# A Shape Optimization Method for Magnetic Induction Tomography

Antoine Laurain^{*}, Michael Hintermueller and Irwin Yousept

Magnetic induction tomography (MIT) is a novel technique for non-contacting measurement of electric properties of conducting materials. Potential applications of MIT can be found in medicine such as in brain imaging. The technology of MIT involves an oscillating magnetic field generated by a transmitter coil, which in turn induces an eddy current inside the conducting materials. Then, the magnetic field arising from the eddy current is detected by a receiver coil, which provides measurements of the corresponding electric conductivity. The use of eddy current model in MIT is justified by the small wavelength for its operating frequencies. Typically, the frequencies lie between 10 and 100 MHz, i.e. in the range of some micrometers, so that the wavelength is small compared to the size of the conductor. \newline We study the inverse problem of reconstructing the electric conductivity from measurements in the receiver coil, for an MIT system governed by time-harmonic eddy current equations. The electric conductivity is assumed to be piecewise constant, so that the problem may be recast as a shape optimization problem, where the unknown is the shape of the inclusion whose conductivity is different from the the background. The evolution of the shape during the optimization process is achieved using a level set method.

Mathematics Subject Classification: 49N45 49Q10

Keywords: inverse problem; shape optimization; magnetic induction tomography

Minisymposion: Optimization of Mechanical Systems