Hybrid nonlinear preconditioning approach for phase-field model with double-obstacle potential

The phase-field (PF) model based on a double-obstacle (DO) potential has been extensively used to simulate free boundary problems in material science similarly to the model based on the double-well (DW) potential. The nonlinear preconditioning method, which transforms the PF variable into a linear interfacial distance function, enables stable computations even with coarse interface resolution in DW model. However, when applied to the DO model, this approach leads to numerical instability, rendering the simulations practically infeasible. To address the issue, we propose a novel hybrid nonlinear preconditioning approach for the PF model with the DO potential. In this approach, the nonlinear preconditioning DO model is applied inside the interface region, while the nonlinear preconditioning DW model is used in the outer region where the DO model becomes numerically unstable. We first derive a single-phase-field equation based on this approach and subsequently extend it to a multi-phase-field equation. The proposed method is validated through two-dimensional simulations of single-grain growth and the triple junction migration. Our results show that the proposed model provides improved numerical accuracy compared to the conventional DO model under low interface resolution conditions and effectively reduces the influence of grid anisotropy. These results demonstrate that the proposed strategy offers a computationally accurate and efficient approach for DO potential PF simulations under coarse grid resolution.