Begell House Inc.
International Journal for Multiscale Computational Engineering
JMC
1543-1649
5
5
2007
Numerical Studies of a Coarse-grained Approximation for Dynamics of an Atomic Chain
351-367
Lin
Ping
Department of Mathematics, The National University of Singapore, Singapore 117543
Pavel
Petr
Brno University of Technology, Brno, Czech Republic
In many applications, materials are modeled by a large number of particles (or atoms) where each particle interacts with all others. Near or nearest-neighbor interaction is considered to be a good simplification of the full interaction in the engineering community. However, the resulting system is still too large to be solved under the existing computer power. In this paper we shall use the finite element and/or quasicontinuum idea to both position and velocity variables in order to reduce the number of degrees of freedom. The original and approximate particle systems are related to the discretization of the virtual internal bond model (continuum model). We focus more on the discrete system since the continuum description may not be physically complete because the stress-strain relation is not monotonically increasing and thus not necessarily well posed. We provide numerical justification on how well the coarse-grained solution is close to the fine grid solution in either a viscosity-demping or a temporal-average sense.
Adaptive Model Selection Procedure for Concurrent Multiscale Problems
369-386
Mohan A.
Nuggehally
Scientific Computation Research Center, Rensselaer Polytechnic Institute, 110 8th St., Troy, NY 12180, USA
Mark S.
Shephard
Rensselaer Polytechnic Institute
Catalin
Picu
Rensselaer Polytechnic Institute
Jacob
Fish
Civil Engineering and Engineering Mechanics, Columbia University, New York, New York
10027, USA
An adaptive method for the selection of models in a concurrent multiscale approach is presented. Different models from a hierarchy are chosen in different subdomains of the problem domain adaptively in an automated problem simulation. A concurrent atomistic to continuum (AtC) coupling method [27], based on a blend of the continuum stress and the atomistic force, is adopted for the problem formulation. Two error indicators are used for the hierarchy of models consisting of a linear elastic model, a nonlinear elastic model, and an embedded atom method (EAM) based atomistic model. A nonlinear indicator , which is based on the relative error in the energy between the nonlinear model and the linear model, is used to select or deselect the nonlinear model subdomain. An atomistic indicator is a stress-gradient-based criterion to predict dislocation nucleation, which was developed by Miller and Acharya [6]. A material-specific critical value associated with the dislocation nucleation criterion is used in selecting and deselecting the atomistic subdomain during an automated simulation. An adaptive strategy uses limit values of the two indicators to adaptively modify the subdomains of the three different models. Example results are illustrated to demonstrate the adaptive method.
A Force-Based Blending Model forAtomistic-to-Continuum Coupling
387-406
Santiago
Badia
Sandia National Laboratories
Pavel
Bochev
Sandia National Laboratories
Richard
Lehoucq
Sandia National Labs
Michael
Parks
Applied Mathematics and Applications, Sandia National Laboratories, Albuquerque, New Mexico 87185-1320, USA
Jacob
Fish
Civil Engineering and Engineering Mechanics, Columbia University, New York, New York
10027, USA
Mohan A.
Nuggehally
Scientific Computation Research Center, Rensselaer Polytechnic Institute, 110 8th St., Troy, NY 12180, USA
Max
Gunzburger
Department of Scientific Computing, Florida State University, Tallahassee, Florida 32306-4120
A method for coupling atomistic and continuum models across a subdomain, or bridge region,
is presented. Coupling is effected through a force-based blending model. The method properly
accounts for the the atomistic and continuum contributions to the force balance at points in
the bridge region. Simple patch tests and computational experiments are used to study the
method and its properties in one dimension. A discussion of implementation issues in higher
dimensions is provided.
Goal-oriented Atomistic-Continuum Adaptivity for the Quasicontinuum Approximation
407-415
Marcel
Arndt
School of Mathematics, University of Minnesota, 206 Church St., SE Minneapolis, MN 55455, USA
Mitchell
Luskin
University of Minnesota
We give a goal-oriented a posteriori error estimator for the atomistic-continuum modeling
error in the quasicontinuum method, and we use this estimator to design an adaptive
algorithm to compute a quantity of interest to a given tolerance by using a nearly minimal
number of atomistic degrees of freedom. We present computational results that demonstrate
the effectiveness of our algorithm for a periodic array of dislocations described by a
Frenkel-Kontorova-type model.
Flow Patterns in the Vicinity of Triple LineDynamics Arising from a Local Surface TensionModel
417-434
J.
Monnier
INPG & INRIA, lab. LJK, BP 53, F-38041 Grenoble Cedex 9, France
A. M.
Benselama
INPG & INRIA, lab. LJK, BP 53, F-38041 Grenoble Cedex 9, France
I.
Cotoi
INPG & INRIA, lab. LJK, BP 53, F-38041 Grenoble Cedex 9, France
We model and simulate numerically a droplet impact onto a solid substrate. The triple line
dynamics modeling is implicit (as opposed to classical explicit mobility relations); it is based on
the Shikhmurzaev equations. These equations include generalized Navier slip type boundary
conditions with extra local surface tension gradient terms. Numerical results when spreading
are presented. A particular attention is paid to flow patterns near the contact line.