A voxel based method of constructing and skinning conformal and functionally graded lattice structures suitable for additive manufacturing

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A

voxel-based

method

of

constructing

and

skinning

conformal

and

functionally

graded

lattice

structures

suitable

for

additive

manufacturing

A.O.

Aremu,

J.P.J.

Brennan-Craddock,

A. Panesar,

I.A.

Ashcroft

∗

,

R.J.M.

Hague,

R.D.

Wildman,

C. Tuck

FacultyofEngineering,UniversityofNottingham,UK

a

r

t

i

c

l

e

i

n

f

o

Articlehistory: Received9October2015

Receivedinrevisedform11October2016 Accepted27October2016

Availableonline9November2016

Keywords: Lattice Voxel Tesselation Net-skin Functionalgrading

a

b

s

t

r

a

c

t

AdditiveManufacturing(AM)enablestheproductionofgeometricallycomplexpartsthataredifﬁcult tomanufacturebyothermeans.However,conventionalCADsystemsarelimitedintherepresentation ofsuchparts.Thisissueisexacerbatedwhenlatticestructuresarecombinedorembeddedwithina complexgeometry.Thispaperpresentsacomputationallyefﬁcient,voxel-basedmethodofgenerating latticescomprisedofpracticallyanycelltypethatcanconformtoanarbitraryexternalgeometry.The methodofconforminginvolvesthetessellationandtrimmingofunitcellsthatcanleave‘hanging’struts atthesurface,whichisapossiblepointofweaknessinthestructure.Amethodofjoiningthesestruts toformanexternaltwodimensionallattice,termeda‘net-skin’isalsodescribed.Traditionalmethods ofmanufacturinglatticestructuresgenerallydonotallowvariationofcellpropertieswithina struc-ture;however,additivemanufacturingenablesgradedlatticestobegeneratedthatarepotentiallymore optimal.Amethodoffunctionallygradinglatticesis,therefore,alsodescribedtotakeadvantageofthis manufacturingcapability.

1. Introduction

The geometric freedom afforded by Additive Manufacturing (AM)allowsthefabricationofcomplexlatticestructureswithout theuseoftooling.Ratherthanformingpartsinasubtractive(e.g., machined)orformative(e.g.,moulded)manner,AM‘prints’parts inlayers.TechniquesclassiﬁedasAMincludepowerbedfusion, vatpolymerization,jetting,materialextrusion,sheetlamination anddirectenergydeposition(ASTM2792-12).Industrial exploita-tionof these techniquesrequires research and developmentin processes,materialsanddesign-systems.Whileprocessand mate-rialshaveadvancedconsiderably,therehasbeenlessprogressto dateondesignissues. Thedevelopmentof designstrategies for AMenablesseveraladvantagesoverconventionalmanufacturing.

夽OneoftheauthorsofthisarticleispartoftheEditorialBoardofthejournal.Full responsibilityfortheeditorialandpeer-reviewprocessforthisarticlelieswiththe journal’sEditor-in-ChiefProf.RyanWickerandDeputyEditorProf.EricMacDonald. Furthermore,theauthorsofthisarticlehadnoanddonotcurrentlyhaveaccessto anyconﬁdentialinformationrelatedtoitspeer-reviewprocess.

∗Correspondingauthor.

E-mailaddress:[email protected](I.A.Ashcroft).

Onesuchadvantageistherealizationofcomplexlight-weighted parts.Bydesigningapartwithmaterialonlyintheregionsthat arerequired,thepart’sstrengthtoweightratioisincreased.Inthe aerospaceandautomotiveindustries,thistranslatestoeconomic savingsfromreducedfuelconsumptionandareducedcarbon foot-print.

Currently,twoparadigmsareprominentlyusedtorealize light-weightedpartsforAM.Theﬁrst,topologyoptimization(TO),seeks theoptimaldistributionofmaterialinapartbyapplying math-ematicalalgorithmstothepart’sdomain[1].Anumberofthese algorithmshave beendeveloped, includinghomogenization[1], evolutionary structural optimization [ESO] [2,3], solid isotropic material with penalization [4] and level sets [5]. According to Brackettetal.[6],theutilizationofthesealgorithmsiscurrently limitedbymesh resolution,manufacturingconstraintsand post processing.Also,Aremuetal.[7]showedthatthecomplexityand performanceofamechanicalpartoptimizedwithabidirectional evolutionary structuraloptimization [BESO]algorithm could be compromisedbyvaluesassumedfortheoptimizationparameters. RecentresearchhasseenthedevelopmentofAMtechniques tomanufacturemulti-functionalcomponents[8].Optimizationof suchpartsrequiresmultipleobjectives,witharesultingseriesof solutionsinaParetooptimalset[9,10].Thecomputationaleffort

http://dx.doi.org/10.1016/j.addma.2016.10.006

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niquesareyettobeconsideredinsuchalgorithms;thereforeitis difﬁculttorealizestructuralfeatures oftheoptimized solutions withoutthe use of supports. Such supports couldcompromise thequalityofthesolutionsinceitisimpracticaltoremove sup-portsfromlattices.UntilasuitableTOalgorithmisdevelopedthat includesAMconstraintsforlatticesolutions,itisbeneﬁcialto con-sider a more pragmatic approach for designing lattices for AM objects.Thereisa broad range ofpredesigned lattices, mostof whichareinspiredbynaturalsystems[15],thatcouldbemade withAMtechniques.Understandingthemultifunctional capabili-tiesoftheselatticeswouldsupportadesignerintheexploitation ofAM’sdesignspace.

Agrowingnumberofpublicationsareinvestigatingthe struc-turalpropertiesofpredesignedlatticeswithAMtechniques;some ofthese canbeseen in [16–24].Razmirez et al.[16] manufac-turedopencelllatticeswithanAMtechniquecalledelectronbeam melting.Yanet al.[17] fabricatedgyroidlattices withselective laser melting (SLM) using stainless steelpowder and observed anincreasein yieldstrength ascellsizewasreduced. Contuzzi etal.[18]determinedthecompressivestrengthofreinforced pil-lartextilelatticeswhileSmithetal.[19]investigatedtheeffects ofreinforcementsonbodycentredcubiclattices.Yan[20]studied thepropertiesofaluminiumlatticesatdifferentvolumefraction andshowed thatthemicro-hardness and compressivestrength decreaseswithincreaseinthecellsize.Mullenetal.[21]usedSLM toproducetitaniumlatticestosupportbonegrowth.McKown[22]

investigatedthebucklingandimpactbehaviourofbodycentred cubiclatticeandobservedcontrastingmodesoffailurefordifferent conﬁgurationsofthelattice.Vesenjaketal.[23]observedthat lat-ticeswithcircularcellshadbetterenergyabsorptionthanquadratic cells,bothofwhichweremanufacturedwithselectivelaser sin-tering.Generally,latticesmanufacturedwithAMtechniquesare self-supportingmakingthemsuitableassupportstructureswhen buildingotherAMobjects.Husseinetal.[24]showedthiswith gyroidanddiamondlattices. Inorder toenhance thequalityof parts,acompromiseisneededbetweenthelatticecellsizesand volumefractionofthesupport.Exploitationof latticeswiththe designfreedomofferedbyAMreliesontheavailabilityofdesign methodsandsoftwaretools,whicharestilllimitedforAM.Guo andLeu[25]calledforconceptualdesignmethods,novelCAD sys-temsandsimulationcapabilitiestoaddressthisneed.Aﬁrststep todevelopinglatticedesignmethodsandsystemsistheselection ofappropriategeometryrepresentationschemes,sincethe capa-bilitiesofcommercialcomputer-aideddesign(CAD)packagesin representinglatticesarelimited.

Conventional(CAD)toolsarenotefficientatrepresentinglattice structuresduetotheunderlyingmethodsbywhichtheyhandleand manipulategeometry.Themaingeometryrepresentationschemes areboundaryrepresentation[Brep],constructivesolidgeometry [CSG],spatialsubdivision,medialsurfacerepresentationand pro-ceduralrepresentation[26].B-repandCSG,themostcommonof these,areimplementedinCADsystems[26,27].B-repstoresthe features of anobject as a series of boundaries defined by ver-tices, edgesand surfaces[28]; while CSGrepresentsthe object asBooleanandtranslationaloperationsbetweenaseriesof sim-plerobjects[29].Definitionofbasicelementsinbothschemesare

Littleworkhasbeencarriedouttodatetoapplytheseschemesto solvingdesignissuesinAM.ChenandWang[33]usedanadaptive contouringmethodbasedonaspatialschemetorepairerroneous B-repmodels.Similarly,thevoxelrepresentationschemecanbe usedtoembedavastrangeoflatticetopologiesincomplexthree dimensionalobjects.

Thereis a need, therefore, for easy to use, computationally efﬁcientandgeometricallyﬂexiblemethodsofgeneratinglattice structurestotakeadvantageoftheextensioninlattice manufac-turingenabledbyAM.Thispaperproposesanovel,voxel-based methodofgeneratinguniformorgradedlatticesthatcanconform toanarbitraryexternalgeometryviatessellationandtrimming. Anartefactofsuchtrimmedstructuresistheexistenceofweak, unconnectedstructuralfeaturesattheboundariesofthedomain; whichcouldinitiatefailureinservice.Amethodofaddressingthis problembygeneratingasurfacelattice,ornet-skin,fromthevoxel modelofthetrimmedstructureanditsdomainisalsodescribedin thepaper.

Thenextsectionof thepaperdiscussessomeof thegeneral issuesconcerningthegenerationofconformallatticestructures. Thisisfollowedbydescriptionsoftheproposedmethodsof gener-atinglattices,trimmingtoanexternalgeometry,creatinganet-skin andfunctionallygradingthelattice.Numerousexamplesareused todemonstratethesemethods.

2. Constructinglatticestructures

Latticestructuresarecomposedofrepeatingelements tessel-latedacrossadomain.ConsiderFig.1a,whichshowsanelementary unit,usuallycalledthe‘unitcell’.Tessellatingeightunitsofthiscell inthexandfourintheydirectiongivesthelatticeshowninFig.1b. Thelatticestructureinthiscasecanbeseentoberectangular.Itis oftendesirabletofitatessellatedlatticetoamuchmorecomplex domain.Themannerinwhichthisisachievedinfluencesthe prop-ertiesofthelattice.ConsiderthearbitrarydomainshowninFig.2a. Thelatticecaneitherbeswepttofitwithinthedomain(Fig.2b), meshandmappedintothedomain(Fig.2c),orthedomaincanbe usedasatemplatetotrimatessellatedlatticestructureasseenin

Fig.2d.Regionsofthelatticeresidingontheinteriorofthedomain areretainedwhilstothersareremoved.

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Fig.1.Atwodimensionallatticestructuregeneratedbytessellatingaunitcellalongxandyaxes:a)UnitCellb)Latticestructure.

strutsofalatticestructure.Themainlimitationofthisapproach isthattheemergentstructurewillbedistorted,thisbeing depen-dentontheexternalgeometryoftheobjectgeometryasinthe sweptlattice,againdeviatingfromtheoriginalpropertiesofthe lattice.Therangeofcelltypesgeneratedbythismethodisalso lim-ited.Trimmingtessellatedlattices(Fig.2d)withthedomainisa betterapproachifthepropertiesoftheunitcellaretobelargely retained.ThisisakintoaBooleanintersectionofthedomainand atessellatedlatticestructure.Themainadvantageofthetrimming methodisthatitisrobust,sinceitisequallyapplicabletoboth sim-pleandcomplicatedcellsanddomains.Themaindisadvantageis theemergenceofweakboundaries,wherethetrimmedunitcells lacksupport.Apossiblesolutiontothisproblemistoprovidea solidskinalongtheboundaries,asshowninFig.3;however,this willlimitaccesstothelattice,whichmaybedetrimentalfor cer-tainapplications,hinderpostprocessing(e.g.unprocessedmaterial removal)andpotentiallyaddunnecessaryweight.Theremayalso beanaestheticreasontohavethelatticestructurevisible.

Asmall number of publicationsdescribe methods toembed trimmedlatticesinthreedimensionaldomains.Paskoetal.[39]

usedperiodicfunctionstomodelregularandirregularlattice struc-tures.Representinglatticestructureswithfunctionsisanefﬁcient androbustapproachtolatticegeneration,allowingdynamic con-troloftopologicalfeatures.However,alimitednumberoflattice topologiescanberepresentedthisway.Nguyenetal.[40] devel-opedamethodthatﬁtsalatticetoadomainbydecomposingthe domainintoeasilymappedregions.ThemethodisbasedonB-rep andwasshowntobesuitableforlatticescomposedoftruss-like structuresbutisyettobeprovenforotherstypes.Generatinglattice structuresviavoxelmodels,however,ispotentiallymoreversatile thanboundaryorfunctionrepresentationandformsthebasisof themethodpresentedinthispaper.

Fig.3. Alatticestructurepartiallycoveredbyasolidskin.

3. Methodology

Inthissectiontheproposedmethodofcreatingconformallattice structurewillbedescribed.Firstly,thevoxelmethodof represent-ingaunitcellisdescribed,followedbythemethodoftessellating andtrimmingtheunitcelltoconformtoanexternalgeometry.A methodofcreatinganet-skintojointhetrimmedstrutsatthe sur-faceisthendescribedtoprovidebetterstructuralintegritytothe latticestructure.Finallyamethodoffunctionallygradinga confor-mallatticeisdescribedthatenablesmoreoptimallatticestructures tobegeneratedforspeciﬁcapplications.

3.1. Trimmedlatticestructure

Thestartingpointtothemethodisageometricdescriptionof thelatticeunitcellandtheexternalgeometrythatwillformthe designdomainforthelattice.Thetopologyofthelatticedomain andunitcellcanconvenientlybeachievedwithconventionalCAD packageswhichrepresentthedomainasab-repmodel,although almostanymethodofrepresentinggeometrycanpotentiallybe used.Thedomainandcellgeometriesarethenvoxelizedusingany

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Fig.4.a)Lowresolutionpixelimageofacircleb)similarlylowresolutionvoxelmodelofasphere.

ofanumberofmethods,suchastheonesdescribedin[41–43]. Voxelmodelsdeﬁnethevolumeofashapeinabitwisemanner, unlikethesurfacerepresentationinmostCADsoftware.Analogous totherectangularpixelsthatrepresenta2Dimagesuchasabitmap, voxelsarediscreteblocksand,assuch,voxelmodelshave‘stepped’ surfaces[44],asillustratedinFig.4.Avoxelmodelisessentiallya 3Dmatrix,witheachelementofthematrixrepresentingavoxel. Thesimplestformofthesemodelsoccurswhenthevoxelsassume alogicvalue(i.e.oneorzero),withoneindicatingasolidspaceand zeroindicatingvoidspace[45].

Awidevarietyoflatticecells cannowbebuiltwithadditive manufacturing,withsomeexamplesshowninTable1.The intrin-sicstructuralpropertiesofthelatticearebuiltintotheunitcell.This mightincludethedimensions,density,thicknessesofsurfacesand diameterofstruts,dependingonthetypeoflatticecell.For exam-pleahelicaldiameterdeﬁnesaspring-likestrut.Ahelicaldiameter ofzerodeﬁnesasimplestraightstrut.Voxelmodelsofthedomain andunitcellmodelsareconsideredasaseriesof2Dslicesstacked inamannerthatpreservesthestructuralfeaturesoftheoriginal b-repmodel.Itshouldbenotedthatmorevoxelsarerequiredto representahighlycomplexunitcellascomparedtocellswith sim-plertopologies.Forexample,thehelicalunitcellsshowninTable1

wouldrequiremorevoxelsthanthecellswithstraightmembers. Inordertogeneratea practicallyusefullatticestructure,the dimensionsoftheunitcellshouldbemuchsmallerthanthatof thedomain,i.e.byat leastanorder ofmagnitude. Togenerate thetrimmedlatticestructure,operationsanalogoustoaBoolean intersectionareperformed.Thevoxelmodeloftheunitcellisﬁrst tessellatedtoencompassthedomain.A2Dsliceofatessellatedunit cellanditscorrespondingdomainisshowninFig.5.For descrip-tivepurposes,voxelssettoonearereferredtoasbeingactiveand thosewithavalueofzeroareinactive.Anentirelydeactivatedarray isinitiatedtocontainthetrimmedlatticestructure;withthesame sizeasthatofthedomainandthetessellatedstructure.Thedomain arrayischecked,element-wise,andvoxelsinthetrimmed struc-tureareactivatediftheelementsincorrespondingpositions of thedomainandtessellatedstructureareactive,asdemonstrated

with‘checka’inFig.6.Otherwise,voxelsinthetrimmedstructure remaindeactivatedasillustratedby‘checkb’and‘checkc’.

Themethodrunsbothdomainandtessellatedstructurethrough anANDgate−ﬁndingthemaximumvalueandpassingthatonto theequivalentelementofthetrimmedstructurearray.Theprocess isanalogoustoaBooleanintersectionbetweenthedomainandthe tessellatedstructureresultinginmaterialretentioninregionsthat hadmaterialinboththedomainandthetessellatedstructure.

Forsomeapplications,incliningthelatticeatanangletothe domaincouldenhanceperformance.Generatingarotatedlattice canbeachievedbyrotatingthedomainpriortovoxelization.Most CADsoftwareisequippedwithtoolsforsuchanoperation.After thedomainshowninFig.6isrotatedby45◦andvoxelized,itis transformedtoFig.7.Thesteppedboundariesareaconsequenceof thevoxelrepresentationandarelesspronouncedwhenﬁnervoxel meshesareused.Also,therotateddomainwouldrequirealarger voxelmodelsincetheboundingboxwouldincrease.Tessellating andtrimminggivesthetrimmedstructureshowninFig.7,which couldbeconsideredasarotatedlatticewithrespecttotheoriginal domain.Rotatingthevoxelizeddomainortessellatedunitcellcan achievesimilarresultsbutatagreatercomputationalexpensethan theformerstrategy;andcouldintroduceprecisionerrorsintothe process.Forathreedimensionaldomain,rotationofthedomainis abouttheaxesofinclination.

Whilevoxelizingandtessellatingthecell,itisimportantto pre-determinethenumberofvoxelsrequiredforbothcellanddomain. Theratiobetweenthesizeoftheunitcellandthedomain deter-minethenumberofcellsthatcanbetessellatedacrossthedomain andthecellsize.Forexample,acubicdomainwith10mmsides andmodelledwith150voxelswillrealizeatrimmedlatticewith 2mmcellsizeiftheunitcellsaremodelledwith30voxels; reduc-ingthenumberofvoxelsemployedforthedomainwhileﬁxing thecellvoxelsat30wouldincreasetheunitcellsize.However, suchareductioncouldleadtoalossofgeometricprecision.This canalsohappenifthenumbersofvoxelsarereducedinthecell.It ispreferabletoutilizetheleastnumberofvoxelsthatcapturethe mostessentialcellularfeaturesandthendeterminethenumberof

Table1

Aselectionofstructuretypesintegratedintothestructuretrimmingprocess.

Cell Kelvin/truncatedoctahedron Cubic Alternatedcubic

Struttype Triangularcross-section Cylindrical Wave Helical Helical Cylindrical

ModelView

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Fig.5.Generationofatrimmedstructureslice−analogoustoaBooleanintersection.

Fig.6.Trimmingstructuretoadomain−analogoustoaBooleanintersection.

Fig.7. RotationofaDomainwithrespecttothetessellatedlattice.

voxelsrequiredforthedomaintoachieveaspeciﬁccellsize.This isillustratedforamechanicalpartinthenextsection.

3.2. Anexampleofacomplexmechanicalpartembeddedwitha latticestructure

Thecapabilityofthevoxellatticegenerationmethodtoconform toanexternalgeometryisillustratedinFig.8,whereregionsofa mechanicalpartarelatticedwhiletheothersremainsolid.Regions oftheparttobelatticedareclassiﬁedasthelatticedomainand arecolouredgreeninFig.8a.Theotherregionsarethereforethe non-designdomainsincetheyinteractwithneighbouringparts (Non-designisgrey).ThecellshowninFig.8bbelongstothefamily ofminimalsurfacesgovernedbymathematicalfunctions,thathas potentiallyusefulpropertiesforAMsystems.Thepartexperiences thermalcyclingin-service.Embeddingalatticeinthepartoffers awaytodissipateheattransferred(sincelatticesusuallyhavea highsurfacearea)whilereducingtheweightofthepart.The lat-ticedomainisreshowninitsboundingboxinFig.8c.Toachievea cellsizeof4mmwithacellularvoxelarrayof30;thesizeofthe domainarraywassetat489by459by88.Thissizeisan approxi-matemultipleoftheboundingbox.Bychangingtheresolutionof

thesearrays,thecellsizeiscontrollable,asmentionedpreviously. Usingthevoxelmethod,thepartwaslatticedandmanufactured viaselectivelasermeltingusinganaluminiumalloy.Fig.8dshows themanufacturedlatticedplateafterpostprocessing.

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Fig.8. Astructuralpartembeddedwithadoublegyroidlattice:(a)partgeometrywithlatticedomaindeﬁned,(b)doublegyroidunitcell,(c)latticedomaininbounding box,(d)latticedpartmanufacturedwithselectivelasermelting.

handlingsurfacesorforaestheticreasons.Amethodofgenerating net-skinsfromthevoxelmodelofthetrimmedlatticeisdescribed inthenextsection.

3.3. Generatinganet-skinviaorthographicvoxelprojection

Itisgenerallypreferabletoconnectcutmembersatthe bound-ariesoftrimmedstrut-basedlatticesastheyprovideaweakpoint inthelatticestructureandarealsoaestheticallydispleasing. Con-siderthetrimmedlatticeshowninFig.9a,withcutmembersatits boundary.Addinganet-skintothetrimmedlattice,resultsinthe structureshowninFig.9b.Theskinisaporousformofthesolid skin,andcanbeconsideredasa2Dsurfacelatticeconformalto theoutergeometry,asshowninFig.9c.Thistypeofskinimproves theintegrityofthestructurewhilstaidingpostprocessing oper-ations,suchastheremovalofpowder.Thissectiondescribesan approachtogeneratenet-skinstoconnectthehangingfeaturesat theboundariesofatrimmedlattice.

Itispossibletogeneratea net-skinfromthevoxelimageof thetrimmedlatticestructure,A.Thisapproachconsistsofthefour stagesshowninFig.10.Themainideaofthemethodistoconstruct

avoxelmodelofthenet-skinbyclassifyingvoxelsonasolidskin aseitheractiveorinactive.Theﬁrststepisthereforetodetermine thevoxelimageofasolidskin.Onceasolidskinhasbeencreated; projectionofactivevoxelsinthetrimmedstructurecanbeusedto determinewhichvoxelsinthesolidskinwillbeactive.The selec-tionofvoxelstoprojectisimportantinterms ofcomputational efﬁciency.Thesetofvoxelsprojectedisthereforelimitedtothose closesttothesolidskin.

Firstly,thesolidskinisdeﬁnedbyalayerofactivevoxels resid-ingontheboundaryofthevoxeldomain,asillustratedinFig.11for atwodimensionalcase.Interiorvoxels(Fig.11a)aredeactivatedby changingtheirvaluetozero;givingthehollowimageofthesolid skinshowninFig.11b.Theboundaryvoxelscanbedetermined byﬁndingthenumberofactiveneighbouringvoxels.Alleight sur-roundingvoxelsofaninteriorvoxelareactive,incontrasttothose forboundaryvoxels.

Assumingthevoxelizeddomainisdenotedbyanarray,Band thesolidskinisC,thiscanberepresentedby

Cij=

1,if

Bij=1

∧(f

Bij

<8)

0,otherwise

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Fig.11.AtwodimensionalsolidofNet-skinforanarbitrarydomaina)VoxelizedDomainb)Solidskin.

whereiandjarerowandcolumnindices,f

Bij

isanoperationon

BijtodeterminethenumberofactivevoxelsconnectedtoBij. MAT-LAB’s“bwperim”functioncanbeusedtodeterminethesolidskin. Forathreedimensionalcase,26voxelsareintheneighbourhood ofasinglevoxel.Therefore,equation1becomes,

Cijk=

1,if

BijK=1

∧(f

Bijk

<26)

0,otherwise

wherekistheindexinthethirddimension.Secondly,active vox-elsresidinginAwhoseindicesareclosetothoseinCareselected forprojection.Thisstageimprovesthecomputationalefficiencyof themethodbylimitingthenumberofvoxelsprojectedtothisset, whichissufficienttoformthenet-skin.Activevoxelsinthegrey regioninFig.12belongtothissetandareusedtoformthenet-skin. Thesizeoftheselectedregionwoulddifferfordifferenttrimmed lattices,therefore,foracompleteskin,thesizeoftheregionmust becontrolledadaptivelyuntilasuitableskinisformed.The selec-tionofvoxelsintheconcernedregionconsistsoftwooperations. ThefirstistheerosionofthevoxelizeddomainBwhichisthen sub-tractedfromthetrimmedlatticestructureA.Theerodeddomain isusedtoremoveactivevoxelsatthecenterofthedomainfroma projectionset.Forathreedimensionallatticeanorthogonalview ofthetrimmed latticefromthetop,bottom,rightorleftwould showthatactivevoxelsatthecenterofthedomainareshieldedby

Fig.12. Anillustrationofselectedvoxelsforprojection.

thoseclosesttothesolidskin.Therefore,thesecentervoxelsdonot inﬂuencethenet-skin.

Ifthedomainshown inFig.11ais eroded,theimageshown in Fig.13a is created.The broken lineillustrates theboundary oftheun-erodedimage.Subtractingtheerodeddomainfromthe trimmedlatticegivesFig.13b.Inordertoincreasethenumberof voxelsprojected,thedomainiserodedanumberoftimestoreduce theregionsubtractedfromthetrimmedstructures.Itispossibleto constructanarrayDcontainingvoxelstoprojectfromAandBusing

Dijk=

1_,_ifA₋(BT)=1 0,otherwise

whereisamorphologicalerosionofBwithastructuringelement T[46].MATLAB’s“imerode”functioncanbeusedtoerodeB.Active voxelsinDareprojectedalongorthogonalaxesinthethirdstage. Theprojectionisachievedbyaddingintegralmultiplesofunit vec-torsalongtheseaxestotheindicesofactivevoxelsinD.Thethree dimensionalCartesiancoordinateaxisshowninFig.14acanbe rep-resentedasthearrayinFig.14b.TheﬁrstrowinIisaunitvector alongthepositivex-axiswhileotherrowsrepresentthoseforthe otheraxes.MultiplyinganyrowinIwithaninteger,nandadding

Fig.14.Cartesiancoordinatessystema)graphicalaxesb)arrayIcontainingvectors ofthecoordinateaxis.

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Fig.15.Firststageinextractingboundaryvoxelsillustratedwithacubea)cubemodelb)SectionalviewX–Xofsolidskinc)BCCUnitcelld)trimmedlatticee)latticeand solidskinf)projectionalongz-axiswithedgevoxelsg)projectionalongz-axiswithoutedgevoxelsh)Net-skin.

toavoxelinDtranslatesthatvoxelinD.Thevalueassumedbyn determinestheextenttowhichavoxelisprojected. Mathemati-cally,theprojectionofthissinglevoxelalongthex-axisisgivenby

V=[ijk]+n[100]

whereVisavectorcontainingtheindicesofthenewvoxel. Sim-ilarly,activevoxelsinDareprojectedintheotherﬁveCartesian directions,withaseriesofnvalues.

Thevalueofnis selectedtoensureallprojections reachthe boundaryofthesolidskin.AssumingVisthesetofvectorsderived fromtheprojection,inthethirdstage,activevoxelsinCwithindices absentinVaredeactivatedtoachievethenetskin.Hence,thenet skinarray,Eisdeterminedby,

Eijk=

1,if

Bijk=1

∧([i,j,k] ∈V)

0,otherwise

Theselectedactivevoxelsmaybeprojectedinall6Cartesian directionsdependingonthetopologyofthedomain.However,for someapplications,itmaybesufﬁcienttoprojectalongasingleaxis. Forexampleiftheboundariesofalatticedomainareincontactwith anon-designsolidwall.Insuchacase,projectionofactivevoxels inDalonganaxisthatisnotincontactwiththeotherregionsof

anobjectissufﬁcient.Suchanoperationrequiresonlytworows inI,i.e.therowsthatreﬂectthepositiveandnegativedirections ofthechosenaxis.Forathreedimensionalcase,considerthe vox-elizedimageofacubeshowninFig.15a;thishasthesolidskinin

Fig.15b.Embeddinga2by2by2bodycentredcubic(BCC)unitcell (Fig.15c)inthecubegivesthegeometryinFig.15d.Thesection viewofthesolidskinissuperimposedonthelatticeandshownin

Fig.15e.Projectingvoxel‘r’,asdenotedbythebrokenline,along thez-axis,marks‘s’foractivation,asillustratedin theenlarged view.Projectingselectedvoxelsalongthez-axisaxisinboth posi-tiveandnegativedirectionsgivestheimageshowninFig.15f.The skinhastworegions,thenetandconnectors.Theconnectorsappear intheskinduetotheinclusionofvoxelsattheedgeofthelattice. ExcludingthesevoxelsgivestheimageinFig.15gwhilecombining projectionsinallthreeCartesiancoordinatesgivesthegeometryin

Fig.15hwithouttheconnectors.Forobjectswithfeaturesthatare generallyorthogonaltoeachother,combiningtheportionofthe skinsinthethreeaxesisrelativelystraightforward.

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Fig.17.Ahighlycomplexdomaintoillustratenetskinconstructionviavoxelprojectiona)Complexgeometryb)Netskin.

theaimofstrengtheningtheboundaryofthetrimmedlatticeby connectinghangingfeaturesinthisregion.Thisisillustratedfora sphericaldomaininFig.16a.A4by4by4BCClatticeis embed-dedinthedomaintogiveFig.16b.Projectingselectedvoxelsalong thex-axisgivesthenet-skinshowninFig.16c.Itispreferableto projectinthedirectionofmaximumsurfaceareatomaximizethe appearanceofthenetandminimizeconnectors.Thewholedomain ofthepartshowninFig.8isillustratedinFig.17aandthenet-skin createdviavoxelprojectionforthisdomainisshowninFig.17b.

Themethoddescribedaboveissuitableforabroadrangeof lat-tices,however,itmightbelessbeneﬁcialforsurface,asopposedto truss,typelattices.Generally,thenet-skinimposesanorthographic projectionofthetrimmedlatticeonthesolidskin.Consideringthe doublegyroidandBCCcellsasrepresentativeofsurfaceandtruss

typelatticesrespectively;asviewedfromtheside,Fig.18aand 18bshowstheorthogonalviewsthatwouldbeseen.These rep-resenttheviewstransferredtothenet-skinduringtheprojection method.Theregionsthatareidentifiedas‘holes’aretheporous partsofthenet-skin.Forthedoublegyroid,thisregionisrelatively smallcomparedtothatoftheBCCcellandcanbeseentobe com-prisedofacirculararray.Athigherdensities,thecircularregions becomesmallerandgeneratinganet-skinwiththevoxel projec-tionmethodsimplyachievesanear-solidskin.Theoretically,the net-skinmethodisapplicabletothedoublegyroidandsimilar lat-tices,however,thebenefitsofsuchaskincouldbeminimal.Fig.8c showsthesideviewofadoublegyroidskintrimmedtoacubic domain.Thenet-skingeneratedbyfittingthedoublegyroidtothe complexdomainshowninFig.17aisillustratedinFig.18d,with

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Fig.19. Combiningthetrimmedstructureandthenet-skinwithaunionoperation.

theskinshowinglowerporositythanthatinFig.17b.Thiscould beusefulsinceitwilloffermoresupporttothetrimmedlattice, butwouldpotentiallyaddunnecessaryweightandrestrictpowder removal.Analternativemethodthatcanbeusedtogeneratethe skinforsurfacetypelatticesisdescribedin[47].

3.4. Combiningstructureelements

Thegenerationofthetrimmedstructureandnet-skinhavebeen describedintheprevioussections.Theseexistasapairofequally sizedvoxelarrays.Tocombinethetwo,aunionoperationis per-formed.Adeactivatedarrayisinitializedwiththesamesizeasthe trimmedstructureandnet-skin.Elementsinthearrayareactivated ifthecorrespondingvoxelsinthetrimmedstructureornetskinare active.Thisoverlaysthenet-skinontopofthetrimmedstructure, essentiallyunitingthetwo,asillustratedinFig.19.

3.5. Conversiontosliceﬁle

Apossibleroutetomanufacturingthecombinedstructure cre-atedbythepreceding stepsis toconvertthe voxelmodelto a commongeometricformforAM,suchasSTL,slicetheresulting geometryandgeneratethemachinefile.However,thenumberof facetsrequiredtodefinethelatticeasanSTLfilemakesthis compu-tationallyexpensiveanditisdifficulttoachievetrimmedlattices withhighlyrefineddetailsviathisroute.It ispreferable, there-fore,toconstructthemachinefilesdirectlyfromthevoxelmodels. ForsomeAMprocessesthatutilizerasterslices(e.g.jetting),each layerofthevoxelmodeliswellsuitedforwritingthebitmapfiles requiredbytheprocess.OtherAMprocesses,suchasselectivelaser sintering,requirevector-basedslicefiles.Toconvertthecombined structuretoavectorslice file,theoutlinesoftheshapes repre-sentedbythearrayaretraced.Thisisdepictedonasmallsectionof anindividualstrutinFig.20.Inthiscase,aMatlabedgedetection algorithmcalled‘bwboundaries’wasused,whichispartoftheimage

Fig.20.Tracingtheboundaryofastrutinavoxelizeddomain.

processingtoolbox[48].Thisalgorithm searchesforaboundary pixel(i.e.asolidpixelwithatleastonevoidpixelasaneighbour,and thensearchesitsneighboursinaclockwiseoranticlockwisefashion forthenextsolidpixel.Thisidentifiesanorderedseriesofboundary pixelsandaseparateseriesisgeneratedforeachindividualshape onthebitmap.Eachboundarypixelisdefinedbyitsrowand col-umnco-ordinates;theseco-ordinatesaremultipliedbyascaling factortoconverttomillimetremeasurements.Thescalingfactor isdependentontheresolutionofthebitmapslicesinputintothe methodology:bydefault,onepixelequals0.14mm.Eachboundary stringcanbewrittenasa‘polyline’,forexampletoaCommonLayer Interface(CLI)fileusedbylasersinteringmachines[49].

3.6. Functionalgrading

Thevoxelbasedapproachtolatticegenerationdescribedabove alsoallowstherealizationoffunctionallygradedstructures.This canbeachievedbyvaryingstrut(orsurface)geometryacrossa part.Asmentionedpreviously,thevoxelrepresentationofthevoxel structureconsidersmatricesofonesandzerostorepresentsolid and voidrespectively.Thisis analogoustotheblackand white bitmapsusedtodescribethevariousmethodsinthispaper:where blackandwhiteareusedtodenotesolidandvoid.However,the methodalso hasthecapability togenerate functionally graded structuresbyoverlayinga greyscaleimageontothedomain.A simpleexamplewouldbetooverlayagradient,asdemonstratedin

Fig.21.Ratherthanthebinaryreasoningof‘ifblack,construct struc-ture’,theprocessbecomes‘ifavoxelisaparticularshadeofgrey, constructaparticularincrementofstructure’.Agradientisdepicted withan8-bitgreyscaleimagethatisoverlaidovereach domain slicewithaBooleanintersection(toallowtheassignmentof differ-entstructuretypestoarangeofvalues);theresultantdomainslice isalsoagreyscale,8-bitbitmap.Tobeabletogeneratea function-allygradedstructure,alibraryofcellsisrequiredthatincrement fromoneendofthegradedrangetoanother.Fig.21gradesfroma

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Fig.22.Functionallygradedstructuresgeneratedfrominputgradients.

straightstruttoahelixstrut.Betweenthetwoextremesarehelical strutsthatgraduallyreduceinamplitudetozero.

Anothersimpleexampleofasuitablerangewouldbetograde fromathinstrutofaparticulardesigntoathickervariant,asshown inFig.22.Therearesixteenincrementsofthestrut design;the differencein amplitudebetweenincrementsislessthan canbe resolvedbythebitmapresolution(i.e.lessthanthepixelwidth: 0.14mm).Inpractice,anynumberofdesignincrementscouldbe implementedforfunctionalgrading.Theoverlaidimageusedto determinestrutincrementisan8-bitbitmap.Thisisa256state greyscalerange(valuesof0–255),sowhendividedintothe six-teenincrements,sixteenshadesofgreycorrespondtoeachstrut increment.8-bitgreyscalebitmapsareastandardbitmapformat andthusmanyprogrammesexistthatcangeneratethem.Many image-manipulationprogrammescangenerategradientsthatcan bewrittenas8-bitbitmaps, makingthegenerationofgradients astraightforwardtask.Any8-bitbitmapcanbeusedasinputto generatefunctionallygradedstructures.

Onceitcanbeseenthatagradedlatticecanbegeneratedfrom agreyscale,thequestionofhowtogeneratethegrey-scalearises. In[50]itisshownhowastressanalysisoradensitybasedtopology optimizationcanbeusedtogenerateagrey-scalethatrepresents theoptimaldistribution of materialdensity within a structure, basedonaspeciﬁcsetofloadsandconstraints.In[50]adithering methodwasusedtogenerateapointcloudofvaryingdensitythat wasthenusedtogeneratealatticethatwasfunctionallygradedby

varyingthecellsizethroughthestructure.Themethodof generat-inggradedlatticestructuresinthispaperprovidesaninteresting comparisonbywhichasimilarstartingpoint,i.e.agreyscale repre-sentingtheoptimaldistributionofmaterial,canbeusedtogenerate alatticewithconstantcellsizebutvariationsinindividualcell def-initionthroughthestructuretoprovidethefunctionalgrading.In bothcases,however,acriticalaspectofthesuccessofthemethod isinmappingtheperformanceofthecellsinthelatticestructure tomaterialdensity(orstiffness)inthecontinuumstructureused inthestressanalysis(ortopologyoptimization).

4. Discussion

Ithasbeenshownthatvoxel-basedmodelsareveryversatile andefﬁcientbuildingblocksforconstructinglatticestructures.As thesemodelsrepresenttheentirevolumeratherthanaseriesof boundaries,asseenwithB-repmodels,thereisnoassociatedriskof generatingmanifolderrorsorinvalidgeometries.Thespeedofthe trimmingmethodislessdependentonthetopologyofthedomain, butcouldbereliantonthecomplexityoftheunitcell,whichwill nowbeconsidered.TheMobiusknotdomainshowninFig.23ais embeddedwiththeunitcellsshowninFig.23band23c.Thesimple cubiccell(Fig.23b)hasthreecylindricalstrutsarrayedorthogonally toeachotherandconnectedatthecenterofthecell.TheF2BCCunit cell(Fig.23c)iscomposedof12struts,fourofwhichconnectthe

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Table2

BreakdownoftimespentateachstageoftrimmingtwolatticetypesnamelythesimplecubicandF2BCC.

SimpleCubic F2BCC

Triangles 49442 171994

MeanTime(s) StandardDeviation(s) MeanTime(s) StandardDeviation(s)

DomainVoxelization 69.34 0.22 69.34 0.22

CellVoxelization 1.62 0.01 5.45 0.03

TrimmingStructure 0.31 0.01 0.31 0.01

NetSkin 99.46 0.40 124.82 0.40

StructureandNetSkin 0.08 0.00 0.08 0.00

CLIﬁle 9.84 0.07 24.53 0.43

Total 180.65 0.71 224.53 1.09

cornersofdifferentfacesofthecellwhileanothereightconnectthe cornerssharingaface.

Halfofthestrutslyingonthefacesresideinadjacentcellswhen tessellated,thereforeonlyhalfofthesefacestrutsexistina sin-glecell.Therelativelyhighernumberofstrutsandconnectivityin theF2BCCtothatinthesimplecubiccellsuggestthatF2BCChas

greatercomplexity.B-repSTLﬁlesofboth cellsandthedomain werevoxelizedwithaMATLABvoxelizationscript[51],andthen subjectedtothelatticegenerationstepsdescribedintheprevious sections.Thecellswerevoxelizedwith30by30by30voxelswhile thedomainwasvoxelizedwith511by480by299voxelstobring thecellsizeto3mm.Thetrimmedlatticesandtheirnet-skinsare showninFig.24withthetimetakenateachstepshowninTable2. Eachofthestepswasrepeated100timeswiththemeantimeand standarddeviationdetermined.AdesktopcomputerwithIntel(R) Core(TM)CPU i7-3820(3.60GHz)and64GBofRAMona 64-Bit windowsoperatingsystemwasusedforallthestepstoensurea faircomparison.

ItcanbeseeninTable2thatthetimefordomainvoxelization

wasequalforbothcelltype,thisislogicalasthecellswereﬁttedto thesamedomain.However,thehighercomplexityassociatedwith theF2BCCrequiresittobemodelledwithagreaternumberof tri-angles,aconsequenceofwhichisahighervoxelizationtime(three times)thanthatofthesimplecubiccell.Muchofthedifferencein voxelizationtimecanbeattributedtothetimerequiredtoreadand processthegreaternumberoffacetsinanF2BCCSTL.

Itcanbeseenthatthesameamountoftimeisspentintrimming thetessellatedlatticeandcombiningthetrimmedlatticewiththe domain.Thelowstandarddeviationsuggeststhetrimmingstep hashighrepeatabilityandislessdependentonthecomplexityof thecell.However,moretimeisspentconstructingthenet-skinfor theF2BCClatticethanforthesimplecubiclattice,astheprojection methodisalsodependentonthecomplexityoftheunderlyingcell. Combiningthetrimmedstructurewiththenet-skinrequiredthe sametimeforbothcelltypesbutCLIfilegenerationdiffered signifi-cantly.Itcanbeinferredfromthetablethatthedomainvoxelization andnet-skingenerationstepsdeterminetherateatwhichthe lat-ticeisgenerated,withonlythelatterbeingsignificantlyaffectedby thecomplexityoftheunitcell.

Anadditionaladvantageoftheproposedmethodisthatit is scalable;resolutioncanbeeasilychangedtogeneratelargeparts

atalowresolution(for‘draftquality’structuresorforcoarserAM processes)orsmallpartsatahighresolution.Generatingalarge part(i.e.onerequiringalargenumberofvoxelstodefinethedesign domain)withhighresolutioncoulddrasticallyincreasethe compu-tationalcost,hence,a‘drafting’capabilityisofsignificantpractical benefit.Highresolutionshouldallowbetterrepresentationofthe latticedetail;howeverthereisaminimumlevelofdetailthatcanbe achievedwithanAMsystem(oranyothermanufacturingprocess), alsothebuildvolumeofmostAMmachinesislimited,hence,there arecalculablelimitstothemaximumresolutionneededandsizeof files.Thevoxelmethodisthereforewellsuitedforthesemachines sincethecomputationalcostofthemethodisreasonableandcan bematchedtomachinecapablity.

Insummary,thispaperhasattemptedtoaddresstheneedfor amethodtogeneratelatticegeometriesthatismorecompatible thancurrentCADmethodswiththestructuresrealizableviaAM. Characterizingtheactualperformanceof suchlatticestructures isstill anactiveareaofresearch,andastheaimofthecurrent paperwastopresent a novelmethodoflatticegeneration,the functionalrequirementsofthetrimmedlatticeswerenotexplicitly considered.Theeffectssolidandnet-skinsonthefunctional perfor-manceofthelatticesrequiresfurtherresearch,however,[50]may bereferredtoforinitialworkinthisﬁeld.

5. Summaryandconclusions

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ﬁnishlimitationsoftheprocessprovidesacalculablelimitbeyond whichfurtherresolutionisunnecessary.

Acknowledgements

Thisworkwasfundedbytheengineeringandphysicalscience researchcouncil(EPSRC)undergrantEP/I033335/1andInnovate UKundergrantTS/J002518/1.

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