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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. [email protected]
Claude Le Bris
Affiliation:
CERMICS, Ecole Nationale des Ponts et Chaussées, 6 et 8 avenue Blaise Pascal, Cité Descartes, 77455 Marne-la-Vallée, France. [email protected]; [email protected]
Frédéric Legoll
Affiliation:
CERMICS, Ecole Nationale des Ponts et Chaussées, 6 et 8 avenue Blaise Pascal, Cité Descartes, 77455 Marne-la-Vallée, France. [email protected]; [email protected]
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Abstract

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

Keywords

Multiscale methodsvariational problemscontinuum mechanicsdiscrete mechanics.
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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