NumericalSimulationMethodofAcousticFieldPositiveProblembasedonMagnetoacousticTomographywithMagneticInductionHuiXia1,GuoqiangLiu1,YanhongLi1,YangZhang1,ShiqiangLi1andLaifuZhang21.InstituteofElectricalEngineering,ChineseAcademyofSciencesBeijing，China2.ShanxiElectricPowerResearchInstituteShanxi,CAbstractMagnetoacousticimpedancetomographywithmagneticinduction(MAT-MI)isanewimagingmethod.Itsimagesreflectconductivitydistribution.Inthispaper,wefirstlyproposedthenumericalsimulationmethodofmulti-physicsfieldscouplingtoobtainthedistributionofacousticfieldinMAT-MIwithoutthestaticmagneticfield.Simpleacousticdetectionexperimentsareconductedtovalidatethealgorithm.Theresultsdemonstrateditsfeasibility,andmayprovidesometheoreticalfoundationforthefurtherresearchonthereal-timedetectionofacousticsignalsandthereconstructionmethodoftheMAT-MI.Keywords-magnetoacoustictomographywithmagneticinduction(MAT-MI),Multi-physicsfieldscoupling,two-dimensionalaxisymmetricmodel,numericalsimulationofacousticfieldI.INTRODUCTIONAsakindoffunctionalimaging,Electricalimpedancetomography(EIT)hasmanypredominancecomparedwithconventionalimagingmeans,suchasnon-invasivediagnose,highimagingqualityandsoon.ButEIThasnotbeenusedinclinicalapplicationbecauseofitslowresolutionnow1-3.Inordertoresolvetheproblem,Magnetoacoustictomographywithmagneticinduction(MAT-MI)isproposedbyBinHeetal4,whichisshowninFig.1.InMAT-MI,imagingtargetisplacedinastaticmagneticfieldwithpulsedmagneticstimulationimposedonit,thepulsedcurrentinduceseddycurrentinthesample,andtheinducededdycurrentinstaticmagneticfieldgeneratesLorentzforce.TheLorentzforcecausesacousticvibration,andthegeneratedacousticwavecanbemeasuredaroundthesampletoreconstructtheconductivitydistributionofthesample.Figure1.TheillustrationofMAT-MI(quotedfrom4)OnthebasisoftheprincipleMentionedabove,weproposeanewnon-static-magnetMAT-MImethod.Inthispaper,weanalyzetheprinciplesofmulti-physicsfieldscoupling,includingthetwo-dimensionalaxisymmetrictransientelectromagneticfield,displacementfield,soundfield,andputforwardthemethodofmulti-physicscalculations.Onthebasicofaboveall,theformulaforcalculatingthevariousfieldsarederivedindetail,andconductthesimpleacousticdetectionexperimentstovalidatethemethod.II.THENUMERICALSIMULATIONMETHODOFMULTI-PHYSICSFIELDSCOUPLINGThemethodadoptsimpulsingpowersourceasthedrivingsource,excitingcoilgeneratesalternatingelectromagneticfieldwhichexcitesLorentzforceinthesample.TheLorentzforcecausesvibrationofsampleboundary,thenacousticwavesisexcitedintheair.Wecaninversethesampleresistivitybydetectingacousticwavesignal.Thesoundfielddistributionofthesamplecanbesimulatedthroughsolvingthemulti-physicalequationwhichincludeselectromagneticequation,wienerequationofelasticsolidsandsoundfieldequationintheair.A.TheequationofaxisymmetricelectromagneticfieldsTheexcitingcoilishollowcylindricalcoil,androundcoppersheetisselectedasthesample,thesimulationmodelhasaxialsymmetry,sothevectormagneticpotentialAKonlyhascircumferentialcomponent,labeledasA,thecorrespondingaxisymmetricelectromagneticequationis:22s2A1AAAAJrrrrzt+=(1)Whereismagneticpermeability,iselectricalconductivity,andsJiscurrentdensityoftheexcitingcoil.Althoughthecurrentdensityoftheexcitingcoilgeneratesonlycircumferentialcomponent,magneticfluxdensityincludesradialandaxialcomponent,wecangetAJt=978-1-4244-4713-8/10/$25.002010IEEErABz=zAABrr=+(2)Inordertoavoidthesingularityattheboundarywhichrequalstozero,sosupposeuistheratioofAandr,thentheEq.(2)becomes222suuuuur3rrrJrrztt+=(3)OnbothsidesoftheEq.(3)aremultipliedby2r,wecanget222323332suuuuur3rrrrJrrrztt+=(4)Ifnotetherandzforxandyrespectively,weget233332s2uuuuxxxxJxxxyytt+=(5)FromtheEq.(5),wecansee+yuxyxuxx33isthe)(3uxunderrectangularcoordinatesystem,wecanget()23332s2uuxuxxJxtt=(6)AccordingtothesolvingrangeoftheFig2a,wecanseethat1istheairrange,2isthesampleposition,3istheexcitingcoilposition.Inthe1area,conductivityequalszero,andthereisnoexcitingsource.Inthe2area,thereisalsonoexcitingsource.Inthe3area,thecurrentinthecoilisthesourcecurrent.Thenequationofthethreesolvingareascanbewroterespectively()3xu0=(71)()33uxux0t+=(72)()32sxuJx=(73)Atthesymmetryaxisandinfinityboundary,theboundaryconditionisthatuequalszero.So,afterobtainingtheu,substitutingrAU/=intoEq.(1),wecangetelectricfieldintensityandmagneticfluxdensityAuErtt=，ruBrz=，zuBr2ur=+(8)123232323112323Figure2.Solvingmodels（a)Electromagneticfieldsolvingmodel（b)displacementfieldsolvingmodel（c)SoundfieldsolvingmodelBasedonEq.(8),wecanget.sF=JBKKK(9)B.AxisymmetricNavierequationsofelasticsolidsAcordingtothetheoryofcontinuummechanics,thewienerequationofelasticsolidcanbederivedthroughusingmomentumconservationprinciple,lawofconservationofmassandconstitutiveequationofmechanicalpropertiesinaninertialreferenceframe.Thevectorformofthewienerequationcanbewroteas222uGGuuFt12v()=+KKKK(10)Whereuurzt=(,)Kisdisplacementfield,FKisunitvolumeforce,isdensityofcoppersheet,Gisshearmodulus,andvisPoissonsratio.Underthecylindricalcoordinates,Eq.(10)canbewrote22rrrr22uuGGuF12rrt+=(101)22zzz2uGGuF12zt+=(102)rrzuuuurrz=+(103)Where2ru、2zu、randzcanbewrote222rrrr22uuu1urzrr=+(111)222zzzz22uuu1urzrr=+(112)rzrr2uuuu1rrrzrrr=+(113)rzruuu1zzrzrz=+(114)Inordertovoidthesingularityattheboundary,supposeorruur=，andsubstitutingroruu=intoEq.(10-1),andOnbothsidesoftheequationmultipliedbythe2r，wecanget()()()22323ororor22223orzr22G1uuur3rGr12rrzuuGrFr12rzt+=(12)Thesolvingrangeisshowninfigure2b,theboundaryconditionscanbewroteatthe2and3sFnp=KK(13)WheresnKisunitnormalvectorwhichpointingtheoutsideofthesampleorcoil.C.AxisymmetricacousticwaveequationIntheexperiment,becausethereisnoLorentzforceintheair,theacousticwaveequationinthesolvingrangeofFig2ccanbewroteas222210ppct=(14)Inthecylindricalcoordinate,wecanget2222222110=ppppctrrrz(15)Wheretheboundaryconditionisr=0attheaxisofsymmetry,andp=0attheinfinitepoint.Onthe2and3,theboundaryconditionareasfollows,22unpnt=KK(16)AccordingtotheEq.(10)Eq.(16),wecansolvethesoundwavedistributioninthesoundfieldofthesample.III.EXPERIMENTSA.SimulationexperimentInthesimulationprocess,thewaveformofexcitingcurrentcanbeshownasfollow0()sin()=tVItetL(17)wheredischargevoltage0V1000V=,inductionL=7.7H,resistanceR=8.06e-3，capacityC=200F，=R/2L，21/()LC=.Inthecourseofpracticalapplication,thecurrentwaveformisinterceptedbyathyristor,andonlyreservesthefirstpositivespike.Theimpulsewidthisabout120S,numericalsimulationresultofsoundfielddistributionat60SisshownbelowinFig.3.Figure3a.Atthetimeof60s,soundfielddistributionoftheexcitingcoilitselfFigure3b.Atthetimeof60s,soundfielddistributionofthesampleFromtheFig.3a,wefindthatthesoundfielddistributionofexcitingcoilcanbeapproximatelyconsideredasacircularringwhosecenteristhecoilstheinsideandoutsideboundaries,andatthesymmetryaxis,thesoundfieldisthestrongest.Atthesametime,wefind,inthedisplacementy=0,theacousticsignalstrengthgeneratedbycoilitselfisweak,itcanbeshieldedbymeansofsomemeasuresthatcaneffectivelyeliminatetheinfluenceofacousticsignalgeneratedbythecoilitself.0.000000.000030.000060.000090.00012-250000-200000-150000-100000-50000050000100000150000Signalintensity/a.uTime/s0.00050.0010.0020.0050.0080.01Figure4a.Atx=0,thesimulationacousticsignal0.000000.000030.000060.000090.00012-200000-1000000100000Signalintensity/a.uTime/s00.00050.0020.0050.010.15Figure4b.Aty=0.0005,thesimulationacousticsignalInFig.3b,wecanseethatsoundfielddistributionconcentratearoundtheaxisofsymmetry.Inordertofurtherunderstandthecharacteristicsofacousticsignals,weselectthedifferentcoordinatepointstosimulatetheacousticsignal,andthetime-stepsetto10S.Afterachievingtheacousticsignalofthevariouspoint,thecontinuous120Sdataweresegmentedinto0.1SepochsforFFTtransformandobtainthesignalfrequency.Intheaxisofx=0,weobtainthesimulationacousticsignalshowninFig.4a,andintheaxisofy=0.0005m,weobtainthesimulationacousticsignalshowninFig.4b.Afteranalysisandcalculation,wefindthatthefrequencyofacousticwavesignalmainlyconcentrateintherangeof3-5KHzinthesphericalsoundfieldrangewhosecenteristhesamplescenterandradiusisapproximately0.005m.B.AcousticdetectionexperimentWeadopttheexperimentalsystemtodetectthesoundfieldofthecoppersheetsample.Withregardtoadetaileddescriptionoftheexperimentcanrefertoliterature5.Inthesphericalsoundfieldrangewhosecenteristhesamplescenterandtheradiusisapproximately0.005m,theacousticsignalunderexcitationisdetected.ThenweprocessthedetectedsoundsignalbyFFT,andobtainsignalspectrum.TheacousticsignalofmeasurementpointtisshowninFig.5.Figure5.DetectedacousticwavesignalanditsspectrumAftermulti-pointmeasurementandanalysis,wefindthatthefrequencyofdetectedsoundwavesignalmainlyconcentrateinthespectrumrangeof3-5KHz,itisconsistentwiththesimulationresults.Itprovesthatthesimulationmethodofmulti-physicalfieldcouplingiscorrect,themethodofMAT-MIisfeasible.IV.CONCLUSIONMedicalimagingisaresearchdomainwithbroaddevelopmentprospect,itisessentialtotheadvancementofmedicineandimprovementofpeopleslife.Inthispaper,ourmethodshowsthatitispossibletocompletetwo-dimensionalaxisymmetricacousticwavepositionproblemofMAT-MIwithoutthestaticmagneticfield.Itcanbeseenasthetheoreticalreferenceforthefuturestud