1,171 research outputs found
Numerical Analysis of Three-dimensional Acoustic Cloaks and Carpets
We start by a review of the chronology of mathematical results on the
Dirichlet-to-Neumann map which paved the way towards the physics of
transformational acoustics. We then rederive the expression for the
(anisotropic) density and bulk modulus appearing in the pressure wave equation
written in the transformed coordinates. A spherical acoustic cloak consisting
of an alternation of homogeneous isotropic concentric layers is further
proposed based on the effective medium theory. This cloak is characterised by a
low reflection and good efficiency over a large bandwidth for both near and far
fields, which approximates the ideal cloak with a inhomogeneous and anisotropic
distribution of material parameters. The latter suffers from singular material
parameters on its inner surface. This singularity depends upon the sharpness of
corners, if the cloak has an irregular boundary, e.g. a polyhedron cloak
becomes more and more singular when the number of vertices increases if it is
star shaped. We thus analyse the acoustic response of a non-singular spherical
cloak designed by blowing up a small ball instead of a point, as proposed in
[Kohn, Shen, Vogelius, Weinstein, Inverse Problems 24, 015016, 2008]. The
multilayered approximation of this cloak requires less extreme densities
(especially for the lowest bound). Finally, we investigate another type of
non-singular cloaks, known as invisibility carpets [Li and Pendry, Phys. Rev.
Lett. 101, 203901, 2008], which mimic the reflection by a flat ground.Comment: Latex, 21 pages, 7 Figures, last version submitted to Wave Motion.
OCIS Codes: (000.3860) Mathematical methods in physics; (260.2110)
Electromagnetic theory; (160.3918) Metamaterials; (160.1190) Anisotropic
optical materials; (350.7420) Waves; (230.1040) Acousto-optical devices;
(160.1050) Acousto-optical materials; (290.5839) Scattering,invisibility;
(230.3205) Invisibility cloak
Reconstruction of generic anisotropic stiffness tensors from partial data around one polarization
We study inverse problems in anisotropic elasticity using tools from
algebraic geometry. The singularities of solutions to the elastic wave equation
in dimension with an anisotropic stiffness tensor have propagation
kinematics captured by so-called slowness surfaces, which are hypersurfaces in
the cotangent bundle of that turn out to be algebraic varieties.
Leveraging the algebraic geometry of families of slowness surfaces we show
that, for tensors in a dense open subset in the space of all stiffness tensors,
a small amount of data around one polarization in an individual slowness
surface uniquely determines the entire slowness surface and its stiffness
tensor. Such partial data arises naturally from geophysical measurements or
geometrized versions of seismic inverse problems. Additionally, we explain how
the reconstruction of the stiffness tensor can be carried out effectively,
using Gr\"obner bases. Our uniqueness results fail for very symmetric (e.g.,
fully isotropic) materials, evidencing the counterintuitive claim that inverse
problems in elasticity can become more tractable with increasing asymmetry.Comment: 39 pages, 4 figures. Computer Code included in the ancillary files
folde
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
Cloaking for a quasi-linear elliptic partial differential equation
In this article we consider cloaking for a quasi-linear elliptic partial
differential equation of divergence type defined on a bounded domain in
for . We show that a perfect cloak can be obtained via a
singular change of variables scheme and an approximate cloak can be achieved
via a regular change of variables scheme. These approximate cloaks though
non-degenerate are anisotropic. We also show, within the framework of
homogenization, that it is possible to get isotropic regular approximate
cloaks. This work generalizes to quasi-linear settings previous work on
cloaking in the context of Electrical Impedance Tomography for the conductivity
equation
- …