1,720,991 research outputs found
A review on the enhancement of near-cloaking using the multilayer structure
No full textThe aim of this article is to review the effective near-cloaking structure for the conductivity problem using multilayer structures. We first consider the multipolar expansion for the voltage perturbation due to the presence of inhomogeneities. We then discuss the enhancement of near-cloaking using the multilayer structure such that their first GPTs vanish
SERIES EXPANSIONS OF THE LAYER POTENTIAL OPERATORS USING THE FABER POLYNOMIALS AND THEIR APPLICATIONS TO THE TRANSMISSION PROBLEM
No full textWe consider the conductivity transmission problem in two dimensions with a simply connected inclusion of arbitrary shape. It is well known that the solvability of the transmission problem can be established via the boundary integral formulation in which the Neumann-Poincare (NP) operator is involved. In this paper, we derive series expansions of the layer potential operators based on geometric function theory and exhibit a novel approach to the transmission problem. We first construct a collection of harmonic basis functions by using the Faber polynomials associated with the simply connected inclusion. We then derive explicit series expansions of the layer potential operators with respect to the constructed basis functions. In particular, the NP operator becomes a doubly infinite, self-adjoint matrix operator, whose entry is explicitly given by the Grunsky coefficients corresponding to the inclusion. By applying the finite section method to this matrix formulation, we obtain an approximation scheme for the spectrum of the NP operator and the solution to the transmission problem. This scheme allows us to numerically compute, in a simple way, the spectrum of the NP operator for a smooth domain when the exterior conformal mapping is known. We additionally derive an explicit integral expression for the coefficients of the exterior conformal mapping so that for an inclusion of arbitrary shape, one can numerically compute the exterior conformal mapping by solving only one boundary integral equation and apply the proposed approach. We provide numerical examples to demonstrate the effectiveness of the proposed method.
Construction of inclusions with vanishing generalized polarization tensors by imperfect interfaces
No full textWe investigate the problem of planar conductivity inclusion with imperfect interface conditions. We assume that the inclusion is simply connected. The presence of the inclusion causes a perturbation in the incident background field. This perturbation admits a multipole expansion of which coefficients we call as the generalized polarization tensors (GPTs), extending the previous terminology for inclusions with perfect interfaces. We derive explicit matrix expressions for the GPTs in terms of the incident field, material parameters, and geometry of the inclusion. As an application, we construct GPT-vanishing structures of general shape that result in negligible perturbations for all uniform incident fields. The structure consists of a simply connected core with an imperfect interface. We provide numerical examples of GPT-vanishing structures obtained by our proposed scheme.
Analytical shape recovery of a conductivity inclusion based on Faber polynomials
No full textA 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.
Geometric multipole expansion and its application to semi-neutral inclusions of general shape
No full textWe 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.
Bayesian optimization approach for tracking the location and orientation of a moving target using far-field data
No full textWe investigate the inverse scattering problem for tracking the location and orientation of a moving scatterer using a single incident field. We solve the problem by adopting the optimization approach with the objective function defined by the discrepancy in far-field data. We rigorously derive formulas for the far-field data under translation and rotation of the target and prove that the far-field data is locally Lipschitz with respect to the orientation of the target at the true angle. By integrating these formulas with the Bayesian optimization approach, we reduce the cost of objective function evaluations. For the instance of an unknown target, machine learning via fully connected neural networks is applied to identify the shape of the target. Numerical simulations for randomly generated shapes and trajectories demonstrate the effectiveness of the proposed method.
SPECTRAL ANALYSIS OF THE NEUMANN--POINCARE\' OPERATOR FOR THIN DOUBLY CONNECTED DOMAINS
No full textWe analyze the spectrum of the Neumann--Poincare'\ (NP) operator for a doubly connected domain lying between two level curves defined by a conformal mapping, where the inner boundary of the domain is of general shape. The analysis relies on an infinite-matrix representation of the NP operator involving the Grunsky coefficients of the conformal mapping and an application of the Gershgorin circle theorem. As the thickness of the domain shrinks to zero, the spectrum of the doubly connected domain approaches the interval [ - 1/2, 1/2] in the Hausdorff distance and the density of eigenvalues approaches that of a thin circular annulus.
Provable wavelet-based neural approximation
No full textIn this paper, we develop a wavelet-based theoretical framework for analyzing the universal approximation capabilities of neural networks over a wide range of activation functions. Leveraging wavelet frame theory on the spaces of homogeneous type, we derive sufficient conditions on activation functions to ensure that the associated neural network approximates any functions in the function space induced by the activation function, along with an error estimate. These sufficient conditions accommodate a variety of smooth activation functions, including those that exhibit oscillatory behavior. Furthermore, by considering the L2-distance between smooth and non-smooth activation functions, we establish a generalized approximation result that is applicable to non-smooth activations, with the error explicitly controlled by this distance. This provides increased flexibility in the design of network architectures.
Blow-up of Electric Fields between Closely Spaced Spherical Perfect Conductors
No full textThe electric field increases toward infinity in the narrow region between closely adjacent perfect conductors as they approach each other. Much attention has been devoted to the blow-up estimate, especially in two dimensions, for the practical relevance to high stress concentration in fiber-reinforced elastic composites. In this paper, we establish optimal estimates for the electric field associated with the distance between two spherical conductors in n-dimensional spaces for n epsilon 2. The novelty of these estimates is that they explicitly describe the dependency of the blow-up rate on the geometric parameters: the radii of the conductors
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