1、AMathematicalModelfortheMechanicalEtchingofGlassJ/H/M/tenThijeBoonkkampTechnischeUniversiteitEindhoven/DepartmentofMathematicsandComputerSciencetenthijewin/tue/nlSummary/Anonlinearfirst-orderPDEdescribingthedisplacementofaglasssur-facesubjecttosolidparticleerosionispresented/Theanalyticalsolutionisd
2、erivedbymeansofthemethodofcharacteristics/Alternatively/theEngquist-Osherschemeisappliedtocomputeanumericalsolution/Keywords/solidparticleerosion/kinematiccondition/singlePDEoffirstorder/characteristic-stripequations/Engquist-Osherscheme1IntroductionSomemoderntelevisiondisplayshaveavacuumenclosure/t
3、hatisinternallysupportedbyaglassplate/Thisplatemaynothinderthedisplayfunction/Forthatreasonithastobeaccuratelypatternedwithsmalltrenchesorholessothatelectronscanmovefreelyfromthecathodetothescreen/Onemethodtomanufacturesuchglassplatesistocoveritwithanerosion-resistantmaskandblastitwithanabrasivepowd
4、er/InSection2wepresentanonlinearfirst-orderPDEmodellingthisso-calledsolidparticleerosionprocess/Next/inSection3/wepresenttheanalyticalsolutionusingthemethodofcharac-teristics/Alternatively/inSection4/webrieflydescribeanumericalsolutionprocedure/2MathematicalModelforPowderErosionInthissectionweoutlin
5、eamathematicalmodelforsolidparticleerosion/toproducethintrenchesinaglassplate；formoredetailssee4/Consideraninitiallyflatsubstrateofbrittlematerial/coveredwithaline-shapedmask/Weintroducean(x/y/z)-coordinatesystem/wherethe(x/y)-planecoincideswiththeinitialsubstrateandthepositivez-axisisdirectedAMathe
6、maticalModelfortheMechanicalEtchingofGlass387intothematerial/Acontinuousfluxofalumina(Al2O3)particles/directedinthepositivez-direction/hitsthesubstrateathighvelocityandremovesmaterial/Thepositionz=(x/t)ofthetrenchsurfaceattimetisgovernedbythekinematicconditiont+(x)f(x)=0/0x0/(1)wherexisthetransverse
7、coordinateinthetrench/andwhere(x)istheparticlemassflux/whichwillbespecifiedlater/Thespatialvariablesandxarescaledwiththetrenchwidthandthetimetwithacharacteristictimeneededtopropagateasurfaceatnormalimpactoverthiswidth/Thefunctionf=f(p)in(1)isdefinedbyf(p)/=parenleftbig1+p2parenrightbigk/2/(2)withkac
8、onstant(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)parenle
9、ftbig1+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(
10、0；)=()/(5e)388J/H/M/tenThijeBoonkkampwheresandaretheparametersalongthecharacteristicsandtheinitialcurve/respectively/Notethatthesolutionof(5b)and(5e)istrivial/andwefindt(s；)=sandq(s；)=()/Inordertomodelthefiniteparticlesize/whichmakesthatparticlesclosetothemaskarelesseectiveintheerosionprocess/weintr
11、oducetransi-tionregionsofthickness/Weassumethat(x)increasescontinuouslyandmonotonicallyfrom0attheboundariesofthetrenchto1atx=/1/Theparameterischaracteristicofthe(dimensionless)particlesizeandatypicalvalueis=0/1/Weadoptthesimplestpossiblechoicefor(x)/i/e//(x)=x/if0x/1ifx1/(1x)/if1x1/(6)Asaresultof(6)
12、/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
13、/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/T
14、hespeedoftheseshocksisgivenbythejumpconditionds/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/A
15、MathematicalModelfortheMechanicalEtchingofGlass389(region1)/therighttransitionregion1x1(region2)/theinteriordomainleftofthefirstshock(region3)/theinteriordomainrightofthesecondshock(region4)andtheregionbetweenthetwoshocks(region5)；seeFig/1/Note/thatthelocationoftheshocksdependsonthesolutionthrough(7
16、)/Wecanderivetheanalyticalsolutionof(5)intheregions1/3and5/coupledwithanumericalsolutionof(7)/Thesolutionintheothertworegionfollowsbysymmetry；formoredetailssee4/TheresultsarecollectedinFig/2/whichgivesthesolutionforandpattimelevelst=0/0/0/1///1/0for=0/1andk=2/33/Thisfigurenicelydisplaysthefeaturesof
17、thesolution/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-ramet
18、ervaluesare=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/Letnjdenotet
19、henumericalapproximationof(xj/tn)/Afinitevolumenumericalschemefor(1)canbewritteninthegenericformn+1j=njt(xj)Fparenleftbigpnj1/2/pnj+1/2parenrightbig/(8)withpnj1/2anumericalapproximationofp(xj1/2/tn)andF=F(plscript/pr)thenumericalfluxfunction/thatweassumetodependontwovaluesofp/ThenumericalfluxFparenleftbigpnj1/2/pnj+1/2parenrightbigisanapproximationoff(p(xj/tn)/Weapproximatepnj1/2bycentraldierencesandtaketheEngquist-Osher
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