TWO SCALE ANALYSIS FOR VERY ROUGH THIN LAYERS AN EXPLICIT CHARACTERIZATION OF THE POLARIZATION

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TWO-SCALE ANALYSIS FOR VERY ROUGH THIN LAYERS. AN EXPLICIT CHARACTERIZATION OF THE POLARIZATION TENSOR IONEL SORIN CIUPERCA, RONAN PERRUSSEL, AND CLAIR POIGNARD Abstract. We study the behaviour of the steady-state voltage potential in a material composed of a two-dimensional object surrounded by a very rough thin layer and embedded in an ambient medium. The roughness of the layer is described by a quasi ?–periodic function, ? being a small parameter, while the mean thickness of the layer is of magnitude ?? , where ? ? (0, 1). Using the two-scale analysis, we replace the very rough thin layer by appropriate trans- mission conditions on the boundary of the object, which lead to an explicit characterization of the polarization tensor as defined in Vogelius and Capde- boscq (ESAIM:M2AN. 2003; 37:159-173). This paper extends the previous works of Poignard (Math. Meth. App. Sci. 2009; 32:435-453) and of Ciuperca et al. (Research report INRIA RR-6812), in which ? ≥ 1. 1. Introduction Consider a material composed of a two-dimensional object surrounded by a very rough thin layer. We study the asymptotic behaviour of the steady-state voltage potential when the thickness of the layer tends to zero. We present approximate transmission conditions to take into account the effects due to the layer without fully modeling it.

• thin layers


• solutions u?


• ?t ?


• scale analysis


• main results


• function ?


• constant thickness


• positive constant


• ?? ∫

Sujets

• Analysis
• Thin layers (oceanography)
• Scale analysis

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 37
• Langue: English

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TWO-SCALEANALYSISFORVERYROUGHTHINLAYERS. ANEXPLICITCHARACTERIZATIONOFTHEPOLARIZATION TENSOR IONELSORINCIUPERCA,RONANPERRUSSEL,ANDCLAIRPOIGNARD Abstract. Westudythebehaviourofthesteady-statevoltagepotentialin amaterialcomposedofatwo-dimensionalobjectsurroundedbyaveryrough thinlayerandembeddedinanambientmedium.Theroughnessofthelayeris describedbyaquasiε–periodicfunction,εbeingasmallparameter,whilethe meanthicknessofthelayerisofmagnitudeε β ,whereβ ∈ (0,1).Usingthe two-scaleanalysis,wereplacetheveryroughthinlayerbyappropriatetrans- missionconditionsontheboundaryoftheobject,whichleadtoanexplicit characterizationofthepolarizationtensorasdefinedinVogeliusandCapde- boscq(ESAIM:M2AN.2003;37:159-173).Thispaperextendstheprevious worksofPoignard(Math.Meth.App.Sci.2009;32:435-453)andofCiuperca etal. (ResearchreportINRIARR-6812),inwhichβ ≥ 1. 1. Introduction Consideramaterialcomposedofatwo-dimensionalobjectsurroundedbyavery roughthinlayer.Westudytheasymptoticbehaviourofthesteady-statevoltage potentialwhenthethicknessofthelayertendstozero.Wepresentapproximate transmissionconditionstotakeintoaccounttheeffectsduetothelayerwithout fullymodelingit.Thispaperendsaseriesof3papersdealingwiththesteady- statevoltagepotentialindomainswiththinlayerwithanonconstantthickness. Unlike[17,18]inwhichthelayerisweaklyoscillating,andunlike[12],whichdeals withtheperiodicroughnesscase,weconsiderherethecaseofaveryroughthin layer.Thismeansthattheperiodoftheoscillationsismuchsmallerthanthe meanthicknessofthelayer.Moreprecisely,weconsideraperiodequaltoε,while themeanthicknessofthelayerisofmagnitudeε β ,whereβisapositiveconstant strictlysmallerthan1. Underthishypothesis,theformalderivationoftheasymptoticexpansionisvery difficult,sincethelimitgeometryoftheproblemisverysingular.Asfarasweknow, thisproblemasnotbeentackledyet,andthegoalofthispaperistoprovideaway inordertoobtainthefullasymptoticdevelopment.Actually,usingatwo-scale analysistechnique,weidentifythetransmissionconditions,whichapproximatethe influenceofthemembraneatthefirstorderontheelectricpotential.Theseap- proximatetransmissionsarequitesurprinsing,sinceunlike[12]theydonotinvolve thetangentialderivativesofthepotential.Moreovertheyinvolvethefunctionsr 1 andr 2 definedby(8),whichmixthemembraneandtheexteriorconductivityin complexway.Hence,unlikethecaseoftheweaklyoscillatingthinmembrane(see [17]),sincethequasiε-periodoftheoscillationsoftheroughlayerisfastcompared Keywordsandphrases.Asymptoticanalysis,Two-scaleconvergence,FiniteElementMethod, Laplaceequations. 1

2I.S.CIUPERCA,R.PERRUSSEL,ANDC.POIGNARD m Ω ε 1 Γ Ω Γ ε 0 Ω ε Ω Figure1.Geometryoftheproblem. toitsthickness,thenthelayerinfluenceonthesteady-statepotentialcannotbe approximatedbyonlyconsideringthemeaneffectoftheroughlayer.Thereforethe mainresultofthispaperisquiteunexpected,andtheapproximatetransmission conditionsnotintuitive. Asfor[12],themotivationcomesfromacollaborativeresearchwithSchlum- bergeronthemodelingofsoils.Inthesimplestmodels,siltysoilsarecomposed ofwater,clayandquartz.Theclayleavesareorganizedonthesurfaceofthe quartzgrains.Becauseoftheirelectricalproperties,thepresenceorthelackof clayleaveschangesconsiderablytheeffectivepropertiesofsoils.Actually,since theyareelectricallycharged,theclayleaveshavethewatermoleculesstuckaround them.Thischangestheelectricpropertiesoftheso-calledboundwaterorhygro- scopicwater[21,20].Thisphenomenonoccursonfewlayersofwatermolecules, thereforethehygroscopicwaterismodelledbyaroughthinlayer.Howeverweare confidentthatourresultisusefulformoredifferentapplications,particularlyinthe electromagneticresearcharea. 1.1. Descriptionofthegeometry. Forsakeofsimplicity,wedealwiththetwo- dimensionalcase,howeverthethree-dimensionalcasecanbestudiedinthesame wayuptofewappropriatemodifications. LetΩbeaboundedsmoothdomainof R 2 withconnectedboundary∂Ω.For ε>0,wesplitΩintothreesubdomains:Ω 1 ,Ω ε m andΩ ε 0 .Ω 1 isasmoothdomain strictlyembeddedinΩ.WedenotebyΓitsconnectedboundary.Thedomain Ω ε m isthethinoscillatinglayersurroundingΩ 1 (seeFig.1).WedenotebyΓ ε the oscillatingboundaryofΩ ε m : Γ ε =∂Ω ε m \Γ. ThedomainΩ ε 0 isdefinedby Ω ε 0 =Ω\(Ω 1 ∪Ω ε m ).