86 research outputs found
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 for or . Using
ideas from transformation optics, we construct an approximate cloak by
``blowing up" a small ball of radius to one of radius . 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 ) 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
approaches , the -norm of the scattered field outside the cloak, and
its far field pattern, approach 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 of the Neumann boundary;
and when (ii) The homogeneous Dirichlet boundary condition is replaced by a
homogeneous Neumann boundary condition on a ``small'' subset
of the Dirichlet boundary. The relevant quantity that
measures the ``smallness'' of the subset differs in the
two cases: while it is the harmonic capacity of 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 , 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 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 for
some
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