Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics
ESAIM: Modélisation mathématique et analyse numérique, Tome 39 (2005) no. 4, pp. 797-826.

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.

DOI : 10.1051/m2an:2005035
Classification : 65K10, 74G15, 74G70, 74N15
Mots clés : multiscale methods, variational problems, continuum mechanics, discrete mechanics
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Blanc, Xavier; Bris, Claude Le; Legoll, Frédéric. Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics. ESAIM: Modélisation mathématique et analyse numérique, Tome 39 (2005) no. 4, pp. 797-826. doi : 10.1051/m2an:2005035. http://www.numdam.org/articles/10.1051/m2an:2005035/

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