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Marek Netušil - Homogenization for dummies

Date: 04/11/2015 14:00
Place: Seminární místnost KNM

In real-life applications one often encounters heterogeneous problems for which the representative dimension of heterogeneity is significantly smaller than the dimension of the studied object. As examples we can mention layered materials composed of different types of metal, fluid flow through porous environment, and soft tissues of living organisms (composed of cells, collagen, elastin, and so on). Direct approach, i.e creating a system of equations for all constituents and their connection via boundary conditions, is inappropriate for several reasons.

1) Because of the small dimension of the heterogeneity one needs a very fine grid for a numerical scheme. This makes a computation to be very demanding, if not impossible.

2) Insight and inference of such model become also very difficult, if not impossible.

Homogenization is a mathematical approach, that can be simply explained as follows:

Is it possible to replace the complicated composite material with a more simple one, that is homogeneous, in such a way that this new material exhibits behavior similar to the original one?

Purpose of the lecture is a very short introduction to basics of the homogenization theory. The approach will be demonstrated on a linear heat equation.