28-03-2014, 02:50 PM
Homogenisation theory for partial differential equations
Homogenisation theory.ppt (Size: 709.5 KB / Downloads: 33)
Overview of my talk
What is homogenisation?
Homogenisation applied to steady state heat conduction
One dimensional case
Some properties of the homogenised coefficients
What is homogenisation?
Problem with two time or length scales: slow/macroscopic and fast/microscopic
Treat these scales as independent variables
Derive a homogenised problem: depends only on slow scale and still has the relevant macroscopic structure
Some remarks at this point
The cell problem satisfies the solvability condition.
Unique first order corrector field if we demand zero average over Y.
Function undetermined at this point, but not needed here.
Summary of homogenisation
Multiple scales expansion ansatz
Derive equations for and .
First equation independent of y.
Second equation gives cell problem.
Third equation gives homogenised equation.
Recap
Homogenised problem for heat conduction.
The effective coefficients in the one dimensional case.
Now: more general properties of the coefficients.
Conclusions
Homogenisation: look at macro scale structure.
Get cell problem, homogenised equation and effective coefficients.
In one dimension we calculated the coefficient.
Homogenisation preserves some properties, not all.