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Analysis of a prototypicalmultiscale method coupling atomistic and continuum mechanics

Published online by Cambridge University Press:  15 August 2005

Xavier Blanc ,
Claude Le Bris  and
Frédéric Legoll
Xavier Blanc
Affiliation:
Laboratoire J.-L. Lions, Université Pierre et Marie Curie, Boîte courrier 187, 75252 Paris, France. blanc@ann.jussieu.fr
Claude Le Bris
Affiliation:
CERMICS, Ecole Nationale des Ponts et Chaussées, 6 et 8 avenue Blaise Pascal, Cité Descartes, 77455 Marne-la-Vallée, France. lebris@cermics.enpc.fr; legoll@cermics.enpc.fr
Frédéric Legoll
Affiliation:
CERMICS, Ecole Nationale des Ponts et Chaussées, 6 et 8 avenue Blaise Pascal, Cité Descartes, 77455 Marne-la-Vallée, France. lebris@cermics.enpc.fr; legoll@cermics.enpc.fr
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Abstract

In order to describe a solid which deforms smoothly in some region, butnon smoothly in some other region, many multiscale methods have recentlybeen proposed. They aim at coupling an atomistic model (discretemechanics) with a macroscopic model (continuum mechanics).We provide here a theoretical ground for such a coupling in aone-dimensional setting. We briefly study the general case of a convexenergy, and next concentrate on a specific example of a nonconvex energy, the Lennard-Jones case. In thelatter situation, we prove that the discretization needs to account inan adequate way for the coexistence of a discrete model and a continuousone. Otherwise, spurious discretization effects may appear.We provide a numerical analysis of the approach.

Keywords

Multiscale methodsvariational problemscontinuum mechanicsdiscrete mechanics.

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Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 39 , Issue 4 , July 2005 , pp. 797 - 826
Copyright
© EDP Sciences, SMAI, 2005

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