Conference proceedings article

?-scale convergence applied to the stationary heat conduction equation with non-periodic thermal conductivity matrix

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Author list: Persson, Jens

Publisher: Vienna University Press (WUV)

Place: Wien

Publication year: 2009

Start page: 2720

End page: 2723

Number of pages: 4

ISBN: 978-3-901608-35-3

Abstract

In this contribution we study the homogenization of non-periodic stationary heat conduction problems with homogeneous Dirichlet boundary data by applying the recently developed ?-scale convergence technique developed by Holmbom and Silfver. ?-scale convergence can be seen as either being a special case of scale convergence (developed by Mascarenhas and Toader) or of “generalized” two-scale convergence (developed by Holmbom, Silfver, Svanstedt and Wellander). From either viewpoint, it is a possibly powerful generalization of Nguetseng’s classical, periodic two-scale convergence method. We give a definition of the concept of ?-scale convergence, which is then used to claim a main theorem on homogenization of certain non-periodic stationary heat conduction problems. The original part of the contribution starts by defining a two-dimensional “toy model”. We show that the “toy model” satisfies the right conditions such that the aforementioned main theorem on the homogenization can be employed. In this way we derive the homogenized problem, i.e. the homogenized thermal conductivity matrix, and the local problem. The contribution is concluded by giving a numerical example where we explicitly compute the homogenized thermal conductivity matrix.

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