Strategies for functionally graded lattice structures derived using topology optimisation for additive manufacturing

Strategies

for

functionally

graded

lattice

structures

derived

using

topology

optimisation

for

Additive

Manufacturing

Ajit

Panesar

_,

_Meisam

_Abdi

_,

_Duncan

_Hickman

_,

_Ian

_Ashcroft

a_{AdditiveManufacturingand3DPrintingResearchGroup,UniversityofNottingham,NG72RD,UK} b_{DepartmentofAeronautics,ImperialCollegeLondon,SW72AZ,UK}

c_{FacultyofTechnology,DeMontfortUniversity,LE19BH,UK}

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received5April2017 Receivedinrevisedform 26September2017 Accepted13November2017 Availableonline14November2017

Keywords:

Latticestructures Functionalgrading Topologyoptimisation

Designforadditivemanufacturing Finiteelementanalysis

a

b

s

t

r

a

c

t

AnumberofstrategiesthatenablelatticestructurestobederivedfromTopologyOptimisation(TO) resultssuitableforAdditiveManufacturing(AM)arepresented.Theproposedstrategiesareevaluatedfor mechanicalperformanceandassessedforAMspecificdesignrelatedmanufacturingconsiderations.From amanufacturingstand-point,supportstructurerequirementdecreaseswithincreasedextentoflatticing, whereasthedesign-to-manufacturediscrepanciesandtheprocessingefforts,bothintermsofmemory requirementsandtime,increase.ResultsfromFiniteElement(FE)analysisforthetwoloadingscenarios considered:intendedloading,andvariabilityinloading,provideinsightintothesolutionoptimalityand robustnessofthedesignstrategies.LatticestrategiesthatcapitalisedonTOresultswerefoundtobe considerably(∼40–50%)superiorintermsofspecificstiffnesswhencomparedtothestructureswhere thiswasnotthecase.TheGradedstrategywasfoundtobethemostdesirablefromboththedesignand manufacturingperspective.Thepresentedpros-and-consforthevariousproposeddesignstrategiesaim toprovideinsightintotheirsuitabilityinmeetingthechallengesfacedbytheAMdesigncommunity.

©2017TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).

1. Introduction

Cellularmaterialshavebeenutilisedforcenturiesinawide vari-etyofapplicationsandarecommoninnaturalmaterialssuchas wood,bone,spongeandcoral.Thesecanberegardedasa struc-turethatconsistsofanetworkofsolidstrutsorplateswhichform theedges and faces of cells [1]. More recently thesematerials havebeenspecificallydesignedtofulfilmulti-functionalmaterial requirementsinweightreduction,energyabsorption,heattransfer andthermalinsulation[2–5].Conventionalmethodsoffabricating metalliccellularmaterialsincludeliquidstateprocessingsuchas directfoamingandsprayfoaming,solidstateprocessing includ-ingpowdermetallurgysinteringofpowdersandtheuseoffibres, electro-depositionandvapourdeposition[6,7].Althoughaltering parametersofthesemanufacturingprocessesallowsforsome con-troloverporeshapeand size,theyremainlimitedtoproducing randomlyorganisedstructures. Thisis in contrasttothe layer-by-layermanufacturingparadigmofAdditiveManufacturing(AM) whichenablesthecreationofcellularmaterialswithapredefined

∗Correspondingauthorat:DepartmentofAeronautics,ImperialCollegeLondon, SW72AZ,UK.

E-mailaddress:a.panesar@imperial.ac.uk(A.Panesar).

externalgeometryandinternalarchitecture.Tangetal.[8]provides anoverviewofthevariousfieldswherecellularmaterialdesigned specificallyforAMarebeingutilised.Cellularstructuresdesigned forfabricationviaAMarecommonlyreferredtoaslattice struc-turesduetobeinginherentlynon-stochastic.Otherterminologyin literatureincludeslightweightstructuresandfoams,whilemany ofthesetermscanbeusedinterchangeably,latticestructureshas beenchosenasthemostaccuratedescriptorofthestructures inves-tigatedherein.

AMenablestheproductionofcomplexgeometriesthatwere previouslydifficultorimpossibletomanufactureandthishasled tothecreationofmanydifferentdesignstrategiesinorderto man-ufacturepartsthat arebothhighquality,withintolerancesand economictoproduce[9].Furthermore,designscannowberealised thathavehighspecificstiffnessandstrengthwhilstreducing(or eliminating)materialwastage,primarilyduetoAM’ssynergywith topologyoptimisation(TO), a structuraloptimizationtechnique thatiterativelyimprovesthemateriallayoutwithinagivendesign space,fora givensetofloadsand boundaryconditions[10,11]. However,itisimportanttoconsidermanufacturingspecific fac-torswhendesigningforAM,theseinclude:requirementforsupport material,limitationsinbuildangle,wallsize,holesizeand dimen-sionalaccuracy[12,13].

https://doi.org/10.1016/j.addma.2017.11.008

Therehavebeenmanyapproachesproposedforcreating opti-misedlatticestructures.Brackettet al.[14] suggestedmapping volumefractions of thelatticeunit cells onto the intermediate densitiesofanun-penalisedSIMPsolution,makingthegreyscale densitysolutionofTOpossibletomanufacturewithAM.Thiswork utilisedthetessellationoftheunitcellforthelatticegeneration,i.e. whereaselectedunitcelltemplateistessellatedacrossthedesign domaininaregularfashion.Chengetal.[15]presentedamethod wherethedensitydistributionfromTOresultswereusedtoobtain theradiivalues(employedforthecreationofstrutmembers)at theverticesoftheunitcellstoallowthematerialgradingofa tes-sellatedlatticestructure.TeufelhartandReinhart[16]optimised thestrutdiameters ofanirregularlatticestructureby capitalis-ingonthefluxofforcewithinasolidstructure.Theyshowedthat enhancementinperformancecanbeachievedwhencomparedtoa regular(uniformlytessellated)counterpart,mainlyduetoamuch smootherdistributionofstressesowingtothestressconformal natureof thelattice.Togenerate latticewithvaryingcellsizes, Brackettetal.[17]offeredanerrordiffusionbasedmethodwhich enabledthemappingofirregularunitcelllatticeontoagreyscale input.Thiswasachievedbygeneratingditheredpointsfromthe greyscaleimagefollowedbyconnectingthem,usingeither Delau-naytriangulationorVoronoitessellation.

Genericgroundtrussstructureapproaches [18–20]havealso beeninvestigatedforlatticeoptimisation,withsomefocusedon AM[21].Heuristicmethodswhere unitcells arechosen froma predefinedlibraryandsubsequently subjectedtosize optimisa-tiontosupportdifferentstressstates(withinastructure)havealso beendevelopedtocreateconformallattices[22].Alzahranietal. [23]progressedtheabove-mentionedheuristicmethodfurtherto utilisetherelativedensityinformationobtainedfromTOto auto-maticallydeterminethediameterofeachindividualstrutinthe structure.Theyfoundthemethodcapableofproducing reliable structuresundermultipleloadingconditionsandalsocapableof reducingthecomputationalcostassociatedwiththedesignofthese structures.

Structuraloptimizationtechniquesrelyingonweighted objec-tivefunctionshavealsobeenemployedfororthopaedicimplant applicationsthattakeintoaccountmultipledesignobjectivesfor lattice structures including stiffness, porosity and surface area [24].Many fields ofresearch includingbiomedical, physicsand materialsscience,havealsoconsideredoptimisationoftheunit cellitself.Thisiseitherdoneforanindividualunitcell,whichis subsequentlytessellatedthroughoutthedesignvolume,orevery unitcellthroughoutthedesignvolumeisoptimisedtoaunique geometry.Cadmanetal. [25]providesa comprehensivereview of these strategies for optimised material design for a broad rangeofmulti-functionalandmulti-disciplinaryapplications,such as,phononic/photonicbandgapmaterials,metamaterials,tailored fluidpermeabilityanddiffusivity,negativePoisson’sratio, conduc-tivepropertiesandfunctionallygradedmaterials.

Recentadvancesinsoftwaretargetedtowardscomputeraided designanddesignforAMhaveprovidednewwaystodesignlattice structures.SomenotabletoolsinthisregardareRhino(Rhinoceros 3D)[26],3-Matic(Materialise)[27],Simpleware(Synopsys)[28], Within(Autodesk)[29]andOptistruct(Altair)[30].Rhino(McNeel) providesaplatformforavarietyofpluginsandadd-onsincluding;a versionofSymvol(Uformia)[31]thatprovidesanF-Repbasedtool thatcancreatecomplexlatticestructures,andGrasshopperaplugin thatprovidesagraphicalscriptingplatformincreasingthe simplic-ityincreatingcomplexdesigns.3-Matic,SimplewareandWithin havefeaturesthataredesignedspecificallytoaidinthedesignof latticestructures,suchasunitcelllibrariesandtheabilityto cre-atestructureswithagradeddensity.Howevercurrentversionsof thesesoftwarelacktheabilitytomaplatticeontoanimported(or generated)3DdensitymapresultingfromaTO.Optistructisthe

onlyonethatofferstheabilitytoincludegradedlatticestructures aspartoftheoptimisationroutine,howeverthecurrent implemen-tationremainslimitedtoonlyonetypeofcell(definedbytheedges Tetrahedron).

It canbeseen,therefore, thatalthough thedesignfreedoms offeredthroughAMallowforthecreationofcomplexlattice struc-tures,thedevelopmentofeffectivestrategies todesign themis anongoingareaofresearch.Latticestructurescanbeexploited inAMformanyreasonsincluding:enablingthefabricationofa TOsolutionwithintermediatedensities(asfoundindensitybased methods);reducingpartdistortions,astheirinherentporosity min-imisesresidualstresses, and asa resultrequire fewersupports (supportneedsarealleviatedwithinclusionofself-supportingunit cells[32]);andimproveddesignrobustness.Thereforeembracing latticestructureindesigncanbeusefulwhenconsideringboththe mechanicalperformanceandmanufacturingaspectsofAMdesign. Thispaperaimstodemonstratethatacloserinterplaybetween latticedesignapproachesandTOisthenextstepinrealising opti-mallatticedesignsforAM.Thereforethisworkfocusesondefining strategiesthatallowfortherealisationoflatticestructuresthatare derivedusingTOsolutions;andbenchmarkingtheirperformance intermsofboththeload-bearingcapabilityandtheAMspecific designrelatedmanufacturingconsiderations.Thenoveltyofthis workisinutilisingtheTO-to-lattice mapping,presented bythe authorsin[14],toderivedifferinglatticestrategiesandindicate theirrelative performance.Thepapertakesthefollowing struc-ture:firstly,differingstrategiesforrealisinglatticestructuresare defined,therationalebehindthemanddetailsoflatticegeneration, ispresented;secondly,criterionforthemechanicalperformance evaluationandthemetricsemployedforassessingtheAMspecific designrelatedmanufacturingconsiderationsaredetailed;thirdly, resultsarepresentedandthestrengthsandweaknessesofthe dif-ferentstrategiesdiscussed;andlastly,withtheaimtobetterequip designersandengineers,insightbackedbyquantitative investiga-tionintothesuitabilityofproposedlatticestrategiesfordifferent objectivesisprovided.

2. Methodology

2.1. Methodology:realisinglatticedesigns

Thissectioncomprisesofthreesubsections:thefirstprovides detailsontheTOprocedurethatunderpinstherealisationoflattice strategies,thesecondintroducesthevariousproposedstrategies, andlastlythedetailsongeneratingalatticestructurewith func-tionallygradingisprovided.

2.1.1. Obtainingoptimaltopologies:discretesolid-voidstructures andgreyscaledensitysolutions

SolidIsotropicMaterialPenalisation(SIMP)method[10,33]– adensity-based approachwherethematerialdistribution prob-lemisparametrisedbythematerialdensitydistributiontoobtain optimaltopologieswaschosenfortheTOprocedurebecausethe SIMPimplementationenablesresultsintwoforms,onethatallows forefficientdiscretesolid-voidrepresentationandtheotherinthe greyscaledensityrepresentation.

TheobjectiveoftheSIMPimplementationusedhereisto iden-tifytheoptimaldistributionofmaterialdensity,subjectedtotarget volumeconstraint,tominimisestructuralcompliance(ameasure oftheinverseofstiffness).Theobjectivefunctionisdefinedas:

C= 1 2U

Fig.1. ResultsfromSIMPforvolumefractionconstraintof0.5whenpenaltyexponent(␸):a)setto‘1_{andb)setto‘3’.}

subjectto:

V∗− N

i=1

Vii=0 (2)

whereCisthecompliance,KistheglobalstiffnessmatrixandUis theglobaldisplacementvector,Viistheelementalvolume,Nisthe totalnumberofelements,V*isthetargetstructuralvolume,and elementaldensity‘␳i’∈[␳min,1].Forthiswork,␳minwassetat 1e-6.Thematerialinterpolationschemeusedisasfollows:

E(_i)=E1ϕ_i (3)

where E1 istheYoung’smodulus for thesolidregion,and ␸is theSIMPpenaltyexponent.IntheiterativeTOdesignprocess,the SIMPpenaltyschemeassumesaninterpolationruletopenalisethe intermediatedensitiessoastorevealpart-solidpart-voidregions whichareamenableforrealisingrealmaterialsorindeedlattice structures.

Thesensitivity of theobjective function withrespecttothe changeofith_{elementdensityis}

∂

∂

T ik

0

iui (4)

wherek0_i denotesthestiffnessmatrixanduirepresentsthe dis-placementvectorfortheithelement.Anumericalimplementation of[34]utilisingtheoptimalitycriterionmethodwasemployedto obtaintheSIMPsolutionsonameshwithequallysizedcubic ele-ments.Thepenaltyexponent(␸)wassetto1toobtaingrey-scale densityrepresentationsandavalueof3for␸waschosenwhen adiscretesolid-voidrepresentationwasdesired(seeFig.1).The discretesolid-voidrepresentationisobtainedbythresholdingthe densityresultswithaniso-value of0.5 utilisingtheiso-surface functioninMATLAB[35].

2.1.2. Introductiontolatticedesignstrategies

Asdiscussedearlier,thereareseveralapproachestorealise lat-ticesolutions.Herein,aunitcelltessellationapproachisadopted, similartothatpresentedbytheauthorsin[36].Sincethiswork focusesonstudyingthestrategiesforlatticegeneration, unpenal-izedSIMPsolutionsarechosentobeinterpretedinsteadofthose obtainedwithaunitcellspecificpenalisationexponent,makingthe materialdensitytolatticemappingcell-typeindependent.Arange oflatticeconcepts,generatedusingeitherthediscretesolid-void orthegreyscalerepresentationsoftheTOresultsareillustrated inFig.2.Thethreeprincipallydifferingstrategiesthatembracea latticestructurewithinatopologyoptimisedsolutionare:

1)“IntersectedLattice”(Fig.2b)–obtainedfromintersectinga dis-cretesolid-voidTOsolutionwitha uniformmaterialdensity latticethatfillstheentiredesigndomain.Auniformmaterial densitylatticecomprisesofunitcellsthathaveaconstant

vol-Table1

Qualitydescriptorsindicatingmosttoleastpromisingcandidateswhenconsidering designsofFig.2forseveralobjectives.

DesignStrategy Solutionoptimality Designeffort Supportrequirements

Solid Most Less Most

Intersected Somewhat Somewhat Somewhat

Graded More Most More

Scaled Less More Less

Uniform Least Least Least

umefraction‘’(ratiobetweenmaterialvolumeandthevolume ofthedesignspace).

2)“GradedLattice”(Fig.2c)–mappingalatticewithvarying mate-rialdensityontothegreyscaledensitysolutionofTOsuchthat densityvaluesbelowtheVlower

f boundattain=0(i.e.nomaterial) andthoseaboveV_fhigherboundattain =1(i.e.fullysolid). 3)“ScaledLattice”(Fig.2d)−mappingalatticewithvarying

mate-rialdensityontoare-scaledgreyscaledensitysolutionofTOsuch thatthedensityvaluesareboundedbetween[Vlower

f ,V

higher

f ].

Atthetwoextremesofthisspectrumarethe“Solid”(Fig.2a) and “Uniform”(Fig.2e)strategies which showa topology opti-misedsolidsolutionandanuniformmateriallatticethatfillsthe entiredesigndomain,respectively.ItisofnotethatbothGraded andScaledstrategiesfalldirectlyunderthecategoryof function-allygradedlatticestructuresandifmaterialgradingisnotthemost importantaspectthenIntersected strategycouldbeincludedas well.

Fig.2.Representativestructureswithavolumefractionof0.5:a)Solid(SIMPsolution),b)IntersectedLattice(derivedbyintersectionSolidSIMPsolutionwithUniform Lattice,wherebothhadasimilarvolumefractionof∼0.707i.e.sqrt(0.5)),c)GradedLattice,d)ScaledLattice,ande)UniformLattice.

organicshapesandnecessitateadditionalmaterialassupportfor asuccessfulbuild.Withtheinclusionofself-supportingunitcells togeneratelatticesderivedfromTOsolutions,liketheIntersected andGradedstrategies,theneedforsupportisalleviated.ForScaled andUniformstrategies,thatfilltheentiredesigndomain,onecan expectno(orbase-lineminimum)needforsupports.

2.1.3. Generationoffunctionallygradedlattices

Broadlyspeaking,latticeunitcellscanbeconstructedusinga) strutbasedmembersasshowninFig.3orb)surfacebased repre-sentation,forexample,theTriplyPeriodicMinimalSurfaces(TPMS) ofFigs.4–6.

Thenomenclatureforthelibraryofstrutbasedunitcells pre-sentedinFig.3isasfollows:BCCisBodyCanteredCubic;BCCzis BCCwith‘z’directionreinforcement;FCCisFaceCentredCubic; FBCCresultsfromtheunionofFCCandBCC;andnameswith‘S’ prefixareself-supportingvariantswhichhavenomembersthat

layparalleltothex−yplane.Strutbasedunitcellscanbe gener-atedbyspecifyingastrutdiameter(oriso-value)forthesigned distancefunctioncreatedoverthe1Dtopologyoftheunitcell(i.e. connectivityofthevertices).

TPMScan bedefinedby implicitfunctions(i.e.f(x,y,z)=t) wheretheisovalue‘t’governstheoffsetfromthelevel-sets(i.e. whenfunctionvalueequalszero)[37].Thenomenclatureforthe libraryofTPMSpresentedin Fig.4is asfollows:‘G’is Schoen’s Gyroid;‘P’is Schwarz’sPrimitive;‘D’isSchwarz’sDiamondand ‘W’isSchoen’siWP.Thereaderisdirectedto[38]forfurther back-groundinformationontheTPMSandtheirgoverningequations.

Inpractice,surfacerepresentationsofTPMSwithinequality con-ditionsareusedtogeneratesolidstructures,forinstance,expressed as

Fig.3. Libraryoftrussbasedunitcells(Vf=0.2):a)BCC,b)BCCz,c)FCC,d)FBCC,e)S-FCC,f)S-FCCz,g)S-FBCC,andh)S-FBCCz.

Fig.4.Libraryof(implicit)surfacebasedunitcells–Networkphase(representatives):a)G(Schoen’sGyroid),b)P(Schwarz’sPrimitive),c)D(Schwarz’sDiamond,d)W (Schoen’siWP),e)Lidinoid(bySvenLidin),f)Neovius(bySchoen’sstudentNeovius),g)Octo(bySchoen),andh)SplitP.

Fig.5.Schwarz’sPsurfaces(forvalueoft=0.37):a)f(x,y,z)≤t,b)f(x,y,z)≤−t,andc)−t≤f(x,y,z)≤t.

Fig.5aandbpresentthesolidstructuresderivedbyutilisingthe inequalityof(5)fortheSchwarz’sPsurfacewhent=0.37and−0.37 respectively.Thegoverningequationof‘P’surfaceis

fp(x,y,z)=cos(xx)+cos(yy)+cos(zz) (6)

Fig.6.Libraryof(implicit)surfacebasedunitcells(Vf=0.2):a)D-G,b)D-P,c)D-D,d)D-W,e)D-Lidinoid,f)D-Neovius,g)D-Octo,andh)D-SplitP.

TPMSofFig.4canbeeasilyextendedbydefiningthesolidstructure as

f2(x,y,z)≤t2 (7)

i.e.f(x,y,z) boundedbetween[−t,t].Theserepresentationsare commonlyreferredtoas“Double”variantofthebaseTPMSand thereforehaveaprefix‘D’totheirnames(seeFig.6).Fig.5cpresents theresultingD-P(i.e.doublevariantofSchwarz’sPrimitive) struc-turewhenusing thevalueoft=0.37.It is ofnotethat theD-P structureisineffectthesubtractionofFig.5bfromFig.5aasthisis whattheinequalityof(7)dictates.

Double variants of TPMS, besides having the matrix phase where₋t_≤f_≤thavetwomoresolidrepresentationsresultingfrom f<−tandf>t.Thesenon-intersectingmanifoldsthatsandwichthe matrixphaseareoftenreferredtoasthenetworkphases.Inthis workonlymatrixphasestructuresareinvestigated,duetotheir resemblancetoloadbearingthin-walledstructures.

Materialgradingofa3Dstructurecanbeachievedbyoperating ona4Drepresentationin (x,y,z,t)-space.Fromanimplementation

standpoint,onecouldconsiderstatingthematerialgradingtobe similarinformtothatofEq.(5)thereforeresultingin

f(x,y,z)≤t(x,y,z) (8)

Here, tis anisovalue matrix in the (x,y,z)-spacethat con-trolsthevolumefraction(orVf)variationforthelatticestructure withinCartesianspace.TherelationshipbetweentandVfwere obtainedforvarious unit cellsby employingcurve-fitting tech-niquestoensurethattheaveragevolumefractionforaunitcell equatedtotheaveragevolumefractionforthecorresponding vol-umetricregionwithinmaterialdensity/gradingmap. Eachvoxel (volumetricpixel)forevaluationassumedhomogenisedproperties withfieldvaluesattheelementcentroid.Fig.7showsanexample oflineargradingforBCCandD-PunitcellswhereVfvariesfrom0.1 atthebottomto0.5atthetopofthelatticestructure.

Fig.8.3DCantileverproblem.

2.2. Methodologyfornumericallyevaluatingthemechanical performanceofdesignstrategies

MoststudiesconcerningTOmethodsconsidertheclassic can-tileverplate(in2D)orbeam(in3D)problemandthereforethis workalsoutilisessuchastructure(seeFig.8withadesigndomain of240×120×60voxels/units)toinvestigatetheproposeddesign strategies.Forobtaininglatticestructuresunitcellsoccupyinga 20_×20_×20 voxelcubewereused,resultinginstructures com-prising of 12×6×3 unit cells. Mechanical performances were evaluatedfortwoloadingscenarios:definedloading(i.e.intended loading),anddefinedloadingwithuncertainties(i.e.variabilityin loading).TotalStrainEnergy(SE)ofthestructureswasusedasthe performancemeasure–alowerSEindicateamoreefficient(stiff) structure.

ThestructuresofFig.9wereusedforthenumerical investiga-tionandhadatotalV_f of0.5.Thelatticevariantsutilisedbounds

Vlower

f of0.3andV

higher

f of0.85.Doingso,ensuredthatthefinite ele-mentmodelshadaminimumof3elementthickwall/strutsinthe lowestV_f regionsandthiswasnoticedtoadequatelycapturethe unitcellbehaviourinthemeshrefinementstudy(convergenceof displacementvaluesunderthecantileverloadingcase).ForFig.9b andc,SIMPsolid-voidsolutionsobtainedwithatotalV_fof0.7were usedtointersectwitha constantV_f latticeof0.715resultingin averagestructuralV_fof0.5.

H).The%x,y,zcomponentvaluesfortheconsideredunitload vec-torsarereportedinTable2.Structureswereloadedunderallof thespecified15casesandineachcasehadthedominantvector componentalongthe‘–yaxis’.Wherethiscontributionishigher than95%oftheloadvectormagnitude(unity),theload variabil-ityisreferredaslowerandisdenotedbyF_L.Conversely,forhigher variabilityassociatedwithcontributionsunder95%along‘–yaxis’, thetermF_Hisused.Toassesstheeffectsofdirectionality,F₁utilises seconddominantloadingalongthe‘–zaxis’,F₃alongthe−xaxis andF₂considersequalcomponentsaboutthe‘–xand−zaxes’.

Allstructureswereanalysedusinghexahedralelementsfilling theentiredesign domainwithanedgelengthofonevoxel/unit resultinginameshcomprisingof1.728millionelements.Finite Elements(FE)representingsolidregionswereassignedamoduliof unityandthoserepresentingvoidregionsassignedthemuchlower valueof1e-6(soft-killapproach).ThevalueforthePoisson’sratio waschosenas0.3.AcommercialFEsolver,specificallyMSCNastran [39],wasusedtocalculatethetotalSEtoinformthemechanical performance.

2.3. Methodologyforassessingthedesignrelatedmanufacturing considerationsforthevariousdesignstrategies

Inaddition tothemechanical performanceevaluationof the presenteddesign strategies,thisstudyinvestigates anumberof designrelatedmanufacturingconsiderations,specifically, manu-facturingissuesthatcanbetackledtosomeextentbymodifying adesign.Thedataforcomparisonwascollectedfromprocessing thesolutionsusing3DprintingsoftwaresuchasMagics[40]and Autofab[41].Thethreecriteriainvestigatedinthisstudyconsist of:i)supportstructurerequirement,ii)processingeffort,andiii) design-to-manufacturediscrepancy.

Table2

%x,y,zcomponentsoftheconsideredunitloadvectorsfortheloadingscenarioII.

−F₁ −F₂ −F₃

x y z x y z x y z

Lowervariability(FL)

0 99 1 1 98 1 1 99 0

0 96.9 3.1 1.9 96.2 1.9 3.1 96.9 0

Highervariability(FH)

0.9 94.1 5 3 93.9 3 5 94.1 0.9

2.1 92.2 5.8 4.2 91.6 4.2 5.8 92.2 2.1

0.5 88.3 11.3 5.6 88.9 5.6 11.3 88.3 0.5

AnimportantconsiderationwhendesigningpartsformetalAM, specificallySelectiveLaserMelting(SLM),isthesupportstructure requirement.Ingeneral,downwardslopingfaceswithanangle of45◦orlessrequiresupportstructures[42],andadditional sup-portreinforcementmaybeneededinregionsthatexperiencehigh residualstressduetothebuildprocessinordertopreventpart distortion.Althoughover-supportingacomponentcanreducethe chanceofabuildfailure,itincreasesthemanufacturingtime,the powderused,andthepost-processingeffortsforsupportremoval. The aimof the supportstructure investigationin this study is tocomparetherelativesupportstructureneedsforthedifferent designstrategies.Similarto[43],afixedbuildorientation,i.e.yaxis (ortheloadingdirection)ischosenasitprovidedgreaterscopefor comparingthesupportstructurerequirementwhencomparedto xandzaxeswhilstkeepingsimilarbuildtimes.Itisofnotethat byidentifyingtheoptimalbuilddirectionforeachdesign,minimal supportneedsspecificforadesigncanbeobtainedprecisely.Such aninvestigationisbeyondthescopeofthisworkandthereforea fixedbuildorientationisemployedfortherelativecomparison.

Toquantifythesupportrequirementforthedifferentdesigns, Magicssupportgenerationmodulewasemployed.Thesurfacearea oftheoverallsupportinastructurewasusedasameasurefor sup-portrequirements.Thedifferentdesignswereaddedtoa scene representingaRealizerSLM50buildplatformandorientedwith the‘yaxis’normaltothebuildplatform(refertoFig.15).The soft-wareautomaticallygeneratedsupportstructuresusingproprietary algorithmsandthisallowedforafaircomparisontobemadeofthe supportstructurerequirementforthevariousdesignstrategies.

Theprocessingeffortsrequiredforgeneratingthesuitablefile formatfora3Dprinter(i.e.AMmachine)fromthedesignfile for-matcanberelatedtothegeometricalcomplexityofthedesigned samplesaswellasthesizeofthefilerepresentingthesamples. STeroLithography (STL)is thede-facto fileformat for manufac-turein theAMsector.The STLfiledescribes apart’s geometry (specificallysurface)usingtriangles,therefore,thesizeofanSTL fileisdirectlydependentonthenumberoftrianglesusedto repre-sentthesurfaceandin-turnagoodmeasureofpartcomplexity.A

largeSTLfilespecifiesagreaternumberoftrianglesandasaresult requiresmorecomputermemorytoperformcomputational oper-ations(suchasBooleanoperations),andalsoresultsinincreased processingeffortsneededtogenerateaslicedfileformatfor3D printing.Tobetterunderstandtheprocessingefforts,twomeasures wereconsidered:Measure-1–thenumberoftrianglesneededto definethegeometry,andMeasure-2–thetimespentongenerating theslicedfilesofthecomponents(inMagics).Doingsoallowsfor investigationastowhetherthereisadirectcorrelationor depen-dencybetweenthetwomeasures.

Inadditiontothesupportrequirementsandprocessingefforts, designcomplexityplaysaroleinthedesign-to-manufacture dis-crepancy, i.e. disparity between the intended design and the manufactured part. For AM components, this disparity can be attributedtotwomainsources:thedeflectionofcomponentdueto ahighresidualstress,andsurfaceroughnesseffects.Lattice struc-tureswhencomparedtosolidsamplesforthesametotalvolume arelessaffectedbyresidualstressduetotheirinherentporosity butcanbedetrimentallyinfluencedbythesurfaceroughnessand ballingeffects[44]duetothedifficultiesassociatedwith perform-ingsurfacefinishoperations.Apreliminarilyinvestigationshowed thattheweightsofthemanufacturedlatticepartswerefoundto benoticeablydifferentfromtheintendeddesignedweights(due tolooselyattachedpowder).Asecondorderpolynomialfunction adequatelycapturedtherelationshipbetweentheerrorpercentage (theabovediscusseddisparity)andthesurfaceareatovolumeratio foranumberofSLMmanufacturedTitaniumlatticecubessamples (densitiesrangingfrom0.1to0.6).

ε=|wb−wd

wd |×

100%=−0.12

V

V

−4.7 (9)

where,␧istheerror%,Aisthesurfacearea,Vthevolumeofthe part,andwbandwdrepresentthebuildanddesignedweightofthe component,respectively.Inthiswork,Eq.(9)isusedtoreportthe disparity/errorbetweendesignandmanufacturefortheexplored designstrategies.

Fig.11.TotalstrainenergyofthevariousdesignstrategiesforthecaseofF1variabilityinloading.

Fig.12.TotalstrainenergyofthevariousdesignstrategiesforthecaseofF₂variabilityinloading.

3. Resultsanddiscussion

3.1. Numericalevaluation

TheresultingtotalSEforthedifferentdesignstrategiesapplied tothecantileverbeamtestcaseunderintendedloadingare pre-sentedinFig.10.Theseresultsareinagreementwiththeintuitively predictedrankingofTable1.TheSolidsolutionderivedfromthe penalisedSIMPoptimisationattainedthelowestSE.Conversely,the UniformlatticestructurepossessedthehighestSE.Intersectedand GradedstructuresgavethelowestSEvaluesforstructures embrac-inglattices.TheScaledlatticedesigngavemuch lower(58%)SE valuethantheUniformlatticeindicatingamoreoptimalsolution althoughthesolutionwasnoticeablyinferior(∼10%)tothe Inter-sectedandGradedstructures.Thisconfirmsthehypothesisthata solutions’extentofdeviationfromtheoriginalTOsolutiondirectly relatestooptimalityfor achosenstrategy.Itis ofnotethatthe performanceofanIntersectedlattice,inprinciple,canrange any-wherebetweenaSolidandUniformsolutionasthestructureitself isderivedbyintersectingaSolidsolutionwithaUniformlatticebut forallpracticalpurposesitissafetoassumethatthisapproachwill employagreatdealofTOtoattainsuperiorperformance.

ThetotalSEvaluesforthedifferentdesignstrategieswhen sub-jectedtotheloadingofF₁,F₂andF₃ arepresentedinFigs.11–13, respectively.Foreachdesignstrategy,threecolumnsareshown: left-most(bluecolour)istheresultfromtheintendedloadingcase

(forreferencepurposes),middle(greycolumn)correspondstothe lowervariabilitycase(F_L)andright-most(orangecolour)thehigher variabilitycase(F_H).Errorbarsindicatethehighestandlowest val-ues,tohighlighttherangeinresponsetoeachloadanddonotto indicateuncertainty.

Fig.13.TotalstrainenergyofthevariousdesignstrategiesforthecaseofF₃variabilityinloading.

Table3

Strainenergyinthebeamunderdifferentmodesofdeformation−asimplifiedanalyticalapproach.

Loadcase GeneralstatementforthestoredSE SEinbeamofFig.8(L=4BandH=2B)

StretchingduetoFx U=

0

Fx.Fx.dl

2EA =

x.Fx E

.

2BH

=

.

B

BendingduetoFy _U=

0

M.M.dl

2EI

=

.

BH3

=

.

B

BendingduetoFz =

E

.

B3H

=

.

B

Table4

Performanceresiliencefordesignstrategies.

F_L F_H

F₁ F₂ F₃ F₁ F₂ F₃

Solid 0.07 0.04 0.1 0.19 0.01 0.18 Intersected 0.08 0.04 0.11 0.19 0.01 0.18 Graded 0.06 0.02 0.08 0.16 0 0.14 Scaled 0.09 0.03 0.11 0.22 0.01 0.18 Uniform 0.06 0.06 0.12 0.12 0.06 0.21

loadingscenarioofTable3andtheareamomentofinertiasmay changemakingthecurrentinterpretationsvoid.

WhencomparingvariousdesignstrategiesforF_LandF_H variabil-ities,theratioofthemeanSEvalueassociatedwiththevariable loading case (i.e. middle and right-most columns) and the SE for intended loading (i.e.left most column) can help estimate PerformanceResilience(PR).Herein,PR isdefined asthe abso-lutedifferencebetweentheratiocomputedaboveandunity.The smallerthevaluethegreatertheresilience,astheperformanceof thestructurevariedtheleastwhensubjecttovariationinloading. Table4reportsthePRvaluesforthefivedesignstrategies investi-gated.TheGradedstrategywasobservedtobemostresilientasit attainedthelowestPRvalues(exceptforF1

Hwhereitwassecond lowest).Conversely,theUniformstrategy wasfoundtobeleast resilient(exceptforF1

HwhereitattainedalowPRvalue).Theother threestrategiesattainedintermediatePRvalues,withtheScaled strategybeingmarginallylessresilientthantheothers.

InadditiontothereportedtotalSEvalues,reviewofthe distri-butionofSEvaluesthroughtheuseofcontourmapsrevealsmore informationabouthowthedifferentdesignsrespondtoloading. ScaledandUniformlatticeswerecomparedastheybothfilledthe entiredesigndomain(acommonfeature)andonlydifferedinthe waythematerialwasdistributedwithinthevolume.Fig.14shows thedifferencesintheSEdistributionbetweenthetwostrategiesfor thecaseofintendedloading.TheUniformlatticeappearstohavea greatervariationinthedistributionofSEandthisisconfirmedby theStandardDeviation(SD)values.TheSDcalculationutilised

val-uesthatwerethresholdedtobelowthecut-offvalueof1.8e-5(the samethresholdisusedfortheillustrationsofFig.14).TheUniform latticewasfoundtohaveastandarddeviationof2.47whereasthe Scaledlatticewasfoundtobeonly1.36.AtighterbandofSEvalues (i.e.lowerSD)should,inprinciple,indicateamoreoptimalstrategy astheloadbearingcapabilityofastructureisbetterutilisedwith mostregionsattainingSEvaluesclosertotheaverage.

Whenconsideringdesignapproaches,differentunitcell geome-triesformanimportantpartofthedecision makingprocess.To highlightthis,theIntersectedlatticewastestedwithbothD-Punit cells(Fig.15aandb)andBCCunitcells(Fig.15candd).Visual com-parisonshowsthattheBCCcellshaveraisedSEattheinterfaceof unitcells.Thiscanbeconsideredasafeatureofusingastrutbased latticestructuresuchasBCCasopposedtothesurfacebasedD-P structure.Ascanbeseen,theD-Pstructurehasasmoother distribu-tionofSE.Thishaspotentialadvantagesinpreventingfailuredueto stressconcentrations.Stressconcentrationsarealsovisiblewhere unitcellshavebeencutinvoidregions,thisislessnoticeableinthe surfacebaseddesignduetothehigherdegreeofconnectivitythat acut‘wall’orsurfacehasincomparisontoacutstrutmember.The trendsseeninthevisualinterpretationofSEvaluesaresupported bythenumericaldistributionofSEvalues.TheBCCstructurehasa SDof1.40whiletheD-PstructurehasaSDof1.06,whenusingthe cut-offvalueof1.2e-5(i.e.thesamethresholdusedforthe illus-trationsofFig.15).ItisofnotethatwiththeIntersectedstrategy, latticescanbegeneratedbyintersectingahigherVfSolidsolution withalowerVfUniformlatticetoresultinastructurecomprised ofthinmembers,leadingtohigherstressconcentrations.

3.2. Manufacturingevaluation

Fig.14.Contourmapsshowingstrainenergyvariationfor:a)ScaledLattice(isometricview),b)ScaledLattice(sectionview),c)UniformLattice(isometricview),andd) UniformLattice(sectionview).

Table5

Designrelatedmanufacturingconsiderationsforthedesignstrategies.

Solid Intersected Graded Scaled Uniform

SupportStructurerequirement Realvalue 3501 2826 2485 1509 1350 Relativemeasure 1 0.807 0.710 0.431 0.386

Processingefforts

Triangles 1716 71986 110476 175794 188058 Relativemeasure 0.009 0.383 0.588 0.935 1

Time 1.8 4.3 4.4 7.3 7.8

Relativemeasure 0.231 0.551 0.5641 0.9359 1

Design-to-Manufacturediscrepancy Estimatederror% 0.4 3.5 2.8 8.2 8.5 Relativemeasure 0.005 0.412 0.329 0.965 1

thesametrianglereductionprocedure(inMagics)afteremploying identicalvaluesfortheminimumdetailsize.

Asdiscussedearlier,SolidsolutionsfromTOcanoften resem-bleorganicshapesandhaveconsiderableoverhangsurfaces(asin Fig.16aandb)necessitatingextensivesupporttoenablea success-fulSLMbuild.Conversely,latticedesignswithself-supportingunit cellsfillingtheentiredesigndomain,i.e.ScaledandUniform strate-giesofFig.16eandf,requirebase-lineminimumsupportswhich amountedto∼40%oftheSolidcase.Intermediatesupport

require-ments(70–80%oftheSolidcase)wereobservedfortheIntersected andGradedstrategies,Fig.16canddrespectively,mainlydueto thepresenceofsomeinternal/externaloverhangsurfacesasthese strategiesonlyfillpartofthedesigndomainwithlattice.

Fig.17presentstherelativemeasuresreportedinTable5inan easytocompare(pictorial)formatwithsurfaceareaincludedfor reference.Whenmovingleft-to-right(i.e.SolidtoUniform strat-egy),ageneralincreasingtrendisobservedforallthedesignrelated manufacturing considerations (except for the support

Fig.17.Relativemeasurereportedforthecomparisonofdesignrelatedmanufacturingconsiderations.

ments).Ingeneral,forthecaseoflatticescomprisedofunitcells thatare notself-supporting,a similarincreasingtrend for sup-portrequirementcanbeexpectedsincethegeometriccomplexity thatnecessitatessupportincreaseswithlatticing.Withthe inclu-sionofself-supportingunit cells,this trendcan bereversed,as showninthisinvestigation(andthepredictiverankingofTable1), givingconfidenceinthehypothesis“capturingthegeometric com-plexitiesthatnecessitatesupportgivesagoodestimateofsupport requirementsforadesign”.

Itisevidentthatthegreatertheextentoflatticing,thehigher thesurfaceareaofthesolution.IntersectedandGradedlattices havenearlytwo-foldthesurfaceareawhereastheScaledand Uni-formlatticeshavealmostthree-foldwhencomparedtotheSolid solution.Itisofnotethatthenumberoftrianglesneededto rep-resentaSolidsolutionincomparisontoUniformlatticeisgreatly reduced(1/100th),duetothemuchsimplersurfacepatchesneeded todefineitsgeometry.ComparingGradedwithIntersectedlattice, althoughtheformerhadmarginallylowersurfacearea(aresultof accommodatingsolidregions),therepresentationoflowerVfunit cellsmeantahighernumberoftriangleswereneededoverall.To putthingsinperspectivethelowerVfinGradedwas0.3whereas theIntersectedlatticeutilisedauniformVfof0.715.

Thereseemstobeadirect correlationbetweenthe“number oftriangles”(Measure-1ofprocessingeffort)andprocessingtime (Measure-2)whenthenumberoftrianglesexceedthe110kmark (i.e.structureachievessufficientgeometriccomplexity).However withtrianglenumbers,asisthecasewithSolidsolutionand Inter-sectedlattice,abaseline/offsetprocessingeffort(time)neededfor slicingcanbeseen(seeFig.17).

Owingto therelatively low surface areaof the Graded lat-ticewhencomparedtootherlatticedesigns,alowvalueforthe design-to-manufacturediscrepancyisfound.Itcanbeinferredthat theGraded,amongstotherlatticestrategies,canproduce manu-facturedpartsthatdeviatetheleastfromtheintendeddesign.It shouldbenotedthatdesign-to-manufacturediscrepancyaddsto theweightofthepart,andpotentiallyeffects mechanical prop-erties, therefore when utilising lattice structures for optimised designstrategies,particularlyforlowdensity,oneneedstotake intoaccountthechangeinthepartweightaftermanufacturingin ordertocompensatefortheextraweightaddedtothecomponent afterthebuild.

InadditiontotheAMspecificdesignrelatedmanufacturing con-siderationsexploredabove,thereareseveralothersthatneedto beconsideredwhen designing latticestructuresforSLM. These includetheminimumwallthicknessthat themachine can pro-duce,andtheminimumporesizeforremovingloosepowderafter thebuild. Therefore, for each cell type and cellsize, there is a minimumandmaximumdensitythatispossibletomanufacture.

Anotherissue,istheself-supportingpropertyofthecellswhichcan beaffectedbyincreasingthecellsizeifthecellhasanyoverhang surfacesgreaterthanthespecifiedlimit.Evidently,thisisnotan issueforaBCCcellwith45◦members,however,foratruss-likecell withhorizontalmembers,e.g.FCCandFBCC,oneshouldtakethis intoaccount.

4. Concludingremarks

Topologyoptimisation(TO)techniquesand generativelattice designapproachesareincreasinglybeingemployedforthedesign ofAdditivelyManufactured(AM)partsandarethefocusofongoing researchintheAMcommunity.Thedevelopmentofstrategiesto createcomplexlatticestructuresutilisingoptimalsolutionsfrom TOhasgreatpotentialthroughoutawiderangeofresearchfields. Thisinvestigationexploredrobustnessandeffectiveness of pro-poseddesignstrategies,bothintermsofmechanicalperformance andAMspecificdesignrelatedmanufacturingconsiderations.From theresultsobtainedinthiswork,thefollowingconclusionscanbe drawn.

Lattice strategies that are informed by TO results (i.e. Intersected/Graded/Scaled lattices) are considerably (∼40–50%) superiorintermsofspecificstiffnesstothestructureswherethisis notthecase(i.e.Uniformlattice).TheGradedlatticestrategywas identifiedasthemostrobuststrategyfromamechanical perfor-mancestand-pointduetoitshighresiliencetoloadingvariabilities. Forcutmembers,asisthecasewiththeIntersectedandGraded strategies, latticesgeneratedfromsurfacebased unitcells were foundtobemoreefficientattransmittingloadswhencompared tostrutbasedunitcells,essentiallybecauseoftheirhigherdegree ofconnectivity.

Inprinciple,ageneralincreasingtrendwithincreasedextentof latticingisobservedforalltheAMspecificdesignrelated manufac-turingconsiderationsinvestigated,specifically,supportstructure requirement, processingefforts and design-to-manufacture dis-crepancy.However, thetrend for support requirementscan be reversedbyemployingself-supportingunitcells,givingconfidence totheproposedhypothesisi.e.“capturingthegeometric complex-ities that necessitate supportgives a good estimate of support requirementsforadesign”.

Table6

Qualitydescriptorsindicatingmosttoleastpromisingdesignstrategiesforseveralobjectives.

DesignStrategy Optimality–intendedloading PerformanceResilient Supportstructurerequirements Processingefforts Design-to-Manufacturingdiscrepancy

Solid Most Somewhat Most Least Least

Intersected More More More Less Somewhat

Graded Somewhat Most Somewhat Somewhat Less

Scaled Less Less Less More More

Uniform Least Least Least Most Most

wastheleastaffectedoverall when assessedfor design related manufacturingconsiderations.

Theresultsobtainedforthecomparativeinvestigationof lat-ticedesignstrategies,thoughemployingthespecifictestcaseofa cantileverbeam,areapplicablemoregenerally.Thisisduetothe resultsbeinglargelyintrinsictothedesignstrategyand indepen-dentofdesignparameterschosen.Theauthorsbelievethatthekey findingsfromthiswork,summarisedqualitativelyinTable6,will helpinformengineersanddesignersaboutthepros-and-consof theproposeddesignstrategies,enablingthemtomakeinformed choicesaboutthesuitabilityofastrategyforagivendesign-spec accommodatingmanufacturingconsiderations.Itisenvisagedthat thiswillprovidethenecessaryknow-howandin-turntheincentive torealisecomplexlatticegeometriesviaAM.

Acknowledgment

ThisworkwassupportedbytheEngineeringandPhysical Sci-encesResearchCouncil[grantnumberEP/N010280/1].

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