONBETWEENFREEREGIONSINDIFFERENTCSPACES,THEPROJECTIONOFAREGIONFROMTHECSPACEOFONECHANNELTOTHATOFANOTHERCHANNELISNEEDEDTHEFOLLOWINGNOTATIONSAREFIRSTINTRODUCEDANDWILLBEUSEDINTHESUBSEQUENTDISCUSSIONSONPROJECTIONSANDTHERESTOFTHEPAPERNOTATIONSUSEDINDESCRIBINGHIGHDIMENSIONALSPACESSNDENOTESANNDIMENSIONALSPACESPANNEDBYTHESETOFAXESXNX1,X2,XNSMDENOTESANMDIMENSIONALSPACESPANNEDBYTHESETOFAXESXMXPRIME1,XPRIME2,XPRIMEMPNDENOTESAPOINTINSNANDPNX1,X2,XN,WHEREXIDENOTESACOORDINATEONTHEITHAXISXIRNDENOTESAREGIONINSNRNSNRNISASETOFPOINTSINSNPROJSMPNDENOTESTHEPROJECTIONOFAPOINTPNFROMSNTOSMPROJSMRNDENOTESTHEPROJECTIONOFAREGIONRNFROMSNTOSMNOTATIONSUSEDINDESCRIBINGACOOLINGSYSTEMNCDENOTESTHENUMBEROFCHANNELSINTHECOOLINGSYSTEMNFDENOTESTHETOTALDEGREESOFFREEDOMOFTHECOOLINGSYSTEMCIDENOTESTHEITHCHANNELOFTHECOOLINGSYSTEMSIDENOTESTHECSPACEOFCIDESIGN402008334349337FRIDENOTESTHEFREEREGIONINSITHATIS,ITISTHEFREEREGIONOFANINDIVIDUALCHANNELCISFDENOTESTHECSPACEOFTHECOOLINGSYSTEMFRFDENOTESTHEFREEREGIONINSFTHATIS,ITISTHEFREEREGIONOFTHECOOLINGSYSTEMCONSIDERTHEPROJECTIONOFAPOINTPNINSNTOAPOINTPMINSMFIG5AILLUSTRATESEXAMPLESOFPROJECTIONUSINGSPACESOFONEDIMENSIONTOTHREEDIMENSIONSPROJECTIONSAREILLUSTRATEDFORTHREECASESIXMXNIIXMXNANDIIIXMNEGATIONSLASHXN,XNNEGATIONSLASHXM,ANDXNXMNEGATIONSLASHFORI,EACHCOORDINATEOFPMISEQUALTOACORRESPONDINGCOORDINATEOFPNTHATISONTHESAMEAXISFORIIANDIII,THEPROJECTIONOFPNISAREGIONRMFOREACHPOINTPMINRM,ACOORDINATEOFPMISEQUALTOTHATOFPNIFTHATCOORDINATEISONACOMMONAXISOFSNANDSMFORTHEOTHERCOORDINATESOFPM,ANYVALUECANBEASSIGNEDTHEREASONFORTHISSPECIFICDEFINITIONOFTHEPROJECTIONS,INPARTICULAR,FORCASESIIANDIII,ISASFOLLOWSCONSIDERTWOADJACENTCHANNELSCNANDCMASTHEYAREADJACENT,THEYMUSTBECONNECTEDANDTHUSTHEIRCSPACESSNANDSMSHARESOMECOMMONAXESASSUMETHATACONFIGURATIONTHATCORRESPONDSTOAPOINTPNINSNHASBEENSELECTEDFORCNTOMAINTAINTHECONNECTIVITY,THECONFIGURATIONFORCMMUSTBESELECTEDSUCHTHATTHECORRESPONDINGPOINTPMINSMSHARESTHESAMECOORDINATESWITHPNONTHEIRCOMMONAXESTHISIMPLIESTHATPMCANBEANYPOINTWITHINTHEPROJECTIONOFPNONSM,WHERETHEMETHODOFPROJECTIONISDEFINEDABOVETHEPROJECTIONSOFAREGIONRNINSNTOSMARESIMPLYTHEPROJECTIONSOFEVERYPOINTINRNTOSMFIG5BILLUSTRATESTHEREGIONPROJECTIONSTHEFORMALDEFINITIONOFPROJECTIONISGIVENBELOWDEFINITION1PROJECTION11IFXMXN,PROJSMPNISAPOINTPMXPRIME1,XPRIME2,XPRIMEM,WHEREFORXPRIMEIXJ,XPRIMEIXJFORALLI1,MTOSIMPLIFYTHENOTATIONSINSUBSEQUENTDISCUSSION,THISPROJECTIONISREGARDEDASAREGIONTHATCONSISTSOFTHESINGLEPOINTPMTHATIS,PROJSMPNPM12IFXMXN,PROJSMPNISAREGIONRMPM|PROJSNPMPN13IFXMNEGATIONSLASHXN,XNNEGATIONSLASHXM,ANDXNXMNEGATIONSLASH,PROJSMPNISAREGIONRMPM|PROJSIPMPROJSIPN,WHERESIISTHESPACESPANNEDBYXNXMIFXNXM,PROJSMPNISDEFINEDASSM14PROJSMRNISDEFINEDASTHEREGIONRMPM|PMPROJSMPN,PNRNASDISCUSSEDINSECTION21,ANYPOINTPFINFRFGIVESAVALUEFOREACHDEGREEOFFREEDOMOFTHECOOLINGSYSTEMSOTHATTHEGEOMETRYOFTHECHANNELSISFREEFROMINTERFERENCEWITHTHEOTHERMOULDCOMPONENTSINOTHERWORDS,THEPROJECTIONOFPFTOEACHSIISINTHEFREEREGIONFRIOFEACHCITHUS,FRFISDEFINEDASFOLLOWSDEFINITION2FREEREGIONINTHECSPACEOFACOOLINGSYSTEMFRFPF|PROJSIPFFRI,I1,NCAIDEDNOTETHATACCORDINGTOTOSIALWAYSCONTAINSONLYTHATSPANSIISALWAYSASUBSETTHECONSTRUCTIONOFTHEALREADYBEENEXPLAINEDINTHEFOLLOWINGTHEOREMISUSEFULTHEOREM1FRFNCINTERSECTIONDISPLAYI1PROJSFFRIINTUITIVELY,THISTHEOREMSAYSFIRSTPROJECTEDTOTHECSPACECANTHENBEOBTAINEDBYPERFORMINGAMONGTHEPROJECTIONSTHEUSEDINTHEPROOFAREGIVENOFTHECSPACEFANDTOFACILITATETHEBETWEENTHEREGIONSCANUSEAKINDOFCELLUSEDIN21,24THEREGIONRFINEACHBOXISDEFINEDBYSFTHEINTERSECTIONOFOFTHETWOSETSOFHIGHDIMENSIONALBOXESINTERVALSOFEACHOFTHEBYMTHREEOJSFFRICANTHENBEBOXESTHECONSTRUCTIONFIG5THEPROJECTIONSOFPOINTSANDREGIONSINSNTOSMDEFINITION11,THEPROJECTIONOFPFASINGLEPOINTBECAUSETHESETOFAXESOFTHEAXESTHATSPANSNFREEREGIONFRIOFEACHCIHASSECTION22TOFINDFRFFROMFRI,THATTOFINDFRF,ALLOFTHEFRIAREOFTHECOOLINGSYSTEMSFFRFTHEBOOLEANINTERSECTIONSPROOFOFTHEOREM1ANDTHELEMMAS24REPRESENTATIONANDCOMPUTATIONTOREPRESENTTHEFREEREGIONFRCOMPUTATIONOFTHEBOOLEANINTERSECTIONSINAHIGHDIMENSIONALSPACE,WEENUMERATIONMETHODSIMILARTOTHEONEBASICIDEAISTOAPPROXIMATEAHIGHDIMENSIONALSFBYASETOFHIGHDIMENSIONALBOXESSPECIFYINGANINTERVALONEACHAXISOFTWOREGIONSISACHIEVEDBYTHEINTERSECTIONBOXESTHEINTERSECTIONBETWEENTWOISSIMPLYTHEINTERSECTIONBETWEENTHEBOXESINEACHAXISASSUMINGTHATEACHFRIISAPPROXIMATEDDIMENSIONALBOXES,THEPROJECTIONPRAPPROXIMATEDBYMNFDIMENSIONAL338CGLI,CLLI/COMPUTERINTHEAPPENDIXDESIGN402008334349OFFRFTHATUSESTHEOREM1THENREQUIRESMNCINTERSECTIONSBETWEENNFDIMENSIONALMAXIMUMOFMNCNFOFBOXESUSEDTOREPRESENTINTERSECTIONSANDFRISANTICIPATEDTHATTHEARESTILLMAJORPROBLEMSIMPROVEDMETHODIS3ANEFFICIENTTECHNIQUETOAVOIDTHEHIGHFORTHEREPRESENTATIONINSTEAD,WEPROCESSTOEXAMPLESHOWNINISASSUMEDINTHISALONGTHEZDIRECTIONHASFOURDEGREESEACHCHANNELCIARESHOWNINFIG6BCHANNELC1FIRST,AAASIMPLECOOLINGSYSTEMWITHFOURCHANNELSANDFOURDEGREESOFFREEDOMBTHEFREEREGIONFRIOFEACHCHANNELINITSCONFIGURATIONSPACESIFIG6ASIMPLIFIEDEXAMPLEOFACOOLINGSYSTEMDESIGNBOXES,ANDFRFISREPRESENTEDBYADIMENSIONALBOXESALTHOUGHTHENUMBERTHEINTERMEDIATERESULTSOFTHEFCANBEREDUCEDBYSPECIALTECHNIQUES,ITMEMORYANDCOMPUTATIONALREQUIREMENTSOFTHISMETHODINTHENEXTSECTION,ANDEVELOPEDFORCSPACECONSTRUCTIONTOREPRESENTANDNOTTOCOMPUTEFRFEXPLICITLYFOCUSONATECHNIQUETHATENABLESTHECOMPUTATIONALWORKONTHECSPACESOFEACHINDIVIDUALCHANNELFIRST,CONSIDERTHESIMPLIFIEDDESIGNFIG6FORTHEPURPOSEOFILLUSTRATION,ITEXAMPLETHATTHEREISNOVARIATIONINFRIOFTHEMOULDINSERTANDTHUSTHECOOLINGSYSTEMOFFREEDOMASSHOWNINFIG6ATHESIOFTWODIMENSIONALANDTHEASSUMEDFRIARECONSIDERASIMPLEMETHODFORDESIGNINGCGLI,CLLI/COMPUTERAIDEDMEMORYANDCOMPUTATIONALREQUIREMENTSANDCONSTRUCTIONOFFRF,WECHOOSENOTDESIGN402008334349339POINTP1CANBESELECTEDFROMWITHINFR1SOTHATC1ISFREEFROMINTERFERENCEWITHANYOBSTACLEHOWEVER,S1ISSPANNEDAIDEDCONTINUEDEVENTHOUGHTHEIRCSPACESOFC1,IE,THEYAREASWELL,BECAUSETHESYSTEMARECONNECTEDHAVEANEFFECTINTHECOOLINGSYSTEMTODEVELOPADESIGNOFEACHINDIVIDUALCHANNELS,SELECTIONOFAPOINTALWAYSEXISTACORRESPONDINGTHATALLOFTHECHANNELSSYSTEMTOADDRESSTHISSIISNEEDEDDEFINITION3PRIISPRIPROJSIFRFOBVIOUSLY,FORANALWAYSACORRESPONDIFR2AGAIN,ASP2X3MUSTHAVEAVALUEFR3ALSO,ASMUSTALSOBEINSIDEP1,P2,P3,ANDP4C1DETERMINETHEVALIDDESIGNSFORC1,THETHEEFFECTOFFR4VALIDREGIONINFR3,FINALLYINS1THEALLOFTHEEFFECTSOFISFORMALLYCHANNELSCIANDOFTHEIRFREEREGIONSDONOTHAVEANAXISCOMMONTOTHATNOTADJACENTTOC1,HAVETOBECONSIDEREDCOOLINGCHANNELSTHATMAKEUPTHECOOLINGACHOICEINONEDEGREEOFFREEDOMWILLCHOICEOFANOTHERDEGREEOFFREEDOMOFTHEPROCESSTHATWORKSONTHECSPACESAMAJORCONCERNISTHATAFTERTHEINTHECSPACEOFONECHANNEL,THEREMUSTPOINTINALLOFTHEOTHERSISUCHCANBECONNECTEDTOFORMAVALIDCOOLINGCONCERN,THEPROJECTIONOFFRFTOEACHDEFINEDASTHEPROJECTIONOFFRFTOSIWHICHWECANFINDAP2X2,X3WITHINHASACOORDINATEX3INX3,THECOORDINATEFORWHICHWECANFINDAP3X3,X4WITHINFR4HASAXESX3ANDX4,P4X3,X4FR4FIG7SHOWSTHESEQUENCEOFPOINTSTHATCONSTITUTESAVALIDDESIGNFORCHANNELTHEABOVEILLUSTRATIONREVEALSTHATTOREGIONINS1THATREPRESENTSALLOFTHEVALIDFREEREGIONFR4SHOULDBECONSIDEREDFIRSTSHOULDBE“REFLECTED”INS3TODETERMINETHEWHICHSHOULDTHENBE“REFLECTED”INS2,ANDRESULTINGVALIDREGIONINS1THENINCLUDESFR4,FR3,FR2,ANDFR1ACOMPOSITIONOPERATIONDEFINEDFORTHISPURPOSEDEFINITION4COMPOSITIONFORTWOADJACENTCI1INACOOLINGSYSTEM,THECOMPOSITION340CGLI,CLLI/COMPUTERCFREEREGIONSINS1AFTER“INTERSECTION”WITHFR2FIG6BYX1ANDX2,ANDX2ISSHAREDBYS2HENCE,THECONSTRAINTSIMPOSEDBYTHOSEOBSTACLESINS2MUSTALSOBECONSIDEREDINANATTEMPTTOFINDALLOFTHEFEASIBLEPOINTSFORDESIGNINGC1,FR1IS“INTERSECTED”WITHFR2THERESULTOFTHIS“INTERSECTION”ISSHOWNINFIG6C,WHICHISOBTAINEDBYREMOVINGTHEREGIONINFR1WHEREX2J,CRJ,IIFIJ,CRJ,IIFIJ,CRJ,IASANEXAMPLE,THATLEADSTOTHECONSTRUCTCR3,4,WHICHFR3,ASSHOWNINCR2,4PROFINALLY,CR1,4ISGISHOWNINFIG8CCR1,4TAKESINTOACCOUNTTHECHANNELSTHATANYPOINTINCR1,4COOLINGSYSTEMCANBYAPPLYINGTHEBEOBTAINEDBYSELECTINGINTOSINOVALIDDESIGNTHECOMPOSITIONDESIGNSTHATMAYOBTAINEDBYTHISITISIMPORTANTPARTOFTHEVALIDVALIDDESIGNSFORWINGTHEOREMOFASEQUENCEOFALLOFTHEBYABOOLEANIMPORTANTFEATUREBYCOMPUTATIONSCRI,1ANDCRI,NCPERFORMEDINSITHEINTERSECTIONFIG7ASEQUENCEOFPOINTSINFRITHATGIVESAVALIDDESIGNOFTHECOOLINGSYSTEMI1FRIFRI1IFR
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