Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics

Xavier Blanc; Claude Le Bris; Frédéric Legoll

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2005)

  • Volume: 39, Issue: 4, page 797-826
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Blanc, Xavier, Bris, Claude Le, and Legoll, Frédéric. "Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 39.4 (2005): 797-826. <http://eudml.org/doc/245383>.

@article{Blanc2005,
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.},
author = {Blanc, Xavier, Bris, Claude Le, Legoll, Frédéric},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {multiscale methods; variational problems; continuum mechanics; discrete mechanics; Multiscale methods},
language = {eng},
number = {4},
pages = {797-826},
publisher = {EDP-Sciences},
title = {Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics},
url = {http://eudml.org/doc/245383},
volume = {39},
year = {2005},
}

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: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2005
PB - EDP-Sciences
VL - 39
IS - 4
SP - 797
EP - 826
AB - 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.
LA - eng
KW - multiscale methods; variational problems; continuum mechanics; discrete mechanics; Multiscale methods
UR - http://eudml.org/doc/245383
ER -

References

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