时域积分方程时间步进算法及其快速算法学士学位论文撰写范例英文.docx

时域积分方程时间步进算法及其快速算法学士学位论文撰写范例英文.docx

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时域积分方程时间步进算法及其快速算法学士学位论文撰写范例英文

摘要

本文基于电磁理论中的Calderόn关系与Calderόn恒等式所揭示的不同积分算子之间的关系，系统地研究了Calderόn预条件技术及其在计算电磁学积分方程方法中的应用。

研究内容全面覆盖了求解理想导电体目标和均匀或分层均匀介质目标电磁散射与辐射问题的积分方程中的Calderόn预条件技术。

在导体积分方程方面，研究了电场积分方程在中频，低频，以及高频区的Calderόn预处理方法。

在介质积分方程方面，则研究了PMCHWT积分方程的Calderόn预处理方法，和N-Müller积分方程的Calderόn技术。

本文对金属问题中的第二类Fredholm积分方程和介质问题中的第二类Fredholm积分方的精度改善进行了深入详尽的研究。

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关键词：

电磁散射，面积分方程方法，Calderόn预条件方法，数值计算精度，第二类Fredholm积分方程

ABSTRACT

RevealedbytheCalderόnrelationandtheCalderόnidentitiesinelectromagnetictheory,thepropertiesandrelationofdifferentintegraloperatorsinthecomputationalelectromagnetics（CEM）areutilizedtoconstructtheCalderόnpreconditioningtechniques,whichareappliedintheintegral-equation-basedmethodsinthisthesis.AthoroughandsystematicresearchhasbeenaccomplishedtocovertheCalderόnpreconditioningtechniquesfortheperfectelectricconductor（PEC）andthedielectriccases.ForthePECcase,theCalderόnpreconditioningfortheelectric-fieldintegralequation（EFIE）atmid,low,andhighfrequenciesareconstructedandstudied.Forthedielectriccases,theCalderόnpreconditioningforthePoggio-Miller-Chang-Harrington-

Wu-Tsai（PMCHWT）integralequationareinvestigated,andtheCalderόntechniquefortheN-Müllerintegralequationisdeveloped.Moreover,theaccuracyimprovingtechniqueforthesecond-kindFredholmintegralequationforbothPECanddielectriccasesisalsostudiedinthisthesis.

…

Keywords:

Electromagneticscatteringandradiation,surface-integral-equation-basedMethods,Calderόnpreconditioningmethods,numericalaccuracy,Fred-

holmintegralequationsofthesecondkind

Contents

Chapter1Introduction1

1.1ResearchBackgroundandSignificance1

1.2StateofArts1

1.3ContentsandInnovationsoftheThesis2

1.4OutlineoftheThesis2

Chapter2TheoreticalBasics3

2.1IntegralEquationsinElectromagnetics3

2.2270MHzPlanWaveExcitation3

2.3TheSolutionofIntegralEquationsinElectromagnetics4

2.3.1GeneralPrincipleoftheMethodofMoments4

2.3.2GeometricalModelingandDiscretizationofObject4

2.3.2.1PlanarTriangularModel4

2.3.2.2CurvilinearTriangularModel4

2.3.3TheChoiceofBasisFunctions5

2.3.3.1PlanarRWGBasisFunctions6

2.3.3.2CurvilinearRWGBasisFunctions6

2.3.4TheSolutionofMatrixEquations6

2.3.4.1DirectAlgorithms6

2.3.4.2IterativeAlgorithms6

2.4Conclusion6

Chapter3CalderόnPreconditioneratMidFrequencies7

3.1Introduction7

3.2CalderόnRelationandCalderόnIdentities7

3.3CalderόnPreconditioneratMidFrequencies7

3.4NumericalExamples7

3.5Conclusion7

Chapter4CalderόnPreconditioningTechniqueforN-Müller8

4.1Introduction8

4.2N-MüllerIntegralEquations8

4.3TheDerivationofN-MüllerEquations8

4.4TheDiscretizationofN-MüllerEquations8

4.5NumericalExamples8

4.6Conclusion8

Chapter5Conclusions9

5.1ConcludingRemarks9

5.2FutureWork9

Acknowledgements10

References11

OriginofForeignLanguageMaterials12

TranslationofForeignLanguageMaterials13

Chapter1Introduction

1.1ResearchBackgroundandSignificance

Integral-equation-basednumericalmethodscombinedwithfastalgorithmsarecapableofsolvingelectromagneticproblemsofcomplexstructuresandmaterialpropertieswithagoodaccuracyandahighefficiency.Theyarewidelyusedinavarietyofengineeringapplications,suchastheefficientanalysisofthreedimensionalradarscatteringproblems,thesimulationoftheinputimpedanceandtheradiationpropertiesofantennasystems,thecalculationoftheinputresponseandthetransmissionefficiencyofmicrowavecircuits,theevaluationoftheelectromagneticinterference（EMI）betweencomplexelectromagneticsystems,andthecomputeraidedelectromagneticcompatibility（EMC）designs.Theversatility,capability,accuracyandefficiencyoftheintegral-equation-basedmethodshavemadethemanimportantandcosteffectiveapproachintheanalysisanddesignofelectromagneticproblemsandapplications.

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1.2StateofArts

Fromthe1960s,thenumericalmethodsofelectromagneticanalysishavebeenfastdevelopedbecauseoftheirversatilityandflexibility.Manywell-knownnumericalmethodshavebeenintroducedduringthattime,includingthefiniteelementmethod（FEM）[1]andthefinitedifferencetimedomainmethod（FDTD）[2,3],whicharebasedonthesolutiontotheMaxwell’sequationsindifferentialform,andthemethodofmoments（MoM）[2,4-6],whichisbasedonthesolutiontotheMaxwell’sequationsinintegralform.Especiallyfrom1990s,withthefastdevelopmentsofhighperformancecomputingsystems,thetheoriesandmethodsofcomputationalelectromagneticshavebeenadvanceddramatically.Theincreasesoftheclockspeedandthememorysizeofcomputersystemsandthedevelopmentsofhighlyefficientelectromagneticcomputingalgorithmsmakethenumericalmethodscapableofsolvingelectromagneticengineeringproblems.

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1.3ContentsandInnovationsoftheThesis

BasedontheCalderόnrelationandtheCalderόnidentities,thisthesishasdevelopedseveralCalderόnpreconditioningtechniquesandinvestigatedtheirapplicationsintheintegral-equation-basedcomputationalelectromagneticmethods.TheresearchcontenthascoveredtheCalderόnpreconditioningtechniquesfortheperfectelectricconductor（PEC）anddielectriccases.ForthePECcase,theCalderόnpreconditionsatmid,low,andhighfrequenciesareinvestigated.Forthedielectriccase,theCalderόnpreconditioningtechniquesforthePMCHWTandN-Müllerintegralequationsaredeveloped.Thenumericalaccuracyofthesecond-kindFredholmintegralequationsareinvestigatedandimprovedinthisthesis.

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1.4OutlineoftheThesis

Thisthesisisorganizedasfollows.

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Chapter2TheoreticalBasics

Inthischapter,thegeneralmethodsofconstructingthecommonlyusedintegralequationsinelectromagneticsareintroducedbasedonthesurfaceequivalenceprincipleandthevolumeequivalenceprinciple.

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2.1IntegralEquationsinElectromagnetics

Intheintegral-equation-basedcomputationalelectromagneticmethods,theunknownfunctionsintheelectromagneticproblemssuchasthescatteringorradiationfieldsaremodeledintermsoftheequivalentsurfaceorvolumeelectric/magneticsourcesbyapplyingthesurfaceorvolumeequivalenceprinciples,respectively.

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2.2270MHzPlanWaveExcitation

Inordertoinvestigatetheitsperformanceinhandlingelectricallyverylargeproblemswithoveronemillionunknowns,thesamenumericalexampleisrepeatedagainbyincreasingthefrequencyto270MHz,andkeepingtheincidentangleandpolarizationoftheplanewaveunchanged.Tohaveabetterinsight,thememoryconsumptionandCPUtimerequirementsoftheEFIE,theCP-CFIE（0.8）,andtheCP-AEFIEalgorithmsaregiveninTable2-1.

Table2-1ComparisonofComputationalDataofDifferentAlgorithms

TotalMemory（Mb）

CPUTime

Setup（h）

SolutionTime

Iter.（m）

Tol.（h）

EFIE

3215.84

1.14

3.18

>63

CP-CFIE（0.8）

6386.12

7.84

7.04

27.69

CP-AEFIE

5750.43

6.71

7.47

19.05

AllthecalculationsarecarriedoutonaHPZ400workstationwithaFedora10operatingsystem.

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2.3TheSolutionofIntegralEquationsinElectromagnetics

2.3.1GeneralPrincipleoftheMethodofMoments

Theintegralequationsconstructedintheprecedingsectioncanbesolvedwithadequatenumericalmethods.Oneofthemostcommonlyusedmethodsinsolvingintegralequationsisthemethodofmoments（MoM）introducedbyR.F.Harringtonin1968[5].ThegeneralprincipleandkeypointsofMoMwillbereviewedinthissection.

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2.3.2GeometricalModelingandDiscretizationofObject

Fromthedescriptionintheprecedingsection,itisclearthatinordertosolvefortheunknownequivalentelectromagneticcurrentsdefinedonthesurfaceorinthevolumeofanobstruction,thedefinitiondomainoftheunknowncurrents,whichisthegeometry,needstobedescribedmathematically.Thisistheso-calledgeometricalmodeling.Incomputationalelectromagnetics,geometricalmodelingisthebasicofelectromagneticmodelingandnumericalcalculation,anditsqualitywillaffecttheaccuracyofthenumericalsolutiondirectly.

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2.3.2.1PlanarTriangularModel

Thesimplestandmostcommonlyusedelementinthegeometricalmodelingistheplanartriangle,whichisdefinedbyitsthreevertices（nodes）.

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2.3.2.2CurvilinearTriangularModel

Thecurvedsurfaceofanobjectcanbebettermodeledwithcurvilineartriangularelementswhicharethesecond-ordercurvedsurfaces.Acurvilineartrianglecanbedefinedbysixnodes,threeofwhicharetheverticesofthetriangle,theotherthreearethemidpointsofthreecurvededges.ShowninFigure2-1isthesketchofacurvilineartriangularelement.

Thecurvedsurfaceofanobjectcanbebettermodeledwithcurvilineartriangularelementswhicharethesecond-ordercurvedsurfaces.Acurvilineartrianglecanbedefinedbysixnodes,threeofwhicharetheverticesofthetriangle,theotherthreearethemidpointsofthreecurvededges.ShowninFigure2-1isthesketchofacurvilineartriangularelement.

Figure2-1Thesketchofacurvilineartriangularelement.（a）Thecurvilineartriangleinthecoordinatesystem;（b）Thecurvilineartriangleinthecoordinatesystem

Usingthefollowingcoordinatetransformation,thecurvilineartriangleintherectangularcoordinatesystem,asshowninFigure2-1（a）,canbemappedontothetriangledefinedinaparametriccoordinatesystem,asshowninFigure2-1（b）

（2-1）

where

denotestherectangularcoordinatesofthesixcontrollingnodesinFigure2-1a,

，

，

aretheparametriccoordinatesvaryingfrom0to1,andtheysatisfytherelation

（2-2）

From（2-2）,itisclearthatonlytwovar