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A decoupled finite element algorithm is developed for simulating the vortex dynamics on an elastic superconductor which couples the time-dependent Ginzburg-Landau equation with the complex-valued superconducting order parameter and the vector-valued magnetic potential, and the elasticity equation. We present an iterative algorithm for the decoupled system arising from the time and spatial discretization using a combination of preconditioner, algebraic multigrid method (AMG) and preconditioned conjugate gradient method (PCG). The iterative algorithm allows us to perform large-scale three-dimensional simulations of mesoscale pattern formation during superconducting phase transitions with arbitrary elastic boundary conditions. The performance and efficiency of the algorithm are numerically verified by several benchmark problems, exhibiting up to two orders of magnitude improvement depending on the scale of discrete system compared to the exact solver. 

ABSTRACT Ferroelectric domain variants that are energetically equivalent are expected to remain preserved during polarization reversal under a symmetry‐preserving electric field. However, recent experiments on relaxor–ferroelectric crystals have revealed irreversible elimination of inclined domain walls during AC poling, while the underlying mesoscale mechanism remains unclear. Here, we investigate domain‐wall motion during AC poling of rhombohedral Pb()– single crystals containing 71 and 109 domain walls within a quasi‐two‐dimensional laminated geometry using phase‐field simulations. Simulations reveal that domain‐wall behavior during polarization reversal depends on the spacing ratio between 71 and 109 domain walls. Closely spaced 71 domain walls undergo irreversible elimination, whereas widely separated walls are preserved. A threshold ratio for domain‐wall elimination is identified and found to depend on mechanical boundary conditions. By tracking domain‐wall trajectories, we attribute this behavior to unsynchronized motion of neighboring 71 domain walls arising from long‐range elastic interactions when walls become strongly coupled. This collective motion breaks the symmetry between domain variants and leads to irreversible domain‐wall elimination. These findings provide mechanistic insight into collective domain‐wall evolution during polarization reversal and suggest that proximity‐driven symmetry breaking may provide a mesoscale mechanism for domain engineering in ferroelectrics with high domain‐wall densities. 

Abstract Knowledge of the thermodynamic equilibria and domain structures of ferroelectrics is critical to establishing their structure–property relationships that underpin their applications from piezoelectric devices to nonlinear optics. Here, we establish the strain condition for strain phase separation and polydomain formation and analytically predict the corresponding domain volume fractions and wall orientations of, relatively low symmetry and theoretically more challenging, monoclinic ferroelectric thin films by integrating thermodynamics of ferroelectrics, strain phase equilibria theory, microelasticity, and phase‐field method. Using monoclinic K_xNa_1 − xNbO₃(0.5 < x < 1.0) thin films as a model system, we establish the polydomain strain–strain phase diagrams, from which we identify two types of monoclinic polydomain structures. The analytically predicted strain conditions of formation, domain volume fractions, and domain wall orientations for the two polydomain structures are consistent with phase‐field simulations and in good agreement with experimental results in the literature. The present study demonstrates a general, powerful analytical theoretical framework to predict the strain phase equilibria and domain wall orientations of polydomain structures applicable to both high‐ and low‐symmetry ferroelectrics and provide fundamental insights into the equilibrium domain structures of ferroelectric K_xNa_1 − xNbO₃thin films that are of technology relevance for lead‐free dielectric and piezoelectric applications.