Search

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 $${N \geq 2}$$ 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 that blows up a subset $${K_\varepsilon}$$ in the virtual space, where $${\varepsilon \ll 1}$$ is an asymptotic regularization parameter. $${K_\varepsilon}$$ will degenerate to K as $${\varepsilon \rightarrow +0}$$ , and in our theory K 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 as $${\varepsilon \rightarrow +0}$$ , with u being the scattered wave field corresponding to a sound-hard K. The convergence result is used to theoretically justify the approximate full and partial invisibility cloaks, depending on the geometry of K. 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. [ABSTRACT FROM AUTHOR]

We consider approximate cloaking from a regularization viewpoint introduced in Kohn et al (2008 Inverse Problems 24 015016) for EIT and further investigated in Kohn et al (2010 Commun. Pure Appl. Math. 63 0973-1016) and Liu (2009 Inverse Problems 25 045006) for the Helmholtz equation. The cloaking schemes given by Kohn et al and Liu are shown to be (optimally) within | ln ρ|-1 in 2D and ρ in 3D of perfect cloaking, where ρ denotes the regularization parameter. In this paper, we show that by employing a soundhard layer right outside the cloaked region, one could (optimally) achieve ρN in RN, N ≥ 2, which significantly enhances the near-cloak. We then develop a cloaking scheme by making use of a lossy layer with well-chosen parameters. The lossy-layer cloaking scheme is shown to possess the same cloaking performance as the one with a sound-hard layer. Moreover, it is shown that the lossy layer could be taken as a finite realization of the sound-hard layer. Numerical experiments are also presented to assess the cloaking performances of all the cloaking schemes for comparisons. [ABSTRACT FROM AUTHOR]

Abstract: We analyze higher order finite elements applied to second order elliptic interface problems. Our a priori error estimates in the - and -norm are expressed in terms of the approximation order p and a parameter δ that quantifies how well the interface is resolved by the finite element mesh. The optimal p-th order convergence in the -norm is only achieved under stringent assumptions on δ, namely, . Under weaker conditions on δ, optimal a priori estimates can be established in the - and in the -norm, where is a subdomain that excludes a tubular neighborhood of the interface of width . In particular, if the interface is approximated by an interpolation spline of order p and if full regularity is assumed, then optimal convergence orders and p for the approximation in the - and the -norm can be expected but not order p for the approximation in the -norm. Numerical examples in 2D and 3D illustrate and confirm our theoretical results. [Copyright &y& Elsevier]

The aim of this paper is to compute the mean field and variance of solutions to three-dimensional Maxwell's equations with random interfaces via shape calculus and pivoted low-rank approximation. Based on the perturbation theory and shape calculus, we characterize the statistical moments of solutions to Maxwell's equations with random interfaces in terms of the perturbation magnitude via the first order shape-Taylor expansion. In order to capture oscillations with high resolution close to the interface, an adaptive finite element method using Nédélec's third order edge elements of the first kind is employed to solve the deterministic Maxwell's equations with the mean interface to approximate the expectation of solutions. For the second moment computation, an efficient low-rank approximation of the pivoted Cholesky decomposition is proposed to compute the two-point correlation function to approximate the variance of solutions. Numerical experiments are presented to demonstrate our theoretical results. [ABSTRACT FROM AUTHOR]