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Abstract:
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Boundary value problems (BVPs) are fundamental in electromagnetic engineering. The aim of this thesis is to introduce mathematical structures that can be exploited in a new way to formulate electromagnetic BVPs. The tools employed come from differential geometry and the theory of manifolds. The structures offer a way to model electromagnetism in a coordinate-free manner, which is independent of the chosen metric. Differentiable manifolds and differential forms are used as models for space and electromagnetic fields, respectively. Together with the pullback, exterior derivative, and wedge product, they can be employed to introduce a formulation of electromagnetism that is invariant under diffeomorphisms.
Differential geometry enables us to formulate general electromagnetic BVPs, including static, initial value, and Cauchy problems, in a unified setting. Furthermore, under diffeomorphisms, equivalence of BVPs arises naturally and provides a unified theoretical setting for many traditional, seemingly different methods and approaches. Because of the diffeomorphism-invariance, in formulations of electromagnetic BVPs the metric of space is needed only to make the first connection between the model and the observations. The thesis introduces also (3 + 1)-decompositions of Maxwell’s equations based on coordinate- and metric-free observer fields. A major results of this thesis is this unified aspect to BVPs and its applications to solution methods.
The structures used are also generic to all dimensions, which makes them natural tools to formulate electromagnetic BVPs of any dimension. In particular, another main result of this thesis is a symmetry-based theory of dimensional reduction of electromagnetic BVPs. It includes a dimensional reduction theorem that gives sufficient conditions for a BVP to be solved as a lower-dimensional BVP and also formulates the lower-dimensional BVP. Because the theory is completely independent of coordinates, metric, and dimension, differential geometric structures are virtually custom-made for it.
The thesis presents several applications and numerical examples, in which the structures offer new insight and benefits. These applications and examples include mesh generation problems, speeding up parametric models that include shape optimization and movement, open-boundary problems, invisibility cloaking, and dimensional reduction of helicoidal geometries. |