Design framework for multifunctional additive manufacturing: coupled optimization strategy for structures with embedded functional systems

(1)

Design

framework

for

multifunctional

additive

manufacturing:

Coupled

optimization

strategy

for

structures

with

embedded

functional

systems

Ajit

Panesar

∗

,

Ian

Ashcroft,

David

Brackett,

Ricky

Wildman,

Richard

Hague

FacultyofEngineering,UniversityPark,UniversityofNottingham,NG72RD,UK

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received4November2016

Receivedinrevisedform24February2017 Accepted21May2017

Availableonline23May2017

Keywords:

Optimization Additivemanufacturing Multi-functional Computationalmethods

a

b

s

t

r

a

c

t

Thedriverforthisresearchisthedevelopmentofmulti-materialadditivemanufacturingprocessesthat providethepotentialformulti-functionalpartstobemanufacturedinasingleoperation.Inorderto exploitthepotentialbeneﬁtsofthisemergenttechnology,newdesign,analysisandoptimization meth-odsareneeded.Thispaperpresentsamethodthatenablesintheoptimizationofamultifunctionalpartby couplingboththesystemandstructuraldesignaspects.Thisisachievedbyincorporatingtheeffectsofa system,comprisedofanumberofconnectedfunctionalcomponents,onthestructuralresponseofapart withinastructuraltopologyoptimizationprocedure.Thepotentialoftheproposedmethodis demon-stratedbyperformingacoupledoptimizationonacantileverplatewithintegratedcomponentsand circuitry.Theresultsdemonstratethatthemethodiscapableofdesigninganoptimizedmultifunctional partinwhichboththestructuralandsystemrequirementsareconsidered.

1. Introduction

Single-materialadditivemanufacturing(AM)processes,suchas

selectivelaser melting,enablethedesign ofgeometrically

com-plexparts.Multimaterialadditivemanufacturing(MMAM)further

expandsthis design freedomto includethe spatialvariation of

materialcompositionandenablemultifunctionalitythroughthe

volumeofapart.Multifunctionality,bydeﬁnition,necessitatesthe

embeddingofactivesub-componentsinordertodeliveradditional

functionalcapability,suchaselectronic,electro-mechanical,

opti-cal,electromagnetic,chemical and thermal [1]. MacDonald and

Wicker[1],ina recentreviewarticle,identiﬁed multifunctional

additivemanufacturing(MFAM)asapivotaltechnologyin

advanc-ingthefuture ofAM.Effortshave beenmadebyresearchersto

develophybridsystemstoachieve MFAM,onesuchexample is

theworkbyLopesetal.[2]wherestereolithographyand direct

printtechnologiesarecombinedtorealizeadditivelymanufactured

electronicdevices.Such hybridapproaches oftenrequire

multi-plemachine/print-restartsandadditional manualor automated

accompanyingprocedures.Conversely,multi-headinkjetprinting,

apromisingMMAMprocessallowsforMFAMdesignstoberealized

withgreaterdegreeofmanufacturingfreedominasingleoperation

∗Correspondingauthor.

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

byco-depositingstructuralandfunctionalinks.Recentworks[3–5]

havedemonstratedthepotentialofjettingforelectronics

appli-cationsand highlighted theconsiderable ongoing researchinto

materialsandprocessdevelopmentforMMAM.

StepstowardsexploitingthedesignfreedomsofAMhavealso

beenmade.OpenFab[6]deﬁnedaproceduretoefﬁcientlygrade

mechanicalpropertiesthroughthevolumeofapartandPonche

etal.[7]proposedathreestepglobaldesignapproachwiththe

aimofbetterintegrationofthedesignrequirements(functional

speciﬁcation)withtheAMprocess.However,therehasbeenlittle

workcarriedouttodateondevelopingthedesignphilosophies

torealize novelMFAM concepts.Theauthors considera closer

interplaybetweentheMMAMandtopologyoptimization(TO−

astructuraloptimizationtechniquethatiterativelyimprovesthe

materiallayoutwithinagivendesignspace,foragivensetofloads

andboundaryconditions[8])keytoprogressingtheMFAMdesign

paradigm.Onedirectbeneﬁciaryofthisistheareaof3Dprinted

electronics,asfabricationofruggedstructuresthatembed

non-traditionalelectronicsystemsinanarbitraryformbecomepossible

[2,9].Thisapproachhasthepotentialtopavethewayfor

light-weight,morecompact,betterintegratedandmoreoptimaldesigns.

TOwithembeddedcomponentshasbeeninvestigated

previ-ouslyforthe“integratedlayoutdesign”problem[10–13],where

theaimhasbeentoﬁndboththeoptimalplacementand

orienta-tionofcomponentsandtheoptimalconﬁgurationofthematerial

simultaneously.Thiswasachievedbyiterativelychangingthe

geo-http://dx.doi.org/10.1016/j.addma.2017.05.009

(2)

Fig.1.Multifunctionaldesign−a)top-leveldiagramshowingcoupledoptimizationofstructureswithembeddedsystem,b)multi-materialjettedconceptprototypeshowing anoptimizedstructuralpartthatembedsaninternalsystem(comprisedofplacedcomponentsandtheassociatedrouting)intendedforthepurposeofstructuralhealth monitoring.

Fig.2.ExampleofaTOautomotivecomponent(transmissionmount)withembeddedsystems−demonstrating3Dplacementofcomponentsandassociatedroutinga) perspectiveview,b)topview.

metricaldesignvariablesdescribingthelocationandorientationof

componentswithinaTOregime.Inallthereportedinstances,the

implementationremainedlimitedtotheinclusionofrigid

compo-nentsthatsolelyintendedtoenhancethestructuralperformance.

Therefore,thisworksseekstomakeastepforwardtowardsthe

inclusionof functionalcomponents(orasystem) ina structure

enablingtheoptimizationofaMFAMdesign.Torealizethisaim,

i.e.optimizethedesignofamultifunctionalpart,couplingboththe systemdesign(whichisbasedonafunctionalperformancei.e.not

limitedtoonlycapturingstructuralormechanicalbehaviour)and

structuraldesign,asillustratedinFig.1,isneeded.

EarlierworkbytheauthorsdetailedaMFAMdesignframework

[9]torealizeafunctionalsystem,speciﬁcallyaimedat3DPrinted CircuitVolume(PCV)applicationsi.e.printedelectronicsintrue 3D−notlimitedto,forexample,printingonsurfacesasin[2]or thestacked2D(i.e.2.5D)PrintedCircuitBoard(PCB)paradigm[14]. Thisworkpresentedmethodsfortheintelligentplacementof func-tionalcomponentsatsuitablesites,andtheassociatedroutingfor

theconductivepathwayswithinapartmanufacturedusing

multi-materialjetting.Moreover,effortsweremadetointegratethese

aspectsofsystemdesignintoaTOproceduresuchthattheﬁnite

elementanalysis(FEA)conductedaspartofaTOaccountedfor

theupdatedmaterialproperties,reﬂectingsystemattributes[15].

Subsequently,thiswasextendedtobeneﬁtfromabi-directional

couplingbetweentheTOandsystemdesignbutthe

implementa-tionremainedlimitedtoaspeciﬁcroutingmethodandsuffered

fromrobustnessissues[16].

ThecapabilityfordesigningMFAMconceptsforPCV

applica-tionisinitsinfancyandthereforethisworkfocusesondeveloping agenericcoupledoptimizationstrategyfortherealisationof

struc-tures with embeddedfunctional systems that are intended for

manufactureusingmulti-materialjetting.Thepapertakesthe

fol-lowing structure: ﬁrstly, a description of design for functional

systemsisprovided;secondly,thestructure-systemcoupling

strat-egy ispresented;thirdly, theheuristic deﬁnitionof thesystem

sensitivities is detailed (so that one can tackle a bi-directional

structure-system coupling);and lastly,theappropriatenessand

robustnessofthisstrategyisdemonstratedbyevaluatingand dis-cussingtheresultsforexampletestcases.

2. Methodology

Avoxelmodelingenvironmentischosenforseamless

transi-tionbetweensystemdesign,numericalanalysisandmanufacture

astheyallrelyondiscretizedvolumetric space[9].Speciﬁcally,

voxelsforsystemdesign,hexahedralelementsforFEAand2D

pix-elswithassociatedlayerthicknessintheraster-based(bmp)ﬁle

formatemployedinjetting.Adoptionofthevoxelmodeling

envi-ronmenteliminatestheneedformanualcomputer-aided-design

operations,includingconversiontothecommon

STereoLithogra-phyﬁleformatandassociatedslicing,whichiswellknowntobe

cumbersomeanderrorprone.

2.1. Functionalsystemdesign

The key enablers for making the functional system design

possibleare:(i)intelligentcomponentplacementand(ii)the

asso-ciatedconnectionsroutedbetweenthem.Forsimplicity,theseare

referredtoasplacementandrouting.Althoughadvancementsin

(3)

Fig.3.Flowchartshowingthecoupledoptimizationprocedure.

Fig.4.The2Dcantileverproblemwithfunctionalsitestobeconnectedtoformthe systemindicatedbycircles.

andmathematicalmethods,asreportedin[17–19],the

implemen-tationshavebeen,atbest,gearedatthe2.5Dparadigm.Manyofthe

PCBdesignstrategieshavebeenadaptedandcoupledwithglobal

evolutionaryoptimizationalgorithmstosolveoptimization

prob-lemsinotherﬁelds.Examplesinclude,GeneticAlgorithms(GA)and

AntColonyOptimization(ACO),employedforpipe/cablerouting

problems[20–23]andoptimumplacementproblemsinstructural

healthmonitoringapplications[24,25].

Tosuccessfully realize structures withembedded functional

systems,oneneedstoexplorethetrue3Ddesignfreedomsoffered

byMFAMinanautomatedsenseandearlierworkbytheauthors[9]

discusseshowthiscanbeachieved.Thekeyelementsofthe

pro-posedmethodsandtheirfoundationalprinciplesaresummarised

below:

i.theintelligentplacementofcomponents− locationselection

basedonaperformanceand/orgeometrycriterion;component

orientationidentiﬁed usingskeletalinformation. Theskeletal

information (or the 1D medial axis) of a part’s topology is

obtainedusingathinningalgorithm,asdetailedin[26,27].

ii.thegenerationofconnectionstoformacircuit,i.e.routing−two

approachesproposedtoachievethisare:i)approximate

rout-ingwhichemploysDijkstra’salgorithm[28]toﬁndtheshortest

pathona1Dmedialaxis(i.e.skeletal)graph,ii)accurate

rout-ingwhichusesanimplementationoftheFastMarching(FM)

method[29].

Fig.2showsanexamplecapturingthefunctionalsystemdesign

capabilityin3D.Here,thehighlevelofaccuracyachievedusingthe

FMimplementationforidentifyingtheshortestroutesbetweena

setofcomponentsthatarenotlimitedtoplanarorientation

con-straints(asisthecasewith2.5DPCBparadigm)isdemonstrated. Inprinciple,itisbesttoperformroutingandplacement

concur-rentlybutastheyhavenesteddependencies,realizingasolution

toapracticalproblemmaybecomeunviable.Thereforethesetwo

stepsareoftentackledindependently.Withregardstothesystem

designconsideredhere,placementofa componentiskeptﬁxed

andthemethodofaccurateroutingisimplemented.Thesystem

optimizationproblemthereforebecomesaroutingoptimization

problemwiththeaimistoimprovethecircuitefﬁciencyby

low-eringenergylosses,whichareproportionaltotheresistance.This

isachievedbyidentifyingtheshortestpathsbetweencomponents

(asresistanceisproportionaltothepathlength)subjectto vari-ousdesignrulesandconstraints.Bydoingso,wealsominimisethe

utilizationoftheexpensivematerialusedforconductivetracks,

providingfurthereconomicbeneﬁt.

2.2. Couplingstrategy

NotableTOmethods,suchasSIMP[8], BESO[30]and

level-set[31],havebeenprincipallydeveloped toidentifythestiffest

structure(throughcomplianceminimization)foragivenmass

con-straint,andthoughtheirimplementationshavebeendemonstrated

(4)

con-Fig.5.Effectsofsystemintegrationincludedintheprocessofstructuraltopologyoptimization−a)Sensitivitiesforstructure,b)sensitivitiesforinternalsystem,c)combined sensitivities,d)resultingcoupledsolution,ande)TOusingjuststructuralsensitivitiesforcomparison.

sideraclassiccantileverplateorbeamproblemandthereforethis workalsoutilizessuchastructuretoinvestigatetheproposed cou-plingstrategy.

BothBESOandlevel-setimplementationsrevealawell-deﬁned

boundaryateveryiterationoftheoptimizationprocess,allowing

forembeddedsystemrealisation(asdiscussedin[9]).However,it isonlytheformerthatprovidesthestructuralelementalsensitivity

withinthevolumeoftheboundaryenablingtheoptimization

pro-cedureofFig.3 tocombinethiswiththecorrespondingsystem

sensitivities to govern the evolution of the combined

(struc-ture+system)solution.InthispapertheTOprocedure(indicated inFig.3)isbasedontherevisedBESOmethod[30,34,35]orwhat onemightcallthediscreteSIMPmethod[36].

Thecompliance(ameasureoftheinverseofstiffness)basedona

purelystructuraloptimizationproblemcanbestatedasminimise:

C= 1 2U

T_KU ₍₁₎

subjectto:

Vf − N

i=1

Vixi=0 (2)

whereCisthetotalstrainenergy(commonlytermedcompliance),

KistheglobalstiffnessmatrixandUistheglobaldisplacement

vector,V_iistheelementalvolumefraction,V_f isthestructural tar-getvolumefractionconstraint,andxi=xminor1,forvoidorsolid

regions,respectively.Forthiswork,xminwassetat1e-6.ThisBESO

formulationusesamaterialinterpolationscheme:

E(xi)=E1xp_i (3)

whereE1istheYoung’smodulusforthesolidregion,andpisthe

penaltyexponent.Asdetailedin[22,23],andexplainedby

Bend-soeandSigmund[8],thesensitivityoftheobjectivefunctionwith respecttotheith_element_is

∂

C

∂

xi

=−px

p−1 i

2 u

T ik

0

iui (4)

wherek0

i denotesthestiffnessmatrixanduirepresentsthe

dis-placementvectorfortheithelement.Therelativerankingofthe

elementalsensitivitiesforbothsolidandvoidregionelementscan

beexpressedas

S_˛ i=

−1

p

∂

C

∂

xi

=

⎧

⎪

⎨

⎪

⎩

1 2u

T ik

0

iui when xi=1

xp_min−1

2 u

T ik

0

iui when xi=xmin

(5)

whereS_˛

iisthestructuralsensitivitynumberforthei

th_element._It

canbeseenthatthesensitivitynumberforthesolidregionisequal totheelementalstrainenergy,andthesensitivitynumberforthe

voidregionisdependentonthevalueofp.Forthiswork,pwas

selectedtobe1,andthereforethestructuralsensitivitynumber

wasequaltothestrainenergyforbothregions.

2.3. Heuristicsensitivitydeﬁnition

Under a generaloptimization framework,one would ideally

statetheobjectivefunctionintermsoftheproblemspeciﬁc vari-ablesinorderthatthesensitivitiesofdesignvariablescanbeeasily

obtained.However,incaseswheredoingthisbecomeschallenging

(5)

Itrlimit _Optimization_iterations_after_which_the_process_is_terminated ₈₀

onecanattempttosolvethetwoproblemsasiftheywereoverlaid. Thisnecessitatesthedeﬁnitionofappropriatesensitivitiesforeach classofproblemandtheirimplementationinacombinedobjective function.Asthisworkfocusesonexampleswheretheplacement locationofcomponentsis pre-determined/speciﬁed,thesystem associatedelementalsensitivitiescanbedeterminedexclusively fromtheroutingaspectofsystemdesignusing

R_˛ i=

1 1+di

(6)

where,diisthedistance(Euclidianmeasure)betweenthe‘ith’ ele-ment(withinthediscretizeddesigndomain)andtheclosestpoint fromitontheroutedpaths.Thisassignsalowervalue(between0 and1)forelementsthatarefurtherawayfromaroutedpathanda valueof‘1tothoseelementswhichformaroute.

Thesystemelementalsensitivitiescanbecombinedwiththose ofthestructuralcounterpartusingaweightedsumapproachas shownin

C_˛

i=ω1×S˛i+(1−ω1)×

_R

˛_i

(7)

where,S_˛

irepresentsthenormalized(datarangebetween0and 1)structural elementalsensitivities(i.e.strainenergydensities) afterthresholdingtheoutliers(i.e.atpointofloadingandboundary conditions)andR_˛

irepresentsthenormalizedsystemelemental sensitivities.ω1 ∈[0,1] isauserdeﬁnedweighttocontrolthe rel-ativeinﬂuenceofthesystemdesignontheoverallcoupledsolution. Preliminaryworkbytheauthors[16]oncouplingsystemdesign

withTOconsideredthecantilever plateproblemof Fig.4.Four

componentlocations(shownascircles)wereidentiﬁedusingthe

resultanttopologyofastructure-onlyoptimization(seeFig.5e) sothattheeffectofthesysteminclusionontheresulting

struc-turewhenemployingacoupledoptimization,canbeinvestigated.

A pair-wiserouting between component locationsutilizing the

approximate routing method[9] (i.e.routes constrained tothe

medialaxisofthestructure)wasconsideredforsystemdesign.

ThecouplingemployedthealgorithmofFig.3andutilizedEqs.

(6)and(7)toassesssolutions.Fig.5presentstheeffectsofsuch

asystemintegrationwhenincludedintheprocessofTO.

How-ever,thisimplementationsufferedfromrobustnessissuesarising

fromtheunstableevolution(poorconvergence)oftheobjective

functionduringtheiterativeprocessoftheoptimization[16].This

wasessentiallyduetothesuddenchangesintheskeletal

topol-ogyresultingfromanydis-connectivityinstructuralmembers.It

isnoteworthythattheskeletaltopologygovernedthesystem

con-ﬁgurationintheaforementionedworkastheapproximateroutes

weremadetoadheretotheskeletaltopology. Inthis work,an

improvedcouplingstrategyisdevelopedthataddressesthis

prob-lemandusesamoreaccurateroutingmethod.

Thetwo key advancesmadein this papertowardsthe

cou-pledoptimizationalgorithmofFig.3are:i)animprovedheuristic deﬁnitionforsystemassociatedelementalsensitivitiesandii)an

improvedmethodofcombiningthesystemelementalsensitivities

Fig.6. Theconsideredextruded2D(2voxeldeep)cantileverproblem.

withthestructuralsensitivities.Fortheformer,theR_˛

ivaluesare

‘bounded’toavoidunwantedinﬂuenceinareasofthetopologynot

pertinenttotherouting,bysettingR_˛

itozerofordigreaterthan

theﬁlterradiusorRmin(seeTable1).

R_˛ i=

1 1+di

(bounded) (8)

andforthelatter,anadaptiveparameterdeﬁnedas

= S_␣ i R_␣ i (9)

whichismultipliedbyR_˛

itoensureanappropriate

contribu-tionofthesystemassociatedelementalsensitivitiestowardsthe

combinedsensitivity,whichisnowdeﬁnedas

C_˛

i=ω1×S˛i+(1−ω1)×

×R_˛

i

(10)

Although, Eqs. (7) and (10) share the same weighted sum

approachonthestructureandsystemsensitivitiesthatisnecessary fortheupdateofdesignvariables,itistheimprovedformulation intheheuristicsthattheauthorsevaluateanddiscussinthenext

sectionthatresultsinasigniﬁcantimprovementontheprevious

formulation.

3. Simulations,resultsanddiscussion

Thisworkraises tworesearch questionswhich allowfor an

assessmentoftheappropriatenessandrobustnessoftheproposed

couplingstrategy.

Researchquestion−I.Whatinﬂuencedoestheheuristic

sensi-tivitydeﬁnition(speciﬁcally,theuseofR_˛

i(bounded)and)have

onthecoupledsolution?

Researchquestion−II.Whetheritisadvantageoustoperform

acoupled(structure+system)optimizationornot?(Asa

compar-ativeuncoupledsolution,theauthorsrefertoperformingaTOto

obtainastructureandsubsequentlyaddingthesystemdesign)

Eachresearchquestionisbesttackledwithhelpofatestcase.

(6)

Fig.7. CoupledoptimizationforatargetVfof0.4witha)R˛_i(unbounded)and(notincluded),b)R˛_i(unbounded)and(included),c)R˛_i(bounded)and(notincluded), d)R_˛

i(bounded)and(included).

Fig.8. CoupledoptimizationforatargetVfof0.25witha)R˛_i(bounded)and(notincluded),b)R˛_i(bounded)and(included).

showninFig.6(withtheleftedgeﬁxedandavertically

down-wardforcebeingappliedtothebottomrightcorner).Theextrusion

extentofthedesigndomainwaskeptto2voxelsasthisallowed

formultipleroutestoexistbetweencomponentswithoutoverlaps.

Thisisbecauseroutesarenolongerlimitedtobeingplanarandcan existin3D.Twocomponentlocations(shownascircles)were iden-tiﬁedusingtheresultanttopologyofastructure-onlyoptimization soastobetterunderstandtheeffectofthesysteminclusiononthe

resultingstructurewhenemployingacoupledoptimizationwith

theimprovedheuristics.

Theparametersusedtosimulatethetestcasesarereportedin

Table1.Theseincludestandardvaluesforthematerialdensityof

solid(structure)andvoidregions,speciﬁcally,1andamuchlower valueof1e-6,respectively.Anintermediatevaluewasemployedfor

thematerialdensitycorrespondingtotherouting(system).This

choicewasmadesothatonlyaveryweakcontributioncouldbe

madebytheﬁniteelementsrepresentingtheroutesonthetotal

strainenergyofthestructure(i.e.measureofstructuralstiffness) asthesystemisnotintendedtobeloadbearingforthisstudy.Equal weightingwaschosenfortheobjectivefunctionformulationasthe focusofthisstudywasnotoninvestigatingtheinﬂuenceof weight-ingparameter(ω1).Theelementalstrainenergiesforthis study

(7)

Fig.9.Comparisoninresultsforuncoupledandcoupledoptimization−a)routingperformedona(converged)topologyoptimizedstructure,b)routingperformedasa coupled(structure+system)optimizationproblem.

Fig.10.Applicationtoamorecomplexproblem−a)coupledsolution,b)combinedsensitivity.

Nastran[37])onauniformmeshhavinganedgelengthof1unit

(Fig.6showsthestructureas128unitsby64unitsindimension).

ResultsforthesimulationofTestCaseI(addressingresearch

questionI)arepresentedinFig.7.Hereweanalyseandattempt

tounderstandtheeffectthatR_˛

i(bounded)andhave

(indepen-dently)onthesolution.Fig.7ashowsacharacteristicstraight-line

routebetweenthetwo placementlocations,even whensimilar

weighting for structure and system contributions (i.e. ω1=0.5)

wereused.Thissystemdominatedsolutioncanbebestunderstood

byinspectingthecombinedsensitivities(C_˛

i)ofthevoxels

repre-sentingtheroutewhichinthiscasehappenstoacquireanearly

constantandnotablyhighervaluefortheentireroute.Witha50%

weighting,thisshouldnotbethecaseasboththestructureand

systemshouldcontributemoreorlessequallytowardsthesolution.

InFig.7b, withtheinclusion oftheadaptive parameter, a

morebalancedcontributionofthesystemsensitivitiestowardsthe combinedsensitivitiesisachieved.Thisisevidentfromthe

vari-abilityin combinedsensitivityvalues forthevoxels comprising

theroute.However,eveninthiscaseacharacteristicstraight-line routeisobtainedfortheﬁnalsolution,indicatingacleardominance ofthesystemsensitivitytowardsthecombinedsensitivity.Thisis duetoR_˛

ibeingunbounded,resultinginanunnecessarilylarge

regionofinﬂuenceoverwhichthesystemsensitivitiesplayarole,

whichmakesthestructureneartheroutes unnecessarilybulky.

Thissystemsensitivitydominancemakesthepossibilityofelement

removalneartherouteverylow,causingthesystemsolutionto

havenonoticeablechangewiththeoptimizationiterations.

Conversely,a more intuitivesolutionwhere thesystem and

structureseemtoevolvecompetingwitheachothercanbeseen

inFig.7c-dwherethevaluesofR_˛

iarebounded.However,only

marginaldifferencescanbeseenwhencomparingthevoxels

rep-resentingtheroutesinthecoupledsolutionsofFig.7candd.This canbeattributedtotherelativelyhighstructuralvolumecompared tothesystemvolume,makingtheeffectof(inFig.7d)less notice-able.Totestthishypothesisfurther,anothercoupledoptimization wasperformedbutthistimethetargetVf wassetas0.25instead

ofthepreviouslyusedvalueof0.4.

Table2

ComparisoninstructuralperformancebetweensolutionsofFig.8(withandwithout theinclusionof␭).

Fig.8a(␭−notincluded)Fig.8b(␭−included)difference

TotalStrainEnergy88.2 67.4 23.6%

Table3

Comparisoninperformancebetweenuncoupledandcoupledoptimization.

Uncoupledsolution Coupledsolution difference

Pathlength(pixels) 196 138 30%

TotalStrainEnergy 39.9 40.2 <1%

Max.displacement(pixels) 76.7 77.4 <1%

Withthelowervaluefortheratioofstructuretosystemvolume inthesolutionofFig.8(wherethecoupledoptimizationproblem

isperformedforatargetVf of0.25)ascompared tothatinthe

caseoftargetVf=0.4,amorenotableinﬂuenceofcanbeseen−

boththeresultingtopologyofthestructureandthepixels compris-ingtherouteareseentodevelopduringtheoptimizationwiththe

improvedheuristics.Itiscommonforsolutionstogettrappedin

localminimainanon-convexproblemspace,suchasthatseenin

discreteSIMPorBESO.Asaconsequence,theevolutionofthe solu-tionishistorydependent,thedifferentcontributionsfromrouting sensitivities(i.e.withorwithouttheincorporationof)playa dom-inantrole.Thisissupportedbythe23.6%differenceintotalstrain energyvalues(ameasureforcomplianceorinverseofglobal stiff-ness)observedforthesolutionsofFig.8aandb(valuesreportedin

Table2).Moreover,itisclearlyevidentfromthewell-distributed andsmoothlytransitioningvaluesforthecombinedsensitivitiesin

Fig.8bthat:i)havingR_˛

ibounded,andii)theinclusionofthe

adap-tiveparameter,isnecessarytoobtainareliablesolutionasonly thencanoneensuretheappropriatecontributionofthesystem sen-sitivitiestowardsthecombinedsensitivitiesforamultifunctional problem.

ResultsforthesimulationofTestCaseIIaddressingresearch

questionII(i.e.whethertoperform(structure+system)coupled

(8)

pre-sentedquantitativelyinTable3.Thepixelscomprisingtheroute

areverydissimilarforcoupledanduncoupledsolutionsandthis

istobeexpectedasthesystemdesignisconstrainedtotheﬁnal

topologyofthestructureintheuncoupledproblem.Byadoptinga

coupledproblemformulation,nearly30%reductioninpathlength

isachieved.Importantly,despitethevisualdifferencesinstructural topology,negligibledifference(<1%)inthevaluesfortotalstrain energyisobserved.Fromtheconsideredexample,itcanbeinferred

thatthesystemdesignbeneﬁtsunderacoupledoptimization

for-mulationwhilsthavinglittleeffectonthestructuralperformance.

Togainabetterassessmentoftheeffects ofcouplingonthe

structuralperformance,amorecomplexroutingscenariowas

con-sidered where two component pairs are chosen such that the

placementlocationslieoutsidetheresultingstructurewhenone

considersthetopologyoptimizedgeometryofanuncoupled

prob-lem (speciﬁcally, topology of Fig. 9a). Fig. 10a shows how the

structuraltopologyhasevolvedtoaccommodatetheseroutes

join-ingthetwo componentpairsincase ofacoupledoptimization.

Fromapurelystructuralperformancestand-point,thetopologyof

Fig.10ahas15%highertotalstrainenergy(i.e.lessstiff)as com-paredtothatofFig.9a.However,itmustbenotedthatthelatter

hasundergoneasingle-objectiveoptimization(whereonlystrain

energyorinverseofstiffnesswasminimised)whilsttheformer

hasbeensubjectedtothecoupledoptimizationwhereaweighted

sumapproachisconsideredforoptimization.Suchacoupling,in

principle,shouldensurethebestmateriallayoutalong-side

opti-malsystemconﬁguration(inthis work−shortestroutes).This

isbecausetheroutingbetweenthecomponentscan’tberealized

onthestructuraltopologyofFig.9aforthespeciﬁedplacement

locations.

Theproposedcouplingmethodhasshownpromisewhen

tack-lingMFAMproblemandas thecapabilityof thesystemdesign

employedfor this work hasalready beendemonstrated for 3D

applications[9],onecan considerthisstrategy withconﬁdence

(initsnativeformorbyextendingitfurtherbyapplyingto

differ-entclassesofproblemand/ordifferentoptimizationalgorithms)

tosolvesimilar(structure+system)coupledproblemsin3D.It

mustbenotedthatthismethodcatersforthemultipleobjectives

viaa singleweightedsumapproach andconsequentlywill

suf-ferfromthelimitationsofthisformulation.Whereweightedsum

approaches mayprovetobeinadequate, amore generic

multi-objectiveconsiderationsuchasthepareto-frontcriterioncanbe

considered.

4. Concludingremarks

Thispaperhaspresentedacoupledoptimizationformulation

whichallowsfortheoptimalmaterialandsystemlay-outtobe

identiﬁedasittacklesasystemdesignproblemoverlaidona struc-turaldesignproblem.Although,theimmediateapplicationforthis

developmentis enablingthe designof additively manufactured

(jetted)multi-materialpartswithembeddedfunctionalsystems,

forexampleastructuralpartwithelectricalcomponentryand con-ductivetracks,nevertheless,thestrategypresentedhereinshould beconsideredfortacklingamoregeneralclassofengineering prob-lems.Forinstance,civilengineeringstructures(buildings/bridges) thatincorporatesystems(pipes/cables).Thiscoupledoptimization

developmentmarksasigniﬁcantsteptowardsbeingabletoexploit

thedesignfreedomofferedbyMMAM.

Themaincontributionofthispaperistheimprovedheuristic

deﬁnitionthatallows fora moreappropriate couplingstrategy,

wherethesystemdesignisperformedconcurrentlywiththe

struc-turaloptimization.Thisisachievedbyaccommodatingtheeffects

ofsystemincorporationonthestructuralresponseofthepartat

everyiterationwithinamodiﬁedbidirectionalevolutionary struc-turaloptimization.

Thesimulationresultsfortheevaluatedextruded2Dcantilever

testcasesshowthesuitabilityoftheproposedcouplingmethod

where thesystem sensitivities,speciﬁcallyrouting sensitivities, arecombinedwiththestructuralsensitivitiesforamultifunctional designproblem.Theauthorsbelievethiscontributionwillprovide

thenecessary design-innovationand in-turn themanufacturing

incentivetorealizemultifunctionalAMorMFAMproducts.

Acknowledgment

ThisworkwassupportedbytheEngineeringandPhysical

Sci-encesResearchCouncil[grantnumberEP/I033335/2].

References

[1]E.MacDonald,R.Wicker,Multiprocess3Dprintingforincreasingcomponent functionality,Science353(6307)(2016)(p.aaf2093-aaf2093).

[2]A.J.Lopes,E.MacDonald,R.B.Wicker,Integratingstereolithographyanddirect printtechnologiesfor3Dstructuralelectronicsfabrication,RapidPrototyp.J. 18(2)(2012)129–143.

[3]J.Vaithilingam,E.Saleh,C.Tuck,R.Wildman,I.Ashcroft,R.Hague,P.Dickens, 3D-inkjetprintingofﬂexibleandstretchableelectronics,Proceedingsofthe 26thSolidFreeformFabricationSymposium(2015).

[4]E.Saleh,J.Vaithilingam,C.Tuck,R.Wildman,I.Ashcroft,R.Hague,P.Dickens, 3Dinkjetprintingofconductivestructuresusingin-situIRsintering, Proceedingsofthe26thSolidFreeformFabricationSymposium(2015). [5]S.-P.Chen,H.-L.Chiu,P.-H.Wang,Y.-C.Liao,Inkjetprintedconductivetracks

forprintedelectronics,ECSJ.SolidStateSci.Technol.4(4)(2015) P3026–P3033.

[6]K.Vidimce,S.-P.Wang,J.Ragan-Kelley,W.Matusik,OpenFab:a

programmablepipelineformulti-Materialfabrication,ACMTrans.Graphics 32(4)(2013)1–11.

[7]R.Ponche,J.Y.Hascoet,O.Kerbrat,P.Mognol,Anewglobalapproachtodesign foradditivemanufacturing,VirtualPhys.Prototyp.7(June(2))(2012)93–105. [8]M.P.Bendsoe,O.Sigmund,TopologyOptimization:Theory,Methodsand

Applications,Springer,2003.

[9]A.Panesar,D.Brackett,I.Ashcroft,R.Wildman,R.Hague,Designframework formultifunctionaladditivemanufacturing:placementandroutingof3D printedcircuitvolumes,J.Mech.Des.137(11)(2015).

[10]J.Zhu,W.Zhang,P.Beckers,Integratedlayoutdesignofmulti-component system,Int.J.Numer.MethodsEng.78(2009)631–651.

[11]J.H.Zhu,W.H.Zhang,Integratedlayoutdesignofsupportsandstructures, Comput.MethodsAppl.Mech.Eng.12(Jan)(2010)557–569.

[12]W.Zhang,L.Xia,J.Zhu,Q.Zhang,Somerecentadvancesintheintegrated layoutdesignofmulticomponentsystems,J.Mech.Des.133(10)(2011) 104503.

[13]T.Buhl,Simultaneoustopologyoptimizationofstructureandsupports,Struct. Multidiscip.Optim.23(5)(2002)336–346.

[14]E.Beyne,3Dsystemintegrationtechnologies,InternationalSymposiumon VLSITechnology,Systems,andApplications(2006)1–9.

[15]A.Panesar,D.Brackett,I.Ashcroft,R.Wildman,R.Hague,Designoptimization strategyformultifunctional3Dprinting,25thInterationalSolidFreeform FabricationSymposium(2014)592–605.

[16]A.Panesar,D.Brackett,I.Ashcroft,R.Wildman,R.Hague,Designoptimization formultifunctional3Dprintedstructureswithembeddedfunctionalsystems, 11thWorldCongressonStructuralandMultidisciplinaryOptimisation(2015). [17]N.Abboud,M.Grötschel,T.Koch,Mathematicalmethodsforphysicallayoutof printedcircuitboards:anoverview,ORSpectr.30(May(3))(2007)453–468. [18]T.F.Chan,J.Cong,J.R.Shinnerl,K.Sze,M.Xie,MultiscaleOptimizationinVLSI

PhysicalDesignAutomation,MultiscaleOptimizationMethodsand Applications(NonconvexOptimizationandItsApplications),vol.82, Springer-Verlag,2006,pp.1–67.

[19]S.Areibi,Z.Yang,EffectivememeticalgorithmsforVLSIdesign=Genetic algorithms+localsearch+multi-levelclustering,Evol.Comput.12(Jan(3)) (2004)327–353.

[20]J.-H.Park,R.L.Storch,Pipe-routingalgorithmdevelopment:casestudyofa shipengineroomdesign,ExpertSyst.Appl.23(Oct(3))(2002)299–309. [21]X.Ma,K.Iida,M.Xie,J.Nishino,T.Odaka,H.Ogura,Ageneticalgorithmforthe

optimizationofcablerouting,SystemsComput.Jpn.37(Jun(7))(2006)61–71. [22]G.I.F.Thantulage,AntColonyOptimizationBasedSimulationof3DAutomatic

Hose/pipeRouting.PhD,BrunelUniversity,2009.

[23]C.Velden,C.Bil,X.Yu,A.Smith,Anintelligentsystemforautomaticlayout routinginaerospacedesign,InnovationsSyst.SoftwareEng.3(May(2)) (2007)117–128.

(9)