Nonlocal operator method for the Cahn-Hilliard phase field model

Abstract

In this paper we propose a Nonlocal Operator Method (NOM) for the solution of the Cahn-Hilliard (CH) equation exploiting the higher order continuity of the NOM. The method is derived based on the method of weighted residuals and implemented in 2D and 3D. Periodic boundary conditions and solid-wall boundary conditions are considered. For these boundary conditions, the highest order in the NOM scheme is 2 and 3, respectively. The proposed NOM makes use of variable support domains allowing for adaptive refinement. The generalized α-method is employed for time integration and the Newton-Raphson method to iterate nonlinearity. The performance of the proposed method is demonstrated for several two and three dimensional benchmark problems. We also implemented a CH equation with 6th order partial differential derivative and studied the influence of higher order coefficients on the pattern evolution of the phase field.