[1] 
M. Arndt and M. Griebel.
Higher order gradient continuum description of atomistic models for
crystalline solids.
In P. Neittaanmäki et al., editors, Proceedings of the
Fourth European Congress on Computational Methods in Applied Sciences and
Engineering, Jyväskylä, Finland, 2004. [ bib  .ps.gz 1  .pdf 1 ] We propose an upscaling scheme for the passage from atomistic to continuum mechanical models of crystalline solids. It is based on a Taylor expansion of the deformation function up to a given order and describes the material properties to a higher extent than commonly used continuum mechanical models. In particular, the discreteness effects of the underlying atomistic model are captured. The qualitative properties of the technique are numerically analyzed for the model problem of a onedimensional atomic chain. The approach is then applied to the real threedimensional physical example of a silicon crystal. The resulting approximation properties are studied in a stationary setting. Finally, a numerical simulation of the time evolution of the elastic response of crystal silicon is presented.
