Parameter identification for a model for multi-functional materials with hysteresis and thermodynamic compatibility

Multifunctional materials have tremendous potential for engineering applications as they are able to convert mechanical to electromagnetic energy and vice-versa. One of the features of this class of materials is that they show significant hysteresis, which needs to be modeled correctly in order to maximize their application potential. A method of modeling multifunctional materials that exhibit the phenomenon of hysteresis and is compatible with the laws of thermodynamics was developed recently. The model is based on the Preisach hysteresis operator and its storage function and may be interpreted as a two-input, two-output neural net with elementary hysteresis operators as the neurons. The difficulty is that the parameters in the model appear in a non-linear fashion, and there are several constraints that must be satisfied by the parameters for thermodynamic compatibility. In this article, we present a novel methodology that uses the rate-independent memory evolution properties of the Preisach operator to split the parameter estimation problem into three numerically well-conditioned, linear least squares problems with constraints. The alternative direction method of multipliers (ADMM) algorithm and accelerated proximal gradient method are used to compute the Preisach weights. Numerical results are presented over data collected from experiments on a Galfenol sample. We show that the model is able to fit not only experimental data for strain and magnetization over a wide range of magnetic fields and stress but also able to predict the response for stress and magnetic fields not used in the parameter estimation.