Asymptotic analysis of an array of closely spaced absolutely conductive inclusions
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Department of Mathematics and Materials Research Institute, Penn State University, University Park, PA 16802
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Department of Engineering–University of Sannio, Benevento
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Department of Mathematics, Pennsylvania State University, University Park, PA 16802
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LaMUSE–University Jean Monnet, Saint Etienne
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Received:
01 February 2006
Revised:
01 June 2006
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Primary: 34E05, 35C20; Secondary: 78M35.
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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:
ε, the ratio of the period of the micro-structure to
the characteristic macroscopic size, and δ, 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
ε and δ tend to zero. This asymptotic expansion
is uniform with respect to ε and δ in the area
{ε=O(δα), δ=O(εβ)} for any positive α,β.
Citation: Leonid Berlyand, Giuseppe Cardone, Yuliya Gorb, Gregory Panasenko. Asymptotic analysis of an array of closely spaced absolutely conductive inclusions[J]. Networks and Heterogeneous Media, 2006, 1(3): 353-377. doi: 10.3934/nhm.2006.1.353
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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:
ε, the ratio of the period of the micro-structure to
the characteristic macroscopic size, and δ, 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
ε and δ tend to zero. This asymptotic expansion
is uniform with respect to ε and δ in the area
{ε=O(δα), δ=O(εβ)} for any positive α,β.
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