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.
Mots-clés : multiscale methods, variational problems, continuum mechanics, discrete mechanics
@article{M2AN_2005__39_4_797_0,
author = {Blanc, Xavier and Bris, Claude Le and Legoll, Fr\'ed\'eric},
title = {Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {797--826},
publisher = {EDP-Sciences},
volume = {39},
number = {4},
year = {2005},
doi = {10.1051/m2an:2005035},
mrnumber = {2165680},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an:2005035/}
}
TY - JOUR AU - Blanc, Xavier AU - Bris, Claude Le AU - Legoll, Frédéric TI - Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2005 SP - 797 EP - 826 VL - 39 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an:2005035/ DO - 10.1051/m2an:2005035 LA - en ID - M2AN_2005__39_4_797_0 ER -
%0 Journal Article %A Blanc, Xavier %A Bris, Claude Le %A Legoll, Frédéric %T Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics %J ESAIM: Modélisation mathématique et analyse numérique %D 2005 %P 797-826 %V 39 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an:2005035/ %R 10.1051/m2an:2005035 %G en %F M2AN_2005__39_4_797_0
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://archive.numdam.org/articles/10.1051/m2an:2005035/
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