The homogenized energy model for characterizing polarization and strains in hysteretic ferroelectric materials: Implementation algorithms and data-driven parameter estimation techniques

Abstract

Ferroelectric materials, such as lead zirconate titanate, lanthanum-doped lead zirconate titanate, lead manganese niobate, and BaTiO3, provide unique actuator and sensor capabilities for applications including nanopositioning, high-speed valves and fuel injectors, camera focusing and shutter mechanisms, ultrasonic devices for biomedical imaging and treatment, and energy harvesting devices. However, to achieve the full potential of the materials, it is necessary to develop and employ models that quantify the creep, rate-dependent hysteresis, and constitutive nonlinearities that are intrinsic to the materials due to their domain structure. The success of models requires that they be highly efficient to implement since real-time applications can require kilo hertz to mega hertz rates. The calibration of models for specific materials, devices, and applications requires efficient and robust parameter estimation algorithms. Finally, control designs can be facilitated by models that admit efficient and robust approximate inversion. The homogenized energy model is a multiscale, micromechanical framework that quantifies a range of hysteretic phenomena intrinsic to ferroelectric, ferromagnetic, and ferroelastic materials. In this article, we present highly efficient implementation and parameter estimation algorithms for the ferroelectric model. This includes techniques to construct analytic Jacobians and data-driven algorithms to determine initial parameter estimates to facilitate subsequent optimization. The efficiency of these algorithms facilitates material and device characterization and provides the basis for constructing efficient and robust inverse algorithms for model-based control design. The model implementation, calibration, and validation are illustrated using rate-dependent lead zirconate titanate data and single-crystal BaTiO3 data.

A modeling and uncertainty quantification framework for a flexible structure with macrofiber composite actuators operating in hysteretic regimes

“…There are a wide range of models quantifying the hysteretic and nonlinear behavior of piezoelectric materials, and we refer readers to Hu et al (2012) and Smith and Hu (2012) for details about existing models. In the context of MFC, previous models that are geared toward structures, dynamics, and control applications have focused on three approaches: Preisach representations, Prandtl-Ishlinskii (PI) models, or the HEM in combination with classical beam or plate theory.…”

“…As detailed by Hu et al (2012) and Smith and Hu (2012), the HEM is a multiscale, micromechanical approach that begins with an energy formulation at the single crystal or domain length scale. Switching processes due to domain wall nucleation and movement are incorporated by tracking the evolution of dipole fractions x a , which serve as internal variables.…”

“…A critical step for model calibration concerns the estimation of model parameters in a manner that is efficient and robust and facilitates statistical quantification of parameter uncertainty. Data-driven deterministic parameter estimation techniques for PZT materials, reported by Hu et al (2012), form the basis for the deterministic least squares techniques employed here. However, we extend that framework significantly by employing adaptive Markov Chain Monte Carlo (MCMC) techniques to construct densities quantifying parameter uncertainty.…”

“…As detailed in Hu et al (2012), the construction of the densities n I (E I ) and n c (F c ), employed in the hysteresis model equation ( 30), requires the determination of the parameters…”

“…In this article, the homogenized energy model (HEM) is employed to quantify this nonlinear and hysteretic behavior. It has been shown in Hu et al (2012) and Smith and Hu (2012) that the HEM accurately characterizes the creep and rate-dependent hysteresis inherent to PZT. A Bayesian framework that facilitates the quantification of parameter uncertainty is then presented.…”