Mathematical analysis and numerical solution of models with dynamic Preisach hysteresis

Abstract: Hysteresis is a phenomenon that is observed in a great variety of physical systems, which leads to a nonlinear and multivalued behavior, making their modeling and control difficult. Even though the analysis and mathematical properties of classical or rate-independent hysteresis models are known, this is not the case for dynamic models where current approaches lack a proper functional analytic framework which is essential to formulate optimization problems and develop stable numerics, both being crucial in practice. This paper deals with the description and mathematical analysis of the dynamic Preisach hysteresis model. Toward that end, we complete a widely accepted definition of the dynamic model commonly used to describe the constitutive relation between the magnetic field H and the magnetic induction B, in which, the values of B not only depends on the present values of H but also on the past history and its velocity. We first analyze mathematically some important properties of the model and compare them with known results for the static Preisach model. Then, we consider a parabolic problem with dynamic hysteresis motivated by electromagnetic field equations. Under suitable assumptions, we show the well posedness of a weak formulation of the problem and solve it numerically. Finally, we report a numerical test in order to assess the order of convergence and to illustrate the behavior of the numerical solution for different configurations of the dynamic Preisach model