Applied Complex Fluid Seminar

Speaker: Prof. Lin Ping
Affiliation: Department of Mathematics, The National University of Singapore

Title: A quasi-continuum approximation and its analysis


In many applications materials are modeled by a large number of particles (or atoms) where any one of particles interacting with all others through a pair potential energy. The equillibrium configuration of the material is the minimizer of the total energy of the system. The computational cost is high since the number of atoms is huge. Recently much attention has been paid to a so-called quasicontinuum (QC) approximation which is a mixed atomistic/continuum model. The QC method solves a fully atomistic problem in regions where the material contains defects (or larger deformation gradients), but uses continuum finite elements to integrate out the majority of the atomistic degrees of freedom in regions where deformation gradients are small. However, numerical analysis is still at its infancy. In this talk we will conduct a convengence analysis of the QC method in the case that there is no serious defect or that the defect region is small. The difference of our analysis from conventional finite element analysis is that our exact solution is not a solution of a continuous partial differential equation but a discrete atomic scale solution which is not simply related to any conventional partial differential equation. We will consider both one dimensional and two dimensional cases. Some thoughts about the dynamical case may be mentioned as well. The QC method may be related to some other fields such as model reduction and pre-conditioning.