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. A subproblem is the absorbing boundary condition for molecular dynamics model. The goal is to derive and implement the appropriate boundary condition so that the phonons generated by local defects can propagation through the boundary, instead of be reflected back into the system to generate numerical error. The method has been applied to brittle crack problems to study the impact of strong shocks. 

2. MoriZwanzig formalism for
coarsegraining molecular models of materials and biological system.
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. The reduction of the number of degrees of freedom can be accomplished under the general framework of MoriZwanzig. In particular, with the help of a projection operator, the MoriZwanzig formalism offers a compact and elegant scheme to derive the coarsegrained models. Compared to the coupling methods, the current approach is more frstprinciple based because the continuum description is not introduced at the beginning. With proper approximations, the procedure yields a set of generalized Langevin equations (GLE) (Li, 2010). Recently, we have made the connection to the conventional Galerkin projection methods. As it turns out, the GLEs can be derived by using a Galerkin projection to an extended subspace. We have developed an algorithm for evaluating the memory functions in the GLEs. More importantly, the random noise in the GLE can be approximated within the framework, and the fluctuationdissipation theorem (Kubo 66) is automatically satisfied. Our applications include dislocation dynamics, crack propagations and protein dynamics. 

3. Analysis of multiscale models.
The goal of this work is to understand the modeling associated with a coarsegrained model. We have considered quasicontinuum type of methods for systems near and beyond bifurcation points. Our current work focuses on dynamic coupling methods. 

4. 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. As a result, the model is drastically simplied. This method will be aplied to various types of material interfaces. 

5. Heat conduction in nanoscale materials.
The distinct heat conduction properties have made nanoscale materials extremely successful in application. The standard heat equation, a textbook example of a partial differential equation, and a traditionally successful heat conduction model, fails to describe the conduction process. In spite of tremendous interest and application, little theory exists to provide a more suitable mathematical model. On the other hand, these problems also pose a much greater challenge for direct nonequilibrium molecular dynamics (NEMD) simulations at the atomic scale. For instance, while it is often relatively straightforward to incorporate quantities such as displacement, velocity, temperature and pressure into MD simulations as constraints, temperature gradient is very difficult to impose. The temperature gradient that can be imposed is usually is too large to model realistic systems. We derive a coarsegrained model for heat conduction in nanoscale mechanical systems. Starting with an allatom description, this approach yields a reduced model, in the form of conservation laws of momentum and energy. The model closure is accomplished by introducing a quasilocal thermodynamic equilibrium, followed by a linear response approximation. Of particular interest is the constitutive relation for the hear flux, which is expressed nonlocally in terms of the spatial and temporal variation of the temperature.


6. 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. 

7. The Mathematical Foundation of
Molecular Dynamics Simulations.
