Consistent discretization of Maxwell's equations on polyhedral grids [Elektronische Ressource] / von Timo Euler

Timo Euler

Sujets

• Mathematics

Informations

• Publié par: technischen_universitat_darmstadt
• Publié le: 01 janvier 2007
• Nombre de lectures: 20
• Langue: English
• Poids de l'ouvrage: 3 Mo

Extrait

Consistent Discretization of Maxwell’s
Equations on Polyhedral Grids
Vom Fachbereich 18
Elektrotechnik und Informationstechnik
der Technischen Universit¨at Darmstadt
zur Erlangung
der Wu¨rde eines Doktor Ingenieurs (Dr.-Ing.)
genehmigte
DISSERTATION
von
Dipl.-Ing. Timo Euler
geboren in Schlu¨chtern
Referent: Prof. Dr.-Ing. Thomas Weiland
Korreferent: Prof. Dr. rer. nat. Jens Lang
Korreferent: Prof. Dr. techn. Romanus Dyczij-Edlinger
Tag der Einreichung: 10.07.2007
Tag der mu¨ndlichen Pru¨fung: 22.10.2007
D17
Darmst¨adter Dissertation
Darmstadt, 2007CONTENTS
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Kurzfassung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2. Continuous Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1 Continuous Mathematical Tools Needed to State Maxwell’s Equations . . . 5
2.1.1 Space-Time Domain . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.2 Differential Forms, Vector Proxies, and Integration . . . . . . . . . 6
2.1.3 Topological Tools and Properties . . . . . . . . . . . . . . . . . . . 9
2.1.4 Metric Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.1 Maxwell’s Topological Laws . . . . . . . . . . . . . . . . . . . . . . 15
2.2.2 Maxwell’s Material Laws . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.3 Maxwell’s Topological Properties . . . . . . . . . . . . . . . . . . . 17
2.2.4 Maxwell’s Metric Properties . . . . . . . . . . . . . . . . . . . . . . 19
2.2.5 Maxwell’s Initial Boundary Value Problem . . . . . . . . . . . . . . 19
2.2.6 Maxwell’s House . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3. The Finite Integration Technique on Consistent Grids . . . . . . . . . . 27
3.1 Discrete Mathematical Tools Needed to State Maxwell’s Grid Equations . . 29
3.1.1 Consistent Grids and their Duals . . . . . . . . . . . . . . . . . . . 29
3.1.2 Chains, Cochains, Grid Integration, and the Grid Discretization
Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.1.3 Discrete Topological Tools and Properties . . . . . . . . . . . . . . 40iv Contents
3.1.4 Discrete Metric Tools . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2 Maxwell’s Spatially Discrete Grid Equations . . . . . . . . . . . . . . . . . 51
3.2.1 Maxwell’s Discrete Topological Laws . . . . . . . . . . . . . . . . . 51
3.2.2 Maxwell’s Discrete Material Laws . . . . . . . . . . . . . . . . . . . 59
3.2.3 Maxwell’s Topological Properties . . . . . . . . . . . . . . . . . . . 62
3.2.4 Maxwell’s Metric Properties . . . . . . . . . . . . . . . . . . . . . . 66
3.2.5 Spatially Discrete Maxwell’s Initial Boundary Value Problem . . . . 67
3.2.6 Spatially Discrete Maxwell’s House . . . . . . . . . . . . . . . . . . 75
3.3 Maxwell’s Spatially and Temporally Discrete Grid Equations . . . . . . . . 76
3.3.1 Maxwell’s Discrete Topological Laws . . . . . . . . . . . . . . . . . 76
3.3.2 Maxwell’s Discrete Material Laws . . . . . . . . . . . . . . . . . . . 79
3.3.3 Maxwell’s Topological Properties . . . . . . . . . . . . . . . . . . . 82
3.3.4 Maxwell’s Metric Properties . . . . . . . . . . . . . . . . . . . . . . 84
3.3.5 TemporallyandSpatiallyDiscreteMaxwell’sInitialBoundaryValue
Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4. Discrete Material Operators for Polyhedral Grids . . . . . . . . . . . . . 91
4.1 Whitney-FEM Material Operators . . . . . . . . . . . . . . . . . . . . . . . 92
4.2 Polyhedral Whitney Basis Functions by a Continuous Construction . . . . 100
4.2.1 General Construction Algorithm . . . . . . . . . . . . . . . . . . . . 102
4.2.2 Specific Construction Algorithm . . . . . . . . . . . . . . . . . . . . 103
4.3 Polyhedral Whitney Basis Functions Constructed by the FIT . . . . . . . . 112
4.4 Polyhedral Whitney Basis Functions by a Galerkin Construction . . . . . . 115
4.4.1 Continuous Variational Formulation of the Construction Boundary
Value Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.4.2 DiscreteGalerkinFormulationoftheConstructionBoundaryValue
Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.4.3 General Construction Algorithm . . . . . . . . . . . . . . . . . . . . 118
4.4.4 Specific Construction Algorithm . . . . . . . . . . . . . . . . . . . . 119
4.5 Discussion of Existing Polygonal and Polyhedral Schemes . . . . . . . . . . 127
5. Application of Polytope Elements in Electromagnetic Simulations . . . 129
5.1 New Gridding Flexibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.1.1 Rectangular Waveguide with Polygonal Grids . . . . . . . . . . . . 130Contents v
5.2 Enhanced Boundary Approximation in Structured Grids . . . . . . . . . . 137
5.2.1 Dielectric Cylinder in Homogeneous Field . . . . . . . . . . . . . . 137
5.3 Subgridding for Structured Grids . . . . . . . . . . . . . . . . . . . . . . . 141
5.3.1 Partially Filled Waveguide . . . . . . . . . . . . . . . . . . . . . . . 141
5.3.2 Homogeneously Filled Resonator . . . . . . . . . . . . . . . . . . . 143
5.3.3 Partially Filled Resonator . . . . . . . . . . . . . . . . . . . . . . . 147
5.3.4 Reflection Analysis from Subgridding Interface . . . . . . . . . . . . 149
5.4 Hybrid Hexahedral/Tetrahedral Grids . . . . . . . . . . . . . . . . . . . . . 158
5.4.1 Partially Filled Resonator . . . . . . . . . . . . . . . . . . . . . . . 158
5.4.2 Dielectric Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6. Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
Appendix 169
A.Enhanced Dual Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
A.1 Definitions and Topological Tools and Properties . . . . . . . . . . . . . . 171
A.2 Example Application for Grid Equations of Maxwell’s Type . . . . . . . . 179
B. Electric Boundary Conditions in the FIT . . . . . . . . . . . . . . . . . . 185
C. Equivalence of the Whitney-FEM to the FIT . . . . . . . . . . . . . . . . 187
D.Error Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
D.1 General Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
D.2 Example Error Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
E. Stability, Consistency, and Convergence of Discrete Approximation
Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
F. Proofs of Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
F.1 Proofs of Theorems for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . 219
F.1.1 Proof of Theorem 3.9 . . . . . . . . . . . . . . . . . . . . . . . . . . 219
F.2 Proof of Theorems from Chapter 4 . . . . . . . . . . . . . . . . . . . . . . 229
F.2.1 Proof of Theorem 4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . 229vi Contents
F.2.2 Proof of Theorem 4.6 . . . . . . . . . . . . . . . . . . . . . . . . . . 231
F.2.3 Proof of Theorem 4.8 . . . . . . . . . . . . . . . . . . . . . . . . . . 237
Abbreviations and Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
Curriculum Vitae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251ACKNOWLEDGEMENTS
Manypeopleandinstitutionshaveprovidedsupportduringthelastyearsandthushelped
me bring my research to a fruitful conclusion. Thanks. I explicitly thank
• Prof. Dr.-Ing. Thomas Weiland for advising my thesis and providing an excellent
research environment,
• Prof. Dr. rer. nat. Jens Lang and Prof. Dr. techn. Romanus Dyczij-Edlinger for
reviewing this thesis,
• theTechnischeUniversit¨atDarmstadtandtheComputationalEngineeringResearch
Center at the Technische Universit¨at Darmstadt for providing scholarships,
• my current and former colleagues at the Institut fu¨r Theorie Elektromagnetischer
FelderoftheTechnischeUniversit¨atDarmstadt,especiallyDipl.-Ing.AndreasPaech
and Dipl.-Ing. Denis Sievers, for the interesting discussions and a nice time,
• Prof. Dr.-Ing. Rolf Schuhmann for guiding the first steps into my research,
• the research community for stimulating discussions about and input for my work,
• my electrical engineering and mathematics teachers for igniting my interests in a
wide field of scientific topics,
• my family and parents.
And I especially thank Esther Kraus for accompanying me during all these years.ABSTRACT
This thesis introduces polyhedral cell shapes into the formalism of the Finite Integration
Technique (FIT) and shows their practicability in electromagnetic simulations. Emphasis
is put on a rigorous mathematical presentation.
The semi-discrete (discrete in space but continuous in time) and fully discrete Maxwell’s
Grid Equations of the FIT are developed from the continuous Maxwell’s equations accen-
tuatingtheconnectionstodifferentialgeometryandtopology. ThederivationofMaxwell’s
Grid Equation