Corrigendum to ``Determining a sound-soft polyhedral scatterer by a single far-field measurement''

In the paper, G. Alessandrini and L. Rondi, ``Determining a sound-soft polyhedral scatterer by a single far-field measurement'', Proc. Amer. Math. Soc. 133 (2005), pp. 1685-1691, on the determination of a sound-soft polyhedral scatterer by a single far-field measurement, the proof of Proposition 3.2 is incomplete. In this corrigendum we provide a new proof of the same proposition which fills the previous gap.Comment: 3 page

Stable determination of a scattered wave from its far-field pattern: the high frequency asymptotics

We deal with the stability issue for the determination of outgoing time-harmonic acoustic waves from their far-field patterns. We are especially interested in keeping as explicit as possible the dependence of our stability estimates on the wavenumber of the corresponding Helmholtz equation and in understanding the high wavenumber, that is frequency, asymptotics. Applications include stability results for the determination from far-field data of solutions of direct scattering problems with sound-soft obstacles and an instability analysis for the corresponding inverse obstacle problem. The key tool consists of establishing precise estimates on the behavior of Hankel functions with large argument or order.Comment: 49 page

Reconstruction of cracks and material losses by perimeter-like penalizations and phase-field methods: numerical results

We numerically implement the variational approach for reconstruction in the inverse crack and cavity problems developed by one of the authors. The method is based on a suitably adapted free-discontinuity problem. Its main features are the use of phase-field functions to describe the defects to be reconstructed and the use of perimeter-like penalizations to regularize the ill-posed problem. The numerical implementation is based on the solution of the corresponding optimality system by a gradient method. Numerical simulations are presented to show the validity of the method.Comment: 15 pages, 12 figure

Continuity properties of Neumann-to-Dirichlet maps with respect to the H-convergence of the coefficient matrices

We investigate the continuity of boundary operators, such as the Neumann-to-Dirichlet map, with respect to the coefficient matrices of the underlying elliptic equations. We show that for nonsmooth coefficients the correct notion of convergence is the one provided by H-convergence (or G-convergence for symmetric matrices). We prove existence results for minimum problems associated to variational methods used to solve the so-called inverse conductivity problem, at least if we allow the conductivities to be anisotropic. In the case of isotropic conductivities we show that on certain occasions existence of a minimizer may fail

Regularized Transformation-Optics Cloaking for the Helmholtz Equation: From Partial Cloak to Full Cloak

We develop a very general theory on the regularized approximate invisibility cloaking for the wave scattering governed by the Helmholtz equation in any space dimensions via the approach of transformation optics. There are four major ingredients in our proposed theory: 1) The non-singular cloaking medium is obtained by the push-forwarding construction through a transformation which blows up a subset $K_\varepsilon$ in the virtual space, where $\varepsilon$ is a small positive asymptotic regularization parameter. $K_\varepsilon$ will degenerate to $K_0$ as $\varepsilon$ goes to $0$, and in our theory $K_0$ could be any convex compact set in $\mathbb{R}^N$, or any set whose boundary consists of Lipschitz hypersurfaces, or a finite combination of those sets. 2) A general lossy layer with the material parameters satisfying certain compatibility integral conditions is employed right between the cloaked and cloaking regions. 3)The contents being cloaked could also be extremely general, possibly including, at the same time, generic mediums and, sound-soft, sound-hard and impedance-type obstacles, as well as some sources or sinks. 4) In order to achieve a cloaking device of compact size, particularly for the case when $K_\varepsilon$ is not ``uniformly small", an assembly-by-components, the (ABC) geometry is developed for both the virtual and physical spaces and the blow-up construction is based on concatenating different components. Within the proposed framework, we show that the scattered wave field $u_\varepsilon$ corresponding to a cloaking problem will converge to $u_0$ as $\varepsilon$ goes to $0$, with $u_0$ being the scattered wave field corresponding to a sound-hard $K_0$. The convergence result is used to theoretically justify the approximate full and partial invisibility cloaks, depending on the geometry of $K_0$. On the other hand, the convergence results are conducted in a much more general setting than what is needed for the invisibility cloaking, so they are of significant mathematical interest for their own sake. As for applications, we construct three types of full and partial cloaks. Some numerical experiments are also conducted to illustrate our theoretical results

Technical and environmental characterisation of recycled aggregate for reuse in bricks

Waste mud coming from an aggregate washing plant was formerly used as filling material for a pond, aimed at the recovery of an abandoned quarry. Once completed the filling capacity of the pond, the need for identifying a possible reuse of mud produced by the plant arose in order to avoid landfill disposal. Therefore, mud has been geometrically, physically and chemically characterised for its recovery as construction material. A variety of tests was carried out on mud samples as required by EN technical specifications and by Italian environmental standards, focusing particularly on leaching behaviour. The tested material showed satisfactory physical and chemical properties and a release of pollutants below the limits set by the Italian code. Many mix-designs for the production of unfired bricks made of waste mud, sand and straw, stabilised and non-stabilised with lime, gypsum or cement, were developed. The bricks were tested in order to evaluate mechanical properties and leaching behaviour. Mud bricks provided remarkable compressive strength, even if not suitable for structural elements. The use as interior design to minimise humidity changes and to facilitate a thermal insulation is fostered, thus strengthening the so-called green building economy