Geometric multipole expansion and its application to semi-neutral inclusions of general shape

Abstract

We consider the conductivity problem with a simply connected or multi-coated inclusion in two dimensions. The potential perturbation due to an inclusion admits a classical multipole expansion whose coefficients are the so-called generalized polarization tensors (GPTs). The GPTs have been fundamental building blocks in conductivity inclusion problems. In this paper, we present a new concept of geometric multipole expansion and its expansion coefficients, named the Faber polynomial polarization tensors (FPTs), using the conformal mapping and the Faber polynomials associated with the inclusion. The proposed expansion leads us to a series solution method for a simply connected or multi-coated inclusion of general shape, while the classical expansion leads us to a series solution only for a single- or multilayer circular inclusion. We also provide matrix expressions for the FPTs using the Grunsky matrix of the inclusion. In particular, for the simply connected inclusion with extreme conductivity, the FPTs admit simple formulas in terms of the conformal mapping associated with the inclusion. As an application of the concept of the FPTs, we construct semi-neutral inclusions of general shape that show relatively negligible field perturbations for low-order polynomial loadings. These inclusions are of the multilayer structure whose material parameters are determined such that some coefficients of geometric multipole expansion vanish.

Construction of inclusions with vanishing generalized polarization tensors by imperfect interfaces

“…with some constants 𝑠 𝑚𝑛 . Following, 31 we call the expansion in ( 14) the geometric multipole expansion of 𝑢. For each 𝑚, both 𝛽 𝑚𝑛 and 𝑠 𝑚𝑛 are determined by 𝛼 𝑚 .…”

“…𝑚𝑛 the FPTs for an inclusion with an imperfect interface, by extending the definition for an inclusion with perfect interface conditions in Ref. [31].…”

“…[47,48] for details of Faber polynomials. The concept of FPTs has been successfully applied to design semi-neutral inclusions 31 (see also Refs. [49][50][51]).…”

“…GPT-vanishing structures of general shape were also obtained using a shape optimization approach [28][29][30] (see also Ref. [31]). We also refer the readers to recent results [32][33][34][35] that explore the geometric dependance and applications of general-shaped inclusions.…”

“…To address this problem, we find GPT-vanishing structures by employing the concept of the Faber polynomial polarization tensors (FPTs; see Ref. [31]). The FPTs are linear combinations of GPTs with coefficients given by Faber polynomials associated with the inclusion.…”