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外文翻译--玻璃侵蚀的机械物理模型 英文版.pdf

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外文翻译--玻璃侵蚀的机械物理模型 英文版.pdf

AMathematicalModelfortheMechanicalEtchingofGlassJ.H.M.tenThijeBoonkkampTechnischeUniversiteitEindhoven,DepartmentofMathematicsandComputerSciencetenthijewin.tue.nlSummary.AnonlinearfirstorderPDEdescribingthedisplacementofaglasssurfacesubjecttosolidparticleerosionispresented.Theanalyticalsolutionisderivedbymeansofthemethodofcharacteristics.Alternatively,theEngquistOsherschemeisappliedtocomputeanumericalsolution.Keywordssolidparticleerosion,kinematiccondition,singlePDEoffirstorder,characteristicstripequations,EngquistOsherscheme1IntroductionSomemoderntelevisiondisplayshaveavacuumenclosure,thatisinternallysupportedbyaglassplate.Thisplatemaynothinderthedisplayfunction.Forthatreasonithastobeaccuratelypatternedwithsmalltrenchesorholessothatelectronscanmovefreelyfromthecathodetothescreen.Onemethodtomanufacturesuchglassplatesistocoveritwithanerosionresistantmaskandblastitwithanabrasivepowder.InSection2wepresentanonlinearfirstorderPDEmodellingthissocalledsolidparticleerosionprocess.Next,inSection3,wepresenttheanalyticalsolutionusingthemethodofcharacteristics.Alternatively,inSection4,webrieflydescribeanumericalsolutionprocedure.2MathematicalModelforPowderErosionInthissectionweoutlineamathematicalmodelforsolidparticleerosion,toproducethintrenchesinaglassplateformoredetailssee4.Consideraninitiallyflatsubstrateofbrittlematerial,coveredwithalineshapedmask.Weintroduceanx,y,zcoordinatesystem,wherethex,yplanecoincideswiththeinitialsubstrateandthepositivezaxisisdirectedAMathematicalModelfortheMechanicalEtchingofGlass387intothematerial.AcontinuousfluxofaluminaAl2O3particles,directedinthepositivezdirection,hitsthesubstrateathighvelocityandremovesmaterial.Thepositionzζx,tofthetrenchsurfaceattimetisgovernedbythekinematicconditionζtΦxfζx0,00,1wherexisthetransversecoordinateinthetrench,andwhereΦxistheparticlemassflux,whichwillbespecifiedlater.Thespatialvariablesζandxarescaledwiththetrenchwidthandthetimetwithacharacteristictimeneededtopropagateasurfaceatnormalimpactoverthiswidth.Thefunctionffpin1isdefinedbyfp−parenleftbig1p2parenrightbig−k/2,2withkaconstant2≤k≤4.Atheoreticalmodelpredictsthevaluek7/3,3.Equation1issupplementedwiththefollowinginitialandboundaryconditionsζx,00,00.3bTheboundaryconditionsin3bmeanthatthetrenchcannotgrowattheendsx0andx1.3AnalyticalSolutionMethodWecanwriteequation1inthecanonicalformFx,t,ζ,p,qq−Φxparenleftbig1p2parenrightbig−k/20,4withpζxandqζt.Thesolutionof4canbeconstructedfromthefollowingIVPforthecharacteristicstripequations1dxdsFpΦxkp1p2k/21,x0σσ,5adtdsFq1,t0σ0,5bdζdspFpqFqΦx1k1p21p2k/21,ζ0σ0,5cdpds−FxpFζΦprimex11p2k/2,p0σ0,5ddqds−FtqFζ0,q0σΦσ,5e388J.H.M.tenThijeBoonkkampwheresandσaretheparametersalongthecharacteristicsandtheinitialcurve,respectively.Notethatthesolutionof5band5eistrivial,andwefindtsσsandqsσΦσ.Inordertomodelthefiniteparticlesize,whichmakesthatparticlesclosetothemaskarelesseffectiveintheerosionprocess,weintroducetransitionregionsofthicknessδ.WeassumethatΦxincreasescontinuouslyandmonotonicallyfrom0attheboundariesofthetrenchto1atxδ,1−δ.Theparameterδischaracteristicofthedimensionlessparticlesizeandatypicalvalueisδ0.1.WeadoptthesimplestpossiblechoiceforΦx,i.e.,Φx⎧⎪⎪⎨⎪⎪⎩x/δif0≤xδ,1ifδ≤x≤1−δ,1−x/δif1−δx≤1.6Asaresultof6,thegrowthrateofthesurfacepositionclosetothemaskissmallerthaninthemiddleofthehole.SinceΦ0Φ10,weobtainfrom5thesolutionsxt0ζt00andxt11,ζt10,implyingthattheboundaryconditions3bforζareautomaticallysatisfied.Byintroducingtransitionregions,wecreateintersectingcharacteristics.Therefore,thesolutionof4canonlybeaweaksolutionanditisanticipatedthatshockswillemergefromtheedgesxδandx1−δ.Letxξs,1tandxξs,2tdenotethelocationoftheshocksattimetoriginatingatxδandx1−δ,respectively.Eachpointξs,it,ti1,2ontheseshocksisconnectedtotwodifferentcharacteristicsthatexistonbothsidesoftheshocks.Thespeedoftheseshocksisgivenbythejumpconditiondξs,idtp−Φx1p2−k/2,i1,2,7wherepdenotesthejumpofpacrosstheshock.Thus,wecandistinguishthefollowingfiveregionsinthex,tplanethelefttransitionregion0≤x≤δ00.20.40.60.8100.10.20.30.40.50.60.70.80.91xtFig.1.Characteristicsandshocksof5,forδ0.1andk2.33.AMathematicalModelfortheMechanicalEtchingofGlass389region1,therighttransitionregion1−δ≤x≤1region2,theinteriordomainleftofthefirstshockregion3,theinteriordomainrightofthesecondshockregion4andtheregionbetweenthetwoshocksregion5seeFig.1.Note,thatthelocationoftheshocksdependsonthesolutionthrough7.Wecanderivetheanalyticalsolutionof5intheregions1,3and5,coupledwithanumericalsolutionof7.Thesolutionintheothertworegionfollowsbysymmetryformoredetailssee4.TheresultsarecollectedinFig.2,whichgivesthesolutionforζandpattimelevelst0.0,0.1,...,1.0forδ0.1andk2.33.Thisfigurenicelydisplaysthefeaturesofthesolutionaslantedsurfaceinthetransitionregions,aflatbottomintheinteriordomainandacurvedsurfaceinbetween.Also,inwardlypropagatingshocksareclearlyvisible.00.20.40.60.8100.10.20.30.40.50.60.70.80.91xζ00.20.40.60.81−3−2−10123xpFig.2.Analyticalsolutionforthesurfacepositionleftanditssloperight.Parametervaluesareδ0.1andk2.33.4NumericalSolutionMethodAlternatively,wewillcomputeanumericalsolutionof1.Tothatpurpose,wecoverthedomain0,1withcontrolvolumesVjxj−1/2,xj1/2ofequalsize∆xxj1/2−xj−1/2.LetxjbethegridpointinthecentreofVj.Furthermore,weintroducetimelevelstnn∆t,with∆tbeingthetimestep.Letζnjdenotethenumericalapproximationofζxj,tn.Afinitevolumenumericalschemefor1canbewritteninthegenericformζn1jζnj−∆tΦxjFparenleftbigpnj−1/2,pnj1/2parenrightbig,8withpnj±1/2anumericalapproximationofpxj±1/2,tnandFFplscript,prthenumericalfluxfunction,thatweassumetodependontwovaluesofp.ThenumericalfluxFparenleftbigpnj−1/2,pnj1/2parenrightbigisanapproximationoffpxj,tn.Weapproximatepnj±1/2bycentraldifferencesandtaketheEngquistOsher

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