Department of Mathematics Colloquium

University of Idaho

Fall 2011
Thursday,  September 29, 3:30-4:20pm, room TLC 030

Refreshments in Brink 305 at 3:00 p.m.

Regularized deconvolution method for modeling
mesoscale continuum equations for particle systems


Lyudmyla Barannyk 

Department of Mathematics

University of Idaho


A general framework for derivation of continuum equations governing mesoscale dynamics of large particle systems is introduced.  The balance equations for spatial averages such as density, linear momentum, and energy were previously derived by a number of authors (Irving, Kirkwood, Hardy, Murdoch and Bedeaux). 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.  A new closure method for approximating fluxes in terms of other mesoscale averages is proposed. The main idea is to rewrite the nonlinear averages as linear convolutions that relate micro- and mesoscale 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 Hertz like 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.