时域有限差分法

3FDTD1966K.S.Yee(FiniteDifference-TimeDomainFDTD)[1]A.Taflove[2]K.S.Kunz[3]R.Holland[4]MaxwellFFT(RCS)FDTDFaradayAmpere-FDTDFDTDFDTDFDTDFDTD3.1YeeMaxwell=⋅∇=⋅∇∂∂−=×∇∂∂+=×∇0),,,(),,,(),,,(),,,(),,,(),,,(),,,(),,,(tzyxBtzyxtzyxDtzyxHttzyxEtzyxEttzyxEtzyxHrrrrrrrρµεσ(3.1.1)∂∂µ∂∂∂∂HtEzEyxyz=−1()(3.1.2a)∂∂µ∂∂∂∂HtExEzyz=−1(x)(3.1.2b)∂∂µ∂∂∂∂HtEyExzxy=−1()(3.1.2c)∂∂ε∂∂∂∂σEtHyHzExzyx=−−1()(3.1.2d)∂∂ε∂∂∂∂σEtHzHxEyxzy=−−1()(3.1.2e)∂∂ε∂∂∂∂σEtHxHyEzyxz=−−1()(3.1.2f)3.1.1x,y,zij(,∆∆∆xyz,,k,,,)ijk(,,)(,,)xyzixjykzijk=∆∆∆(3.1.3)ijk∆x∆yzoyx∆zExEzHyHxEyEzEzEyEyExExHz3.1.13.1.2Yeen∆t(,,)ijkFijkFixjykzntn(,,)(,,,)=∆∆∆∆(3.1.4)FijnYee3.1.2k,,Yee(3.1.4)3.1.2][)21,,()21,1,(),21,()1,21,()21,21,()21,21,(2121ykjiEkjiEzkjiEkjiEtkjiHkjiHnznznynynxnx∆+−++−∆+−++⋅∆+++=++−+µ(3.1.5a)][),,21()1,,21()21,,()21,,1()21,,21()21,,21(2121zkjiEkjiExkjiEkjiEtkjiHkjiHnxnxnznznyny∆+−++−∆+−++⋅∆+++=++−+µ(3.1.5b)][),21,(),21,1(),,21(),1,21(),21,21(),21,21(2121xkjiEkjiEykjiEkjiEtkjiHkjiHnynynxnxnznz∆+−++−∆+−++⋅∆+++=++−+µ(3.1.5c)][)21,,21()21,,21(),21,21(),21,21(211),,21(2121),,21(212121211zkjiHkjiHykjiHkjiHttkjiEttkjiEnynynznznxnx∆−+−++−∆−+−++⋅∆+⋅∆++⋅∆+∆−=++++++εσεεσεσ(3.1.5d)][),21,21(),21,21()21,21,()21,21,(211),21,(2121),21,(212121211xkjiHkjiHzkjiHkjiHttkjiEttkjiEnznznxnxnyny∆+−−++−∆−+−++⋅∆+⋅∆++⋅∆+∆−=++++++εσεεσεσ(3.1.5e)][)21,21,()21,21,()21,,21()21,,21(211)21,,(2121)21,,(212121211ykjiHkjiHxkjiHkjiHttkjiEttkjiEnxnxnynynznz∆+−−++−∆+−−++⋅∆+⋅∆++⋅∆+∆−=++++++εσεεσεσ(3.1.5f)Out(,∆∆22)∆∆∆∆uxyz=max(,,)(3.1.5)σεµ3.23.1.13.1.2Yee(3.1.5)3.2.1FDTD3.2.1EyExEzHyHxEyEzEzEyExExHz(a)EzExHxEyEzEyEzHyEyExExHz(b)3.2.1(a),(b)3.2.2MaxwellYeeMaxwellMaxwellµεYeeε(3.1.5a-c)Yeeσ3.2.2FDTDεσEx∆yEx(i,j,k)∆zHyzHCSdε1,σ1ε2,σ2ε3,σ3ε4,σ4yHzH3.2.2xE∫∫∫∫∫⋅+⋅=⋅ScSSdESdtEldHrrrrrrσ∂∂ε(3.2.1)CSykjiHzkjiHykjiHzkjiHldHnynznynzc∆−+∆−−∆+−∆+=⋅++++∫)21,,(),21,()21,,(),21,(21212121rr(3.2.2)∆−∆∆≅⋅=⋅+==∑∑∫∫∫∫tkjiEkjiEzySdtESdtEnxnxmmmmSSm),,(),,(414141ε∂∂ε∂∂εrrrr(3.2.3)∑∑∫∫∫∫==∆∆≅⋅=⋅4141),,(4mnxmmmSSkjiEzySdESdEmσσσrrrr(3.2.4)Smm,(,)=145.1.4(3.2.1)∆−−+⋅∆+⋅∆+⋅∆+∆−=+++ykjiHkjiHttkjiEttkjiEnznznxnx),,(),,(211),,(2121),,(212112121εσεεσεσ−+−−++HijkHijkzynyn12121212(,,)(,,)∆(3.2.5)εεεεεσσσσσ=+++=+++123412344,4EyEzHxHyHz3.2.3TerminatedBoundaryCondition(AbsorbingBoundaryCondition)Mur[5][6-9]Mur1981EngquistMajdaMurc0FF0)(220222=−++−Fctzyx∂∂∂∂(3.2.6)x=0x≥00222220222220=∂∂−∂∂−∂+∂⋅∂∂−∂∂−∂−∂−−Fcctztytxtztytx(3.2.7)[]0=∂−∂Ftxν(3.2.8a)[]0=∂+∂Ftxν(3.2.8b))()(12202220210222220tztytztycccc∂∂−∂∂−=∂∂−∂∂−=−−−−ν(3.2.9)(3.2.8))(),,,(xttxyxFν+Ψ=(3.2.10a))(),,,(xttxyxFν−Ψ=(3.2.10a)-xxx=0-x+x(3.2.8a)(3.2.10a)-x(3.2.8a)x=0(+x)(3.2.8a)x=0νTaylor+∂∂−∂∂−=−−−LL)()(2112202220210tztycccν(3.2.11)Mur0|)(010=−=−xFtcx∂∂(3.2.12)Mur0|)](21[022220210=++−=−−xzytxtFcc∂∂∂∂(3.2.13)MurMurzE)]2/1,,0()2/1,,1([)2/1,,1()2/1,,0(1001+−+∆+∆∆−∆++=+++kjEkjExtcxtckjEkjEnznznznz(3.2.14))]2/1,,1()2/1,,1(2)2/3,,1()2/1,,0()2/1,,0(2)2/3,,0()2/1,1,1()2/1,,1(2)2/1,1,1()2/1,1,0()2/1,,0(2)2/1,1,0([)(2)()]2/1,,1()2/1,,0{[2)]2/1,,0()2/1,,1([)2/1,,1()2/1,,0(0200110011−++−++−++−+++−++−++++−++−++∆+∆∆∆++++∆+∆∆++++∆−∆∆−∆++−=+−+−+kjEkjEkjEkjEkjEkjEkjEkjEkjEkjEkjEkjExtcxtckjEkjExtcxkjEkjExtcxtckjEkjEnznznznznznznznznznznznznznznznznznz(3.2.15)Mur3.3(3.1.5)(3.2.5)Fouriert∆3.3.1FDTD20yzx3.3.1FDTD+zgtzttzzvT(,)exp()=−−−−0022(3.3.1)vttzz==0,0Fourier[]GfTf()exp∝−π222(3.3.2)GG10%)(f)(fmaxf21maxTf=(3.3.3)t0t=5WWz∆≥20vzT∆⋅≥1031(3.3.4)0t0,0zzt==0.001%(3.3.5)t03393≥.TTEMTEMTEMtzTEMTEMTEM3.3.1yEyeEyE][),21,21(),21,21()21,21,()21,21,(211),21,(),21,(2121),21,(212121211xkjiHkjiHzkjiHkjiHttkjiEkjiEttkjiEenzenzenxenxenyeenyeny∆+−−++−∆−+−++⋅∆+⋅∆++++⋅∆+∆−=++++++εσεεσεσ(3.3.6)(3.1.5)(3.1.5)(5.1.20)t∆3.43.4.1FDTD∆t∆x∆y∆zFDTD21222max)111(−∆+∆+∆≤∆zyxtv(3.4.1)vmax∆t∆tmaxminminmin2),,min(vzyxt∆∆∆=∆(3.4.2)∆t∆t3.4.2FDTD∆x,∆y,∆z∆tminmax101λ∆h(3.4.3)∆hmax∆x,∆y∆z,λmin11K.S.Yee,“Numericalsolutionofinitialboundaryvalueproblemsinvolvingmaxwell’sequationsinisotropicmedia,”IEEETrans.AntennasPropagation,vol.AP-14,pp.302-307,May,1966.2A.Taflove,andM.E.Brodwin,“Numericalsolutionofsteady-stateelectromagneticscatteringproblemsusingthetime-dependentMaxwell’sequations,”IEEETrans.MicrowaveTheoryandTechniques,vol.23,pp.623-630,1975.3K.S.Kunz,andK.M.Lee,“Athree-dimensionfinite-differencesolutionoftheexternalresponseofanaircrafttoacomplextransientEMenvironmentI:Themethodanditsimplementation,”IEEETrans.ElectromagneticCompatibility,vol.20,pp.328-333,1978.4R.Holland,“Thread:afree-fieldEMPcouplingandscatteringcode,”IEEETrans.NuclearScience,vol.24,pp.2416-2421,1977.5G.Mur,“Absorbingboundaryconditionsforthefinite-differenceapproximationofthetime-domainelectromagneticfieldequation,”IEEETrans.ElectromagneticCompatibility,vol.23,pp.377-382,1981.6J.P.Berenger,“Aperfectlymatchedlayerfortheabsorptionofelectromagneticwave,”J.Comput.Phys.,1994,185-200.7P.Zhao,J.Litva,“AnewstableandverydispersiveboundaryconditionfortheFDTDmethod,”InProc.1994IEEEMTT-sint.symp.,1:35-38.8Z.P.Liao,H.L.Wong,B.P.Yang,andY.F.Yuan,“Atransmittingboundaryfortransientwaveanalysis,”Scien