A Qualitative-analysis of the Behavior of the Galerkin Equations Relevant To Nonlinear Eddy-current Problems

The Galerkin equations relevant to the eddy currents induced in an iron body are considered. These equations are obtained by formulating the field problem in terms of the magnetic vector potential and by applying the Galerkin method. They are shown to have a unique steady-state solution if a certain condition on the magnetic constituitive relationship is satisfied. In particular a T-periodic source gives rise to a unique T-periodic solution to which all other solutions converge asymptotically independently from the initial conditions. Under the same condition the exponential decay of the 'transients' is shown, and an explicit lower and upper bound for its rate is given. These structural properties allow us to excluse a priori that qualitatively different asymptotic behaviours, including even chaotic solutions, may occur. Numerical simulation, when based on qualitiative information of this type, enables us to obtain the quantitative properties in an efficient manner. In order to demonstrate the practical use of these results some numerical experiments are presented.