J. Marian, G. Venturini, B. L. Hansen, J. Knap, M. Ortiz and G. H. Campbell
Modelling and Simulation in Materials Science and Engineering, 18, 015003 (2010).
The concurrent bridging of molecular dynamics and continuum thermodynamics presents a number of challenges, mostly associated with energy transmission and changes in the constitutive description of a material across domain boundaries. In this paper, we propose a framework for simulating coarse dynamic systems in the canonical ensemble using the quasicontinuum method (QC). The equations of motion are expressed in reduced QC coordinates and are strictly derived from dissipative Lagrangian mechanics. The derivation naturally leads to a classical Langevin implementation where the timescale is governed by vibrations emanating from the finest length scale occurring in the computational cell. The equations of motion are integrated explicitly via Newmark's (beta = 0; gamma = 1/2) method, which is parametrized to ensure overdamped dynamics. In this fashion, spurious heating due to reflected vibrations is suppressed, leading to stable canonical trajectories. To estimate the errors introduced by the QC reduction in the resulting dynamics, we have quantified the vibrational entropy losses in Al uniform meshes by calculating the thermal expansion coefficient for a number of conditions. We find that the entropic depletion introduced by coarsening varies linearly with the element size and is independent of the nodal cluster diameter. We rationalize the results in terms of the system, mesh and cluster sizes within the framework of the quasiharmonic approximation. The limitations of the method and alternatives to mitigate the errors introduced by coarsening are discussed. This work represents the first of a series of studies aimed at developing a fully non-equilibrium finite-temperature extension of QC.