An Optimal Parameters Determination for Ferroelectric’s Polarization Model
Authors: Scaliukh A. , Soloviev A. , Dmitrieva E. , Shevtsova M. 
Published: January 2011

In the presented article we propose a mathematical model for nonlinear response of the polycristalline ferroelectrics, an efficient numerical algorithm for its parameters identification, and finally we deal with the ways of their using in practice. Piezoelectrics and ferroelectrics constitute an important class of materials known owing theirs wide application as sensors and actuators in a large number of devices and components. The control, description, and understanding of piezoelectrics and ferroelectrics behavior present thus an important and difficult undertaking from both the practical and theoretical point of view. The piezoelectric hysteresis is an important property of piezoelectric and ferroelectric materials caused by different physical processes that take place in ferroelectric materials, e.g. domain-wall pinning, defect ordering, etc. Most often hysteresis is undesired in high-precision sensor, actuator and capacitor applications. But leaving out of framework the origin and mechanisms of the piezoelectric hysteresis the necessity of devices rational design forces to develop the specialized CAE systems to be able to simulate and optimize an efficiency of the ferroelectric-based devices, considering both useful and undesirable phenomena. So, finite-element modeling of sensor and actuator devices requires knowledge of the supplementary constitutive relations that are valid at broad range of electric fields, including the cases of irreversible polarization or depolarization process. There are several approaches to deriving the governing relationships, particularly, orientation Jiles–Atherton model, where governing relationships are formulated by increments between some intrinsic and target parameters. At low mathematical complexity this model is based on the transparent physical meanings, allowing to describe the work of electric field rotating the domains, and energy destructing the fixed domain walls. The proposed model represents a nonlinear ordinary differential equation relative to polarization and driving by electric field. Together with incremental theory this model allows to describe the real behavior of the physical object and to determine all needed field features at list for quasistatic process. As many models describing a nonlinear behavior of whole class of polycristalline materials our model depends on the five intrinsic parameters which have a different physical nature, and have influence on the nonlinear hysteretic response of material. Settings of model imply an unambiguous determination of such parameters. Identification of these parameters is a coefficient inverse problem, and for its resolving we have used the experimentally obtained hysteretic loops. At numerical implementation the set of these five parameters minimize a discrepancy functional square depended on experimentally observed and calculated points of hysteretic loop. Due to complexity of the minimized functional behavior on the space of identified parameters the minimization procedure was realized by means of Genetic Algorithm Toolbox MATLAB. The developed numerical method for hysteresis differential operator parameters identification has shown the good efficiency, robustness, and speedy convergence. These parameters then have used for static and modal analysis by finite element package ACELAN that utilizes an incremental theory for describing of irreversible polarization process. Finally we demonstrate some calculation results for non-uniformly polarized piezoceramic elements.