Sensitivity analysis of an eddy current problem related to induction heating

We study a mathematical model for the inductive heating of steel.It consists of a vector potential formulation of Maxwells equations coupled with a heat equation and an evolution equation for the volume fraction of high temperature phase in steel. An important task for practical applications of induction heating it to find the optimal coupling distance between inductor and workpiece. To this end, we employ the speed method to investigate the sensitivity of solutions to the state equations with respect to perturbations of the inductor coil. We show the existence of strong material derivatives for the state variables and apply the structure theorem to characterize the Eulerian derivative of the cost functional