# Homogenization

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.

#### Schedule (Thursday from 15H45 to 17H45 at the lecture room of CMAP)

Classes by G. Allaire on September 26, October 10, 17, 24 and 31, November 7, December 5.

• Two-scale asymptotic expansions for a diffusion equation
• Two-scale convergence
• Derivation of Darcy's law by homogenization of Stokes equations in a porous media
• Homogenization of a nonlinear convex energy
• Homogenization of convection-diffusion equations and two-scale asymptotic expansions with drift
• Classes by F. Alouges on October 3, November 14, 21 and 28, December 12 and 19.

• Numerical analysis of the homogenization method
• Multiscale finite element methods
• Homogenization of micromagnetic model

• #### Bibliography

• ALLAIRE G., Shape optimization by the homogenization method, Springer Verlag, New York (2002).
• BAKHVALOV N., PANASENKO G., Homogenization : averaging processes in periodic media, Mathematics and its applications, vol.36, Kluwer Academic Publishers, Dordrecht (1990).
• BENSOUSSAN A., LIONS J.L., PAPANICOLAOU G., Asymptotic analysis for periodic structures, North-Holland, Amsterdam (1978).
• CIORANESCU D., DONATO P., An introduction to homogenization, Oxford Lecture Series in Mathematics and Applications 17, Oxford (1999).
• JIKOV V., KOZLOV S., OLEINIK O., Homogenization of differential operators and integral functionals, Springer, Berlin, (1995).
• TARTAR, L., The general theory of homogenization. A personalized introduction. Lecture Notes of the Unione Matematica Italiana, 7. Springer-Verlag, Berlin; UMI, Bologna (2009).

• Some lecture notes partly covering the topic of the course:

A reference paper on the topic of the course:

Some slides partly covering the topic of the course:

Some problems and exercises for the students who want to practice before the final exams (sorry, they are in French !):

Final exams: