Molecular and multiscale methods for the numerical simulation of materials

Frédéric 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.