Mathematische Theorie direkter und inverser transienter Wirbelstromprobleme

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Transient (e.g. pulsed) excitation currents generate electromagnetic fields which in turn induce electric currents in proximal conductors. Mathematically, this can be described by partial differential equations, the eddy-current equations, which are obtained by neglecting the dielectric displacement currents in Maxwell’s equations. The eddy-current equations are of parabolic-elliptic type: In insulating regions, the field instantaneously adapts to the excitation (quasistationary elliptic behaviour), while in conducting regions, this adaptation takes some time due to the induced eddy currents (parabolic behaviour). Eddy current effects are used for remotely detecting conducting objects (e.g. in the context of land mine detection) and to non-invasively identify flaws inside a conductor (so-called eddy-current testing). In mathematical terms this leads to the inverse problem of reconstructing the conductivity coefficient in the eddy current equations from (partial) knowledge of the solution(s). In the proposed project we aim to utilize a unified variational theory for the parabolic-elliptic equations to theoretically study identifiability questions in the inverse problem and derive rigorously justified reconstruction strategies.