《计算电磁学》第八讲

《计算电磁学》PartII：矩量法Dr.PingDU(杜平)SchoolofElectronicScienceandAppliedPhysics,HefeiUniversityofTechnologyE-mail:[e-mail]1DeterministicProblemsNov.24,20112Outline§1.1Introduction(介绍)§1.2FormulationofProblems(问题的描述）§1.3Methodofmoments(矩量法）§1.4PointMatching(点匹配或点选配)§1.5SubsectionalBases(子域基）§1.6ApproximateOperators(近似算子）§1.7ExtendedOperators(扩展算子）3§1.1IntroductionConsiderequationsoftheinhomogeneoustype(非齐次型)()Lfg(1-1)whereLisanoperator(算子),fisthefieldorresponse(unknownfunctiontobedetermined),andgisthesourceorexcitation(knownfunction).Bythetermdeterministicwemeanthatthesolutionto(1-1)isunique.Thatis,onlyonefisassociatedwithagiveng.4Twoterminologies:Analysis(分析)&Synthesis(综合)1)AproblemofAnalysisinvolvesthedeterminationoffwhenLandgaregiven.2)AproblemofSynthesisinvolvesthedeterminationofLwhenfandgarespecified.Antennaarraysynthesis天线阵列综合Generallyspeaking,thesolutionisnotunique.Thesolutionisunique.Electromagneticinverseproblems电磁逆问题Twoexamples:5whereandarescalarsand*denotesacomplexconjugate.0if0*,0if0ffff(1-4),,,fghfhgh(1-3),,fggf(1-2)Aninnerproductisascalardefinedtosatisfy1.Innerproduct(内积)§1.2Formulationofproblems62.OperatoranditspropertiesAnadjointoperator(伴随算子)anditsdomainaL,,aLfgfLg(1-5)forallfinthedomainofL.If,anoperatorisself-adjoint(自伴的).aLLThedomainofisthatofL.aLPropertiesofthesolutiondependonpropertiesoftheoperator.AnoperatorisrealifLfisrealwheneverfisreal.7Anoperatorispositivedefiniteif0*,00positivedefinitefLfpositivesemidefinitenegativedefinite(1-6)forallinitsdomain.0f3.SolutionIfthesolutiontoexistsandisuniqueforallg,thentheinverse()Lfgoperatorexistssuchthat1L1()fLg(1-7)Ifgisknown,then(1-7)representsthesolutiontotheoriginalproblem.(1-7)isaninhomogeneousequationforgiffisknown.Anditssolutionis.()Lfg8Landareapairofoperators,eachofwhichistheinverseoftheother.1LExample1.Giveng(x),findf(x)intheintervalsatisfying01x22()dfgxdx(0)(1)0ffThisisaboundaryproblemforwhich22dLdxTherangeofListhespaceofallfunctionsgintheintervalthatwewishtoconsider.01x(1-8)(1-9)(1-11)9Thesolutionto(1-8)isnotunique(不唯一)unlessappropriateboundaryConditionareincluded.Inotherwords,boththedifferentialoperatoranditsdomainarerequiredtodefinetheoperator.Defineaninnerproductforthisproblemis10,()()fgfxgxdx(1-11)(1-11)satisfiesthepostulates(条件)(1-2)to(1-4),asrequired.Thedefinition(1-11)isnotunique.Forexample,10()()()wxfxgxdx(1-12)wherew(x)0isanarbitraryweightingfunction(加权函数),isalsoanacceptableinnerproduct.10However,theadjointoperatordependsontheinnerproduct,whichcanoftenbechosentomaketheoperatorself-adjoint.Tofindtheadjointofadifferentialoperator,weformtheleftsideof(1-5),andintegratebyparts(分部积分)toobtaintherightside.Forthepresentproblem,12112000121200,dfdfdgdfLfggdxdxdxdxdxdxdgdgdffdxfgdxdxdx(1-12)Thelasttermsareboundaryterms,andthedomainofmaybechosensothatthesevanish.aL11Thefirstboundarytermsvanishby(1-9),andthesecondvanishif(0)(1)0gg(1-14)Itisevidentthattheadjointoperatorto(1-10)fortheinnerproduct(1-11)is22adLLdx(1-15)SinceandthedomainofisthesameasthatofL,theoperatorisaLLaLself-adjoint(自伴的).ItcanbeobservedthatLisarealoperator,sinceisLfrealwhenfisreal.12ThatLisapositivedefiniteoperatorshownfrom(1-6)asfollows:12*11***2000210,dfdfdfdffLffdxdxfdxdxdxdxdfdxdx(1-16)NotethatLisapositivedefiniteoperatoreveniffiscomplex.TheinverseoperatortoLis110()(,')(')'LgGxxgxdx(1-17)whereGistheGreen’sfunction13(1')'(,')(1)'xxxxGxxxxxx(1-18)Thatisself-adjointfollowsfromtheproofthatLisself-adjoint,since1L11212,,LffgLg(1-19)ThatispositivedefinitewheneverLispositivedefinite,andviceversa.1L14§1.3MethodofMomentsLet’sdiscussageneralprocedureforsolvinglinearequations,calledthemethodofmoments(矩量法).Considertheinhomogeneousequation()Lfg(1-20)whereLisalinearoperator,gisknown,andfistobedetermined.LetfbeexpandedinaseriesoffunctionsinthedomainofLas123,,,fffnnnff(1-21)wheretheareconstants.Weshallcalltheexpansionfunctionsorbasisfunctions.nnf15Substituting(1-21)in(1-20),andusingthelinearityofL,wehave()nnnLfg(1-22)Itisassumedthatasuitableinnerproducthasbeendeterminedfor,fgDefineasetofweightingfunctions(权函数)ortestingfunctions(测试函数),123,,,(1-22)witheach.mwTheresultis,,nmnmnwLfwg(1-23)m=1,2,3,…16Thissetofequationscanbewritteninmatrixformismnnmlg(1-24)where11122122,,,,.......mnwLfwLflwLfwLf12n12,,mwggwg(1-25)(1-26)17Ifthematrix[l]isnonsingularitsinverseexists.1[]lThearethengivenbyn1nnmmlg(1-27)andthesolutionforfisgivenby(1-21).Forconciseexpressionofthisresult,definethematrixoffunctions123nffff(1-28)andwrite1nnnmnmffflg(1-29)18Thissolutionmaybeexactorapproximate,dependingonthechoiceofthenfand.nwThematrix[l]maybeeitherofinfiniteorder(无限阶)orfiniteorder(有限阶).Theformeronecanbeinvertedonlyinspecialcases,forexample,ifitisdiagonal(对角线的).Ifthesetsandarefinite,thematrixisoffiniteorder,andcanbeinverted.Choic