The Nanometric and Micrometric Scales of the Structure and Mechanics of Materials Revisited: An Introduction to the Challenges of Fully Deterministic Numerical Descriptions

Nano-science and nano-technology as well as the fine modelling of the structure and mechanics of materials from the nanometric to the micrometric scales use descriptions ranging from the quantum to the statistical mechanics. This paper revisits the modelling at these scales and points out the main challenges related to the numerical solution of such models that some times are discrete but involves an extremely large number of particles (as it is the case of molecular dynamics simulations or coarse-grained molecular dynamics) and other times are continuous but they are defined in highly multidimensional spaces leading to the well known curse of dimensionality issues. The curse of dimensionality concerned by some of these deterministic models will be emphasized and their numerical implications will be addressed in the second part of this work. 1 ha l-0 03 48 45 0, v er si on 1 19 D ec 2 00 8 Author manuscript, published in "An introduction to the challenges of fully deterministic numerical descriptions 6, 3 (2008) 191-213"