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Hyeonbae Kang

Publications and source records attributed to Hyeonbae Kang.

At least 73 records · Page 4Linked to original sources
arXiv2012

Strong convergence of the solutions of the linear elasticity and uniformity of asymptotic expansions in the presence of small inclusions

We consider the Lamé system of linear elasticity when the inclusion has the extreme elastic constants. We show that the solutions to the Lamé system converge in appropriate $H^1$-norms when the shear modulus tends to infinity (the other modulus, the compressional modulus is fixed), and when the bulk modulus and the shear modulus tend to zero. Using this result, we show that the asymptotic expansion of the displacement vector in the presence of small inclusion is uniform with respect to Lamé parameters.

math.AP↗
arXiv2012

Enhancement of near-cloaking for the full Maxwell equations

In this paper, we consider near cloaking for the full Maxwell equations. We extend the recent results, where the quasi-static limit case and the Helmholtz equation are considered, to electromagnetic scattering problems. We construct very effective near cloaking structures for the electromagnetic scattering problem at a fixed frequency. These new structures are, before using the transformation optics, layered structures and are designed so that their first scattering coefficients vanish. Inside the cloaking region, any target has near-zero scattering cross section for a band of frequencies. We analytically show that our new construction significantly enhances the cloaking effect for the full Maxwell equations.

math.AP↗
arXiv2012

Spectral theory of a Neumann-Poincaré-type operator and analysis of anomalous localized resonance II

If a body of dielectric material is coated by a plasmonic structure of negative dielectric constant with nonzero loss parameter, then cloaking by anomalous localized resonance (CALR) may occur as the loss parameter tends to zero. The aim of this paper is to investigate this phenomenon in two and three dimensions when the coated structure is radial, and the core, shell and matrix are isotropic materials. In two dimensions, we show that if the real part of the permittivity of the shell is -1 (under the assumption that the permittivity of the background is 1), then CALR takes place. If it is different from -1, then CALR does not occur. In three dimensions, we show that CALR does not occur. The analysis of this paper reveals that occurrence of CALR is determined by the eigenvalue distribution of the Neumann-Poincaré-type operator associated with the structure.

math.AP↗
arXiv2012

Tracking of a Mobile Target Using Generalized Polarization Tensors

In this paper we apply an extended Kalman filter to track both the location and the orientation of a mobile target from multistatic response measurements. We also analyze the effect of the limited-view aspect on the stability and the efficiency of our tracking approach. Our algorithm is based on the use of the generalized polarization tensors, which can be reconstructed from the multistatic response measurements by solving a linear system. The system has the remarkable property that low order generalized polarization tensors are not affected by the error caused by the instability of higher orders in the presence of measurement noise.

math.NA↗
arXiv2012

Localization, Stability, and Resolution of Topological Derivative Based Imaging Functionals in Elasticity

The focus of this work is on rigorous mathematical analysis of the topological derivative based detection algorithms for the localization of an elastic inclusion of vanishing characteristic size. A filtered quadratic misfit is considered and the performance of the topological derivative imaging functional resulting therefrom is analyzed. Our analysis reveals that the imaging functional may not attain its maximum at the location of the inclusion. Moreover, the resolution of the image is below the diffraction limit. Both phenomena are due to the coupling of pressure and shear waves propagating with different wave speeds and polarization directions. A novel imaging functional based on the weighted Helmholtz decomposition of the topological derivative is, therefore, introduced. It is thereby substantiated that the maximum of the imaging functional is attained at the location of the inclusion and the resolution is enhanced and it proves to be the diffraction limit. Finally, we investigate the stability of the proposed imaging functionals with respect to measurement and medium noises.

math.AP↗
arXiv2012

Boundary perturbations due to the presence of small linear cracks in an elastic body

In this paper, Neumann cracks in elastic bodies are considered. We establish a rigorous asymptotic expansion for the boundary perturbations of the displacement (and traction) vectors that are due to the presence of a small elastic linear crack. The formula reveals that the leading order term is ε^2 where εis the length of the crack, and the ε^3-term vanishes. We obtain an asymptotic expansion of the elastic potential energy as an immediate consequence of the boundary perturbation formula. The derivation is based on layer potential techniques. It is expected that the formula would lead to very effective direct approaches for locating a collection of small elastic cracks and estimating their sizes and orientations.

math.AP↗
arXiv2012

Spectral analysis of the Neumann-Poincaré operator and characterization of the gradient blow-up

When perfectly conducting or insulating inclusions are closely located, stress which is the gradient of the solution to the conductivity equation can be arbitrarily large as the distance between two inclusions tends to zero. It is important to precisely characterize the blow-up of the gradient. In this paper we show that the blow-up of the gradient can be characterized by a singular function defined by the single layer potential of an eigenfunction corresponding to the eigenvalue ${1}/{2}$ of a Neumann-Poincaré type operator defined on the boundaries of the inclusions. By comparing the singular function with the one corresponding to two disks osculating to the inclusions, we quantitatively characterize the blow-up of the gradient in terms of explicit functions.

math.AP↗
arXiv2012

Bounds on the volume fractions of two materials in a three dimensional body from boundary measurements by the translation method

Using the translation method of Tartar, Murat, Lurie, and Cherkaev bounds are derived on the volume occupied by an inclusion in a three-dimensional conducting body. They assume electrical impedance tomography measurements have been made for three sets of pairs of current flux and voltage measurements around the boundary. Additionally the conductivity of the inclusion and surrounding medium are assumed to be known. If the boundary data (Dirichlet or Neumann) is special, i.e. such that the fields inside the body would be uniform were the body homogeneous, then the bounds reduce to those of Milton and thus when the volume fraction is small to those of Capdeboscq and Vogelius.

math.AP↗
arXiv2012

Spectral theory of a Neumann-Poincare-type operator and analysis of cloaking due to anomalous localized resonance

The aim of this paper is to give a mathematical justification of cloaking due to anomalous localized resonance (CALR). We consider the dielectric problem with a source term in a structure with a layer of plasmonic material. Using layer potentials and symmetrization techniques, we give a necessary and sufficient condition on the fixed source term for electromagnetic power dissipation to blow up as the loss parameter of the plasmonic material goes to zero. This condition is written in terms of the Newtonian potential of the source term. In the case of concentric disks, we make the condition even more explicit. Using the condition, we are able to show that for any source supported outside a critical radius CALR does not take place, and for sources located inside the critical radius satisfying certain conditions CALR does take place as the loss parameter goes to zero.

math.AP↗
arXiv2012

Bounds on the volume fraction of the two-phase shallow shell using one measurement

We study the size estimate problem for the two-phase shallow shell equations in this paper. Our aim is to derive bounds on the volume fraction of each phase assuming that the material properties of the two phases are given. The approach in this paper is based on the translation method. One of the key steps is to connect the shallow shell equations to the thin plate equation.

math.AP↗
arXiv2012

Equivalence of inverse problems for 2D elasticity and for the thin plate with finite measurements and its applications

In this paper, we prove that the inverse problems for 2D elasticity and for the thin plate with boundary data (finite or full measurements) are equivalent. Having proved this equivalence, we can solve inverse problems for the plate equation with boundary data by solving the corresponding inverse problems for 2D elasticity, and vice versa. For example, we can derive bounds on the volume fraction of the two-phase thin plate from the knowledge of one pair of boundary measurements using the known result for 2D elasticity \cite{ml}. Similarly, we give another approach to the size estimate problem for the thin plate studied by Morassi, Rosset, and Vessella \cite{mrv07, mrv09}.

math.AP↗
arXiv2011

Sharp estimates for the Neumann functions and applications to quantitative photo-acoustic imaging in inhomogeneous media

We obtain sharp $L^p$ and Hölder estimates for the Neumann function of the operator $\nabla \cdot γ\nabla - ik$ on a bounded domain. We also obtain quantitative description of its singularity. We then apply these estimates to quantitative photo-acoustic imaging in inhomogeneous media. The problem is to reconstruct the optical absorption coefficient of a diametrically small anomaly from the absorbed energy density.

math.AP↗
arXiv2011

Enhancement of near-cloaking. Part II: the Helmholtz equation

The aim of this paper is to extend the method of improving cloaking structures in the conductivity to scattering problems. We construct very effective near-cloaking structures for the scattering problem at a fixed frequency. These new structures are, before using the transformation optics, layered structures and are designed so that their first scattering coefficients vanish. Inside the cloaking region, any target has near-zero scattering cross section for a band of frequencies. We analytically show that our new construction significantly enhances the cloaking effect for the Helmholtz equation.

math.AP↗
arXiv2011

Asymptotics and Computation of the Solution to the Conductivity Equation in the Presence of Adjacent Inclusions with Extreme Conductivities

When inclusions with extreme conductivity (insulator or perfect conductor) are closely located, the gradient of the solution to the conductivity equation can be arbitrarily large. And computation of the gradient is extremely challenging due to its nature of blow-up in a narrow region in between inclusions. In this paper we characterize explicitly the singular term of the solution when two circular inclusions with extreme conductivities are adjacent. Moreover, we show through numerical computations that the characterization of the singular term can be used efficiently for computation of the gradient in the presence adjacent inclusions.

math.AP↗
arXiv2011

Sharp bounds on the volume fractions of two materials in a two-dimensional body from electrical boundary measurements: the translation method

We deal with the problem of estimating the volume of inclusions using a finite number of boundary measurements in electrical impedance tomography. We derive upper and lower bounds on the volume fractions of inclusions, or more generally two phase mixtures, using two boundary measurements in two dimensions. These bounds are optimal in the sense that they are attained by certain configurations with some boundary data. We derive the bounds using the translation method which uses classical variational principles with a null Lagrangian. We then obtain necessary conditions for the bounds to be attained and prove that these bounds are attained by inclusions inside which the field is uniform. When special boundary conditions are imposed the bounds reduce to those obtained by Milton and these in turn are shown here to reduce to those of Capdeboscq-Vogelius in the limit when the volume fraction tends to zero. The bounds of this paper, and those of Milton, work for inclusions of arbitrary volume fractions. We then perform some numerical experiments to demonstrate how good these bounds are.

cond-mat.mtrl-sci↗
arXiv2011

Enhancement of Near Cloaking Using Generalized Polarization Tensors Vanishing Structures. Part I: The Conductivity Problem

The aim of this paper is to provide an original method of constructing very effective near-cloaking structures for the conductivity problem. These new structures are such that their first Generalized Polarization Tensors vanish. We show that this in particular significantly enhances the cloaking effect. We then present some numerical examples of Generalized Polarization Tensors vanishing structures.

math-ph↗
arXiv2010

Layer Potential Techniques for the Narrow Escape Problem

The narrow escape problem consists of deriving the asymptotic expansion of the solution of a drift-diffusion equation with the Dirichlet boundary condition on a small absorbing part of the boundary and the Neumann boundary condition on the remaining reflecting boundaries. Using layer potential techniques, we rigorously find high-order asymptotic expansions of such solutions. We explicitly show the nonlinear interaction of many small absorbing targets. Based on the asymptotic theory for eigenvalue problems developed in \cite{book}, we also construct high-order asymptotic formulas for eigenvalues of the Laplace and the drifted Laplace operators for mixed boundary conditions on large and small pieces of the boundary.

math.AP↗