Scottish Mathematical Sciences Training Centre

SMSTC Semester 1 2018-19, Wednesdays 11-12:30

This course is run by MIGSAA. It is open to everyone, including non-MIGSAA students, as an SMSTC supplementary module. 

Homogenization theory I: multiscale modelling and analysis of physical and biological processes 

The aim of homogenization theory is to determine the macroscopic behaviour of a system comprising microscopic heterogeneities, e.g. transport processes in a porous medium, signalling processes in a cell tissue, deformations of composite materials. This means that the mathematical model defined in a heterogeneous medium is replaced by equations posed in a homogeneous one, which provide a good approximation of properties of the original microscopic system.

In this course we will learn the main methods of homogenization theory, which are used to prove that solutions of microscopic problem, depending on a small parameter, converge to a solution of the corresponding macroscopic problem, as the small parameter (determined by the characteristic size of the microscopic structure) goes to zero.     

Syllabus:

• main methods of periodic homogenization: formal asymptotic expansion, two-scale convergence, unfolding operator
• derivation of compactness results for two-scale convergence and periodic unfolding operator
• multiscale modelling and analysis of fluid flow in porous media
• multiscale modelling and analysis of transport and reaction processes in perforated and partially perforated domains (i.e. signalling and transport processes in biological tissues, plant root growth)
• dual-porosity: modelling and multiscale analysis (transport and reaction processes in fractured media, in soil with porous particles, in cell tissues)
• multiscale analysis of equations of linear elasticity and viscoelasticity
• main ideas of Gamma-, G- and H- convergences 

Selected references:

Cioranescu D., Donato P. An introduction to homogenization, Oxford University Press, 1999

Bensoussan A., Lions J.L., Papanicolaou G. Asymptotic analysis for periodic structures, North-Holland, 1978

Hornung U. Homogenization and Porous Media, Springer, 1997

Braides A. Gamma-Convergence for beginners, Oxford University Press, 2002

Braides A. A handbook of Gamma-convergence, online

Pavliotis G.A., Stuart A. Multiscale methods: averaging and homogenization, Springer, 2008

Allaire G. Shape optimization by the homogenization method, Springer, 2002.

Assessment: (20 hrs, 15 credits)

7 credits for active participation
8 credits for solving tutorial/homework questions (under guidance of instructor)

Recommended prerequisites: some knowledge of Sobolev spaces and PDEs.