The problem is set up as follows: One starts with an atomistic system with realistic size. Then a subdomain is selected as the actual computational domain. This is typically around critical areas where the deformation is large due to the presence of local lattice defects. Such truncation creates an artificial boundary, where boundary conditions has to be imposed, to model implicitly the effect of the surrounding atoms.

The exact form of the boundary condition can be derived under the assumption that the surrounding atoms are initially at mechanical or thermal equilibrium. The derivation employs a projection formalism, first developed by Mori (1965) and Zwanzig (1973) for studying nonequilibrium processes. The exact boundary condition is expressed as a generalized Langevin equation governing the motion of the atoms at the boundaries. The distinct features include the history dependence term, as a result of the projection, and the random noise introducing phonons from the surrounding environment. It satisfies the second fluctuation-dissipation theorem (Kubo 1966) .

• Molecular statics models.

The full model minimizes the total energy of the entire system. The boundary condition is found by solving an exterior problem, and it is in the form that is similar to Dirichlet to Neumann map.

• Molecular dynamics models at zero temperature.

The purpose is to absorb phonon generated by the local defects, and prevent boundary reflections.

• Molecular dynamics models at finite temperature.

In this case, the surrounding system is initially at thermal equilibrium. This gives rise to a random force in the boundary condition, which has to be sampled in the computation.

• Molecular dynamics model with remote loading condition.

Loading conditions at the remote boundary can be added by solving the elastodynamics model in the surrounding region.

• The calculation of the boundary kernels.

Although the exact boundary kernels can be expressed analytically, the numerical computation is quite involved, especially because the surrounding region is too large. The main idea is to use phonon normal mode, or lattice Green's functions as test functions to obtain the kernels, both in the statics and dynamics cases.

• Efficient representation of the kernel.

Another difficulty in the implementation of the boundary condition is due to the nonlocal nature of the memory functions: They are nonlocal in both space and time, making the computation extremely expensive. Our approach aims to find local representations without compromising the accuracy.

• The second fluctuation-dissipation thorem.

This is a necessary condition for the system to approach to the desired distribution.

• Stability of the boundary condition.

This is particularly important in this context because the system is usually integrated for a long time period.

The main purpose of this project is to develop a coupled atomsitic and continuum models, which is not limited by the temporal and length scales of the atomistic model, or the inaccuracy of the continuum models.

The main component of the multiscale method is a coupling condition at the interface of the two models. The starting point is the observation that both the continuum and atomistic models can be recast into the form of conservation laws. As a result, the coupling condition can be accomplished at the level of fluxes. Such a formulation also allows us to use existing numerical methods for solving conservation laws to better capture elastic waves with sharp front.

The method has been applied to brittle crack problems to study the impact of strong shocks.

In this project, we consider the following problem: Given a subset of atomic degrees of freedom of full MD, called coarse-grain (CG) variables, can we derive a set of equations that only involve the CG variables from the full MD model? Such question is of considerable theoretical and practical interest. This project aims to reduce the dimension of the problem and derive coarse scale models. The models will have a lot of applications.

In (Li 2009), we have proposed a projection method to derive such effective models. In these models, the nonlinear atomic interaction near critical regions, e.g. the neighborhood of defects, is retained while the interaction in the bulk region is simplified so that the effective models are computationally amenable. Comparing to the coupling methods, the current approach is more frst-principle based because the continuum description is not introduced at the beginning. However, many important issues still remain.