83 research outputs found
Team organization may help swarms of flies to become invisible in closed waveguides
We are interested in a time harmonic acoustic problem in a waveguide
containing flies. The flies are modelled by small sound soft obstacles. We
explain how they should arrange to become invisible to an observer sending
waves from and measuring the resulting scattered field at the same
position. We assume that the flies can control their position and/or their
size. Both monomodal and multimodal regimes are considered. On the other hand,
we show that any sound soft obstacle (non necessarily small) embedded in the
waveguide always produces some non exponentially decaying scattered field at
for wavenumbers smaller than a constant that we explicit. As a
consequence, for such wavenumbers, the flies cannot be made completely
invisible to an observer equipped with a measurement device located at
Oscillating behaviour of the spectrum for a plasmonic problem in a domain with a rounded corner
We investigate the eigenvalue problem in a 2D domain divided into two regions
. We are interested in situations where takes positive
values on and negative ones on . Such problems appear
in time harmonic electromagnetics in the modeling of plasmonic technologies. In
a recent work [15], we highlighted an unusual instability phenomenon for the
source term problem associated with : for certain
configurations, when the interface between the subdomains
presents a rounded corner, the solution may depend critically on the value of
the rounding parameter. In the present article, we explain this property
studying the eigenvalue problem . We provide an asymptotic
expansion of the eigenvalues and prove error estimates. We establish an
oscillatory behaviour of the eigenvalues as the rounding parameter of the
corner tends to zero. We end the paper illustrating this phenomenon with
numerical experiments.Comment: Mathematical Modelling and Numerical Analysis (ESAIM: M2AN),
09/12/2016. arXiv admin note: text overlap with arXiv:1304.478
Generalized impedance boundary conditions with vanishing or sign-changing impedance
We consider a Laplace type problem with a generalized impedance boundary
condition of the form on a flat part
of the boundary. Here is the outward unit normal vector to
, is the impedance parameter and is the coordinate
along . Such problems appear for example in the modelling of small
perturbations of the boundary. In the literature, the cases or
have been investigated. In this work, we address situations where
contains the origin and or
g(x)=-\mbox{sign}(x)|x|^\alpha with . In other words, we study
cases where vanishes at the origin and changes its sign. The main message
is that the well-posedness in the Fredholm sense of the corresponding problems
depends on the value of . For , we show that the
associated operators are Fredholm of index zero while it is not the case when
. The proof of the first results is based on the reformulation as 1D
problems combined with the derivation of compact embedding results for the
functional spaces involved in the analysis. The proof of the second results
relies on the computation of singularities and the construction of Weyl's
sequences. We also discuss the equivalence between the strong and weak
formulations, which is not straightforward. Finally, we provide simple
numerical experiments which seem to corroborate the theorems
Non-scattering wavenumbers and far field invisibility for a finite set of incident/scattering directions
We investigate a time harmonic acoustic scattering problem by a penetrable
inclusion with compact support embedded in the free space. We consider cases
where an observer can produce incident plane waves and measure the far field
pattern of the resulting scattered field only in a finite set of directions. In
this context, we say that a wavenumber is a non-scattering wavenumber if the
associated relative scattering matrix has a non trivial kernel. Under certain
assumptions on the physical coefficients of the inclusion, we show that the
non-scattering wavenumbers form a (possibly empty) discrete set. Then, in a
second step, for a given real wavenumber and a given domain D, we present a
constructive technique to prove that there exist inclusions supported in D for
which the corresponding relative scattering matrix is null. These inclusions
have the important property to be impossible to detect from far field
measurements. The approach leads to a numerical algorithm which is described at
the end of the paper and which allows to provide examples of (approximated)
invisible inclusions.Comment: 20 pages, 7 figure
A curious instability phenomenon for a rounded corner in presence of a negative material
We study a 2D scalar harmonic wave transmission problem between a classical
dielectric and a medium with a real-valued negative permittivity/permeability
which models a metal at optical frequency or an ideal negative metamaterial. We
highlight an unusual instability phenomenon for this problem when the interface
between the two media presents a rounded corner. To establish this result, we
provide an asymptotic expansion of the solution, when it is well-defined, in
the geometry with a rounded corner. Then, we prove error estimates. Finally, a
careful study of the asymptotic expansion allows us to conclude that the
solution, when it is well-defined, depends critically on the value of the
rounding parameter. We end the paper with a numerical illustration of this
instability phenomenon
Spectrum of the Laplacian with mixed boundary conditions in a chamfered quarter of layer
We investigate the spectrum of a Laplace operator with mixed boundary
conditions in an unbounded chamfered quarter of layer. The geometry depends on
two parameters gathered in some vector which
characterizes the domain at the edges. We identify the essential spectrum and
establish different results concerning the discrete spectrum with respect to
. In particular, we show that for a given , there is some
such that discrete spectrum exists for
whereas it is empty for
. The proofs rely on classical arguments of
spectral theory such as the max-min principle. The main originality lies rather
in the delicate use of the features of the geometry
Compact imbeddings in electromagnetism with interfaces between classical materials and meta-materials
21 pagesIn a meta-material, the electric permittivity and/or the magnetic permeability can be negative in given frequency ranges. We investigate the solution of the time-harmonic Maxwell equations in a composite material, made up of classical materials, and meta-materials with negative electric permittivity, in a two-dimensional bounded domain Ω. We study the imbedding of the space of electric fields into LÂČ(Ω)ÂČ. In particular, we extend the famous result of Weber, proving that it is compact. This result is obtained by studying the regularity of the fields. We first isolate their most singular part, using a decomposition Ă la Birman-Solomyak. With the help of the Mellin transform, we prove that this singular part belongs to HËą(Ω)ÂČ, for some s > 0. Finally, we show that the compact imbedding result holds as soon as no ratio of permittivities between two adjacent materials is equal to â1
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