648 research outputs found
Nearly cloaking the elastic wave fields
In this work, we develop a general mathematical framework on regularized
approximate cloaking of elastic waves governed by the Lam\'e system via the
approach of transformation elastodynamics. Our study is rather comprehensive.
We first provide a rigorous justification of the transformation elastodynamics.
Based on the blow-up-a-point construction, elastic material tensors for a
perfect cloak are derived and shown to possess singularities. In order to avoid
the singular structure, we propose to regularize the blow-up-a-point
construction to be the blow-up-a-small-region construction. However, it is
shown that without incorporating a suitable lossy layer, the regularized
construction would fail due to resonant inclusions. In order to defeat the
failure of the lossless construction, a properly designed lossy layer is
introduced into the regularized cloaking construction . We derive sharp
asymptotic estimates in assessing the cloaking performance. The proposed
cloaking scheme is capable of nearly cloaking an arbitrary content with a high
accuracy
Determination of singular time-dependent coefficients for wave equations from full and partial data
We study the problem of determining uniquely a time-dependent singular
potential , appearing in the wave equation in with and a
bounded domain of , . We start by considering the unique
determination of some singular time-dependent coefficients from observations on
. Then, by weakening the singularities of the set of admissible
coefficients, we manage to reduce the set of data that still guaranties unique
recovery of such a coefficient. To our best knowledge, this paper is the first
claiming unique determination of unbounded time-dependent coefficients, which
is motivated by the problem of determining general nonlinear terms appearing in
nonlinear wave equations
Uniqueness and factorization method for inverse elastic scattering with a single incoming wave
The first part of this paper is concerned with the uniqueness to inverse
time-harmonic elastic scattering from bounded rigid obstacles in two
dimensions. It is proved that a connected polygonal obstacle can be uniquely
identified by the far-field pattern over all observation directions
corresponding to a single incident plane wave. Our approach is based on a new
reflection principle for the first boundary value problem of the Navier
equation. In the second part, we propose a revisited factorization method to
recover a rigid elastic body with a single far-field pattern
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