Electromagnetics - Modelling, Simulation, Control and Industrial Applications - Abstract
von Harrach, Bastian
Transient excitation currents generate electromagnetic fields which, in turn, induce electric currents in proximal conductors. For slowly varying fields, this can be described by the eddy current equations, which are obtained by neglecting the dielectric displacement currents in Maxwell's equations: [ partial_t (sigma E) curl (1/mu curl E)=-partial_t J ] 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). In this talk, we derive a new unified variational formulation for the parabolic-elliptic eddy current equation, that is uniformly coercive with respect to the conductivity. This allows us to study the case when the conductivity approaches zero, and rigorously linearize the eddy current equations around a non-conducting domain with respect to the introduction of a conducting object. In other words, we characterize how the solution of an elliptic equation changes if the equation becomes a little bit parabolic. We then turn to the inverse problem of detecting a conducting object (i.e., the support of $sigma$) from eddy current measurements, and show how this problem can be attacked using the Linear Sampling and the Factorization Method.