Time history kernel functions for square lattice

Abstract

We propose an exact non-reflecting boundary condition for simulations of a square lattice in finite atomic sub-domain. The boundary condition is based on the series expansion of the single source kernel functions. The time history kernel Laplace transform is first derived in a closed form, where the Fourier transform inversion is performed analytically. An efficient numerical procedure is also developed for the inverse Laplace transform. Differences in the finite boundary condition and the infinite boundary approximation elucidate the cause of the long-lasting difficulty of corner effects. Numerical tests verify accuracy of the proposed approach.

Accurate artificial boundary conditions for the semi-discretized linear Schrödinger and heat equations on rectangular domains

“…To deal with the Schrödinger and heat equations on unbounded domains, we propose first to introduce a second-order spatial discretization of the associated operator on a uniform rectangular grid. Inspired by our previous work in lattice dynamics [28], an efficient algorithm is designed to accurately compute the Green's function of the semi-discretized Schrödinger equation with a single-source. Based on this computation, we are able to design some accurate boundary conditions that eliminate the corner reflection when a rectangular boundary is considered.…”

Advanced Boundary Conditions for the Simulations of Laser Ablation

“…The supposedly most efficient technique for absorption of sound and shock waves in atomistic simulations is based on the so-called "time-history-kernel" method [12,13,7]. In this method the equations of motion of the atoms at the free surface are modified as if they were surrounded by neighboring atoms from all sides.…”

Matching boundary conditions for lattice dynamics

“…Corner reflections appear in most calculations and become prominent if the reflections at other artificial boundary atoms have been well reduced. A recent work made some progress under the framework of the time history kernel treatment .…”

“…It has been systematically studied and implemented by Liu et al to multiple dimensional lattices under the framework of the bridging scale method . This treatment involves a heavy numerical cost in time convolution and requires accurate calculations of the kernel functions, which is not easy in multiple dimensions . In addition, its nonlocality in time reduces the accuracy when it is applied to nonlinear lattices.…”