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

Ferroelectric materials, such as lead zirconate titanate, lanthanum-doped lead zirconate titanate, lead manganese niobate, and BaTiO ₃ , 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 BaTiO ₃ data.