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**Molecular and multiscale methods for the numerical simulation of
materials**

**Frederic LEGOLL**

**University Paris 6 PhD, defended on August 31, 2004**

We investigate in this thesis some molecular models and some multiscale
methods for the numerical simulation of materials.

The first part (chapters 2, 3 and 4) is devoted to an atomistic modelling. Statistical
physics shows that the relevant quantities at the macroscopic scale are
phase space averages. Molecular dynamics can be used to compute these
averages. The time evolution of the system is simulated, that allows one
to compute time averages along the trajectories of the system. Under the
ergodic assumption, these averages converge in the long time limit
to the phase space averages. We study here the convergence rate of the time
averages, and provide a numerical analysis of several schemes.

In a second part, we study some multiscale approaches. The chapter
6 is devoted to the numerical analysis of a method that
couples an atomistic model with a continuum model: the computational
domain is split into two subdomains, one described by a
continuum model, the other one described by an atomistic model. In
particular, we study the criterion that governs the choice, at each
material point, of the model (discrete or continuous).

In the chapter 7, we study the numerical homogenization
of some polycrystal models, that describe matter at the micrometric
scale.

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**Last update: september 2005.**