Geometric series expansion of the Neumann–Poincaré operator: Application to composite materials
The Neumann–Poincaré (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 geometr...
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| Published in | European journal of applied mathematics Vol. 33; no. 3; pp. 560 - 585 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Cambridge
Cambridge University Press
01.06.2022
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| Subjects | |
| Online Access | Get full text |
| ISSN | 0956-7925 1469-4425 |
| DOI | 10.1017/S0956792521000127 |
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| Abstract | The Neumann–Poincaré (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. |
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| AbstractList | The Neumann–Poincaré (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. |
| Author | LIM, MIKYOUNG KIM, MINWOO CHERKAEV, ELENA |
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| Snippet | The Neumann–Poincaré (NP) operator, a singular integral operator on the boundary of a domain, naturally appears when one solves a conductivity transmission... |
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| SubjectTerms | Applied mathematics Composite materials Conformal mapping Domains Inclusions Operators (mathematics) Polarization Series expansion Tensors |
| Title | Geometric series expansion of the Neumann–Poincaré operator: Application to composite materials |
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