《计算电磁学》第八讲课件

《计算电磁学》PartII：矩量法Dr.PingDU(杜平)SchoolofElectronicScienceandAppliedPhysics,HefeiUniversityofTechnologyE-mail:pdu@Chapter1DeterministicProblemsNov.24,2011《计算电磁学》PartII：矩量法Dr.PingDUOutline§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）

2Outline§1.1Introduction(介绍)§1.1IntroductionConsiderequationsoftheinhomogeneoustype(非齐次型)(1-1)whereL

isanoperator(算子),

f

isthefieldorresponse(unknownfunctiontobedetermined),andg

isthesourceorexcitation(knownfunction).Bythetermdeterministicwemeanthatthesolutionto(1-1)isunique.Thatis,onlyonefisassociatedwithagiveng.3§1.1IntroductionConsiderequTwoterminologies:Analysis(分析)&Synthesis(综合)AproblemofAnalysisinvolvesthedeterminationoff

whenLandgaregiven.2)AproblemofSynthesisinvolvesthedeterminationofLwhenfandgarespecified.Antennaarraysynthesis天线阵列综合Generallyspeaking,thesolutionisnotunique.Thesolutionisunique.Electromagneticinverseproblems电磁逆问题Twoexamples:4Twoterminologies:Analysis(分whereandarescalarsand*denotesacomplexconjugate.(1-4)(1-3)(1-2)Aninnerproductisascalardefinedtosatisfy1.Innerproduct(内积)§1.2Formulationofproblems5whereandare2.OperatoranditspropertiesAnadjointoperator(伴随算子)anditsdomain(1-5)forallfinthedomainofL.If,anoperatorisself-adjoint(自伴的).ThedomainofisthatofL.Propertiesofthesolutiondependonpropertiesoftheoperator.AnoperatorisrealifLfisrealwhenever

fisreal.62.OperatoranditspropertiesAnoperatorispositivedefiniteif(1-6)forallinitsdomain.3.SolutionIfthesolutiontoexistsandisuniqueforallg,thentheinverseoperator

existssuchthat(1-7)Ifgisknown,then(1-7)representsthesolutiontotheoriginalproblem.(1-7)isaninhomogeneousequationforgiffisknown.Anditssolutionis.7AnoperatorispositivedefiniLandareapairofoperators,eachofwhichistheinverseoftheother.Example1.Giveng(x),findf(x)intheintervalsatisfyingThisisaboundaryproblemforwhichTherangeofListhespaceofallfunctionsgintheintervalthatwewishtoconsider.(1-8)(1-9)(1-11)8LandareapairofThesolutionto(1-8)isnotunique(不唯一)unlessappropriateboundaryConditionareincluded.Inotherwords,boththedifferentialoperatoranditsdomainarerequiredtodefinetheoperator.Defineaninnerproductforthisproblemis(1-11)(1-11)satisfiesthepostulates(条件)(1-2)to(1-4),asrequired.Thedefinition(1-11)isnotunique.Forexample,(1-12)wherew(x)>0isanarbitraryweightingfunction(加权函数),isalsoanacceptableinnerproduct.9Thesolutionto(1-8)isHowever,theadjointoperatordependsontheinnerproduct,whichcanoftenbechosentomaketheoperatorself-adjoint.Tofindtheadjointofadifferentialoperator,weformtheleftsideof(1-5),andintegratebyparts(分部积分)toobtaintherightside.Forthepresentproblem,(1-12)Thelasttermsareboundaryterms,andthedomainofmaybechosensothatthesevanish.10However,theadjointoperThefirstboundarytermsvanishby(1-9),andthesecondvanishif(1-14)Itisevidentthattheadjointoperatorto(1-10)fortheinnerproduct(1-11)is(1-15)Sinceandthedomainofisthesameasthatof

L,theoperatorisself-adjoint(自伴的).ItcanbeobservedthatLisarealoperator,sinceisLfrealwhenfisreal.11ThefirstboundarytermsvanisThatLisapositivedefiniteoperatorshownfrom(1-6)asfollows:(1-16)Notethat

L

isapositivedefiniteoperatoreveniffiscomplex.TheinverseoperatortoLis(1-17)whereGistheGreen’sfunction12ThatLisapositivedefinite(1-18)Thatisself-adjointfollowsfromtheproofthatLisself-adjoint,since(1-19)ThatispositivedefinitewheneverLispositivedefinite,andviceversa.13(1-18)Thatisself-ad§1.3MethodofMomentsLet’sdiscussageneralprocedureforsolvinglinearequations,calledthemethodofmoments(矩量法).Considertheinhomogeneousequation(1-20)whereLisalinearoperator,gisknown,andfistobedetermined.Letf

beexpandedinaseriesoffunctionsinthedomainofLas(1-21)wheretheareconstants.Weshallcalltheexpansionfunctionsorbasisfunctions.14§1.3MethodofMomentsLet’sSubstituting(1-21)in(1-20),andusingthelinearityofL,wehave(1-22)ItisassumedthatasuitableinnerproducthasbeendeterminedforDefineasetofweightingfunctions(权函数)ortestingfunctions(测试函数),intherangeofL.theproblem.Taketheinnerproductof(1-22)witheach.Theresultis(1-23)m=1,2,3,…15Substituting(1-21)in(1-20),Thissetofequationscanbewritteninmatrixformis(1-24)where(1-25)(1-26)16ThissetofequationscanbewIfthematrix[l]isnonsingularitsinverseexists.Thearethengivenby(1-27)andthesolutionfor

fisgivenby(1-21).Forconciseexpressionofthisresult,definethematrixoffunctions(1-28)andwrite(1-29)17Ifthematrix[l]isnonsingulThissolutionmaybeexactorapproximate,dependingonthechoiceoftheand.Thematrix[l]maybeeitherofinfiniteorder(无限阶)orfiniteorder(有限阶).Theformeronecanbeinvertedonlyinspecialcases,forexample,ifitisdiagonal(对角线的).Ifthesetsandarefinite,thematrixisoffiniteorder,andcanbeinverted.Choicesoftheweightingfunctionandthebasisfunctionsareveryimportant.Somefactorsneedtobeconsidered:(1)accuracyofsolutiondesired,18Thissolutionmaybeexactor(4)realizationofawell-conditionedmatrix(好条件矩阵).(3)sizeofthematrix,and(2)easeofevaluationofthematrixelements,Whenanalyzingthe3DscatteringproblemwithRWGbasisfunction,thedoublesurfaceintegralsareneeded.Itisverytimeconsuming.Characteristicfunctioncanbeusedtocalculatethematrixelements.ForaPCwith1GB,theorderofthematrixcannotbelargerthan5000.Otherwise,“Outofmemory”willappear.Iftheconditionisbad,theconvergencewillbeveryslow.Toaddressthisissues,thepreconditioningtechniquescanbeapplied.19(4)realizationofawell-condExample2.ConsiderthesameequationasintheexampleofSection1-2,butwithspecificsource.Ourproblemis(1-30)(1-31)Thisisasimpleboundary-valueproblem.Itssolutionis(1-32)20Example2.ConsiderthesThisproblemcanbesolvedbyusingthemethodofmoments.Forapowerseriessolution,letuschoose(1-33),sothattheseries(1-21)is(1-34)Notethatthetermxisneededin(1-33),elsethewillnotbeinthedomainofL.Thatis,theboundaryconditionwillnotbesatisfied.21ThisproblemcanbesolvedbyFortestingfunctions,choose(1-35)ThemethodisthatofGalerkin(伽辽金法).Evaluationofthematrices(1-25)and(1-26)fortheinnerproduct(1-11)andisstraightforward,andresultsin(1-36)(1-37)22Fortestingfunctions,chooseForanyfixedN(numberofexpansionfunctions),thearegivenby(1-27)andtheapproximationtofby(1-34).ForN=1,wehave,,andfrom(1-24)ForN=2,thematrixequation(1-24)becomes(1-38)Solvingit,weget(1-39)23ForanyfixedN(numberofexForN=3,thematrixequation(1-24)becomes(1-40)Solvingit,weget(1-41)Thus,weobtain,whichistheexactsolution.24ForN=3,thematrixequation(Figure1-1.SolutionsusingandGalerkin’smethod.Fig.1-1showstherelationbetweentheNandtheaccuracy.25Figure1-1.Solutionsusing§1.4PointMatching(点匹配或点选配)Theintegrationinvolvedinevaluatingtheof(1-25)isoftendifficulttoperforminproblemsofpracticalinterest.Asimplewaytoobtainapproximatesolutionsistorequirethatequation(1-22)besatisfiedatdiscretepointsintheregionofinterest.Thisprocedureiscalledapoint-matchingmethod(点选配法）.Intermsofthemethodofmoments,itisequivalenttousingtheDiracdeltafunc-tionsastestingfunctions.Example3.ReconsidertheproblemofSection1-3,statedby(1-30)and(1-31).26§1.4PointMatching(点匹配或点选配Againwechooseexpansionfunctions(1-33),sothat(1-22)becomes(1-42)Forapoint-matchingsolution,letustakethepoints(1-43)whichareequispacedintheinterval.Requiring(1-42)tobesatisfiedateachgivesusthematrixequation(1-24),withelements27Againwechooseexpansionfunc(1-44)(1-45)Thisresultisidenticaltochoosingweightingfunctions(加权函数)(1-46)whereistheDiracdeltafunctionandapplyingthemethodofmomentswithinnerproduct(1-11).Thenlet’sanalyzetheaccuracyofthepointmatchingmethod.28(1-44)(1-45)ThisresultisideConsiderthesolutionasNisincreased.ForN=1,wehave,,andfrom(1-27).ForN=2,thematrixequationis(1-47)Solvingit,weget(1-48)29ConsiderthesolutionasNisForN=3,theexactsolutionmustbeobtainedagainsincetheexactsolutionisalinearcombinationoftheandweareapplyingNindependenttests.Fig.1-2plotstherelationbetweentheNandtheaccuracy.Figure1-2.Solutionsusingandpointmatchingmethod.30ForN=3,theexactsolution§1.5SubsectionalBases(分域基函数)

Example4.AgainconsidertheproblemofSection1-3,statedby(1-30)and(1-31).Nequispacedpointsontheintervalaredefinedbytheof(1-43).Asubintervalisdefinedtobeofwidth1/(N+1)centeredonthe.ThisisshowninFig.1-3(a).Intheabovediscussion,thebasisfunctionisdefinedovertheentireinterval.Infact,thebasisfunctioncanalsobedefinedoverasubsection,whichiscalledthesub-sectionalbasisfunction(分域基函数).Itisreferredtoasentiresectionbasisfunction(全域基函数).31§1.5SubsectionalBases(分域基函数Fig.1-3.Subsectionalbasesandfunctionalapproximations.32Fig.1-3.SubsectionalbasesaAfunctionexistsoveronlyonesubintervalisthepulsefunction(脉冲函数).(1-49)Abetterbehavedfunctionisthetrianglefunction(三角形函数),definedas(1-50)Notethatthepulsefunctioncannotbeusedasthebasisfunctionunlessthattheextendedorapproximationoperatorisapplied.33AfunctionexistsoveronlyonAfunctionexistsoveronlyonesubintervalisthepulsefunction(脉冲函数).(1-49)Abetterbehavedfunctionisthetrianglefunction(三角形函数),definedas(1-50)ForthecaseN=5thefunctionisshowninFig.1-3(d).34AfunctionexistsoveronlyonAlinearcombinationoftrianglefunctionsoftheform(1-51)givesapiecewiselinearapproximationtof,asrepresentedbyFig,1-2(e).For,wehave(1-52)whereistheDiracdeltafunction.35AlinearcombinationoftriangTofollowthroughthemethodofmoments,letbethebasisfunction.ischosenasthetestingfunction.Forinnerproduct(1-11),thematrixelementsof(1-25)and(1-26)are(1-53)(1-54)36Tofollowthroughthemethodo§1.6ApproximateOperators(近似算子)Example5.Considertheproblem(1-30)and(1-31)byafinite-differenceapproximation.Thisinvolvesreplacingallderivativesbyfinitedifferences.Incomplexproblems,itissometimesconvenienttoapproximatetheoperatortoobtainapproximatesolutions.Fordifferentialoperators,thefinite-differenceapproximationhasbeenwidelyused.Forintegraloperators,anapproximateoperatorcanbeobtainedbyapproximatingthekernel(核)oftheintegraloperator.37§1.6ApproximateOperators((1-56)(1-55)Foragiven,Forthepresentproblem,considertheintervaldividedintosegmentswithendpoints(seeFig.1-3(a)).Forequaltoonesegment,.Afinitedifferenceapproximationtois38(1-56)(1-55)ForagivenasforallfinthedomainofL.Applythemethodofmomentstotheapproximateequation(1-57)subjecttoboundaryconditions.Thepoint-matchingprocedureatisused.Thematrixelementsare(1-58)(1-59)39as§1.7ExtendedOperators(扩展算子)Operator:operation(运算)andadomain(定义域)Waysofextendedoperators:(1)extendthedomain,and(2)extendtheoriginaldomainofL.40§1.7ExtendedOperators(扩展算子)Example6.Supposewewishtousepulsefunctionsforanexpansionoffinamomentsolutionfortheoperator.ThesearenotintheoriginaldomainofL.However,foranyfunctionswandfintheoriginaldomain,(1-60)obtainedfrom(1-11)byintegrationbyparts(分部积分).IfLfdoesnotexist,butdf/dxdoesexist,(1-60)canbeusedtodefineanextendedoperator.41Example6.SupposewewishtoThisextendsthedomainofLtoincludefunctionsfwhosesecondderivativesdonotexist,butthefirstderivativesdoexist.Itisstillassumedthat.Toapplythemethodofmomentsusingthepulsefunctionsandtheextendedoperator,let(1-61)whereParethepulsefunctionsdefinedby(1-49).Thetestingfunctionsare,whereTarethetrianglefunctionsdefinedby(1-50).42ThisextendsthedomainofThematrixelementsof[l]are(1-62)Thematrixelementsof[g]are(1