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Last update July, 2012.  

The focus of my recent research is on the modeling of solids systems. The main objectives is to study mechanical properties using both atomistic models, e.g. molecular mechanics and molecular dynamics, and continuum descriptions, e.g. elasticity and elastodynamics.

Currently, I am working on the following projects:
  • Coupled Atomistic and Continuum Models for Solids.
  • Coarse-grained Molecular Dynamics Models.
  • Atomistic-based Finite Element and Boundary Element Methods for the Reduction of Molecular Statics Models.
  • Physics and Mechanics for Crack Propagation and Kinking.
  • Calculation of Stress in Molecular Models.
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. Coarse-grained 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 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.

Recently, we have extended the conventional Galerkin projection methods to derive a new class of coarse-grained 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. Atomistic-based Boundary Element Method.

We are developing a new reduced computational model, called atomistic-based 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 summation-by-parts 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. In particular, we study the bifurcation behavior as the load parameters are varied. Both mode-I and mixed mode-I/II have been considered.

5. The Calculation of Stress in Molecular Models.

We develop highly accurate formulae for calculation continuum quantities, e.g. stress and heat fluxes, from non-equilibrium molecular dynamics or molecular mechanics models for non-homogeneous systems. The formulae map discrete particle trajectories to continuum field quantities. In addition, it is a way of averaging in space and time. We have developed a Irving-Kirkwood type of formalism to systematically incorporate time averaging into the expressions of the continuum stress. We are also developing high accurate formulae for cases when the atomic displacement is smooth.