AMathematicalModelfortheMechanicalEtchingofGlassJ.H.M.tenThijeBoonkkampTechnischeUniversiteitEindhoven,DepartmentofMathematicsandComputerSciencetenthijewin.tue.nlSummary.Anonlinearfirst-orderPDEdescribingthedisplacementofaglasssur-facesubjecttosolidparticleerosionispresented.Theanalyticalsolutionisderivedbymeansofthemethodofcharacteristics.Alternatively,theEngquist-Osherschemeisappliedtocomputeanumericalsolution.Keywords:solidparticleerosion,kinematiccondition,singlePDEoffirstorder,characteristic-stripequations,Engquist-Osherscheme1IntroductionSomemoderntelevisiondisplayshaveavacuumenclosure,thatisinternallysupportedbyaglassplate.Thisplatemaynothinderthedisplayfunction.Forthatreasonithastobeaccuratelypatternedwithsmalltrenchesorholessothatelectronscanmovefreelyfromthecathodetothescreen.Onemethodtomanufacturesuchglassplatesistocoveritwithanerosion-resistantmaskandblastitwithanabrasivepowder.InSection2wepresentanonlinearfirst-orderPDEmodellingthisso-calledsolidparticleerosionprocess.Next,inSection3,wepresenttheanalyticalsolutionusingthemethodofcharac-teristics.Alternatively,inSection4,webrieflydescribeanumericalsolutionprocedure.2MathematicalModelforPowderErosionInthissectionweoutlineamathematicalmodelforsolidparticleerosion,toproducethintrenchesinaglassplate；formoredetailssee4.Consideraninitiallyflatsubstrateofbrittlematerial,coveredwithaline-shapedmask.Weintroducean(x,y,z)-coordinatesystem,wherethe(x,y)-planecoincideswiththeinitialsubstrateandthepositivez-axisisdirectedAMathematicalModelfortheMechanicalEtchingofGlass387intothematerial.Acontinuousfluxofalumina(Al2O3)particles,directedinthepositivez-direction,hitsthesubstrateathighvelocityandremovesmaterial.Thepositionz=(x,t)ofthetrenchsurfaceattimetisgovernedbythekinematicconditiont+(x)f(x)=0,0x0,(1)wherexisthetransversecoordinateinthetrench,andwhere(x)istheparticlemassflux,whichwillbespecifiedlater.Thespatialvariablesandxarescaledwiththetrenchwidthandthetimetwithacharacteristictimeneededtopropagateasurfaceatnormalimpactoverthiswidth.Thefunctionf=f(p)in(1)isdefinedbyf(p):=parenleftbig1+p2parenrightbigk/2,(2)withkaconstant(2k4).Atheoreticalmodelpredictsthevaluek=7/3,3.Equation(1)issupplementedwiththefollowinginitialandboundaryconditions:(x,0)=0,0x0.(3b)Theboundaryconditionsin(3b)meanthatthetrenchcannotgrowattheendsx=0andx=1.3AnalyticalSolutionMethodWecanwriteequation(1)inthecanonicalformF(x,t,p,q):=q(x)parenleftbig1+p2parenrightbigk/2=0,(4)withp:=xandq:=t.Thesolutionof(4)canbeconstructedfromthefollowingIVPforthecharacteristic-stripequations1dxds=Fp=(x)kp(1+p2)k/2+1,x(0；)=,(5a)dtds=Fq=1,t(0；)=0,(5b)dds=pFp+qFq=(x)1+(k+1)p2(1+p2)k/2+1,(0；)=0,(5c)dpds=(Fx+pF)=prime(x)1(1+p2)k/2,p(0；)=0,(5d)dqds=(Ft+qF)=0,q(0；)=(),(5e)388J.H.M.tenThijeBoonkkampwheresandaretheparametersalongthecharacteristicsandtheinitialcurve,respectively.Notethatthesolutionof(5b)and(5e)istrivial,andwefindt(s；)=sandq(s；)=().Inordertomodelthefiniteparticlesize,whichmakesthatparticlesclosetothemaskarelesseectiveintheerosionprocess,weintroducetransi-tionregionsofthickness.Weassumethat(x)increasescontinuouslyandmonotonicallyfrom0attheboundariesofthetrenchto1atx=,1.Theparameterischaracteristicofthe(dimensionless)particlesizeandatypicalvalueis=0.1.Weadoptthesimplestpossiblechoicefor(x),i.e.,(x)=x/if0x,1ifx1,(1x)/if1x1.(6)Asaresultof(6),thegrowthrateofthesurfacepositionclosetothemaskissmallerthaninthemiddleofthehole.Since(0)=(1)=0,weobtainfrom(5)thesolutionsx(t；0)=(t；0)=0andx(t；1)=1,(t；1)=0,implyingthattheboundaryconditions(3b)forareautomaticallysatisfied.Byintroducingtransitionregions,wecreateintersectingcharacteristics.Therefore,thesolutionof(4)canonlybeaweaksolutionanditisanticipatedthatshockswillemergefromtheedgesx=andx=1.Letx=s,1(t)andx=s,2(t)denotethelocationoftheshocksattimetoriginatingatx=andx=1,respectively.Eachpoint(s,i(t),t)(i=1,2)ontheseshocksisconnectedtotwodierentcharacteristicsthatexistonbothsidesoftheshocks.Thespeedoftheseshocksisgivenbythejumpconditionds,idtp=(x)(1+p2)k/2,(i=1,2),(7)wherepdenotesthejumpofpacrosstheshock.Thus,wecandistinguishthefollowingfiveregionsinthe(x,t)-plane:thelefttransitionregion0x00.20.40.60.8100.10.20.30.40.50.60.70.80.91xtFig.1.Characteristicsandshocksof(5),for=0.1andk=2.33.AMathematicalModelfortheMechanicalEtchingofGlass389(region1),therighttransitionregion1x1(region2),theinteriordomainleftofthefirstshock(region3),theinteriordomainrightofthesecondshock(region4)andtheregionbetweenthetwoshocks(region5)；seeFig.1.Note,thatthelocationoftheshocksdependsonthesolutionthrough(7).Wecanderivetheanalyticalsolutionof(5)intheregions1,3and5,coupledwithanumericalsolutionof(7).Thesolutionintheothertworegionfollowsbysymmetry；formoredetailssee4.TheresultsarecollectedinFig.2,whichgivesthesolutionforandpattimelevelst=0.0,0.1,.,1.0for=0.1andk=2.33.Thisfigurenicelydisplaysthefeaturesofthesolution:aslantedsurfaceinthetransitionregions,aflatbottomintheinteriordomainandacurvedsurfaceinbetween.Also,inwardlypropagatingshocksareclearlyvisible.00.20.40.60.8100.10.20.30.40.50.60.70.80.91x00.20.40.60.813210123xpFig.2.Analyticalsolutionforthesurfaceposition(left)anditsslope(right).Pa-rametervaluesare=0.1andk=2.33.4NumericalSolutionMethodAlternatively,wewillcomputeanumericalsolutionof(1).Tothatpurpose,wecoverthedomain0,1withcontrolvolumesVj=xj1/2,xj+1/2)ofequalsizex=xj+1/2xj1/2.LetxjbethegridpointinthecentreofVj.Furthermore,weintroducetimelevelstn=nt,withtbeingthetimestep.Letnjdenotethenumericalapproximationof(xj,tn).Afinitevolumenumericalschemefor(1)canbewritteninthegenericformn+1j=njt(xj)Fparenleftbigpnj1/2,pnj+1