Pinning boundary conditions for phase-field models

In this paper, we present pinning boundary conditions for two- (2D) and three-dimensional (3D) phase-field models. For the 2D and axisymmetric domains in the neighborhood of the pinning boundaries, we apply an odd-function-type treatment and use a local gradient of the phase-field for points away from the pinning boundaries. For the 3D domain, we propose a simple treatment that fixes the values on the ghost grid points beyond the discrete computational domain. As examples of the phase-field models, we consider the Allen–Cahn and conservative Allen–Cahn equations with the pinning boundary conditions. We present various numerical experiments to demonstrate the performance of the proposed pinning boundary treatment. The computational results confirm the efficiency of the proposed method.