Abstract
We consider a generalized version of the standard checkerboard and discuss the difficulties of finding the corresponding field by standard numerical treatment. A new numerical method is presented which converges independently of the local conductivities.
Highlights
Very few microstructures yield explicit formulae for their effective conductivity
The explicit solution of the corresponding temperature-field was later found by Berdichevskii [1]
In order to illustrate the theorem presented above we have computed the effective conductivity by a standard finite element method (FEM) program in the classical case, l(r) = 1, kg = k and kw = 1/k
Summary
Very few microstructures yield explicit formulae for their effective conductivity. One type of such structures is checkerboards. The explicit solution of the corresponding temperature-field was later found by Berdichevskii [1]. In particular he found that the heat-flux is infinitely high in the corners of the squares. Due to the behavior of the solutions near the corner points it is difficult to solve the corresponding variational problems by usual numerical methods, even for the standard checkerboard. We present a new numerical method for determining the corresponding field which converges in the energy norm independent of the local conductivities
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