Space-Time Discretization of Maxwell's Equations in the Setting of Geometric Algebra Article Swipe
Mariusz Klimek
,
Ulrich Römer
,
Sebastian Schöps
,
Thomas Weiland
·
YOU?
·
· 2018
· Open Access
·
· DOI: https://doi.org/10.26083/tuprints-00007232
YOU?
·
· 2018
· Open Access
·
· DOI: https://doi.org/10.26083/tuprints-00007232
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.
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- Type
- article
- Language
- en
- Landing Page
- http://tuprints.ulb.tu-darmstadt.de/view/person/Klimek=3AMariusz=3A=3A.html>
- https://tuprints.ulb.tu-darmstadt.de/7232/13/TUDthesisFromMath.pdf
- OA Status
- green
- Cited By
- 2
- References
- 7
- Related Works
- 20
- OpenAlex ID
- https://openalex.org/W1525798650
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https://openalex.org/W1525798650Canonical identifier for this work in OpenAlex
- DOI
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https://doi.org/10.26083/tuprints-00007232Digital Object Identifier
- Title
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Space-Time Discretization of Maxwell's Equations in the Setting of Geometric AlgebraWork title
- Type
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articleOpenAlex work type
- Language
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enPrimary language
- Publication year
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2018Year of publication
- Publication date
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2018-01-01Full publication date if available
- Authors
-
Mariusz Klimek, Ulrich Römer, Sebastian Schöps, Thomas WeilandList of authors in order
- Landing page
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https://tuprints.ulb.tu-darmstadt.de/view/person/Klimek=3AMariusz=3A=3A.html>Publisher landing page
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https://tuprints.ulb.tu-darmstadt.de/7232/13/TUDthesisFromMath.pdfDirect link to full text PDF
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YesWhether a free full text is available
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greenOpen access status per OpenAlex
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https://tuprints.ulb.tu-darmstadt.de/7232/13/TUDthesisFromMath.pdfDirect OA link when available
- Concepts
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Minkowski space, Discretization, Maxwell's equations, Geometric algebra, Algebra over a field, Mathematics, Cartesian coordinate system, Space (punctuation), Applied mathematics, Mathematical analysis, Pure mathematics, Computer science, Mathematical physics, Geometry, Algebra representation, Operating systemTop concepts (fields/topics) attached by OpenAlex
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2Total citation count in OpenAlex
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2016: 2Per-year citation counts (last 5 years)
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20Other works algorithmically related by OpenAlex
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| publication_date | 2018-01-01 |
| publication_year | 2018 |
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