Upscaling Techniques for the Passage from Atomistic to Continuum Mechanical Models
Marcel Arndt, Michael Griebel
In this project we analyze the relationship between atomistic and comtinuum
mechanical models for crystalline solids. Of special interest are upscaling
techniques, i.e. the systematic derivation of continuum models from a given
We developed the so-called inner expansion technique to derive a model of the
specimen within the quasi-continuum regime. It captures the microscopic and the
discreteness properties of the original system up to a given order and thus
allows for a better description of the specimen than common upscaling schemes
such as the so-called scaling technique. Furthermore it retains convexity of the
atomistic potential. Consequently the according evolution equations on the
continuum mechanical level are well-posed.
As a first simple example to study the direct expansion technique, we choose a
one-dimensional atomic chain. The following pictures show the time evolution of
an initial perturbation in the center. For the left and the middle picture, the
continuum models obtained by the scaling technique and the direct expansion
technique have been used. The right image gives the solution of the underlying
atomistic system as a reference.
One can clearly observe that the direct expansion captures the microscopic
properties to a high extent, whereas the scaling technique only gives a rough
description of the discrete model.
We furthermore applied the direct expansion technique to the more realistic
potential of Stillinger and Weber for silicon. The following pictures of a
silicon crystal show snapshots of the elastic response to an external
deformation. For the full time evolution see this movie
M. Arndt, M. Griebel.
Derivation of higher order gradient continuum models from atomistic models for crystalline solids.
Multiscale Model. Sim. 4(2):531-562, 2005.
M. Arndt. Upscaling from Atomistic Models to Higher Order Gradient Continuum Models
for Crystalline Solids. Dissertation, Institute for Numerical Simulation,
University of Bonn, 2004.
M. Arndt. Higher order gradient continuum description of atomistic models for crystalline solids.
Proceedings of the Fourth European Congress on Computational Methods in Applied Sciences
and Engineering (ECCOMAS 2004), Finland, 2004.
F. H. Stillinger, T. A. Weber.
Computer simulation of local order in condensed phases of silicon.
Phys. Rev. B 31(8):5262-5271, 1985.