9,829 research outputs found
A transmission problem across a fractal self-similar interface
We consider a transmission problem in which the interior domain has
infinitely ramified structures. Transmission between the interior and exterior
domains occurs only at the fractal component of the interface between the
interior and exterior domains. We also consider the sequence of the
transmission problems in which the interior domain is obtained by stopping the
self-similar construction after a finite number of steps; the transmission
condition is then posed on a prefractal approximation of the fractal interface.
We prove the convergence in the sense of Mosco of the energy forms associated
with these problems to the energy form of the limit problem. In particular,
this implies the convergence of the solutions of the approximated problems to
the solution of the problem with fractal interface. The proof relies in
particular on an extension property. Emphasis is put on the geometry of the
ramified domain. The convergence result is obtained when the fractal interface
has no self-contact, and in a particular geometry with self-contacts, for which
an extension result is proved
Photonic band-gap engineering for volume plasmon polaritons in multiscale multilayer hyperbolic metamaterials
We theoretically study the propagation of large-wavevector waves (volume
plasmon polaritons) in multilayer hyperbolic metamaterials with two levels of
structuring. We show that when the parameters of a subwavelength
metal-dielectric multilayer ("substructure") are modulated ("superstructured")
on a larger, wavelength scale, the propagation of volume plasmon polaritons in
the resulting multiscale hyperbolic metamaterials is subject to photonic band
gap phenomena. A great degree of control over such plasmons can be exerted by
varying the superstructure geometry. When this geometry is periodic, stop bands
due to Bragg reflection form within the volume plasmonic band. When a cavity
layer is introduced in an otherwise periodic superstructure, resonance peaks of
the Fabry-Perot nature are present within the stop bands. More complicated
superstructure geometries are also considered. For example, fractal Cantor-like
multiscale metamaterials are found to exhibit characteristic self-similar
spectral signatures in the volume plasmonic band. Multiscale hyperbolic
metamaterials are shown to be a promising platform for large-wavevector bulk
plasmonic waves, whether they are considered for use as a new kind of
information carrier or for far-field subwavelength imaging.Comment: 12 pages, 10 figures, now includes Appendix
Short time heat diffusion in compact domains with discontinuous transmission boundary conditions
We consider a heat problem with discontinuous diffusion coefficientsand
discontinuous transmission boundary conditions with a resistancecoefficient.
For all compact -domains with a
-set boundary (for instance, aself-similar fractal), we find the first term
of the small-timeasymptotic expansion of the heat content in the complement
of, and also the second-order term in the case of a regularboundary.
The asymptotic expansion is different for the cases offinite and infinite
resistance of the boundary. The derived formulasrelate the heat content to the
volume of the interior Minkowskisausage and present a mathematical
justification to the de Gennes'approach. The accuracy of the analytical results
is illustrated bysolving the heat problem on prefractal domains by a finite
elementsmethod
A Finite Element Approach to Modelling Fractal Ultrasonic Transducers
Piezoelectric ultrasonic transducers usually employ composite structures to improve their transmission and reception sensitivities. The geometry of the composite is regular with one dominant length scale and, since these are resonant devices, this dictates the central operating frequency of the device. In order to construct a wide bandwith device it would seem natural therefore to utilize resonators that span a range of length scales. In this article we derive a mathematical model to predict the dynamics of a fractal ultrasound transducer; the fractal in this case being the Sierpinski gasket. Expressions for the electrical and mechanical fields that are contained within this structure are expressed in terms of a finite element basis. The propagation of an ultrasonic wave in this transducer is then analyzed and used to derive expressions for the non-dimensionalised electrical impedance and the transmission and reception sensitivities as a function of the driving frequency. Comparing these key performance measures to an equivalent standard (Euclidean) design shows some benefits of these fractal designs
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