FEM–CM as a hybrid approach for multiscale modeling and simulation of ferroelectric boundary value problems

Constitutive modeling of ferroelectrics is a challenging task, spanning physical processes on different scales from unit cell switching and domain wall motion to polycrystalline behavior. The condensed method (CM) is a semi-analytical approach, which has been efficiently applied to various problems in this context, ranging from self-heating and damage evolution to energy harvesting. Engineering applications, however, inevitably require the solution of arbitrary boundary value problems, including the complex multiphysical constitutive behavior, in order to analyze multifunctional devices with integrated ferroelectric components. The well-established finite element method (FEM) is commonly used for this purpose, allowing sufficient flexibility in model design to successfully handle most tasks. A restricting aspect, especially if many calculations are required within, e.g., an optimization process, is the computational cost which can be considerable if two or even more scales are involved. The FEM–CM approach, where a numerical discretization scheme for the macroscale is merged with a semi-analytical methodology targeting at material-related scales, proves to be very efficient in this respect.