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Theoretical and numerical analysis for the quasi-continuum approximation of a material particle model

Author: Ping Lin
Journal: Math. Comp. 72 (2003), 657-675
MSC (2000): Primary 65C20, 65K10, 65M15, 65M60, 74N15, 74G65
Published electronically: June 4, 2002
MathSciNet review: 1954960
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Abstract: In many applications materials are modeled by a large number of particles (or atoms) where any one of particles interacts with all others. Near or nearest neighbor interaction is expected to be a good simplification of the full interaction in the engineering community. In this paper we shall analyze the approximate error between the solution of the simplified problem and that of the full-interaction problem so as to answer the question mathematically for a one-dimensional model. A few numerical methods have been designed in the engineering literature for the simplified model. Recently much attention has been paid to a finite-element-like quasicontinuum (QC) method which utilizes a mixed atomistic/continuum approximation model. No numerical analysis has been done yet. In the paper we shall estimate the error of the QC method for this one-dimensional model. Possible ill-posedness of the method and its modification are discussed as well.

References [Enhancements On Off] (What's this?)

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Additional Information

Ping Lin
Affiliation: Department of Mathematics, The National University of Singapore, 2 Science Drive 2, Singapore 117543

Keywords: Lattice statics, particle motion, Lennard-Jones potential, global minimization, finite element method, error estimation, ill-posedness, quasi-continuum approximation, material modeling
Received by editor(s): June 9, 1998
Received by editor(s) in revised form: May 29, 2001
Published electronically: June 4, 2002
Article copyright: © Copyright 2002 American Mathematical Society