An efficient space-time phase field discretization for ferroelectrics

Recent developments of phase field model based on the Ginzburg-Landau theory have provided an unprecedented look at the formation of polarization domain structures and rich phenomena of polarization behaviors in nanoscale ferroelectrics under electrical and mechanical multi-fields. However, the phase field simulations are often computationally expensive. One of the major reasons behind this inefficiency is due to the complex spatio-temporal effects on the dynamical behavior of polarization. In this work, an efficient scheme with error control and adaptive time-stepping is introduced to the phase field model in the context of Ginzburg-Landau theory. The proposed time adaptivity algorithm is based on the discrete maximum norm of the difference in numerical solutions at three consecutive time steps. In addition, the energy stability of the proposed scheme is demonstrated. Several benchmarks of convergence tests are presented to validate the model. The performance of proposed technique is illustrated through numerical examples involving behaviors of polarization in complex ferroelectric nanostructures in three dimensions.