Keywords: hysteresis, stability, finite element method.

Hysteretic nonlinearities appear in many fields of engineering from ferro-magnetics to two-phase flow models with phase transition. In order to solve a problem in which hysteresis arises, an adequate model of hysteresis is required. Since nowadays almost every complex problem is solved by the aid of numerical methods, which are mostly implemented in advanced numerical packages such as MatlabR, or COMSOLRit is advantageous to apply a model of hysteresis which can be smoothly integrated in this software.

The aim of the paper is the comprehensive examination of a powerful hysteresis model in which the hysteretic nonlinear behavior appears as a solution of an ordinary differential equation (ODE), and its implementations into field calculation problems in the finite element method (FEM) numerical scheme. The construction of the model is based on a statistical approach of quasi-random behaviour of clusters of molecules or domains [1], which can be represented by elementary hysteresis operators [2] with random switching field values. One of the main advantages of the model is that the numerical solution of the model equation can be acquired easily and the model implementation into field problems can be handled by means of coupling the "hysteresis" equation to the equations of the problem. On the other hand since the model has the form of an ODE, there are many advanced tools for the investigation of the structure and the solutions of the equation.

The approach to investigating the periodic solutions is based on representing the hysteresis operator by an autonomous system of ODEs, in this way a set of hysteresis loops correspond to a smooth trajectory in the phase-space of the system. It is shown that the model solutions are stable for any type of excitations. In order to examine the exact location of the stable periodic orbits in the phase-space, where transient accommodation ends up and the loop becomes stable, the method of Poincare-sections is applied [3].

The model has been implemented into a magnetic diffusion and a non-isothermal flow problem with phase transition and it proved to be numerically stable. The results suggest that introducing a hysteresis model of the vapor-liquid phase transition can improve other macroscale heat transfer models as well that are associated with boiling and condensation phenomena and are treated until now as virtually isothermal heat transfer processes.

The results of the model implementations are encouraging in a sense that hysteretic behaviour can be described effectively with the aid of continuous differential equations (or autonomous systems of ODEs) and their solutions, and this kind of representation can be interesting in many engineering fields (magnetic materials, two-phase flow, adsorption etc.) where hysteresis arises.

1

Z. Sari, A. Ivanyi, "Statictical Approach of Hysteresis", Phisica B 372, 45-48, 2006.

2

A. Visintin, "Differential Models of Hysteresis", Springer-Verlag, 1994.

3

S. Wiggins, "Introduction to Applied Nonlinear Dynamical Systems and Chaos", Springer, 2003.

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