Variational boundary conditions for molecular dynamics simulations: Treatment of the loading condition

This paper aims to extend the variational boundary conditions for molecular dynamics simulation [X. Li, W. E, Variational boundary conditions for molecular dynamics simulations of solids at low temperature, Commun. Comp. Phys. 1 (2006) 136-176; X. Li, W. E, Boundary conditions for molecular dynamics simulations at finite temperature: treatment of the heat bath, Phys. Rev. B 76 (2007) 104107], to take into account external loading conditions. Two derivations of the exact boundary conditions are presented, one with Mori-Zwanzig projection procedure, and the other using lattice Green's functions. Approximate boundary conditions, which are more efficient in practice, are then discussed. Finally several numerical experiments are presented to demonstrate the effectiveness of these methods.