Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics
Blanc, Xavier ; Bris, Claude Le ; Legoll, Frédéric
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 39 (2005), p. 797-826 / Harvested from Numdam

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.

Publié le : 2005-01-01
DOI : https://doi.org/10.1051/m2an:2005035
Classification:  65K10,  74G15,  74G70,  74N15
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     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: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     volume = {39},
     year = {2005},
     pages = {797-826},
     doi = {10.1051/m2an:2005035},
     mrnumber = {2165680},
     zbl = {pre02213940},
     language = {en},
     url = {http://dml.mathdoc.fr/item/M2AN_2005__39_4_797_0}
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Blanc, Xavier; Bris, Claude Le; Legoll, Frédéric. Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 39 (2005) pp. 797-826. doi : 10.1051/m2an:2005035. http://gdmltest.u-ga.fr/item/M2AN_2005__39_4_797_0/

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