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Atomistic to Continuum limits for computational materials science

Published online by Cambridge University Press:  16 June 2007

Xavier Blanc ,
Claude Le Bris  and
Pierre-Louis Lions
Xavier Blanc
Affiliation:
Laboratoire J.-L. Lions, Université Pierre et Marie Curie, Boîte courrier 187, 75252 Paris, France. blanc@ann.jussieu.fr
Claude Le Bris
Affiliation:
CERMICS, École Nationale des Ponts et Chaussées, 6-8 avenue Blaise Pascal, Cité Descartes, 77455 Marne-la-Vallée. MICMAC, Inria Rocquencourt, 78153 Le Chesnay, France. lebris@cermics.enpc.fr
Pierre-Louis Lions
Affiliation:
Collège de France, 11 place Marcelin Berthelot, 75231 Paris Cedex 05, France. CEREMADE, Université Paris Dauphine, place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France. lions@ceremade.dauphine.fr
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Abstract

The present article is an overview of some mathematical results, which provide elements of rigorous basis for some multiscale computations in materials science. The emphasis is laid upon atomistic to continuum limits for crystalline materials. Various mathematical approaches are addressed. The setting is stationary. The relation to existing techniques used in the engineering literature is investigated.

Keywords

Problems of mechanicsvariational problemsdiscrete to continuum limitmultiscale modelshomogenization theoryΓ-limitquasiconvexitygradient flowsquasicontinuum methodadaptivity.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 41 , Issue 2: Special issue on Molecular Modelling , March 2007 , pp. 391 - 426
Copyright
© EDP Sciences, SMAI, 2007

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