# 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

Abstract
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.