Asymptotic analysis of an array of closely spaced absolutely conductive inclusions

Leonid Berlyand; Giuseppe Cardone; Yuliya Gorb; Gregory Panasenko

doi:10.3934/nhm.2006.1.353

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2006, Volume 1, Issue 3: 353-377. Doi: 10.3934/nhm.2006.1.353

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Asymptotic analysis of an array of closely spaced absolutely conductive inclusions

1.
Department of Mathematics and Materials Research Institute, Penn State University, University Park, PA 16802
2.
Department of Engineering–University of Sannio, Benevento
3.
Department of Mathematics, Pennsylvania State University, University Park, PA 16802
4.
LaMUSE–University Jean Monnet, Saint Etienne

Received: January 31, 2006

Revised: May 31, 2006

Published: August 2006

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Abstract

We consider the conductivity problem in an array structure with square closely spaced absolutely conductive inclusions of the high concentration, i.e. the concentration of inclusions is assumed to be close to 1. The problem depends on two small parameters: $\varepsilon$, the ratio of the period of the micro-structure to the characteristic macroscopic size, and $\delta$, the ratio of the thickness of the strips of the array structure and the period of the micro-structure. The complete asymptotic expansion of the solution to problem is constructed and justified as both $\varepsilon$ and $\delta$ tend to zero. This asymptotic expansion is uniform with respect to $\varepsilon$ and $\delta$ in the area $\{\varepsilon=O(\delta^{\alpha}),~\delta =O(\varepsilon^{\beta})\}$ for any positive $\alpha, \beta.$

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Mathematics Subject Classification: Primary: 34E05, 35C20; Secondary: 78M35.

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