Lars Erik Persson, "Homogenization Method: An Introduction"
1993 | pages: 90 | ISBN: 086238334X | PDF | 11,2 mb
Several problems in e.g. Engineering Sciences, Physics or Chemistry can be modelled by using partial differerential equations with periodically (or almost periodically) and rapidly oscillating coefficients. Let us only mention problems such as heat, sound, current and stress distribution in composite materials, flow in porous media, macroscopic properties of crystalline or polymer structures or optimal design of e.g. plates consisting of several materials, where all these problems are assumed to have a fine and complicated microstructure.
Around fifteen years ago A. Bensoussan, J. L. Lions and G. Papanicolaou developed a homogenization method well suited for treating such problems. This method is based on convergence results for linear differential operators given by e.g. De Giorgi, Spagnolo and Tartar and multiple scales asymptotic expansions. This method provides not only effective "macroscopic" properties but also information about the variation on the microscale level.
This book is meant to be a self contained introduction to the homogenization method for linear partial differential equations. Therefore we focus our interest on a few but hopefully instructive problems and results. At the end of each of the sections 1, 2 and 3 we present concrete homogenization procedures well suited for implementation on e.g. work stations.