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# Space-Time Discretization of Maxwell's Equations in the Setting of Geometric Algebra

## Abstract

In this dissertation a FIT-like discretisation of Maxwell's equations is performed directly in four-dimensional space-time using the mathematical formalism of Clifford's Geometric Algebra. The thesis extends the Finite Integration Technique (FIT) to 4D space-time without introducing any non-relativistic assumptions. The coordinate-free formulation in terms of geometric algebra enhances explicitly relativistic, i.e., without splitting space and time, treatment, which reveals in the fact that any non-relativistic assumptions are not made. The relation of geometric algebra to the existing concepts from differential geometry in the language of differential forms is established in the context of electromagnetic field description. An alternative to the existing approaches formula for the discretisation of material laws on non-orthogonal mesh pairs is derived, investigated and applied. The developed theory is applied to obtain the condition for 3D problems when material matrices are diagonal, and due to quantitative nature of this condition a mesh optimisation procedure is proposed, as well as its limitations in 3D case, which do not occur in 2D, are derived. The other application is simulation of electromagnetic wave propagation in a rotating reference frame. Due to coordinate-free formalism and encoding the movement of the observer in 4D mesh's geometry, derivation of the numerical scheme for rotating observer's resembles the one for inertial (stationary) observers. In other words, relativistic coordinate-free treatment includes inertial and non-inertial observers as special cases, which do not need to be diversified. The comparison of the obtained numerical results with the ones known from literature is performed in order to validate the theoretical results.

Item Type: Ph.D. Thesis
Erschienen: 2018
Creators: Klimek, Mariusz
Title: Space-Time Discretization of Maxwell's Equations in the Setting of Geometric Algebra
Language: English
Abstract:

In this dissertation a FIT-like discretisation of Maxwell's equations is performed directly in four-dimensional space-time using the mathematical formalism of Clifford's Geometric Algebra. The thesis extends the Finite Integration Technique (FIT) to 4D space-time without introducing any non-relativistic assumptions. The coordinate-free formulation in terms of geometric algebra enhances explicitly relativistic, i.e., without splitting space and time, treatment, which reveals in the fact that any non-relativistic assumptions are not made. The relation of geometric algebra to the existing concepts from differential geometry in the language of differential forms is established in the context of electromagnetic field description. An alternative to the existing approaches formula for the discretisation of material laws on non-orthogonal mesh pairs is derived, investigated and applied. The developed theory is applied to obtain the condition for 3D problems when material matrices are diagonal, and due to quantitative nature of this condition a mesh optimisation procedure is proposed, as well as its limitations in 3D case, which do not occur in 2D, are derived. The other application is simulation of electromagnetic wave propagation in a rotating reference frame. Due to coordinate-free formalism and encoding the movement of the observer in 4D mesh's geometry, derivation of the numerical scheme for rotating observer's resembles the one for inertial (stationary) observers. In other words, relativistic coordinate-free treatment includes inertial and non-inertial observers as special cases, which do not need to be diversified. The comparison of the obtained numerical results with the ones known from literature is performed in order to validate the theoretical results.

Place of Publication: Darmstadt
Divisions: 18 Department of Electrical Engineering and Information Technology
18 Department of Electrical Engineering and Information Technology > Institute of Electromagnetic Field Theory (from 01.01.2019 renamed Institute for Accelerator Science and Electromagnetic Fields)
18 Department of Electrical Engineering and Information Technology > Institute of Electromagnetic Field Theory (from 01.01.2019 renamed Institute for Accelerator Science and Electromagnetic Fields) > Computational Engineering (from 01.01.2019 renamed Computational Electromagnetics)
18 Department of Electrical Engineering and Information Technology > Institute of Electromagnetic Field Theory (from 01.01.2019 renamed Institute for Accelerator Science and Electromagnetic Fields) > Finite Methods of Electrodynamics (until 31.12.2018)
Date Deposited: 11 Feb 2018 20:55
Official URL: http://tuprints.ulb.tu-darmstadt.de/7232
URN: urn:nbn:de:tuda-tuprints-72325
Referees: Schoeps, Prof. Dr. Sebastian and Kurz, Prof. Dr. Stefan
Refereed / Verteidigung / mdl. Prüfung: 19 January 2018
Alternative Abstract:
Alternative abstract Language

In dieser Dissertation wird mit Hilfe der formalen mathematischen Methoden von Cliffords geometrischer Algebra eine Diskretisierung der Maxwell Gleichungen in vier-dimensionaler Raumzeit vorgenommen, die der Finiten Integrationstechnik ähnelt. Die Erweiterung kommt ohne Hinzunahme von nicht-relativistischen Annahmen aus. Die koordinatenfreie Formulierung im Formalismus der geometrischen Algebra ermöglicht explizite relativistische Rechnungen. Das Verhältnis zwischen geometrischer Algebra und differentialgeometrischen Konzepten wird im Rahmen von Differentialformen und mit Blick auf die Beschreibung elektromagnetischer Felder, erläutert. Eine alternative Formulierung zur Diskretisierung der Materialgesetze auf nicht-orthogonalen Gittern wird eingeführt, analysiert und getestet. Die dadurch entwickelte Theorie zeigt für den wichtigen drei-dimensionalen Sonderfall, unter welchen Bedingungen die Diskretisierung auf diagonale Material-matrizen führt. Dies erlaubt eine Gitteroptimierungsstrategie und zeigt die Grenzen der Strategie für drei-dimensionale Probleme, die in zwei Dimensionen nicht auftreten. Mit der Simulation der Ausbreitung elektromagnetischer Wellen in einem rotierenden Referenzgebiet wird eine Anwendung diskutiert. Durch die koordinatenfreie Formulierung und die spezielle Behandlung der Bewegung des Beobachters in vier Dimensionen wird eine numerische Methode entwickelt, die analog zu Methoden mit stationären Beobachtern angewendet werden kann. Hierdurch wird deutlich, dass die relativis- tische, koordinatenfreie Beschreibung inertialer und nicht inertialer als Spezialfälle in der erarbeiteten Theorie enthalten sind. Die vorgestellten Methoden werden mit numerischen Methoden aus der Literatur verglichen und getestet.

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