Sandians Pavel Bochev and Rich Lehoucq received a DOE
three year award
A Mathematical Analysis of Atomistic-to-Continuum (AtC) Coupling
Methods starting fiscal year 2006. The award is a collaborative proposal
with
Don Estep (CSU),
Jacob Fish and
Mark
Shephard (RPI), and
Max Gunzburger (FSU).
The research is funded under the Office of Science’s
Multiscale
Mathematics program. The program addresses those science problems
that span many time scales--from femtoseconds to years--and many length
scales--from the atomic level to the macroscopic.
Materials, and in particular, nanostructured materials are governed by
processes that are often controlled by the coupling of structures and dynamics
spanning many length and time scales. Theoretical and computational treatment
of multi-scale phenomena, including the development of predictive
capabilities, is therefore of fundamental importance. Synthesizing, or
coupling, atomistic and continuum descriptions of physical phenomena is an
attempt to ameliorate the overwhelming computational costs associated with an
all atomistic simulation.
Atomistic-to-Continuum (AtC) coupling enables a continuum
calculation to be performed over the majority of a domain of interest while
limiting the more expensive atomistic simulation over a subset of the domain.
Unfortunately, combining atomistic and continuum calculations is challenging
because the former is based on individual non-local force interactions between
atoms or molecules while continuum calculations deal with bulk quantities that
represent the average behavior of millions of atoms or molecules.
Past research in Atomistic-to-Continuum (AtC) models and algorithm development
has paid off in the formulation of procedures that address specific
applications. This previous research has also begun to lead to some degree of
generalization. However, much less effort has been directed at the fundamental
mechanics and/or mathematical theory of AtC methods. For example, a rigorous
mechanical formulation with error, stability, convergence analysis and
uncertainty quantification of coupling atomistic and continuum models is
lacking. As a result, a mathematical and mechanical framework that can provide
a unified theoretical foundation for the formulation, analysis, and
implementation of AtC coupling methods is needed. The goal of our research is
to understand and quantify the limits in AtC coupling methods and the
resulting impact on multiscale simulations. The CSRI at Sandia National Labs
held AtC coupling methods
workshop page where many of these issues were discussed. See the SIAM News, volume 39,
Number 7, September 2006 for the report Researchers Discuss Atomistic-to-Continuum (AtC) Coupling
by Michael L. Parks and
Rich Lehoucq.
The impact of our research is to improve the efficiency and fidelity of
multiscale simulation efforts. For example,
carbon nanotubes
(CNTs) possess unique properties arising from their structure but unavoidable
defects have a major influence. The need to understand the
multiscale
behavior of nanotubes prompted the development of equivalent continuum
theories, which are only valid for perfect structures. Various methods couple
molecular and continuum models in an attempt to account for defects in CNTs.
Our research is to examine these AtC techniques and to place them within a
rigorous mathematical framework.
