Advances in Mathematical Methods for Electromagnetics (Electromagnetic Waves)

748 Pages·2021·20.118 MB·English

by  Kazuya Kobayashi (editor), Paul Denis Smith (editor)

Advances in
Mathematical Methods
for Electromagnetics
TheACESSeriesonComputationalandNumericalModellinginElectricalEngineering
AndrewF.Peterson,PhD–SeriesEditor
The volumes in this series will encompass the development and application of numerical
techniques to electrical and electronic systems, including the modelling of electromagnetic
phenomenaoverallfrequencyrangesandcloselyrelatedtechniquesforacousticandoptical
analysis.Thescopeincludestheuseofcomputationforengineeringdesignandoptimization,
aswellastheapplicationofcommercialmodellingtoolstopracticalproblems.Theserieswill
includetitlesforseniorundergraduateandpostgraduateeducation,researchmonographsfor
reference,andpractitionerguidesandhandbooks.
TitlesintheSeries
K.Warnick,“NumericalMethodsforEngineering,”2010
W.Yu,X.YangandW.Li,“VALU,AVXandGPUAccelerationTechniquesforParallelFDTD
Methods,”2014.
A.Z. Elsherbeni, P. Nayeri and C.J. Reddy, “Antenna Analysis and Design Using FEKO
ElectromagneticSimulationSoftware,”2014.
A.Z.ElsherbeniandV.Demir,“TheFinite-DifferenceTime-DomainMethodinElectromag-
neticswithMATLABSimulations2ndEdition,”2015.
M. Bakr, A.Z. Elsherbeni and V. Demir, “Adjoint Sensitivity Analysis of High Frequency
StructureswithMATLAB,”2017.
O.Ergul,“NewTrendsinComputationalElectromagnetics,”2019.
D.Werner,“NanoantennasandPlasmonics:Modelling,designandfabrication,”2020.
Advances in
Mathematical Methods
for Electromagnetics
Edited by
Kazuya Kobayashi and Paul Denis Smith
TheInstitutionofEngineeringandTechnology
PublishedbySciTechPublishing,animprintofTheInstitutionofEngineeringand
Technology,London,UnitedKingdom
TheInstitutionofEngineeringandTechnologyisregisteredasaCharityin
England&Wales(no.211014)andScotland(no.SC038698).
©TheInstitutionofEngineeringandTechnology2021
Firstpublished2020
ThispublicationiscopyrightundertheBerneConventionandtheUniversalCopyright
Convention.Allrightsreserved.Apartfromanyfairdealingforthepurposesofresearch
orprivatestudy,orcriticismorreview,aspermittedundertheCopyright,Designsand
PatentsAct1988,thispublicationmaybereproduced,storedortransmitted,inany
formorbyanymeans,onlywiththepriorpermissioninwritingofthepublishers,orin
thecaseofreprographicreproductioninaccordancewiththetermsoflicencesissued
bytheCopyrightLicensingAgency.Enquiriesconcerningreproductionoutsidethose
termsshouldbesenttothepublisherattheundermentionedaddress:
TheInstitutionofEngineeringandTechnology
MichaelFaradayHouseSix
HillsWay,StevenageHerts,
SG12AY,UnitedKingdom
www.theiet.org
Whiletheauthorsandpublisherbelievethattheinformationandguidancegiveninthis
workarecorrect,allpartiesmustrelyupontheirownskillandjudgementwhenmaking
useofthem.Neithertheauthorsnorpublisherassumesanyliabilitytoanyoneforany
lossordamagecausedbyanyerrororomissioninthework,whethersuchanerroror
omissionistheresultofnegligenceoranyothercause.Anyandallsuchliability
isdisclaimed.
Themoralrightsoftheauthorstobeidentifiedasauthorsofthisworkhavebeen
assertedbytheminaccordancewiththeCopyright,DesignsandPatentsAct1988.
BritishLibraryCataloguinginPublicationData
AcataloguerecordforthisproductisavailablefromtheBritishLibrary
ISBN978-1-78561-384-5(hardback)
ISBN978-1-78561-385-2(PDF)
TypesetinIndiabyMPSLimited
PrintedintheUKbyCPIGroup(UK)Ltd,Croydon
Contents
Abouttheeditors xxi
Introduction 1
KazuyaKobayashiandPaulD.Smith
References 16
1 Newinsightsinintegralrepresentationtheoryforthe
solutionofcomplexcanonicaldiffractionproblems 17
J.M.L.Bernard
1.1 Representationsofspectralfunctioninfrequencyandtime
domain,forthescatteringbyapolygonalregion 18
1.1.1 BasicelementsinSommerfeld–Maliuzhinets
representationandproperties 18
1.1.2 Spectralfunctions f±attachedtotheradiationofasingle
faceandsimplerelationto f 20
1.1.3 Spectralfunction f fromfarfieldradiationofoneface
witharbitraryshape 22
1.1.4 Exactcausaltimedomainrepresentationofafield
aboveadispersivewedge-shapedregion 23
1.2 Severalordersasymptoticrepresentationforscattering
byacurvedimpedancewedge 25
1.2.1 Asymptoticrepresentationinaregionwithout
creepingwaves 25
1.2.2 Severalordersasymptoticexpressionsinaregionwith
creepingwavesterms 27
1.2.3 Some(cid:2)validationsconcerningtheexpressionof
f = f /knforarbitrarywedgeangle 30
n≥0 n
1.3 Anovelexpressionofthefieldforarbitraryboundedsources
aboveapassiveoractiveimpedanceplane 32
1.3.1 Formulationoftheproblem 32
1.3.2 Anexpressionofpotentials(E ,H )for
inc inc
boundedsourcesJ andM 34
1.3.3 Expressionofthepotentials(E ,H )for
s s
animpedanceplane 34
vi AdvancesinMathematicalMethodsforElectromagnetics
1.4 Spectralrepresentationofthefieldfor3Dconicalscatterers 36
1.4.1 Formulation 36
1.4.2 ExpressionofthepotentialswithKontorovich–Lebedev
integrals 38
1.4.3 Potentialsandpropertiesforanincidentplanewave 41
References 44
2 Scatteringofelectromagneticsurfacewaveson
imperfectlyconductingcanonicalbodies 47
MikhailA.LyalinovandNingYanZhu
2.1 Introductionandsurveyofsomeknownresults 47
2.1.1 Electromagneticsurfacewavesonimpedancesurfaces 49
2.1.2 Electromagneticsurfacewavesonarightcircular
conicalsurface 54
2.2 Excitationofanelectromagneticsurfacewavebyadipole
locatednearaplaneimpedancesurface 56
2.3 Scatteringofaskewincidentsurfacewavebytheedgeon
animpedancewedge 59
2.3.1 Integralequationsforthespectra 61
2.3.2 Far-fieldexpansion 62
2.3.3 Reflectionandrefractionofanincidentsurfacewaveat
theedgeofanimpedancewedge 63
2.3.4 Beyondthecriticalangleofedgediffraction 65
2.4 Conclusion 67
AppendixA 68
A.1 Brewsterangles 69
Acknowledgement 69
References 69
3 Dielectric-wedgeFourierseries 73
SvendBerntsen
3.1 Introduction 73
3.2 Thediffractionproblem 75
3.2.1 TheHilbert-spaceproblem 75
3.2.2 Thesingular-fieldproblem 76
3.3 Integralequations 77
3.4 Thesolutionoftheintegralequation 79
3.5 TheBessel–HankelFourierseries 81
3.6 Incidentplanewaves 87
3.7 Numericalresults 88
3.8 Summary 92
AppendixA 92
Acknowledgment 93
References 94
Contents vii
4 Green’stheorem,Green’sfunctionsandHuygens’
principleindiscreteelectromagnetics 97
JohnM.Arnold
4.1 Introduction 97
4.2 Green’stheoremforadjacencymatrices 99
4.2.1 Adjacencymatrix 99
4.2.2 Weightedadjacencymatrix 100
4.2.3 Matrixofadjacency1 100
4.2.4 ThevertexLaplacianmatrix 100
4.2.5 Incidencematrices 101
4.3 Green’stheoremontopologicalvectorspace 101
4.4 Differenceformsanddiscreteexteriorcalculus 104
4.4.1 Simplicialdecomposition 104
4.4.2 Dualforms 105
4.4.3 Hypercubedecomposition 106
4.4.4 Contextualalgebraicnotationofforms 106
4.4.5 Manifolds,graphsandlattices 107
4.4.6 Essentialsofcelldecomposition 108
4.5 HigherorderGreen’stheoremandGreen’sfunctions 108
4.5.1 Green’stheoremforr-forms 108
4.5.2 Kirchhoff’stheoremforr-forms 109
4.5.3 Green’sfunctionforr-forms 109
4.6 Dynamicalsystemsontopologicalvectorspaces:
Maxwell’sequations 109
4.6.1 DiscreteMaxwellequations 110
4.6.2 Electromagneticfieldsasdifferentialforms 111
4.7 Time-domainGreen’sfunctionsfordynamicalsystems 112
4.8 Discretetime 114
4.9 DiscreteGreen’stheoremandGreen’sfunctionsin
computationalfieldtheory 116
4.9.1 Exterior–interiorconnection 116
4.9.2 Diakoptics 117
4.10 Conclusion 117
References 117
5 Theconceptofgeneralizedfunctionsanduniversalproperties
oftheGreen’sfunctionsassociatedwiththewaveequationin
boundedpiece-wisehomogeneousdomains 119
M.Idemen
5.1 Ashorthistoricalbackground 119
5.1.1 Aremark 123
5.2 Somebasicpropertiesoftheδdistribution 124
5.2.1 Distributionsinvolvingδ(R)andδ(R −R ) 125
1 2
viii AdvancesinMathematicalMethodsforElectromagnetics
5.3 Green’sfunctionsassociatedwiththewaveequationin
boundedpartiallyhomogeneousdomains 128
5.3.1 TheoutgoingGreen’sfunction 130
5.3.2 TheingoingGreen’sfunction 131
5.3.3 Someuniversalpropertiesoftheoutgoingandingoing
Green’sfunctions 132
5.4 Proofsofthetheorems 133
5.4.1 ProofofTheorem1 133
5.4.2 ProofofTheorem2 133
5.4.3 ProofofTheorem3 134
5.4.4 ProofofTheorem4 134
5.4.5 ProofofTheorem5 135
5.4.6 ProofofTheorem6 136
5.5 Application:Aninverseinitial-valueproblemconnected
withthephotoacoustictomographyinbounded
non-homogeneousdomains 138
5.5.1 Extensionoftheinverseinitialvalueproblem
totherange(−∞)<t <∞ 139
5.5.2 Solutionoftheextendedproblem 140
5.5.3 Proofof(5.64) 142
References 143
6 Ellipticcylinderwithastronglyelongatedcross-section:
highfrequencytechniquesandfunctiontheoreticmethods 145
FrédéricMolinetandIvanAndronov
6.1 Introduction 145
6.2 Asymptoticcurrentsonanellipticcylinderwithatruncated
stronglyelongatedcross-section 147
6.2.1 Analysisoftheinteractions 147
6.2.2 Asymptoticfieldintheboundarylayerduetoamagnetic
linecurrent 148
6.2.3 Radiatedfieldandtotalfield 152
6.2.4 Spectraldecompositionofthefieldintheboundarylayer 154
6.2.5 Diffractionbytheedgeofatruncatedellipticcylinder 156
6.2.6 Asymptoticfieldintheboundarylayerduetoanincident
planewave 157
6.2.7 Asymptoticcurrents 160
6.3 Asymptotic currents on a cylinder with an ogival cross-section
composedoftwosymmetricarcsofastronglyelongatedellipse 161
6.3.1 Presentationofthegeometryandanalysisoftheproblem 161
6.3.2 Asymptoticcurrentsoutsidegrazingincidence 162
6.3.3 Grazingincidence 163
6.4 Conclusion 166
References 168
Contents ix
7 High-frequencyhybridray–modetechniques 169
HiroshiShirai
7.1 Introduction 169
7.2 Ray–modeconversiontechnique 170
7.3 Modalexcitationattheaperture 171
7.3.1 Formulation 171
7.3.2 Numericalresults 174
7.4 Diffractionbyaslitonathickconductingscreen 177
7.4.1 Background 177
7.4.2 Formulation 177
7.4.3 Diffractionbyathinslit 182
7.4.4 Diffractionbyathickandloadedslit 182
7.4.5 Diffractionbyatrough 185
7.5 Conclusions 186
Acknowledgments 186
References 186
8 Scatteringanddiffractionofscalarandelectromagneticwavesusing
spherical-multipoleanalysisanduniformcomplex-sourcebeams 189
LudgerKlinkenbuschandHendrikBrüns
8.1 Introduction 189
8.2 SolutionofMaxwell’sequationsinsphero-conalcoordinates 191
8.2.1 Sphero-conalcoordinates 191
8.2.2 SolutionoftheHelmholtzequationinsphero-conal
coordinates 192
8.2.3 Vectorspherical-multipoleexpansionofthe
electromagneticfieldinthepresenceof
aPECsemi-infiniteellipticcone 196
8.3 Complex-sourcebeams 198
8.3.1 ConverginganddivergingCSB 198
8.3.2 UniformCSB 200
8.4 Green’sfunctionofthesemi-infiniteellipticconeforan
incidentuniformcomplex-sourcebeam 202
8.4.1 ScalarGreen’sfunction 202
8.4.2 DyadicGreen’sfunction 203
8.5 Numericalevaluation 204
8.5.1 Convergenceanalysis 204
8.5.2 Numericalresultsforanacousticallysoftor
hardsemi-infiniteellipticcone 206
8.5.3 Numericalresultsforaperfectlyconducting
semi-infiniteellipticcone 208
8.6 Conclusions 212
References 213