Perfectly matched multiscale simulations for discrete lattice systems: Extension to multiple dimensions

Abstract

Extending the technique of the perfectly matched layer (PML) to discrete lattice systems, a multiscale method was proposed by To and Li [Phys. Rev. B 72, 035414 (2005)], which was termed the perfectly matched multiscale simulation (PMMS). In this paper, we shall revise the proposed PMMS formulation, and extend it to multiple dimensions. It is shown in numerical simulations that the perfectly matched layer between the fine scale region and the coarse scale region can provide an efficient remedy to reduce spurious phonon reflections. We have found (i) the bridging projection operator stems from minimization of the temperature of an equilibrium system; (ii) for discrete lattice systems, the perfectly matched layer (PML) can be constructed by stretching the lattice constant, or the equilibrium atomic spacing, in the Fourier domain; (iii) the dispersive relation in the PML zone is significantly different from the one in the original lattice system, and the PML usually behaves like a low-frequency pass filter. This may be one of the mechanisms to eliminate the reflective waves at the multiscale interface. Moreover, we apply the multidimensional PMMS algorithm to simulate a screw dislocation passing through different scales.