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MULTISCALE MODEL. SIMUL. c 2005 Society for Industrial and Applied Mathematics Vol. 4, No. 2, pp. 531562
 

Summary: MULTISCALE MODEL. SIMUL. c 2005 Society for Industrial and Applied Mathematics
Vol. 4, No. 2, pp. 531­562
DERIVATION OF HIGHER ORDER GRADIENT CONTINUUM
MODELS FROM ATOMISTIC MODELS FOR CRYSTALLINE SOLIDS
M. ARNDT AND M. GRIEBEL
Abstract. We propose a new upscaling scheme for the passage from atomistic to continuum
mechanical models for crystalline solids. It is based on a Taylor expansion of the deformation function
and allows us to capture the microscopic properties and the discreteness effects of the underlying
atomistic system up to an arbitrary order. The resulting continuum mechanical model involves higher
order terms and gives a description of the specimen within the quasi-continuum regime. Furthermore,
the convexity of the atomistic potential is retained, which leads to well-posed evolution equations.
We numerically compare our technique to other common upscaling schemes for the example of an
atomic chain. Then we apply our approach to a physically more realistic many-body potential of
crystalline silicon. Here the above-mentioned advantages of our technique hold for the newly obtained
macroscopic model as well.
Key words. continuum limit, quasi-continuum approximation, upscaling, multiscale simulation,
molecular dynamics, continuum mechanics, higher order gradient, hyperbolic, silicon
AMS subject classifications. 35L75, 65M06, 70F10, 74B20, 82C21
DOI. 10.1137/040608738
1. Introduction. The behavior of material often involves quite different length

  

Source: Arndt, Marcel - School of Mathematics, University of Minnesota
Bebendorf, Mario - Institut für Numerische Simulation, Universität Bonn

 

Collections: Computer Technologies and Information Sciences; Mathematics