Numerical approximation of the square phase-field crystal dynamics on the three-dimensional objects

In the natural world, the phase field transition on the surface of a three-dimensional (3D) object is common. The square phase-field crystal (SPFC) equation is an effective model for describing the phase transition from liquid to solid with lattice structure. This study aims to present an efficient and practical computational scheme to simulate the SPFC dynamics on arbitrary surfaces. The surface is represented by the zero level-set of a singed distance function. The closest point method and appropriate pseudo-Neumann boundary condition are used to make the numerical solution to be constant along the normal direction to the arbitrary surface. In this sense, the calculations are transformed into a 3D discrete narrow computational band domain embedding the arbitrary surface and the finite difference method (FDM) is adopted to discretize the governing equation in 3D space. To maintain second-order spatial accuracy, the standard seven-point discrete Laplacian operator is applied to replace the discrete surface Laplace–Beltrami operator. We treat the nonlinear term in a totally explicit form and utilize the stabilization technique to suppress the stiffness of the numerical scheme. Therefore, we avoid solving any nonlinear equation or linear equation with variable coefficients. In each time step, we only need to update the linear system with constant coefficients. The proposed numerical scheme is highly efficient and its implementation is simple. Various numerical experiments demonstrate that the proposed numerical scheme not only achieves the expected accuracy but also works well for the simulations of phase transition and crystallization on surfaces with smooth and sharp shapes.