Master Paris 6----------------------
Master Ecole Polytechnique
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.
Classes by F. Alouges on October 3, November 14, 21 and 28, December 12 and 19.
Some lecture notes partly covering the topic of the course:
1) the theory of periodic homogenization,
2) homogenization in porous media,
3) numerical methods of homogenization.
A reference paper on the topic of the course:
Homogenization and two-scale convergence
Some slides partly covering the topic of the course:
1) introduction to periodic homogenization (Darcy's law, kinetic equations),
2) convergence theory,
3) transport in porous media.
Some problems and exercises for the students who want to practice before the final exams (sorry, they are in French !):
homogenization in a porous medium,
homogenization of Stokes flows,
January 2011 (in French),
January 2011 (in English),
January 2012 (in French),
January 2012 (in English),
January 2013 (in French),
January 2013 (in English),