A variational approach to coarse graining of equilibrium and non-equilibrium atomistic description at finite temperature
The aim of this paper is the development of equilibrium and non-equilibrium extensions of the quasicontinuum (QC) method. We first use variational mean-field theory and the maximum-entropy (max-ent) formalism for deriving approximate probability distribution and partition functions for the system. The resulting probability distribution depends locally on atomic temperatures defined for every atom and the corresponding thermodynamic potentials are explicit and local in nature. The method requires an interatomic potential as the sole empirical input. Numerical validation is performed by simulating thermal equilibrium properties of selected materials using the Lennard–Jones (LJ) pair potential and the embedded-atom method (EAM) potential and comparing with molecular dynamics results as well as experimental data. The max-ent variational approach is then taken as a basis for developing a three-dimensional non-equilibrium finite-temperature extension of the QC method. This extension is accomplished by coupling the local temperature-dependent free energy furnished by the max-ent approximation scheme to the heat equation in a joint thermo-mechanical variational setting. Results for finite-temperature nanoindentation tests demonstrate the ability of the method to capture non-equilibrium transport properties and differentiate between slow and fast indentation.
© 2007 Elsevier. Received 23 April 2007, Revised 16 August 2007, Accepted 6 September 2007, Available online 19 September 2007. We gratefully acknowledge the support of the Department of Energy through Caltech's ASCI ASAP Center for the Simulation of the Dynamic Response of Materials, the support received from the National Science Foundation through an ITR grant on Multiscale Modeling and Simulation and Caltech's Center for Integrative Multiscale Modeling and Simulation, and the support of the Army Research Laboratory through the CHSSI II Project on Scalable Quasi-Continuum Software for Advanced Mixed Atomistic-Continuum Simulation of Material Behavior at the Nanoscale. J.K. gratefully acknowledges the support of the US Department of Energy through the Lawrence Livermore National Laboratory under Contract W-7405-Eng-48.