A Green's function approach to deriving non‐reflecting boundary conditions in molecular dynamics simulations

Abstract

SUMMARYComputer simulations of atomic scale processes in solids are often associated with the issue of spurious reflection of elastic waves at the boundaries of a molecular dynamics domain. In this paper, we propose an approach to emulate non-reflecting boundary conditions in atomistic simulations of crystalline solids. Harmonic response of the outer, non-simulated, region is accurately represented by a memory function, related to the lattice dynamics Green's function. The outward wave flow is cancelled due to work done by the corresponding response forces. Performance of method, dependent on a series of method parameters, is illustrated on a benchmark problem.

A mesoscopic bridging scale method for fluids and coupling dissipative particle dynamics with continuum finite element method

“…Differential equations of motion have the form (31), where the interaction force f ij (see Eq. (32)) is (38) where and f̃i j are the conservative (repulsive), viscous and random forces, respectively. These forces can be expressed as follows (see, for example, [3,10,12,21]): (39) (40) (41) where a is a material coefficient, r i and r j are the position vectors of particles "i" and "j", r ij = r i − r j , r ij = ||r i − r j ||, and ; r max is the radius of interaction domain between particles; v ij = v i − v j is the relative velocity; γ is the friction coefficient or normal damping coefficient; w γ and w̃ are the weight functions for viscous and random forces, with the relation w γ = w̃2 [8,21]; σ is the random force amplitude (for unit mass) σ = (2γk B T ) 1/2 [8,21]; ζ ij is a random number with zero mean and unit variance (chosen independently for each pair of DPD particles and time step); and Δt is a time step size used in the integration of differential equations of motion.…”

“…We implement boundary conditions, such as, for example, no-slip, Maxwellian or specular boundary conditions at the walls, or periodicity conditions at surfaces where particles are entering or leaving the flow domain [50][51][52][53][54][55]. Similar boundary conditions have been introduced in the MD, MD-FE multiscale modeling [26,38,56,57]. Boundary conditions at the common FE-DPD boundary are described in Section 3.4.…”

“…From this form of kinetic energy follow differential equations of motion in both scales which are coupled through the force terms only. Boundary conditions between the MD and FE domains are handled by a history kernel matrix implemented using analytical or numerical procedures [29,34,38].…”

Atom-to-continuum methods for gaining a fundamental understanding of fracture.

“…The THK is the optimal solution for propagation in linear media and, with perfect precision arithmetic and an arbitrarily long history, it is reflectionless. For more complex, three-dimensional (3D) systems the kernel can be determined numerically [22] or analytically [91], although evaluation of the analytical expressions for typical 3D systems remains a challenge. In all cases, the kernel unfortunately has a very slow and oscillatory decay in time and therefore is very expensive computationally.…”

A phonon heat bath approach for the atomistic and multiscale simulation of solids

“…strongly coupled lattice structures, are the following: (1) energy dissipation is modelled with the viscous friction model, where the frictional force is proportional to the instantaneous atomic velocities; however, energy dissipation in lattice structures is determined by a time history of the atomic motion [5][6][7][8][9]; (2) initial conditions are sampled as independent random quantities, though there is a statistical correlation in the motion of adjacent lattice atoms.…”