Peridynamics as an Upscaling of Molecular Dynamics

Abstract

Abstract. Peridynamics is a formulation of continuum mechanics based on integral equations. It is a nonlocal model, accounting for the effects of long-range forces. Correspondingly, classical molecular dynamics is also a nonlocal model. Peridynamics and molecular dynamics have similar discrete computational structures, as peridynamics computes the force on a particle by summing the forces from surrounding particles, similarly to molecular dynamics. We demonstrate that the peridynamics model can be cast as an upscaling of molecular dynamics. Specifically, we address the extent to which the solutions of molecular dynamics simulations can be recovered by peridynamics. An analytical comparison of equations of motion and dispersion relations for molecular dynamics and peridynamics is presented along with supporting computational results.

Peridynamic State-Based Models and the Embedded-Atom Model

“…We refer to these types of material models as structureless. In Section 3.1, we demonstrate how to upscale a structureless MD EAM to a NCM, following similar derivations for the continuous upscaling of a Lennard-Jones MD model presented in [7]. In Section 3.2, we present numerical results for one-dimensional wave propagation showing a speedup of 15× over MD.…”

“…2(a). Similar simulations for the continuous upscaling of nonlocal linear springs and Lennard-Jones MD models were presented in [7]. The reference atomistic system is chosen to have 1,001 atoms interacting through the MD EAM (2.3) with the functions (3.6)-(3.9).…”

“…One such approach, based on higher-order gradient continuum models, is given in [6]. A different approach, using nonlocal continuum models (NCMs) appears in [7]. These methods attempt to use continuum models for the continuous upscaling † of MD.…”

“…By replacing MD models with their corresponding upscaled continuum models, we can simulate length scales not tractable by MD alone. In [7], the continuous upscaling of pairwise MD models is derived within the peridynamics theory of solid mechanics [8,9], recovering characteristic dispersion relations of discrete models at the continuum level. That work is related to coarse-graining, but differs from the phase-space average approach presented in [10].…”

“…That work is related to coarse-graining, but differs from the phase-space average approach presented in [10]. In this paper, we extend the techniques of [7] to multibody potentials of the form of the MD EAM. We derive upscaled nonlocal continuum models inspired by peridynamics, and we demonstrate that all of the nonlocal continuum models we derive are instances of the peridynamics theory.…”