Last update July, 2013.  
The focus of my recent research is on the modeling of materials and biological systems. Currently, I am working on the following projects:


1. Coupled Atomistic and Continuum Models for Solids
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. 

2. Coarsegrained Molecular Dynamics Models for Solids
In this project, we consider the following problem: Given a subset of atomic degrees of freedom of full MD, called coarsegrain (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 frstprinciple based because the continuum description is not introduced at the beginning. However, many important issues still remain. Recently, we have extended the conventional Galerkin projection methods to derive a new class of coarsegrained models. The main idea of this approach is to expand the approximation space near the interface, while still retaining the Galerkin projection. This formulation led to an extended system with additional auxillary variables, which can be viewed as an approximaiton of the GLEs. 

3. Atomisticbased Boundary Element Method.
We are developing a new reduced computational model, called atomisticbased boundary element model (ABEM), derived from a full atomistic model for a crystalline solid system. The procedure is based on a domain decomposition method, where atoms near crystal defects are separated from the surrounding region, and a reduction method, which similar to the boundary integral method for continuum models, removes the atoms in the surrounding region. The reduction procedure relies on the lattice Green's function and a summationbyparts procedure, and it gives rise to a system of equations only involving the atoms at the remote boundary and the interfaces with local defects. 

4. Mechanics and Physics of Crack Propagation.
We study the transition of the stability for the problem of crack propagation based on nonlinear lattice models. In particular, we study the bifurcation behavior as the load parameters are varied. We make connections to the traditional fracture mechanics theory. More importantly, we study the effect of the strain rate in fracture initiation and propagation. 

5. The Mathematical Foundation of
Molecular Dynamics Simulations.
