Geometric series expansion of the Neumann-Poincare operator: Application to composite materials

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The Neumann-Poincare (NP) operator, a singular integral operator on the boundary of a domain, naturally appears when one solves a conductivity transmission problem via the boundary integral formulation. Recently, a series expression of the NP operator was developed in two dimensions based on geometric function theory [34]. In this paper, we investigate geometric properties of composite materials using this series expansion. In particular, we obtain explicit formulas for the polarisation tensor and the effective conductivity for an inclusion or a periodic array of inclusions of arbitrary shape with extremal conductivity, in terms of the associated exterior conformal mapping. Also, we observe by numerical computations that the spectrum of the NP operator has a monotonic behaviour with respect to the shape deformation of the inclusion. Additionally, we derive inequality relations of the coefficients of the Riemann mapping of an arbitrary Lipschitz domain using the properties of the polarisation tensor corresponding to the domain.
Publisher
CAMBRIDGE UNIV PRESS
Issue Date
2022-06
Language
English
Article Type
Article
Citation

EUROPEAN JOURNAL OF APPLIED MATHEMATICS, v.33, no.3, pp.560 - 585

ISSN
0956-7925
DOI
10.1017/S0956792521000127
URI
http://hdl.handle.net/10203/298508
Appears in Collection
MA-Journal Papers(저널논문)
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