Deconvolution closure for mesoscopic continuum models of particle systems

TL;DR

This paper develops a closure method for mesoscopic continuum models of particle systems, enabling the approximation of fluxes through linear convolutions and regularization, validated on Fermi-Pasta-Ulam chains.

Contribution

It introduces a novel convolution-based closure approach for meso-scale continuum equations, avoiding the need to solve underlying ODEs.

Findings

Accurately approximates stress in particle systems

Validates method on Lennard-Jones and granular potentials

Shows good agreement with exact stress calculations

Abstract

The paper introduces a general framework for derivation of continuum equations governing meso-scale dynamics of large particle systems. The balance equations for spatial averages such as density, linear momentum, and energy were previously derived by a number of authors. These equations are not in closed form because the stress and the heat flux cannot be evaluated without the knowledge of particle positions and velocities. We propose a closure method for approximating fluxes in terms of other meso-scale averages. The main idea is to rewrite the non-linear averages as linear convolutions that relate micro- and meso-scale dynamical functions. The convolutions can be approximately inverted using regularization methods developed for solving ill-posed problems. This yields closed form constitutive equations that can be evaluated without solving the underlying ODEs. We test the method numerically on Fermi-Pasta-Ulam chains with two different potentials: the classical Lennard-Jones, and the purely repulsive potential used in granular materials modeling. The initial conditions incorporate velocity fluctuations on scales that are smaller than the size of the averaging window. The results show very good agreement between the exact stress and its closed form approximation.