KimiyaAoki,NgoHaiDong,ToyohisaKaneko,andShigeruKuriyama
ToyohashiUniversityofTechnology,DepartmentofInformationandComputerSciences
1-1,Tempaku-cho,Toyohashi,Aichi,Japan441-8580{aoki,dong,kaneko,kuriyama}@vcl.ics.tut.ac.jp
Abstract
Thispaperdescribesanewphysicallybasedmethodforsynthesizing3Dcrackingpatternsinclaysolidsbyincor-poratingamoisturemodel.Wemeasuredtemporaltransi-tionsofthephysicalparametersusedinthemodel.Thenwedescribeanewapproachtosynthesizecrackingpatternsina3Dfreeformobjectbyitsapproximatequantizedshapeandthemeshrelationofitsoriginalshape.Withthisnewmethod,animatedsynthesisofnotonlyexternalcracksbutalsointernalcracksof3Dfreeformsolidsispossible.
zationalongthex,y,andzdirection.Eachsmallcubicele-mentwillbecalledacell.(Inourexperimentsweemployedδ=2[mm].)Theentirephysicalsimulationiscarriedoutbasedonthisquantizedmodel.However,aftersimulationbasedoncellsiscarriedout,theresultingvisualrepresenta-tionisderivedbasedontheoriginalsmoothshapederivedbyusingtherelationshipbetweenthequantizedmodelandtheoriginalsmoothmodel.2.2
MechanicalModel
1Introduction
Crackingisanaturalphenomenoncommonlyobservedonthesurfaceofsuchobjectsasconcrete,claywalls,mud,pottery,andtreebarks.Thesecracksmaybecreatednat-urallyorinsomecasesbyexternalforce.Cracksonclaywalls,mudetc.areduetoshrinkingvolumeastheyaredryingup.Successfulmodelingofcrackswillenablethevisualizationofnotonlystaticcrackingscenesbutalsody-namicscenesasanimations.Crackingobjectsneedtobeofanyfreeformshapes.Theobjectiveofthispaperistosim-ulatecracksfromaphysicallybasedapproachformeetingtheseobjectives.
Thesynthesisofcrackpatternshasattractedattentionofresearchersincomputergraphics(CG)[1,2,3,4,5,6,7].Herewewillbeconcernedwithmodelingandvisualizingcracksonthesurfaceof3Ddryingclaysolids.Themate-rialusedhereisflour-basedclay.Allthepastworksdidnottakeintoaccountthefactthatcracksappearduetoshrink-ingvolumeofclaycausedbywaterevaporation.Withusingwaterevaporation,crackscanbeobservedatthecrosssec-tionofobject.Thispromptedustointroduceamoisturemodel.
Eachcubiccellcanbebrokenintofivetetrahedrons.Weallocateaspringtoeachedgeofatetrahedron.Themassofanelementisdividedintoeachnode.mirepresentsthemassofnodei.Thefinalmechanicalmodelisrepresentedbyaspringnetworkmodelwhosenodesareonthegridsofthequantizedmodel.Thesimulationisbasedupontheentirecollectionofthesetetrahedrons.
Thevelocityandthedisplacementofeachnodearecal-culatedbysolvingthemotionequationonnodes.Inthiscase,theforcecommittedtoeachnode(Fi)isexpressedwiththefollowingequation:
Fi=−
j
kij
xi−xj
(|xi−xj|−loij)+mig
|xi−xj|
(1)
wherexiandxjarethepositionsofnodei,neighboringnodej,loijandkijarethenaturallengthandspringcon-stantofthespringconnectedtothenodeiandj,gisthegravityacceleration.
Eachspringiscutwhenitisstretchedbeyondthemax-imumstrain,andthencracksarevisualized.Onceanedgeofatetrahedroniscut,itisassumedthatthetetrahedronisdividedintotwo,three,orfourparts.2.3
MoistureModel
2
2.1
Method
GeometricalModel
Asthewaterevaporatesfromasurfacenode,watermovesfromitsinternalneighboringnodestowardthesur-facenode.Inordertodescribethemovementofwater,wedefinemoisturecontentωiatnodei.Letωrepresenttherateofthemassofthewater.Thenthemotionofwater
Givena3Dobject,weobtainitsgeometricalmodelbyquantization.Hereweemployauniformsizeδforquanti-
Proceedings of the 10 th Pacific Conference on Computer Graphics and Applications (PG’02) 0-7695-1784-6/02 $17.00 © 2002 IEEE
betweeneachnodeisgovernedbythefollowingdiffusionequation:
∂2ωi∂ωi
(2)=Dω2
∂t∂xi
wherethediffusionconstantDωistobedeterminedexper-imentally.Inthispapertheaboveequationissolvedbythefinitedifferencemethod(FDM).2.4
LinkbetweenMechanicalModelandMoistureModel
Actual cracksSimulated cracksMoisture model8h8h8hItisunderstoodthatsomeparametersinthemechani-calmodelaregovernedbythemoistureconditionderivedfromthemoisturemodel.Specificallythreeparameters:thespringconstant(k(ω)),maximumstrain(Γ(ω)),andcontractionratio(γ(ω))includedinthemechanicalmodelareconsideredtobefunctionsofthemoisturecontent(ω)ofanode.Thesefunctionsaredefinedonthebasisofmea-surementsonrealclaybeforehand.
18h18hDry
18hWet
Figure1:Crackpatternsandmoisturemodels
3SimulationandExperiment
Thecracksimulationiscarriedoutasfollows.Therearetwotimescales:longtime(t)andshorttime(τ).First,westartbysettingthelongtimetobezero.Assumethatattheendoflongtimet,alltheparametersareset.Thenwegotothemoisturemodelandcomputethemoisturecontentateachnodeofthenetworkmodel.Fromthecom-putedmoisturecontent,wethenderivethespringconstantandthemaximumstrain.Next,westarttheshorttimeiterationswiththenewvaluesforeachnodeandeachedgeinthemechanicalmodeluntilthesystembecomesstable.Aftertheshorttimeiterationsend,thenweincrementthelongtimebyoneas(t+1).Inthisfashion,wealternativelyrepeattheprocessofreadingthemoisturecontentandthatofinitiatingtheshorttimeiterations.
Figure1showsatemporalchangeofsimulatedcracksonacube(4×4×4[cm]).Fromthisfigure,itiscon-firmedthatthetemporalagreementbetweenthesimula-tionandrealisexcellent.Thenextexamplesareasphere(diameter:2.2[cm])andabell(bottomradius:1.45[cm],height:2.9[cm])shapeobjectshowninFigure2.There-sultingcracksarereasonable.Therefore,thetechniquefora3Dfreeformobjectasdescribedaboveisshowntobeef-fective.Thetotalcomputationtimeforthe4×4×4[cm]cubewas8hoursonaPCwithaPentium2GCPU.Thesphereandthebellshapemodelrequired0.7and1hours,respectively.
(a) Sphere(b) Bell
Figure2:Crackpatternonfreeformobject
strengthsofthisnewmethodisexcellenttemporalagree-mentincrackpatternformation.Wealsodevisedanewtechniquetosimulateforafreeformsolidwithitsquan-tizedversion.
Futureworkincludestheeffectofcracksonthemoisturemodel,whichisnotconsideredinthispaper.Anotherworkistoemploytexturemappingonthesurfaceinordertoillustrateusefulapplications.
References
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representationmethodofcracksonaplateglas.TransIPS,Vol.33,No.10,pp.1235–1244,1992.[3]A.T.SkjeltorpandP.Meakin.Fractureinmicrospheremono-layersstudiedbyexperimentandcomputersimulation.Nature,Vol.335,pp.424–426,1988.[4]K.Hirota,Y.Tanoue,andT.Kaneko.Simulationofthree-dimensionalcracks.VisualComputer,Vol.16,pp.371–378,2000.[5]S.GobronandN.Chiba.3dsurfacecellularautomataandtheir
applications.J.ofVisualizationandComputerAnimation,Vol.10,pp.143–158,1999.[6]A.Norton,G.Turk,B.Bacon,J.Gerth,andP.Sweeney.Ani-mationoffracturesbyphysicalmodel.VisualComputer,Vol.7,pp.210–219,1991.[7]J.F.O’BrienandJ.K.Hodgin.Graphicalmodelingandan-imationofbrittlefracture.ComputerGraphics,Vol.34,pp.137–146,1999.
4ConclusionandFutureWork
Weproposedanewcomprehensivecrackmodelwhichconsistsofamechanicalmodelandamoisturemodel.Thesimulationrunswithalternatingturnsofexecutingthetwomodels.Thisnewmodelcanprovidecrackpatternswhichresembleactualcracksreasonablywell.Oneofthe
Proceedings of the 10 th Pacific Conference on Computer Graphics and Applications (PG’02) 0-7695-1784-6/02 $17.00 © 2002 IEEE
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