Near support-free multi-directional 3D printing via global-optimal decomposition

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Delft University of Technology

Near support-free multi-directional 3D printing via global-optimal decomposition

Gao, Yisong; Wu, Lifang; Yan, Dong Ming; Nan, Liangliang

DOI

10.1016/j.gmod.2019.101034

Publication date

2019

Document Version

Final published version

Published in

Graphical Models

Citation (APA)

Gao, Y., Wu, L., Yan, D. M., & Nan, L. (2019). Near support-free multi-directional 3D printing via

global-optimal decomposition. Graphical Models, 104, [101034]. https://doi.org/10.1016/j.gmod.2019.101034

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GraphicalModels104(2019)101034

ContentslistsavailableatScienceDirect

Graphical

Models

journalhomepage:www.elsevier.com/locate/gmod

Near

support-free

multi-directional

3D

printing

via

global-optimal

decomposition

Yisong

Gao

a

_,

_Lifang

_Wu

a,∗

_,

_Dong-Ming

_Yan

b

_,

_Liangliang

_Nan

c

a Faculty of Information Technology, Beijing University of Technology, Beijing, China b NLPR-LIAMA, Institute of Automation, Chinese Academy of Sciences, Beijing, China c Delft University of Technology, Delft, Netherlands

a

r

t

i

c

l

e

i

n

f

o

Keywords: 3D Printing Multi-directional Support-free Model decomposition Global optimization

a

b

s

t

r

a

c

t

In3Dprinting,itiscriticaltouseasfewaspossiblesupportingmaterialsforefficiencyandmaterialsaving. Multiplemodeldecompositionmethodsandmulti-DOF(degreesoffreedom)3Dprintershavebeendeveloped toaddressthisissue.However,mostsystemsutilizemodeldecompositionandmulti-DOFindependently.Only afewexistingapproachescombinethetwo,i.e.partitioningthemodelsformulti-DOFprinting.Inthispaper, wepresentanovelmodeldecompositionmethodformulti-directional3Dprinting,allowingconsistentprinting withtheleastcostofsupportingmaterials.Ourmethodisbasedonaglobaloptimizationthatminimizesthe surfaceareatobesupportedfora3Dmodel.Theprintingsequenceisdeterminedinherentlybyminimizinga singleglobalobjectivefunction.Experimentsonvariouscomplex3Dmodelsusingafive-DOF3Dprinterhave demonstratedtheeffectivenessofourapproach.

1. Introduction

3Dprinting,alsoknownasadditivemanufacturing,hasbeenwidely usedforbothrapidprototypingandsmallbatchproduction,producing objectsrangingfrommicrostructuresusedinbiomedicaltolarge-scale complexpartsappliedintheaerospaceindustry.Theproliferationof3D printingtechniquesisdrivenbytherequirementsofﬂexibilityand eﬃ-ciency.Comparedtotraditionalmanufacturingmethodssuchascarving andmolding,3Dprintingiscapableoffabricatingobjectswith arbitrar-ilysophisticatedshapessuchashollowstructures.

Conventional3Dprintersarelimitedto3degreesoffreedom(DOFs), i.e.,X,Y,andZ.Themajorlimitationofsuchprintersisthatthe mate-rialaccumulatingdirection(i.e.,fabricationdirection)oncesetcannot bechangedineverysingleprintingtask.Therefore,ittypicallyrequires auxiliarysupportingstructurestobeaddedandprintedtopreventthe growingobjectsfromdeformingandevencollapsingundertheeffect ofgravity.Introducingsupportingstructureshaveafewdrawbacks.For example,itconsumesmorematerials(andthuslesscost-effective), pro-longsprintingtime,restrictsprintingflexibility,andrequiresadditional post-processingtoremovethesupportingstructuresthatmayharmthe surfacequalityoftheprintedobjects.

Tobeﬂexibleforobjectswithincreasingcomplexity,various fabri-cationsystemswithmoreDOFshavebeendeveloped.TheextraDOFs providethecapabilitytoadjustthedirectionofmaterialaccumulation duringthefabrication,whichprovidesmoreﬂexibilitytotheprinting

∗_{Corresponding}_author.

E-mailaddresses:ysgao@emails.bjut.edu.cn(Y.Gao),lfwu@bjut.edu.cn(L.Wu),yandongming@gmail.com(D.-M.Yan), liangliang.nan@gmail.com(L.Nan).

systemandmeanwhilereducesbothmaterialandenergyconsumption. Inprinciple,moreDOFsleadtomoreflexibilitybutloweraccuracyin theprintedobjectsduetotheaccumulationofthepositioningerrors oftheservomotors.AssuggestedbyDaietal.[1],a5-DOFcomputer numericalcontrol(CNC)systemcanprovidehigherprintingaccuracy thanusinga6-DOFroboticarmsystem.Inthiswork,weexploitaCNC 3Dprintingsystem(i.e.,translationinX,Y,andZdirectionstodrive theprintheadsplustwoadditionalrotationalaxestochangethe ori-entationoftheplatform)designedbyourselves.Insteadofexploiting thecapabilityoftheextrafreedoms(i.e.,thetwocontinuousrotational axes),werelyonsmartmodelpartitiontechniquesanddiscrete rota-tionoftheextrafreedomstoensurebothflexibilityandhighprinting accuracy.Thesystemworksina“discrete-rotational” style:themodel is decomposedintoseveralpartsbyplanesandthepartsareprinted onebyone(thesubsequentpartsareprinteddirectlyontopofthe pre-viouslyprintedparts).Theprintingprocessof eachpartissimilar to theconventional 3-axis3Dprintingwhiletheplatformcanrotate to thedirectionsuitingthenextpartwhenthepreviouspartisfinished.

Fig.1showstheprintingprocessoftheStanfordbunnymodelusingour system.Withthediscrete-rotationalprintingstrategy,highly-accurate traditionalCNCsubtractivemanufacturingmovementsandexisting ma-turedtool-pathplanningmethodscanbedirectlyappliedtoeachpart of the model.So boththemoving velocityof thesystem and print-ingprecisioncanreachthatofconventional3Dprintersandmachine

https://doi.org/10.1016/j.gmod.2019.101034

Received24February2019;Receivedinrevisedform8May2019;Accepted28May2019 Availableonline11June2019

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Fig.1. OursystemprintingtheStanfordbunnymodel.

tools.Moreover,unidirectionalgeometryoptimizationmethodssuchas support-freehollowing[2],andtopologyoptimizationmethodssuchas surfaceremeshingwithsegmentation[3],canalsobeappliedtoeach partofthemodel.

Whenamulti-DOFprintingsystemisenhancedbymodel decomposi-tion,achievingcollision-freemotionbecomesmorechallengingbecause bothprintingorder andtool-path planningarehighlydependent on eachother.Existingapproachesdecomposemodelsintomultiplesolid parts[4]orlinesegments[5,6]andprinteachpartinaspecificorder. Thisallowseachparttobeprintedinawaysimilartotheconventional printerswithonly3DOFs.Effortshavealsobeenmadetodecompose amodelintocurvedsurfacelayers[1],furtherincreasingtheflexibility andmakingsupportingstructuresunnecessary.Inourwork,wefocus onplanar-layeredfabricationandourgoalistoachievebetter decom-positionresultsformulti-directionalprinting.

Givenanarbitrary3Dmodel,ourmethoddecomposesitintoseveral partswithanoptimalprintingorder,withwhichthemodelpartscanbe printedonebyoneandfinallybuildsupthecompleteobjectinasingle printingpassusingtheleastsupportingmaterials.Wecomprehensively analyzethecollisionsituationsandconvertthesupport-freeconditions fromaconstantdirectiontovaryingdirections.Weproposea heuris-ticdecompositionmethodusingcuttingplanestoavoidthecollision. Weminimizethesurfaceareatobesupportedusingglobal optimiza-tionwithwhichtheprintingsequencecanbedeterminedinherently.In additiontotheprintabilitywithleastsupports,ourmethodimproves surfacequalityaswellbyreducingthenumberofmodelpartsand com-pensatingrefinementalongthepartorientations.Wetestouralgorithm onvariouscomplexmodelstodemonstrateitseffectivenessandwe com-pareourmethodtothepreviouswork[4]ofthesamefabricationstyle toshowitsmerits.

Inshort,ourworkmakesthefollowingcontributions:

• Asetofcollision-freeandnear support-freeconditions for multi-directionalfabrication;

• Aheuristicdecompositionmethodusingcuttingplanestoavoid col-lisionbetweentheextruderandtheprintedparts;

• Amethodbasedonasingle-objectiveglobaloptimizationthatcan simultaneouslyoptimizeforboththedecompositionandprinting or-der.

2. Relatedwork

2.1. Multi-DOFhardwaresystem

Intermsofhardware,multi-DOFprintersarebeingconstantly ex-ploredstarting withanattempton proof-of-concept printingusing a 6-DOFroboticarm[7].A5-axismotionsystemsimilarto5-axisCNC machiningwasproposedtofabricatesimpleshapeonanexistingmodel

[8].Themulti-directionaladditivemanufacturingwaspresentedinthe work[9]using6-DOFparallelkinematicStewartplatform.Afterthat, multi-DOFsystemswith shape decomposition tool-path planningfor multi-directional fabricationwas built up one after anothersuch as

[4–6,10,11],tremendouslyincreasingtheﬂexibilityofprintingand free-domofdesign.

2.2. Modeldecomposition

Onthesoftwareside,modeldecompositionanddeformationarethe twomainapproachestoimproveprintability.Modeldecompositionis favored for its multifunctionandpreservation of theoriginal shape. Shapiraetal.[12]decomposedmodelsaccordingtotheshapeand vol-umediametertoachievecoherentresultswithdifferentgestures.Luo etal.[13]focusedondecomposinglargemodelsintoprintingvolume withaconsiderationofprintability,numberofparts,feasibility, struc-turalsoundness,andaesthetics.Huetal.[14]decomposedmodelsinto approximatepyramidalparts,whichcouldbeprintedwithoutsupports. Herholzetal.[15]presentedmethodsusingaheightfieldforthe print-abilityinasupport-freemanner.Chenetal.[16],catertoprinting effi-ciencyproblem,decomposedmodelintopartsandpackedtheminthe printingvolumesoallthepartscouldbeproducedinonepass.Song etal.[11]builtcoarseinternalbasestructureswithinthegiven3D ob-jectandattachthin3D-printedpartsontothebasetorecoverthefine surfacedetails.Wangetal.[17]improvedoverallsurfacequalityby de-composingandoptimizingtheprintingdirectionsofeachparttoavoid thestaircaseeffectthatharmssurfacequality.Weietal.[18] decom-posedmodelsguidedbytheskeletonsforsupport-free printing espe-ciallytowardsshellmodels.Chenetal.[19]manufacturedtheinnerof theobjectsusinguniversalbuildingblocksandfabricatedoutershells withpyramidaldecompositionwhichcanrealizesupport-freeprinting. Most of the methods mentioned above produce objects in a decompose-and-assemblemanner,whichconsumesextratimeto assem-blethepartstogether.Wuetal.[4]firstlydecomposedsolidmodelinto support-freepartswhichcanbefabricatedonebyone(directly accumu-latedontheprintedparts)inonepassbya6-DOFroboticsystem. Mod-elsarefinallyslicedintoplanarlayersalongdifferentorientationsand printedina‘discrete-rotational’style.Wuetal.[20]improvedtheir pre-viousmethod[4]bypartitioningmodelsusingplanessothatthemethod couldworkwithmodelswithcomplextopologyoneither4-DOFor 5-DOFsystems.Oneofthemoststate-of-the-artmethodsisproposedto decomposeamodelintocurvedsurfacelayers[1],furtherincreasing theflexibilityandmakingsupportstructuresunnecessaryinmostcases. Thisnovelworkfirstlyrealized“3Dprinting” withthefulluseofDOFs.

3. Methodology

Considering printability, material consumption,quality, and effi-ciency, achieving optimalmodeldecomposition is challenging.Hard constraints,i.e.,collision-freeandsupport-free,areunlikelytobefully satisfiedandwithwhichthetoolpathplanningbecome computation-allyexpensive.Inthiswork,wetransformthenecessaryconstraintsto aformthatsimplifiesthecalculation,andwerelaxunnecessarilystrict constraintsforrobustnessconsiderations.

Wedecomposea3Dmodelusingasetofplanes(i.e.,cuttingplanes) forplanar-layeredfabrication.Everytimeacuttingplaneisappliedtoa modelcomponent,themodelcomponentispartitionedintotwoparts. Duringprinting,thepartwillbeprintedontopofthecuttingplanelayer bylayeralongthenormaldirectionofthecuttingplane.

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Y. Gao, L. Wu and D.-M. Yan et al. Graphical Models 104 (2019) 101034 Fig.2.Extruder-platformcollision.(a)Collisionsmayoccurbetween theextruder(inpink)andtheplatform(indarkblue)whenprinting theobjectparts.(b)Theexpandedplatformboxisusedforsimpler collisiondetection.(Forinterpretationofthereferencestocolourin thisﬁgurelegend,thereaderisreferredtothewebversionofthis article.)

Fig.3. Extruder-objectcollision.(a)Anexampleofthe extruder-objectcollision.(b)(c)Theexpandedconvex hullofthemodelpartisusedtosimulatetheworkspace oftheextruderwhenprintingthepart.

3.1. Constraintanalysis

ConstraintI:Collision-free.Toensureprintability,nocollisioncan occurduringtheprintingprocess.Normally,thefollowingtwotypesof collisionsareconsidered:

Extruder-Platform.Intheprevious work[4], theworkspaceofthe multi-DOFprinterisrestrictedtotheupperhalfspaceabovethe hori-zontalplane.Weobservethattherestrictionofrotationalangularrange isunnecessarilystrictandmaymisscollision situationswhen consid-eringtheactualvolumeoftheextruder(i.e.,thenozzlesthatmeltand extrudethematerials).Weseekforanapproximatesolutionthat guaran-teescollision-freebetweentheextruderandtheplatform.Speciﬁcally, wecheckifcollisionoccursbetweeneachmodelpartandtheexpanded platformbox(i.e.,acuboidobtainedbyexpandingtheplatformtoa dis-tanceoftheradiusoftheextruderalongallthemajoraxes.SeeFig.2

(b)),allowingtheplatformtorotatewithoutlimitingtheangularrange (e.g.,[−90◦,90◦]in[4]whiletheplatformcanrotateupto180∘_from itshorizontalpositionwiththeextrarotationalDOFs)withoutcollision. Inourwork,wesettheangularrangetobe[−135◦,135◦](reduced45∘ consideringtheself-supportingangleofthematerials).

Extruder-Object.Decomposition of themodel usingcutting planes

typicallyintroducessharpcornersintheprintingprocess(seeFig.2(a)). Forthe cornerregions, collision usuallycannot be avoidedeven by switchingtheprintingorderoftheparts.Thepreviouswork[4]does notallowintersectionsbetweencuttingplanestooccurwithinthe vol-umeoftheobjecttobeprinted.Wefoundittoostrictandrelaxitto allowacertaintypeofcuttingplaneintersections.Ourobservationis thatcollisioncanbeavoidedwhenpartswithhigherprintingpriority lieunderthebaseplaneofthesubsequentparts.Withthisobservation, weproposeaheuristicmethodtodecomposea3Dmodelintopartswith planesandmeanwhileobtainproperprintingprioritiesforcollision-free printing(seeFig.7foranexampleandSection3.2fordetails).

Inthegeneralcase,extruder-objectcollisionscanbedetectedby us-inganexpandedconvexhullofeachpartofthemodel.Similartothe expandedplatformbox,wecalculatetheconvexhullofeachpartand ex-panditbytheradiusoftheextruder(expandeveryvertexalongits

nor-Fig.4.Inclinationangle(illustratedin2D).

mal’scomponentperpendiculartotheprintingdirection.SeeFig.3(b)). Intersectiondetectionisthenappliedbetweentheexpandedhulland thepartswithhigherprintingpriorityforcollisiondetection.

ConstraintII:Support-free.Giventhefactthatthereisstillno the-oreticalproofoftheexistenceofastrictsupport-freedecompositionof arbitrary3Dshapes[1],werelaxthesupport-freerequirementtoasoft constraint.Insteadofminimizingthematerialcostforthesupporting structure,weattempttominimizetheareaoftheregionsthatrequire supportstoensureprintability.Inourproblemsetting,thesupporting analysisismuchmoresophisticatedthanconventionalmodel decompo-sitionmethodsbecauseofthevaryingprintingorientationsforthe de-composedparts.Weconsiderthreesituationsinmulti-directional print-inganddiscussthemindetailsasfollows.

Overlargeinclinationangle:Wedeﬁnetheanglebetweenthemodel

surfaceanditsprintingorientationasinclinationangle,asshownin

Fig.4.InFDM-basedfabrication,materialscanbeaccumulatedwithout extrasupportingstructures(i.e.,themodelcansupportitselfwithout causingdeformation)whenthesurfaceinclinationangleissmallerthan acertainvalue.Thisthresholdangleiscalledthemaximalself-support angleasusedin [1,2,4,14],andthisthresholdangleisusuallysetto be45∘_or_larger_depending_on_the_stiﬀness_of_the_material._In_printing, deformationcouldbeobservedandtheprintingobjectmayeven col-lapseundertheeﬀectofgravityiftheactualinclinationangleisgreater thanthis threshold.Normally,thesurfaceregions withanexceeding

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Fig.5.Twotypesoflocalminimumpoint.ForthelocalminimumpointA, sup-portingstructures(shadedregions)areneededforitsneighboringfaces.ForB, supportisneededonlyatpointB(withacertainradius)whileitisnotnecessary foritsneighboringfacesbecauseofthesmallinclinationangles.

inclinationangle(calledoverhangingregions)shouldbesupported us-ingadditionalstructurestoensureprintability.Inthiswork, weaim tominimizetheareaoftheoverhangingregions.Whentheareaofthe overhangingregionscanbereducedtozero,thissupport-freeconstraint canbesatisﬁed.

Localminimumpoints:Thesurfacepointwiththesmallestdistance totheplatform(orthebaseplaneofeachpartinthemulti-directional case),comparedwithallitsneighbors(exceptthoselyingonthebase plane),iscalledthelocalminimumpoint.Obviously,regionsarounda minimumpointareoverhangingandsupportsarerequired.Toreduce supportingstructures,wedecomposethemodelinawaysuchthatthe leastnumberofminimumpointspresents.Iftheadjacentfacesofalocal minimumpointdonotneedsupports(i.e.,theinclinationangleissmall. SeethepointBinFig.5),weonlyneedacylinder-likestructurethat supportstheminimumpoint.Insuchcases,thesupportingsurfacearea isdeﬁnedasthecross-sectionalareaofthesupportingcylinder.

Fragileregions:In3Dprintingwithmodeldecomposition,partsare printedinapredefinedorderandonepartisalways(exceptforthefirst one)printedontopofotherpreviouslyprintedparts.Suchaprocess re-quirestocheckifthereexisttoofragileregionsthatcannotaffordother partstobe printedontopofthem.In3Dprinting,typicalfragile re-gionsarethinfinsorbridgesonthepreviouslyprintedparts.Inspired bytheworkofLuoetal.[13],wedetectfragileregionsbycheckingif thereexistsaregionwhosedistancetothebaseplaneofthefollowing partissmallerthanaspecifiedthresholdandmeanwhilewhosenormal issufficientlyclosetothenormalofthebaseplanes(seeFig.6).Here, wesetthedistance thresholdto5mmandthemaximalallowed an-gledeviationissetto10∘_._Fragile_regions_can_be_detected_and_handled moreefficientlyinsuchanapproximatemanner,comparedtoaccurate analysis-basedapproachessuchasfiniteelementanalysis(FEA,which isfairlytime-consuming).Inaddition,preciselydeterminingwhethera regionrequiressupportsandhowmuchmaterialitwillhaveconsumed arerathercomplicated.Duetothesereasons,westrictlydisallowany fragileregions,forwhichwetrysimplymovingthecuttingplanea cer-taindistancealongitsnormaldirectionincaseoffragileregions.

3.2. Modeldecomposition

Avoidingextruder-object collision. Intheprevious workofWu etal.[4],collisionisavoidedbystrictlyprohibitingintersectionsthat occurwithintheinteriorofthemodel,whichiscomputationally expen-sive.Besides,duetosuchastrategyexposestoomuchconstraintonthe locationandorientationofacuttingplane,betterdecompositionsare likelytobeignored.Weobservethatifnoprintedobjectexistsontop ofacutting plane,therewouldnot beanycollision betweenthe ex-truderandtheobjectbecausethemotionoftheextruderiscompletely on/abovethecuttingplane.Basedonthisfact,weutilizeaheuristic methodtosegmentamodelintocollision-freeparts.Giventhecutting planes,themodelispartitionedusingamethodsimilartobinaryspace partitioning(BSP)[21](wecallit‘planecut’).Speciﬁcally,thecutting planesareappliedreverselyinatop-downsegmentationorder,which

iscontrarytotheprintingpriority(seeFig.7foranexample).This de-compositionstrategyensuresthatthenumberofpartsincreasesbyone foreverypartition,leadingtoasimplerprintingprioritydetermination. Asaresult,themodelisdecomposedinto𝑁+1partswithout extruder-objectcollisionsbygivenNcutting planes.However,inanother cut-tingcase thatonly onebranchis chosentodecomposethemodelif theplaneintersectsthemodelatseveralbranches(called‘branchcut’), extruder-objectcollisiondetectionisstillindispensable.Asweobserved, the‘branchcut’ismoresuitablefortree-likemodelswhilethe‘planecut’ suitsmodelswithring-likestructuresbetter(applyingonlyone‘branch cut’onring-likestructuresmayleadtounseparatedresults).

Overhangingdetection.Inpractice,duetothecohesiveandelastic forcesoftheplasticmaterialitself,theobjectwouldnotcollapse im-mediatelywhenprintingtheseverelyslopingregionswithoutsupports.

Fig.8showssuchanexample.However,deformationshouldnotbe ne-glectedastheinclinationangleincreasesforacertaindistance.Sucha distance(wecallitsafedistance)deﬁnesasaferegionforwhich support-ingstructureisnotnecessary.Experimentsshowthatthesafedistance canbeupto10mmfora60∘_inclination_angle._Due_to_the_wide distri-butionoftheinclinationanglesandthecuttingplaneorientations,itis diﬃcult(andalsonotnecessary)toaccuratelycomputeasafedistance foreveryfaceinthemodel.Inourwork,wesetthesafedistancetobe halfofthemaximumdistanceobservedinourexperiments,i.e.,5mm. Withthissafedistance,thedetectionofoverhangingregionsisdetailed inAlgorithm1andillustratedinFig.9.

Algorithm1 Detectionofoverhangingsurfaceregions.

Input:

Modelpart𝑝𝑎𝑟𝑡𝑖anditsorientation𝑂𝑟𝑖𝑒𝑛𝑡𝑎𝑡𝑖𝑜𝑛𝑖;

Output:

Theoverhangingregionsin𝑝𝑎𝑟𝑡𝑖;

1: Computetheinclinationangles,relativeto𝑂𝑟𝑖𝑒𝑛𝑡𝑎𝑡𝑖𝑜𝑛𝑖,forallfaces

in𝑝𝑎𝑟𝑡𝑖 andmarkthepotentialoverhangingfaces(ignoreregions

withsmallareas);

2: Projecttheborderlinesofeachoverhangingregion𝑎𝑟𝑒𝑎𝑗 ontothe

plane𝑃𝑙𝑎𝑛𝑒𝑖(thatpassesthroughthemasscenteroftheregionand

isorthogonalto𝑂𝑟𝑖𝑒𝑛𝑡𝑎𝑡𝑖𝑜𝑛_𝑖);

3: Calculatethemasscenter𝐶𝑒𝑛𝑡𝑒𝑟𝑗 of thelowerboundaryofeach

regionandprojectitonto𝑃𝑙𝑎𝑛𝑒𝑖;

4: Computeanenlargedoﬀsetoftheprojectedcloserboundary(the sidethatis closertotheprojectregioncenter) withthespeciﬁed

safedistanceon𝑃𝑙𝑎𝑛𝑒𝑖.Theoﬀsetregionsarewithinthesafedistance

andaresupport-free,whiletheremainingregionsrequiresupporting structures.

5: Mapthe2Dregions thatrequiresupportsbacktothe3Dmodel, resultingintheoverhangingregionsfor𝑝𝑎𝑟𝑡𝑖.

4. Implementation

4.1. Objectivefunction

Withtheaforementionedanalysisoftheconstraints,collision detec-tionandapproximation,andoverhangingdetection,belowwedetail ourobjectivefunction.AsstatedinSection3.1,werelaxthe support-freeconstraintsrequirementtoasoftrequirementandweminimizethe areaoftheregionsthatrequiresupports(insteadofminimizingthe ac-tualmaterialcostforthesupportingstructure)foranyarbitraryinput models.GivenNcuttingplanes,wepartitiontheinputmodelinto𝑁+1 componentsbyminimizingthefollowingobjectivefunction

min 𝑁 ∑ 𝑖=0 𝑎𝑟𝑒𝑎𝑖 s.t. 𝑂𝑟𝑖𝑒𝑛𝑡𝑎𝑡𝑖𝑜𝑛𝑖⋅ 𝑂𝑟𝑖𝑒𝑛𝑡𝑎𝑡𝑖𝑜𝑛0≥−0.5,𝑖=1,2,…,𝑁; 𝑝𝑎𝑟𝑡𝑖∩𝐵𝑜𝑥𝑝𝑙𝑎𝑡𝑓𝑜𝑟𝑚=∅,𝑖=1,2,…,𝑁;

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Y. Gao, L. Wu and D.-M. Yan et al. Graphical Models 104 (2019) 101034 Fig. 6. (a) An example of fragile regions in the hanging-ballmodel.RegionAisaﬁnandregionBisa bridge.(b)Illustrationoffragileregionsin2D.

Fig.7. ThedecompositionoftheStanfordbunnymodelintothreepartsusing twocuttingplanes.Therednumbersdenotetheactualcuttingorderandthe blacknumbersdenotetheprintingpriority.(Forinterpretationofthereferences tocolourinthisﬁgurelegend,thereaderisreferredtothewebversionofthis article.) 𝑇ℎ𝑖𝑛𝑅𝑒𝑔𝑖𝑜𝑛𝑖=∅,𝑖=0,1,…,𝑁−1; 𝐻𝑢𝑙𝑙𝑖∩ 𝑖−1 ⋃ 𝑗=0𝑝𝑎𝑟𝑡𝑗 =∅,𝑖=1,2,…,𝑁(𝑜𝑝𝑡𝑖𝑜𝑛𝑎𝑙). (1)

whereareaidepictstheprojectedareaoftheoverhangingregionsofparti

thatrequiresupportingstructure.∑𝑁_𝑖₌₀𝑎𝑟𝑒𝑎𝑖denotesthetotalareaof

thesupportingregions.part_iisthedecomposedpartwiththeithplane asitsbaseplaneandwillbe printedalongtheplane’snormal direc-tionOrientationi.Boxplatformrepresentstheexpandedplatformbox,and ThinRegion_i denotesthefragileregions.Hull_i is theexpandedconvex hullofparti and⋃_𝑗=0𝑖−1𝑝𝑎𝑟𝑡𝑗 representstheprintedobjectparts,which

areonlyneededwhenapplying‘branchcut’.ByminimizingEq.(1),we obtaintheparametersforeachcuttingplanes.

4.2. Optimization

Wecombinegeneticalgorithm(GA)withelitism,adaptive probabil-itiesofcrossoverandmutationandsimulatedannealingforbetter con-vergence rateandconstraintsimplementation. Thegeneticalgorithm canbeseenasaMarkovprocessandGAwiththeelitistmodel(i.e., pre-servingthebestindividualineachgeneration)hasbeenproventobe probabilisticconvergent[22],whichguaranteethesolutionis approxi-matelyglobaloptimal.Adaptiveprobabilitiesofcrossoverandmutation realizethetwingoalsofmaintainingthediversityofthepopulationand sustainingconvergencecapacity[23].Simulatedannealingisusedto refusethesolutionsout ofconstraintsandimproveconvergence eﬃ-ciency.SeeAlgorithm2forabetterunderstandingoftheoptimization process.

Parametersetting.ThenumberofcuttingplanesNisspeciﬁedbythe usertakingintoaccountthecomplexityofthemodel.Thesizeofthe populationNpopissetto200andtheuser-speciﬁedterminating

thresh-oldN_termissetto100.

We use 5 variablesto determinethe parameters of each cutting planeandthebranchthatwillbecutbythisplane,includingapoint

𝐩=(𝑎,𝑏,𝑐)andtworotationalangles𝛼,𝛽.Thepointpisusedto deter-minethespeciﬁcbranchtobecut(i.e.,if‘branchcut’isallowed,the branchwiththeshortestdistancetothepointischosen)andthetwo ro-tationalanglesdeterminetheorientationoftheplane.Weconcatenate thevariablesofallcuttingplanesintoahighdimensionalvector

𝑖𝑛𝑝𝑢𝑡_𝑠𝑒𝑡={𝑎1,𝑏1,𝑐1,𝛼1,𝛽1,𝑎2,𝑏2,𝑐2,𝛼2,𝛽2,…}. This vector is then encodedinbinarymodefortheconvenienceofgeneticoperations.

Fitness.Wesetthenegativevalueofthetotalprojectedareaofthe overhangingregions asfitness(GAalwaysaimstomaximizefitness). Thetotalprojectedareaincludestheprojectedareaofeachfinal over-hangingregionandthecross-sectionalareaofthesupportingcylinder (i.e.,𝑐𝑠𝑎𝑟𝑒𝑎=𝜋𝑅2_._R_represents_the_radius_of_the_cylinder._Here_we_use

𝑅=1mm)ofeachlocalminimumpoint.Foreachiterationinthe op-timization, planesthat donotsatisfythecollisionsandfragility con-straintsarerejectedbyusingsimulatedannealing.

Fig.8. Aprintedobjectwithincreasinginclination an-gles.

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Fig. 9.Overhanging detection. (a)Potential faces(markedinorange)withoverlarge incli-nationangles.(b)Theprojectionofthe poten-tialregionsin2D.Theredregionsdenotethe areasrequiresupportingstructures.(c) Map-pingtheoverhangingregionsto3D.(For in-terpretationofthereferencestocolourinthis ﬁgurelegend,thereaderisreferredtotheweb versionofthisarticle.)

Algorithm2 Minimizingthetotaloverhangingarea∑𝑁_𝑖₌₀𝑎𝑟𝑒𝑎_𝑖.

Input:

Theinputmodel𝑀andnumberofcuttingplanes𝑁;

Output:

Thedecompositionofthemodel𝑝𝑎𝑟𝑡𝑖(0≤𝑖≤𝑁)withprinting

or-der; 1: 𝑔𝑒𝑛max⇐ 100 2: 𝑔𝑒𝑛⇐ 0

3: Initializationreturnsthebestﬁtnessoftheinitialpopulation; 4: while(𝑔𝑒𝑛<𝑔𝑒𝑛max)

and(𝑓𝑖𝑡𝑛𝑒𝑠𝑠_𝑔𝑒𝑛∕𝑓𝑖𝑡𝑛𝑒𝑠𝑠_𝑔𝑒𝑛−20>0.99) and(𝑓𝑖𝑡𝑛𝑒𝑠𝑠_𝑔𝑒𝑛≠ 0)do

5: gen++;

6: Preservetheindividualwiththebestﬁtness; 7: Selection;

8: Crossover;

9: Mutation;

10: Calculateﬁtnessofthenewgeneration;

11: SimulatedAnnealing;

12: Replacetheworstindividualbythepreservedone; 13: endwhile

14: Decomposethemodelusingthesolutionwiththebestﬁtness.The solutionrevealsboththecuttingplanesandthecuttingsequence.

Initialization.GiventhenumberofpartitionplanesN(speciﬁedby theuser),solutionsareinitializedbyrandomlysamplingtheparameter spaceundercollision-freeconstraintsandoverhangingtest.

Selection.Theclassicalroulettewheelselectionisapplied,i.e., indi-vidualswithhigherﬁtnessvalueshavehigherprioritytobeselected. TheprobabilityPs ofeachindividual𝐼𝑛𝑑𝑣𝑖(𝑖=1,2,…,𝑁𝑝𝑜𝑝)is

calcu-latedas

𝑃𝑠(𝐼𝑛𝑑𝑣𝑖)=_∑_𝑁𝑓𝑖𝑡𝑛𝑒𝑠𝑠_𝑝𝑜𝑝 (𝐼𝑛𝑑𝑣𝑖) 𝑗=1 𝑓𝑖𝑡𝑛𝑒𝑠𝑠(𝐼𝑛𝑑𝑣𝑗)

. (2)

Every twoselected individuals arethenpairedas parents𝑝𝑘(𝑘=

1,2,…,𝑁_𝑝𝑜𝑝∕2).

Crossover:Uniformcrossoverisused.Eachpairofbitsintheencoded genesofthetwopairedindividualsisexchangedbythecrossover prob-abilityPc.HerePcissetadaptivelyforfasterconvergence:

𝑃𝑐(𝑝𝑘)= { 1.0×𝑓𝑖𝑡𝑛𝑒𝑠𝑠𝑚𝑎𝑥−𝑓𝑖𝑡𝑛𝑒𝑠𝑠(𝑝𝑘) 𝑓𝑖𝑡𝑛𝑒𝑠𝑠𝑚𝑎𝑥−𝑓𝑖𝑡𝑛𝑒𝑠𝑠𝑎𝑣𝑔, 𝑓𝑖𝑡𝑛𝑒𝑠𝑠(𝑝𝑘)≥𝑓𝑖𝑡𝑛𝑒𝑠𝑠𝑎𝑣𝑔 1.0, 𝑓𝑖𝑡𝑛𝑒𝑠𝑠(𝑝_𝑘)<𝑓𝑖𝑡𝑛𝑒𝑠𝑠_𝑎𝑣𝑔 (3)

where𝑓𝑖𝑡𝑛𝑒𝑠𝑠maxandﬁtnessavgdenotethemaximalandtheaverage

fit-nessofthegenerationandfitness(p_k)isthehigherfitnessofp_k,meaning thattheindividualswithhigherfitnessvaluesarelikelytobepreserved.

Mutation:Adaptivemutationisutilizedtoaccelerateconvergence. TheprobabilityofmutationPmisdeﬁnedsimilarlytothatofcrossover:

𝑃𝑚(𝐼𝑛𝑑𝑣𝑖)= ⎧ ⎪ ⎨ ⎪ ⎩ 0.5×𝑓𝑖𝑡𝑛𝑒𝑠𝑠𝑚𝑎𝑥−𝑓𝑖𝑡𝑛𝑒𝑠𝑠(𝐼𝑛𝑑𝑣𝑖) 𝑓𝑖𝑡𝑛𝑒𝑠𝑠𝑚𝑎𝑥−𝑓𝑖𝑡𝑛𝑒𝑠𝑠𝑎𝑣𝑔 , 𝑓𝑖𝑡𝑛𝑒𝑠𝑠(𝐼𝑛𝑑𝑣𝑖)≥𝑓𝑖𝑡𝑛𝑒𝑠𝑠𝑎𝑣𝑔 0_.5_, _{𝑓𝑖𝑡𝑛𝑒𝑠𝑠}(_{𝐼𝑛𝑑𝑣}_𝑖)_<_{𝑓𝑖𝑡𝑛𝑒𝑠𝑠}_𝑎𝑣𝑔 0.05, 𝑃_𝑚(𝐼𝑛𝑑𝑣_𝑖)<0.05 (4)

Simulated Annealing: The probabilistic acceptance processis

sim-ilar to that of conventional simulated annealing, which is applied whenﬁtness(Indvi)≤ﬁtnessavg.Inaddition,anextrarejectioncondition

isaddedtoexcludethemarkedindividualsthatdonotmeetthe con-straints.

Elitism:Aftersimulatedannealing,wereplacetheworstindividual withthebestonefromthepreviousgenerationtopreventtheﬁtness

fromdecreasingasthenumberofgenerationsincreases.

Terminationconditions:Ifthegenerationhasnotexceededthe maxi-mumterminationgenerationNterm,theoptimizationterminatesonlyif

theﬁtnessreaches0(oranacceptablethreshold),ortherelativechange in consecutive20generationsdoesnotexceedaspeciﬁedpercentage (1%inourimplementation),i.e.,|𝑓𝑖𝑡𝑛𝑒𝑠𝑠𝑖−𝑓𝑖𝑡𝑛𝑒𝑠𝑠𝑖−20

𝑓𝑖𝑡𝑛𝑒𝑠𝑠𝑖−20 | ≤0.01).Forthe

lat-tercase,itrequiresthattheuserincreasesthenumberofcuttingplane

N(andrepeattheoptimization)orintroducessupportingstructuresto ensureprintability.

Supportingstructure:Toreducethematerialconsumption,weusethe tree-likecolumnsupportingstructuressimilartoVaneketal.[24].A columnstartsfromanoverhangingsurfaceregionandendsupatother printedpartsortheplatform.TheworkspaceispartitionedusingBinary SpacePartitioning(BSP)bythecuttingplanes.Weallowthecolumns toinclinenomorethan30∘_from_the_cutting_plane’s_normal_direction. ColumncandidatesstartingandendingupwithinthesameBSPspace aregivenhigherprioritytobechosentoreducethecomplexityof sup-portingstructures.

5. Resultsanddiscussions

Wehavetestedourmulti-directionalprintingsystemonvarious chal-lenging3Dmodels.Fig.1showsoursystemintheprintingofthe Stan-fordbunnymodel.

Modeldecomposition.Figs.7,10(right),and15(right)show dif-ferentdecompositionresultsoftheStanfordbunnymodel.The advan-tageofourmodeldecompositionstrategyisthatitprovidescontrolover thenumberoftheresultedpartsandmeanwhileminimizesthematerial consumption.AnegativecorrelationbetweenNandthetotalareatobe supportedcanbediscovered,i.e.,thebesttotalareadecreases(or re-mainsconstant)whenNincreases.TodetermineN,weuseabrute-force strategy(iterativeincrementofN)withaglobal-optimalmethodtogain thebestresultsofNplanes.Theiterationterminateswhenthearea be-comessmallenough(asasuggestion,nooverhangingregionbutmaybe severallocal minimumpoints.Seetheoverhangingareas inTable1,

Figs. 12and13forexamples)ortheNexceedsanexpectingnumber (setbytheuser).Ifthedecompositionresultscannotreachnear support-free,anobjectivefunctioncanbeusedtoﬁndabalancebetweenNand theareatobesupported

min 𝛼 ⋅ 𝑎𝑟𝑒𝑎𝑁𝑜𝑟𝑚𝑎𝑙𝑖𝑧𝑒𝑑+(1−𝛼) ( 𝑁 𝑁𝑒𝑥𝑝𝑒𝑐𝑡 )2 (5)

whereweuseaquadratic(orevenhigher)termtodescribeN,simulating theboomingcalculationalcostwhenincreasingN.Theparameter𝛼 can

besetaccordingtotheexpectingnumberNexpect.

HardwaresettingsandSurfacequality.Forthehardware parame-ters,thepositioningaccuracyofeachtranslationalaxisis0.005mmfor

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Y. Gao, L. Wu and D.-M. Yan et al. Graphical Models 104 (2019) 101034 Fig.10. DecompositionresultoftheStanfordbunnymodel.Left:the resultofWuetal.(5cuttingplanesand6parts).Right:Ourresult(2 cuttingplanesand3parts).

Fig.11. Surfacequalityoftheprintedbunnyobject usingourprintingsystem.Left:Theregionswith over-largeinclinationanglesaremarkedinyellow.Middle andright:thezoom-insofthedetailsinthetwoprinted regions.(Forinterpretationofthereferencestocolour inthisﬁgurelegend,thereaderisreferredtotheweb versionofthisarticle.)

Fig.12. (a)Decompositionandsupport-freeanalysis ofthedragonmodel.Thesaferegionsaremarkedin or-angeandthreelocalminimumpointsaremarkedinred pointsandlines(indicatingtheorientation).(b)The surfacequalityoftheprintedobjectusingour print-ingsystem.Left:Theprintedobject.Onesaferegion ismarkedinblueandoneregionwithtwolocal min-imumpointsismarkedinyellow.Right:thezoom-ins ofthedetailsinthetwoprintedregions.(For interpre-tationofthereferencestocolourinthisﬁgurelegend, thereaderisreferredtothewebversionofthisarticle.)

Table1

Statisticofourpartitionresults.

Model Faces N Initialization (in sec ) Optimization time(in sec ) Overhanging area ( mm 2 )

Kitten 10000 1 93 463 20.85 2 87 60 0 Bunny 13026 1 80 240 87.28 2 81 491 0 H-ball 11540 2 53 355 0 Dragonstand 24598 1 182 780 4 𝜋 Armadillo 9998 1 99 442 7 𝜋

therepeatabilityand0.058mmper300mmforthepositioningerror, thepositioningaccuracyofeachrotationalaxisis0.01∘_for_the repeata-bility(noexactoverallaccuracy).Thediameterofthenozzleis0.4mm, thelinewidthandthelayerthicknessofmaterialextrusionare0.2mm. Thehighestprintingspeedissetto50mm/s.

Toreducethematerialconsumption,ourmethoddoesnotintroduce thesupportingstructureatthesaferegions(withoverlargeinclination anglesbutcanbeignored).Experimentsshowthattheprintingquality ofthesaferegions(evenwithlargeinclinationangles)aresatisfactory. AscanbeseenfromFig.11andtheregionsmarkedinblueinFig.12(b), thesurfacequalitylossinthesaferegionscanbeneglected.

Besides,onthedragonmodel(seeFig.12),weonlyaddsupporting structuretoonelocalminimumpointunderthemouthwhileleaving thepointsonthetailunsupported.Itcanbeseenthatwithoutadding supportstothelocalminimumpoints,thedeformationbecomesserious (theregionmarkedinyellowinFig.12(b)),whichindicatesthe neces-sityoftheanalysisonthelocalminimumpoints.

Comparetothecurved-surface-layeredprintingstyleofDaietal.[1], theoverallsurfacequalityoftheobjectsgeneratedbytheplanar-layered manner printingperforms betterthankstotheﬂatnessandthesame thicknessofeachlayer(whichcontributetothematerialaccumulation). Ourresultsaredecomposedbyplanesandslicedintoplanarlayersso

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Fig. 13. (a) Decomposition of the ar-madillomodel.(b)(c)Theprintedobject intwoviews.Onlyseveralsupporting struc-turesareusedtoﬁnishtheobject.The over-hangingpointsandregionscannotbe op-timizedduetotheextruder-platform colli-sionrestriction.

Fig.14. Diﬀerent typesof supporting struc-tures.Left:cylinder-likesupportingstructures. Middle: Branch structures [25]. Right: Our methodresultsinsupport-freeprinting.

Fig.15. Decompositionresultsofthehanging-ballmodel.Left:theresultofWu etal.(5planes,6parts).Right:outresult(2planes,3parts).

thattheprintingqualitycouldbesuperiortothecurved-surface-layered printingresultsunderthesamehardwareconditions.

Materialsaving. The motivationof ourwork is toachieve near support-free3Dprinting.Toevaluatethis,wemeasurethevolumeof theprintedobjectstorecordthematerialcostbyassumingthatallthe modelsareprintedinthesolidform.Wechoosetwocommonsupport structuregenerationstrategy, onecompletelyfilling theoverhanging regionsandtheotheronegeneratingbranchstructures.Fig.14shows thebunnymodelprintedwithdifferenttypesofsupportingstructures. Itcosts8.8%extramaterialwiththefirststrategy(Fig.14(left))and

3.4%forthebranchtypestructures.Byminimizingthetotalsupporting areaoftheoverhangingregions,ouralgorithmcanachievesupport-free printingforthismodel,whichmeansthereductionofsupportmaterial usedis100%).

Timing.Ouralgorithmswereimplemented inC++withOpenMP parallelization enabled on a laptopwith Intel Corei7-7770HQCPU (with4cores/8threads)and8GBRAM.Therunningtimesand over-hangingareasusingdiﬀerentnumbersofcuttingplanesarereportedin

Table1.Sincetherotationalmotionoftheplatformbetweenthe print-ingprocessesoftwopartsonlytakesupseveralseconds,theprinting timeismainlyinﬂuencedbythetotallengthofthetool-paths,whichis increasedbyaddingextrasupportingstructures.Therefore,theprinting timecanbesavedbyreducingtheuseofsupports.Inourexperiment, whilethematerialcostreducesabout10%,thetimeconsumptioncan reduceabout20%ifthedecompositionresultreachessupport-free(the savingpercentagedependsonmodels).

Comparison.Wefirstlycompareouralgorithmwiththeprevious workofWuetal.[4]sincethedecompositionandprintingstyleoftwo methodsaresimilar.ForthebunnymodelinFig.10andthe hanging-ballmodelinFig.15,bothmethodssuccessfullydecomposedthemodel. However,ourdecompositionresultshaveafewernumberofpartsand shorterseamlength(seeFig.10).Withfewerpart,boththecumulative error ofthemulti-DOFsystemandthetotallengthoftheconnecting seamsbetween partscanbe reduced.Besides,theircoarse decompo-sitionmethodisbasedontheshape-diameteranalysis.Thusitisonly applicabletomodelsthatcanbeabstractedbyaskeletonstructureand mayfailformoregeneralobjects,especiallyforring-likemodels.For ex-ampleinthekittenmodelshowninFig.16,theshape-diameter-based methodresultsinunfeasibledecompositionresults(seethegreenpartin theleftsub-figure)evenwithasubsequentfinetuningstep.Incontrast, ouroptimization-basedmethodis capableofdecomposing themodel intopartsrespectingprintability.

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Y. Gao, L. Wu and D.-M. Yan et al. Graphical Models 104 (2019) 101034 Fig. 16. Decomposition results of the kitten model. Left: the coarse decomposition result of Wuetal. [4](thegreenpartishardtomodifyfor printability). Middle and right: our results in two views.(Forinterpretationofthereferencestocolour inthisﬁgurelegend,thereaderisreferredtotheweb versionofthisarticle.)

Fig. 17. The printed bunny: (a) The result of Wu et al. [4] (5 planes). (b) The result of Dai etal.[1](curved-layered).(c)Ourresult(2planes).

Fig.17showsacomparisonoftheprintedbunnymodel.Thenumber ofpartsandtotalseamlengthofourresultaresuperiortothecompeting methodofWuetal.[4].

ComparetothemethodofDaietal.[1],ourschemeisinferioron themotionﬂexibilitybutsuperioronthecomplexity of implementa-tion.Methodstowardconventional3Dprinting,suchashollowing, slic-ingandtool-pathplanning,canbedirectlyappliedoneachpartofour results.Wehaverealizedsupport-freeornearsupport-freeprintingon manymodels(seetheresultsofarmadillomodelinFig.13andbunny modelinFigs.10and17ascomparisons).Ourprintedobjectsperform betterbeneﬁttingfrombothhigheraccuracyoftheprintingsystemand planar-layeredprintingstyle.

6. Conclusionsandlimitations

We propose a global optimization-based model decomposition methodfordiscretemulti-directional3Dprinting,whichiscapableto achievenearsupport-freeprinting.Weproposedaheuristicmethodto partitionthemodelintopartsfreeofcollisions.Thecuttingplanesand decompositionorderaredeterminedbysolvinganoptimization prob-lemthatisformulatedtominimizethetotalsurfaceareathatrequires supportingstructures.Theadvantageofourmodeldecomposition algo-rithmisthatitalwaysresultsinafewernumberofpartsandtherefore reducestheaccumulationofmechanicalsystemerroraswellasthetotal lengthofseamsbetweenpart.

Thoughourmethodcanachievenearsupport-free3Dprinting,itstill haslimitations.First,theoptimalcuttingplanesaredeterminedusinga geneticalgorithmwhichiscomputationallyineﬃcient,especiallywhen thenumberofcuttingplanesislarge.Second,theuserhastospecifythe numberofcuttingplanes,whichistypicallyatrialanderrorprocess re-lyingonuserexperiences.Theoretically,theoptimalnumberofcutting planescanbeautomaticallydeterminedbybruteforcesearch(i.e.,

run-ningthesameoptimizationwithdiﬀerentNvalues).Thecomputation maybecomeunaﬀordablewhenNbecomeslarge.

Acknowledgments

ThisworkispartiallysupportedbytheBeijingNaturalScience Foun-dation(J170001andL182059),theNationalNaturalScience Founda-tion of China (61772523, 61702022, 61802011 and61620106003),

ChinaPostdoctoralScienceFoundation(2018T110019),“Rixin” Train-ingProgrammeFoundationfortheTalentsbyBeijingUniversityof Tech-nology,andtheConstructionProjectforNationalEngineering Labora-toryforIndustrialBig-dataApplicationTechnology(312000522303).

References

[1] C. Dai, C.C.L. Wang, C. Wu, S. Lefebvre, G. Fang, Y.-J. Liu, Support-free volume printing by multi-axis motion, ACM Trans. Graph. 37 (4) (2018) 134:1–134:14, doi: 10.1145/3197517.3201342 .

[2] W. Wang, Y.-J. Liu, J. Wu, S. Tian, C.C.L. Wang, L. Liu, X. Liu, Support- free hollowing, IEEE Trans. Visual. Comput. Graph. 24 (10) (2017) 2787–2798, doi: 10.1109/TVCG.2017.2764462 .

[3] D. Khan, D.-M. Yan, F. Ding, Y. Zhuang, X. Zhang, Surface remeshing with robust user-guided segmentation, Comput. Vis. Media 4 (2) (2018) 113–122, doi: 10.1007/s41095-018-0107-y .

[4] C. Wu, C. Dai, G. Fang, Y.-J. Liu, C.C.L. Wang, RoboFDM: a robotic system for support-free fabrication using FDM, in: 2017 IEEE International Conference on Robotics and Automation, 2017, pp. 1175–1180, doi: 10.1109/ICRA.2017.7989140 .

[5] R. Wu, H. Peng, F. Guimbretière, S. Marschner, Printing arbitrary meshes with a 5D of wireframe printer, ACM Trans. Graph. 35 (4) (2016) 101:1–101:9, doi: 10.1145/2897824.2925966 .

[6] Y. Huang, J. Zhang, X. Hu, G. Song, Z. Liu, L. Yu, L. Liu, FrameFAB: robotic fabrication of frame shapes, ACM Trans. Graph. 35 (6) (2016) 224:1–224:11, doi: 10.1145/2980179.2982401 .

[7] S. Keating, N. Oxman, Compound fabrication: a multi-functional robotic platform for digital design and fabrication, Rob. Comput.-Integr. Manuf. 29 (6) (2013) 439–448, doi: 10.1016/j.rcim.2013.05.001 .

[8] Y. Pan, C. Zhou, Y. Chen, J. Partanen, Multitool and multi-axis computer numerically controlled accumulation for fabricating conformal features on curved surfaces, J. Manuf. Sci. Eng. 136 (3) (2014) 031007, doi: 10.1115/1.4026898 .

(11)

[9] X. Song, Y. Pan, Y. Chen, Development of a low-cost parallel kinematic machine for multidirectional additive manufacturing, J. Manuf. Sci. Eng. 137 (2) (2015) 021005, doi: 10.1115/1.4028897 .

[10] W. Gao, Y. Zhang, D.C. Nazzetta, K. Ramani, R.J. Cipra, RevoMaker: enabling multi- directional and functionally-embedded 3d printing using a rotational cuboidal platform, in: Proceedings of the 28th Annual ACM Symposium on User Interface Soft- ware & Technology, 2015, pp. 437–446, doi: 10.1145/2807442.2807476 .

[11] P. Song, B. Deng, Z. Wang, Z. Dong, W. Li, C.-W. Fu, L. Liu, Coﬁfab: coarse-to- ﬁne fabrication of large 3d objects, ACM Trans. Graph. 35 (4) (2016) 45:1–45:11, doi: 10.1145/2897824.2925876 .

[12] L. Shapira, A. Shamir, D. Cohen-Or, Consistent mesh partitioning and skele- tonisation using the shape diameter function, Vis. Comput. 24 (4) (2008) 249, doi: 10.1007/s00371-007-0197-5 .

[13] L. Luo, I. Baran, S. Rusinkiewicz, W. Matusik, Chopper: partitioning models into 3d-printable parts, ACM Trans. Graph. 31 (6) (2012) 129:1–129:9, doi: 10.1145/2366145.2366148 .

[14] R. Hu, H. Li, H. Zhang, D. Cohen-Or, Approximate pyramidal shape decomposition, ACM Trans. Graph. 33 (6) (2014) 213:1–213:12, doi: 10.1145/2661229.2661244 .

[15] P. Herholz, W. Matusik, M. Alexa, Approximating free-form geometry with height ﬁelds for manufacturing, in: Computer Graphics Forum, vol. 34, 2015, pp. 239–251, doi: 10.1111/cgf.12556 .

[16] X. Chen, H. Zhang, J. Lin, R. Hu, L. Lu, Q. Huang, B. Benes, D. Cohen-Or, B. Chen, Dapper: decompose-and-pack for 3d printing, ACM Trans. Graph. 34 (6) (2015) 213:1–213:12, doi: 10.1145/2816795.2818087 .

[17] W.M. Wang, C. Zanni, L. Kobbelt, Improved surface quality in 3d printing by optimizing the printing direction, Comput. Graph. Forum 35 (2) (2016) 59–70, doi: 10.1111/cgf.12811 .

[18] X. Wei, S. Qiu, L. Zhu, R. Feng, Y. Tian, J. Xi, Y. Zheng, Toward support-free 3d printing: a skeletal approach for partitioning models, IEEE Trans. Visual. Comput. Graph. 24 (10) (2018) 2799–2812, doi: 10.1109/TVCG.2017.2767047 .

[19] X. Chen, H. Li, C.-W. Fu, H. Zhang, B. Chen, D. Cohen-Or, 3D fabrication with univer- sal building blocks and pyramidal shells, ACM Trans. Graph. 37 (6) (2018) 189:1– 189:15, doi: 10.1145/3272127.3275033 .

[20] C. Wu, C. Dai, G. Fang, Y.-J. Liu, C.C.L. Wang, General support-eﬀective decomposition for multi-directional 3d printing, CoRR (2018) arXiv: 1812.00606 .

[21] H. Fuchs, Z.M. Kedem, B.F. Naylor, On visible surface generation by a priori tree structures, 1980, pp. 124–133, doi: 10.1145/965105.807481 .

[22] D. Bhandari, C.A. Murthy, S.K. Pal, Genetic algorithm with elitist model and its convergence, Int. J. Pattern Recognit. Artif. Intell. 10 (06) (1996) 731–747, doi: 10.1142/S0218001496000438 .

[23] M. Srinivas, L.M. Patnaik, Adaptive probabilities of crossover and mutation in genetic algorithms, IEEE Trans. Syst. Man. Cybern. 24 (4) (1994) 656–667, doi: 10.1109/21.286385 .

[24] J. Vanek, J.A.G. Galicia, B. Benes, Clever support: eﬃcient support structure generation for digital fabrication, Comput. Graph. Forum 33 (5) (2015) 117–125, doi: 10.1111/cgf.12437 .

[25] R. Schmidt, N. Umetani, Branching support structures for 3d printing, ACM SIG- GRAPH 2014 Studio, 2014, doi: 10.1145/2619195.2656293 . 9:1–9:1