A computational algorithm for crack determination: The multiple crack case

An algorithm for recovering a collection of linear cracks in a homogeneous electrical conductor from boundary measurements of voltages induced by specified current fluxes is developed. The technique is a variation of Newton's method and is based on taking weighted averages of the boundary data. The method also adaptively changes the applied current flux at each iteration to maintain maximum sensitivity to the estimated locations of the cracks

Singular solutions to a nonlinear elliptic boundary value problem originating from corrosion modeling

We consider a nonlinear elliptic boundary value problem on a planar domain. The exponential type nonlinearity in the boundary condition is one that frequently appears in the modeling of electrochemical systems. For the case of a disk we construct a family of exact solutions that exhibit limiting logarithmic singularities at certain points on the boundary. Based on these solutions we develop two criteria that we believe predict the possible locations of the boundary singularities on quite general domains

Effective Behavior of Clusters of Microscopic Cracks Inside a Homogeneous Conductor

We study the effective behaviour of a periodic array of microscopic cracks inside a homoge­neous conductor. Special emphasis is placed on a rigorous study of the case in which the corresponding effective conductivity becomes nearly singular, due to the fact that adjacent cracks nearly touch. It is heuristically shown how thin clusters of such extremely close cracks may macroscopically appear as a single crack. The results have implications for our earlier work on impedance imaging

Far field broadband approximate cloaking for the Helmholtz equation with a Drude-Lorentz refractive index

This paper concerns the analysis of a passive, broadband approximate cloaking scheme for the Helmholtz equation in ${\mathbb R}^d$ for $d=2$ or $d=3$. Using ideas from transformation optics, we construct an approximate cloak by ``blowing up" a small ball of radius $\epsilon>0$ to one of radius $1$. In the anisotropic cloaking layer resulting from the ``blow-up" change of variables, we incorporate a Drude-Lorentz-type model for the index of refraction, and we assume that the cloaked object is a soft (perfectly conducting) obstacle. We first show that (for any fixed $\epsilon$) there are no real transmission eigenvalues associated with the inhomogeneity representing the cloak, which implies that the cloaking devices we have created will not yield perfect cloaking at any frequency, even for a single incident time harmonic wave. Secondly, we establish estimates on the scattered field due to an arbitrary time harmonic incident wave. These estimates show that, as $\epsilon$ approaches $0$, the $L^2$-norm of the scattered field outside the cloak, and its far field pattern, approach $0$ uniformly over any bounded band of frequencies. In other words: our scheme leads to broadband approximate cloaking for arbitrary incident time harmonic waves

Small perturbations in the type of boundary conditions for an elliptic operator

In this article, we study the impact of a change in the type of boundary conditions of an elliptic boundary value problem. In the context of the conductivity equation we consider a reference problem with mixed homogeneous Dirichlet and Neumann boundary conditions. Two different perturbed versions of this ``background'' situation are investigated, when (i) The homogeneous Neumann boundary condition is replaced by a homogeneous Dirichlet boundary condition on a ``small'' subset $\omega_\varepsilon$ of the Neumann boundary; and when (ii) The homogeneous Dirichlet boundary condition is replaced by a homogeneous Neumann boundary condition on a ``small'' subset $\omega_\varepsilon$ of the Dirichlet boundary. The relevant quantity that measures the ``smallness'' of the subset $\omega_\varepsilon$ differs in the two cases: while it is the harmonic capacity of $\omega_\varepsilon$ in the former case, we introduce a notion of ``Neumann capacity'' to handle the latter. In the first part of this work we derive representation formulas that catch the structure of the first non trivial term in the asymptotic expansion of the voltage potential, for a general $\omega_\varepsilon$, under the sole assumption that it is ``small'' in the appropriate sense. In the second part, we explicitly calculate the first non trivial term in the asymptotic expansion of the voltage potential, in the particular geometric situation where the subset $\omega_\varepsilon$ is a vanishing surfacic ball

On the Regularity of Non-Scattering Anisotropic Inhomogeneities

In this paper we examine necessary conditions for an anisotropic inhomogeneous medium to be non-scattering at a single wave number and for a single incident field. These conditions are expressed in terms of the regularity of the boundary of the inhomogeneity. We assume that the coefficients, characterizing the constitutive material properties of the medium, are sufficiently smooth, and the incident wave is appropriately non-degenerate. Our analysis utilizes the Hodograph transform as well as regularity results for nonlinear elliptic partial differential equations. Our approach requires that the boundary a-priori is of class $C^{1,\alpha}$ for some $0<\alpha<1$