This course is an introduction to homogenization theory with a view on multiscale modelling and numerical simulation. Mathematically, homogenization can be defined as a theory for averaging partial differential equations. It has many potential applications, including the derivation of effective properties for heterogeneous media, the rigorous definition of composite materials, the macroscopic modelling of microscopic systems and the design of multiscale numerical algorithms. We shall illustrate these issues by considering various examples from continuum mechanics, physics or porous media engineering.

We first deal with the homogenization of periodic structures by the method of two-scale asymptotic expansions which will be rigorously justified by the notion of two-scale convergence. We shall discuss issues related to correctors, boundary layers, error estimates as well as some generalizations to the non-periodic case. We then use this periodic homogenization theory as a modelling tool for deriving macroscopic models for heterogeneous media. Finally we introduce so-called multiscale finite element methods for performing numerical homogenization.

Dernière mise à jour : mardi 7 février 2012