305 research outputs found
Cloaking via mapping for the heat equation
This paper explores the concept of near-cloaking in the context of
time-dependent heat propagation. We show that after the lapse of a certain
threshold time instance, the boundary measurements for the homogeneous heat
equation are close to the cloaked heat problem in a certain Sobolev space norm
irrespective of the density-conductivity pair in the cloaked region. A
regularised transformation media theory is employed to arrive at our results.
Our proof relies on the study of the long time behaviour of solutions to the
parabolic problems with high contrast in density and conductivity coefficients.
It further relies on the study of boundary measurement estimates in the
presence of small defects in the context of steady conduction problem. We then
present some numerical examples to illustrate our theoretical results.Comment: 31 pages, 7 figures, 1 tabl
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
Enhanced Microscale Hydrodynamic Near-cloaking using Electro-osmosis
In this paper, we develop a general mathematical framework for enhanced
hydrodynamic near-cloaking of electro-osmotic flow for more complex shapes,
which is obtained by simultaneously perturbing the inner and outer boundaries
of the perfect cloaking structure. We first derive the asymptotic expansions of
perturbed fields and obtain a first-order coupled system. We then establish the
representation formula of the solution to the first-order coupled system using
the layer potential techniques. Based on the asymptotic analysis, the enhanced
hydrodynamic near-cloaking conditions are derived for the control region with
general cross-sectional shape. The conditions reveal the inner relationship
between the shapes of the object and the control region. Especially, for the
shape of a deformed annulus or confocal ellipses cylinder, the cloaking
conditions and relationship of shapes are quantified more accurately. Our
theoretical findings are validated and supplemented by a variety of numerical
results. The results in this paper also provide a mathematical foundation for
more complex hydrodynamic cloaking
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