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# Consistent Discretization of Maxwell's Equations on Polyhedral Grids

### Euler, Timo (2007):Consistent Discretization of Maxwell's Equations on Polyhedral Grids.Darmstadt, Technische Universität, [Ph.D. Thesis]

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Item Type: Ph.D. Thesis
Title: Consistent Discretization of Maxwell's Equations on Polyhedral Grids
Language: English
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 accentuating the connections to differential geometry and topology. The derivation of Maxwell's Grid Equations is valid for a set of arbitrary dual consistent grids allowing also for curved polyhedral cell shapes. This possibility has been known for quite some time, but material relations were only known for special cell shapes like hexahedra, tetrahedra, prisms, pyramids, or dual orthogonal grids. In this thesis, material relations for arbitrary polyhedral grid cells with straight edges and planar faces are derived. Examples from a wide range of electromagnetic applications show the practicability of these polyhedral grid cells in numerical simulations.

Alternative Abstract:
Alternative AbstractLanguage

In dieser Arbeit werden Polyederelemente in die Methode der finiten Integration (FIT) eingeführt. Die praktische Anwendbarkeit in elektromagnetischen Simulationen wird gezeigt. Ein weiterer Schwerpunkt ist die rigorose mathematische Einbindung. Die Gitter-Maxwell-Gleichungen der FIT werden in semi-diskreter (diskret im Raum und kontinuierlich in der Zeit) und in voll diskreter Form aus den kontinuierlichen Maxwellschen Gleichungen hergeleitet. Verbindungen zu den Disziplinen der Differentialgeometrie und Topologie werden ersichtlich. Die semi-diskreten und voll diskreten Gitter-Maxwell-Gleichungen gelten in der hergeleiteten Form für beliebige, duale, konsistente Gitter, einschließlich gekrümmten Polyedergittern. Obwohl diese Möglichkeit seit einiger Zeit bekannt ist, konnten die benötigten Materialbeziehungen bisher nur für spezielle Elementformen wie Hexaeder, Tetraeder, Prismen, Pyramiden oder dual orthogonale Gitter hergeleitet werden. In dieser Arbeit werden Materialbeziehungen für beliebige Polyeder mit planaren Flächen und geraden Kanten eingeführt. Beispiele aus verschiedenen elektromagnetischen Bereichen zeigen die praktische Anwendbarkeit in numerischen Simulationen.

German
Place of Publication: Darmstadt
Uncontrolled Keywords: Finite-Integrations-Methode, Finite-Elemente-Methode, Polyedergitter, diskrete Differentialformen, Whitney-Formen
Alternative keywords:
Alternative keywordsLanguage
Finite-Integrations-Methode, Finite-Elemente-Methode, Polyedergitter, diskrete Differentialformen, Whitney-FormenGerman
Finite Integration Technique, Finite Element Method, polyhedral grid, discrete differential forms, Whitney formsEnglish
Classification DDC: 600 Technik, Medizin, angewandte Wissenschaften > 620 Ingenieurwissenschaften
Divisions: 18 Department of Electrical Engineering and Information Technology
Date Deposited: 17 Oct 2008 09:22
URN: urn:nbn:de:tuda-tuprints-8955
Referees: Lang, Prof. Dr. Jens ; Dyczij-Edlinger, Prof. Dr. Romanus
Advisors: Weiland, Prof. Dr.- Thomas
Refereed: 22 October 2007