TU CE GSC CE People at the Graduate School CE Alumni Mariusz Klimek Research Project of Mariusz Klimek

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

Considering abstract formulations is not common. However, this may lead, after some theoretical derivations, to more efficient and reliable simulation techniques which are important in any industrial application. Moreover, theoretical descriptions of recent devices, e.g., global positioning system, Mössbauer spectroscope or optical gyroscope, require considering relativistic effects.

The approach presented here is an attempt to extend applicability of computational engineering to this area.

Clifford's Geometric Algebra (GA) in the context of discretization of Maxwell's equations is closely related to the Cell Method and the Finite Integration Technique (FIT).

However, it differs from FIT as GA enables to perform discretization directly in space-time. Compared to the exterior algebra of differential forms GA gives more algebraic tools at the cost of explicit introduction of the metric.

1. Mesh Optimization with Respect to Anisotropic Materials

GA's clear geometric interpretation was used to derive a goal function, whose minimization results in Hodge-optimized material matrices being diagonal or diagonal-dominant. Effectively it is an optimization of the primal/dual mesh pair of a finite difference based discretization scheme taking into account the material properties. As a research example a standing wave in 2D cavity filled with an anisotropic material was investigated. Convergence of the scheme for various choices of mesh pairs was discussed. The limitations of the method in the 3D case were investigated.

The classical orthogonal FIT mesh pair is a proper choice for scalar and diagonal material tensors. However, in the case of deformed primal grids, the orthogonal dual results in diagonal material matrices only for scalar material coefficients. When electric permittivity is a tensor and non-diagonal entries in material matrix are disregarded, the convergence is lost. However, adapting the dual mesh according to our criterion fixes the problem and allows to treat arbitrary material tensor consistently. In 3D setting we have shown that the permittivity tensor being proportional to the permeability tensor implies existence of the mesh pair guaranteeing diagonal material matrices. However, proportionality of material tensors is very restrictive for physical applications.

2. Description in Non-Inertial Reference Frames

GA provides an elegant formulation of Maxwell's equations in the space-time setting. Especially the integral form in space-time allows to directly discretize the equations on a given 4D space-time mesh pair. Therefore, given a mesh describing the motion of the system one can create a numerical scheme to simulate electromagnetic fields in the reference frame in which the system is at rest.

Computational Electromagnetism

Prof. Dr.-Ing. Thomas Weiland (TEMF, Computational Electromagnetic Laboratory)

Mariusz Klimek

M.Sc.

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