A conductivity inclusion, inserted in a homogeneous background, induces a perturbation in the background potential. This perturbation admits a multipole expansion whose coefficients are the so-called generalized polarization tensors (GPTs). GPTs can be obtained from multistatic measurements. As a modification of GPTs, the Faber polynomial polarization tensors (FPTs) were recently introduced in two dimensions. In this study, we design two novel analytical non-iterative methods for recovering the shape of a simply connected inclusion from GPTs by employing the concept of FPTs. First, we derive an explicit expression for the coefficients of the exterior conformal mapping associated with an inclusion in a simple form in terms of GPTs, which allows us to accurately reconstruct the shape of an inclusion with extreme or near-extreme conductivity. Secondly, we provide an explicit asymptotic formula in terms of GPTs for the shape of an inclusion with arbitrary conductivity by considering the inclusion as a perturbation of its equivalent ellipse. With this formula, one can non-iteratively approximate an inclusion of general shape with arbitrary conductivity, including a straight or asymmetric shape. Numerical experiments demonstrate the validity of the proposed analytical approaches.
This paper concerns the inverse problem of determining a planar conductivity inclusion. Our aim is to analytically recover from the generalized polarization tensors (GPTs), which can be obtained from exterior measurements, a homogeneous inclusion with arbitrary constant conductivity. The primary outcome of recovering a homogeneous inclusion is an inversion formula in terms of the GPTs for conformal mapping coefficients associated with the inclusion. To prove the formula, we establish matrix factorizations for the GPTs.
Using a distributed representation formula of the Gateaux derivative of the Dirichlet to Neumann map with respect to movements of a polygonal conductivity inclusion, [11], we extend the results obtained in [8] proving global Lipschitz stability for the determination of a polygonal conductivity inclusion embedded in a layered medium from knowledge of the Dirichlet to Neumann map.
We consider the problem of determining a polyhedral conductivity inclusion embedded in a homogeneous isotropic medium from boundary measurements. We prove global Lipschitz stability for the polyhedral inclusion from the local Dirichlet-to-Neumann map extending in a highly nontrivial way the results obtained in [18] and [19] in the two-dimensional case to the three-dimensional setting.
We consider an insulated conductivity model with two neighboring inclusions of $m$-convex shapes in $\mathbb{R}^{d}$ when $m\geq2$ and $d\geq3$. We establish the pointwise gradient estimates for the insulated conductivity problem and capture the gradient blow-up rate of order $\varepsilon^{-1/m+β}$ with $β=[-(d+m-3)+\sqrt{(d+m-3)^{2}+4(d-2)}]/(2m)\in(0,1/m)$, as the distance $\varepsilon$ between these two insulators tends to zero. In particular, the optimality of the blow-up rate is also demonstrated for a class of axisymmetric $m$-convex inclusions.
In this paper, the insulated conductivity model with two touching or close-to-touching inclusions is considered in $\mathbb{R}^{d}$ with $d\geq3$. We establish the pointwise upper bounds on the gradient of the solution for the generalized $m$-convex inclusions under these two cases with $m\geq2$, which show that the singular behavior of the gradient in the thin gap between two inclusions is described by the first non-zero eigenvalue of an elliptic operator of divergence form on $\mathbb{S}^{d-2}$. Finally, the sharpness of the estimates is also proved for two touching axisymmetric insulators, …
We study the insulated conductivity problem with inclusions embedded in a bounded domain in $\mathbb R^n$, for $n \ge 3$. The gradient of solutions may blow up as $\varepsilon$, the distance between inclusions, approaches to $0$. We established in a recent paper optimal gradient estimates for a class of inclusions including balls. In this paper, we prove such gradient estimates for general strictly convex inclusions. Unlike the perfect conductivity problem, the estimates depend on the principal curvatures of the inclusions, and we show that these estimates are characterized by the first non-ze…
Ellipsoidal characterization of neutral inclusions for imperfect bonding of high-conductivity type
Gunma University（群马大学）；Korea Institute for Advanced Study（韩国高等研究院）；Inha University（仁荷大学）；Zhejiang University of Technology（浙江工业大学）；Tohoku University（东北大学）
Optimal estimates for the perfect conductivity problem with inclusions close to the boundary
当一个凸形理想导电夹杂靠近矩阵域边界时，一个较大的包含该夹杂的凸区域，电场可以任意大。利用能量方法，我们建立了该理想电导率问题梯度估计的逐点上界和下界。这些结果给出了任意形状和任意尺寸导体的最佳电场爆破率。Ammari，Kang，Lee，Lee and Lim（2007）研究了一个特殊的情况，当圆形夹杂靠近二维圆形矩阵域的边界时。从方法论的角度来看，本文所发展的技术与以往局限于圆形情形的技术有很大的不同，这使得我们可以进一步研究具有散度型的一般椭圆型方程.
In the perfect conductivity problem of composite material, the electric field concentrates in a narrow region in between two inclusions and always becomes arbitrarily large when the distance between inclusions tends to zero. To characterize such singular behavior, we capture the leading term of the gradient and reveal that the blow-up rates are determined by their relative convexity of the two adjacent inclusions. On the other hand, a blow-up factor, which is a linear functional of boundary data, is found to determine the blow-up will occur or not.
In high-contrast composite materials, the electric field concentration is a common phenomenon when two inclusions are close to touch. It is important from an engineering point of view to study the dependence of the electric field on the distance between two adjacent inclusions. In this paper, we derive the upper and lower bounds of the gradient of solutions to the conductivity problem where two perfectly conducting inclusions are located very close to each other. To be specific, we extend the known results of Bao-Li-Yin (ARMA 2009) in two folds: First, we weaken the smoothness of the inclusion…
We investigate the homogenization of inclusions of infinite conductivity, randomly stationary distributed inside a homogeneous conducting medium. A now classical result by Zhikov shows that, under a logarithmic moment bound on the inter-particle distance, an effective model with finite homogeneous conductivity exists. Relying on ideas from network approximation, we provide a relaxed criterion ensuring homogenization. Several examples not covered by the previous theory are discussed.
In the perfect conductivity problem of composite material, the gradient of solutions can be arbitrarily large when two inclusions are located very close. To characterize the singular behavior of the gradient in the narrow region between two inclusions, we capture the leading term of the gradient and give a fairly sharp description of such asymptotics.
We study the insulated conductivity problem with inclusions embedded in a bounded domain in $\mathbb{R}^n$. The gradient of solutions may blow up as $\varepsilon$, the distance between inclusions, approaches to $0$. It was known that the optimal blow up rate in dimension $n = 2$ is of order $\varepsilon^{-1/2}$. It has recently been proved that in dimensions $n \ge 3$, an upper bound of the gradient is of order $\varepsilon^{-1/2 + β}$ for some $β> 0$. On the other hand, optimal values of $β$ have not been identified. In this paper, we prove that when the inclusions are balls, the optimal valu…
We provide an analytical solution for the elastic fields in a two-dimensional unbounded isotropic body with a rigid inclusion. Our analysis is based on the boundary integral formulation of the elastostatic problem and geometric function theory. Specifically, we use the coordinate system provided by the exterior conformal mapping of the inclusion to define a density basis functions on the boundary of the inclusion, and we use the Faber polynomials associated with the inclusion for a basis inside the inclusion. The latter, which constitutes the main novelty of our approach, allows us to obtain a…
We study the insulated conductivity problem with inclusions embedded in a bounded domain in $\mathbb{R}^n$. The gradient of solutions may blow up as $\varepsilon$, the distance between inclusions, approaches to $0$. An upper bound for the blow up rate was proved to be of order $\varepsilon^{-1/2}$. The upper bound was known to be sharp in dimension $n = 2$. However, whether this upper bound is sharp in dimension $n \ge 3$ has remained open. In this paper, we improve the upper bound in dimension $n \ge 3$ to be of order $\varepsilon^{-1/2 + β}$, for some $β> 0$.
The Neumann-Poincaré 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 Neumann-Poincaré operator was developed in two dimensions based on geometric function theory. In this paper, we investigate geometric properties of composite materials by using this series expansion. In particular, we obtain explicit formulas for the polarization tensor and the effective conductivity for an inclusion or a periodic array of inclusions of arbitrary…
We deal with the problem of determining the shape of an inclusion embedded in a homogenous background medium. The multifre-quency electrical impedance tomography is used to image the inclusion. For different frequencies, a current is injected at the boundary and the resulting potential is measured. It turns out that the potential solves an elliptic equation in divergence form with discontinuous leading coefficient. For this inverse problem we aim to establish a logarithmic type stability estimate. The key point in our analysis consists in reducing the original problem to that of determining an…
We consider a boundary value problem for the conductivity equation in a bounded domain containing an inclusion which is nearly touching to the domain's boundary. We assume that the domain and the inclusion are disks with conductivity jump on the boundary of the inclusion. By using the layer potential technique and adopting the bipolar coordinates, we derive the asymptotic formulas which explicitly describe the gradient blow-up of the solution as the distance between the inclusion and the domain's boundary tends to zero. It turns out that the gradient blow-up term can be identified with the ele…
