Higher-order FDTD Schemes for Waveguides and Antenna Structures (Synthesis Lectures on Computational Electromagnetics)

226 Pages·2006·3.69 MB·English

by  Theodoros D. Tsiboukis

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Higher Order FDTD Schemes
for Waveguide and Antenna
Structures
i
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Copyright©2006byMorgan&Claypool
Allrightsreserved.Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,ortransmittedin
anyformorbyanymeans—electronic,mechanical,photocopy,recording,oranyotherexceptforbriefquotations
inprintedreviews,withoutthepriorpermissionofthepublisher.
HigherOrderFDTDSchemesforWaveguideandAntennaStructures
N.V.KantartzisandT.D.Tsiboukis
www.morganclaypool.com
1598290282 paper Kantartzis/Tsiboukis
1598290290 ebook Kantartzis/Tsiboukis
DOI10.2200/S00018ED1V01Y200604CEM003
APublicationintheMorgan&ClaypoolPublishers’series
SYNTHESISLECTURESONCOMPUTATIONALELECTROMAGNETICS
Lecture#3
FirstEdition
10987654321
PrintedintheUnitedStatesofAmerica
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Higher Order FDTD Schemes
for Waveguide and Antenna
Structures
NikolaosV.KantartzisandTheodorosD.Tsiboukis
AppliedandComputationalElectromagneticLaboratory
DepartmentofElectricalandComputerEngineering
AristotleUniversityofThessaloniki
Thessaloniki,Greece
SYNTHESISLECTURESONCOMPUTATIONALELECTROMAGNETICS#3
M
&C &
Morgan Claypool Publishers
iii
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ABSTRACT
This publication provides a comprehensive and systematically organized coverage of higher
order finite-difference time-domain or FDTD schemes, demonstrating their potential role as
a powerful modeling tool in computational electromagnetics. Special emphasis is drawn on
theanalysisofcontemporarywaveguideandantennastructures.Acknowledgedasasignificant
breakthroughintheevolutionoftheoriginalYee’salgorithm,thehigherorderFDTDoperators
remainthesubjectofanongoingscientificresearch.Amongtheirindisputablemerits,onecan
distinguishtheenhancedlevelsofaccuracyevenforcoarsegridresolutions,thefastconvergence
rates, and the adjustable stability. In fact, as the fabrication standards of modern systems get
stricter, it is apparent that such properties become very appealing for the accomplishment of
elaborateandcredibledesigns.
KEYWORDS
finite-difference time-domain methods, FDTD, computational electromagnetics, Yee’s algo-
rithm,waveguideandantennastructureandanalysis,electromagneticmodeling,
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Contents
1. Introduction..................................................................1
1.1 Time-DomainModelinginComputationalElectromagnetics ................ 1
1.2 TheFDTDMethodforWaveguideandAntennaAnalysis...................2
1.2.1 EssentialPrinciplesandEvolutionAspects ..........................2
1.2.2 InherentShortcomingsandEfficiencyEnhancement.................2
1.3 TheHigherOrderFDTDFormulation.....................................4
1.3.1 MeritsandPerformanceIssues.....................................4
1.3.2 BasicClassification................................................5
References...............................................................6
2. ConventionalHigherOrderFDTDDifferentiation.............................9
2.1 Introduction.............................................................9
2.2 Fundamentals...........................................................10
2.3 DevelopmentoftheBasicConventionalAlgorithm.........................13
2.4 HigherOrderFDTDModelingofBoundariesandMaterialInterfaces.......17
2.4.1 MaximumAccuracyandOptimizedCompactSchemes..............17
2.4.2 ImplicitMethodsforComplexCartesianDomains..................20
2.4.3 One-SidedDifferenceOperatorsforHomogeneous
DielectricBoundaries............................................23
2.4.4 TreatmentofInhomogeneousDielectricBoundaries ................ 26
2.4.5 DerivativeMatchingwithFictitiousPoints.........................29
2.5 Dispersion-OptimizedHigherOrderFDTDTechniques...................31
2.5.1 Modified(2,4)SchemewithImprovedPhaseAccuracy.............32
2.5.2 Angle-OptimizedFDTDAlgorithms ............................. 35
2.5.3 Dispersion-Relation-PreservingFDTDSchemes...................38
2.5.4 DesignofControllableDispersion-ErrorTechniques................42
2.5.5 Coefficient-ModificationAlgorithms..............................47
2.6 HigherOrderFDTDSchemesinCurvilinearCoordinates..................49
References..............................................................52
3. HigherOrderNonstandardFDTDMethodology..............................57
3.1 Introduction............................................................57
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3.2 TheNonstandardFinite-DifferenceAlgorithm ............................ 58
3.2.1 DiscretizationoftheWaveEquation...............................60
3.2.2 DiscretizationofMaxwell’sEquations ............................. 64
3.3 DevelopmentoftheHigherOrderNonstandardForms
inCartesianCoordinates.................................................67
3.3.1 TheoryofthePreciseSpatial/TemporalOperators..................67
3.3.2 HigherOrderNonstandardLeapfrog-TypeIntegrators..............71
3.3.3 AlternativeApproaches...........................................73
3.4 GeneralizedHigherOrderCurvilinearFDTDMethod.....................74
3.4.1 FormulationoftheNonorthogonalDiv-CurlProblem...............75
3.4.2 TheTopologicallyConsistentHigherOrder
NonstandardFramework.........................................77
3.4.3 SystematicConstructionofMultidirectionalTessellations............80
3.4.4 ExtensiontoNon-CartesianandUnstructuredGrids................85
3.5 DispersionErrorandStabilityAnalysis....................................87
3.5.1 TheEnhancedDispersionRelation................................87
3.5.2 TheModifiedStabilityCriterion..................................89
3.6 IssuesofPracticalImplementation........................................89
References..............................................................89
4. AbsorbingBoundaryConditionsandWidenedSpatialStencils ................. 95
4.1 Introduction............................................................95
4.2 HigherOrderFDTDFormulationofAnalyticalABCs.....................96
4.3 HigherOrderPMLAbsorbers...........................................98
4.3.1 ConventionalRealizationinCartesianCoordinates................. 99
4.3.2 NonstandardCurvilinearImplementation.........................104
4.4 WidenedSpatialStencilsandDissimilarMediaInterfaces ................. 110
4.4.1 TheGhost-NodeProcess........................................110
4.4.2 Self-AdaptiveCompactOperators ............................... 111
4.4.3 TreatmentofArbitrarilyCurvedMaterialInterfaces ............... 113
References.............................................................115
5. StructuralExtensionsandTemporalIntegration..............................121
5.1 Introduction...........................................................121
5.2 ModelingofLossyandDispersiveMediawithHigherOrder
FDTDSchemes.......................................................121
5.2.1 LossyFormulations.............................................122
5.2.2 DispersiveMedia...............................................123
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5.3 ImprovementviatheCorrectionofMaterialParameters ................... 128
5.4 EnhancedSpatialApproximations ...................................... 131
5.4.1 AlternativeStencilManagement .................................131
5.4.2 ImprovementThroughtheDispersionRelation....................133
5.5 GeneralizingTemporalIntegration ......................................138
5.5.1 HigherOrderLeapfrogSchemes.................................138
5.5.2 HigherOrderMultistageRunge–KuttaIntegrators................139
References.............................................................140
6. HybridandAlternativeHigherOrderFDTDSchemes........................143
6.1 Introduction...........................................................143
6.2 HybridSecond-OrderandHigherOrderFDTDTechniques...............143
6.3 DiscreteSingularConvolutionandSymplecticOperators..................146
6.4 TheHigherOrderADI-FDTDMethod.................................150
6.4.1 MainMotivesandObjectives....................................150
6.4.2 GeneralizedTheoreticalEstablishment...........................150
6.4.3 StabilityandDispersion-ErrorAnalysis...........................154
6.5 HigherOrderWeightedEssentiallyNonoscillatorySchemes
intheTimeDomain ................................................... 157
6.5.1 DevelopmentoftheThree-DimensionalAlgorithm................157
6.5.2 ModelingofMediaDiscontinuities...............................161
References.............................................................163
7. SelectedApplicationsinWaveguideSystems..................................167
7.1 Introduction...........................................................167
7.2 ExcitationSchemesandOpen-EndTruncation...........................167
7.3 MultimodalHigherOrderFDTDAnalysis...............................169
7.4 Applications–NumericalResults........................................171
7.4.1 BasicWaveguideProblems ...................................... 171
7.4.2 Three-DimensionalRealisticDevices.............................174
References.............................................................186
8. SelectedApplicationsinAntennaStructures..................................189
8.1 Introduction...........................................................189
8.2 ExcitationIssuesandFeedingModels....................................189
8.3 AnalysisofEssentialStructures..........................................191
8.4 ContemporaryAntennaConfigurations..................................193
8.4.1 MicrostripandCavity-BackedAntennas..........................194
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8.4.2 DielectricResonatorAntennas...................................200
8.4.3 FractalArrays..................................................204
8.4.4 Metamaterial-LoadedApplications...............................207
References.............................................................210
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Preface
The present book provides a comprehensive and systematically organized coverage of higher
orderFDTDschemes,demonstratingtheirpotentialroleasapowerfulmodelingtoolincom-
putationalelectromagnetics.Specialemphasisisdrawnontheanalysisofcontemporarywaveg-
uide and antenna structures. Acknowledged as a significant breakthrough in the evolution of
the original Yee’s algorithm, the higher order FDTD operators remain the subject of an on-
going scientific research. Among their indisputable merits, one can distinguish the enhanced
levelsofaccuracyevenforcoarsegridresolutions,thefastconvergencerates,andtheadjustable
stability. In fact, as the fabrication standards of modern systems get stricter, it is apparent
that such properties become very appealing for the accomplishment of elaborate and credible
designs.
InChapter1,thebookbeginswithabrief,yetinformative,introductionontime-domain
algorithms, the most characteristic shortcomings of Yee’s technique, and the classification of
higherorderFDTDschemesinconventionalandnonstandardcounterparts.Theinvestigation
oftheformer–bothexplicitandimplicit–alongwithathoroughanalysisofinterfacetreatment
and dispersion-error optimization is conducted in Chapter 2. Chapter 3 covers the theoreti-
calformulationofhigherordernonstandardoperatorsandintroducesageneralizedcurvilinear
covariant/contravariantmethodologyforbroadbandsimulations.Thetopicofabsorbingbound-
ary conditions for open-region termination is carefully examined in Chapter 4, which also
presentsseveraltechniquesforthemanipulationoftheinevitablywidenedspatialstencilsnear
dissimilar material interfaces or lattice ends. Chapter 5 discusses diverse structural extensions
and temporal integration approaches focusing on lossy, frequency-dependent and dispersion-
reduction approaches. Moreover, the hybridization of higher order FDTD forms with other
efficientalgorithmsandthepossibilityofincreasingtheorderofaccuracyofseveralalternative
methods are explored in Chapter 6. Finally, Chapters 7 and 8 are devoted to the numerical
modeling of up-to-date waveguides and antennas, respectively. A lot of indicative examples
are presented and various complicated configurations are successfully solved. All applications
are realistic, while most of the results are compared with measurements or reference data. It
is stressed that every chapter closes with an extensive list of references that offer additional
evidenceanddetailsonthecompetenceofhigherorderschemestothemoreinterestedreader.
Conclusively,theflavorofthebookistheprofitablelinkbetweentheunderlyingphysical
backgroundandthecomputationalpracticeofthehigherorderFDTDmethod.Therefore,itis
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x PREFACE
thehopeoftheauthorsthatitwillbecontributiveasabasistotheresearcherpursuingeffective
substitutes to traditional ideas or serve as a state-of-the-art review for this rapidly growing
numericaltechnique.
Thessaloniki N.V.Kantartzis
April2006 T.D.Tsiboukis