288 research outputs found
High frequency homogenisation for elastic lattices
A complete methodology, based on a two-scale asymptotic approach, that
enables the homogenisation of elastic lattices at non-zero frequencies is
developed. Elastic lattices are distinguished from scalar lattices in that two
or more types of coupled waves exist, even at low frequencies. Such a theory
enables the determination of effective material properties at both low and high
frequencies. The theoretical framework is developed for the propagation of
waves through lattices of arbitrary geometry and dimension. The asymptotic
approach provides a method through which the dispersive properties of lattices
at frequencies near standing waves can be described; the theory accurately
describes both the dispersion curves and the response of the lattice near the
edges of the Brillouin zone. The leading order solution is expressed as a
product between the standing wave solution and long-scale envelope functions
that are eigensolutions of the homogenised partial differential equation. The
general theory is supplemented by a pair of illustrative examples for two
archetypal classes of two-dimensional elastic lattices. The efficiency of the
asymptotic approach in accurately describing several interesting phenomena is
demonstrated, including dynamic anisotropy and Dirac cones.Comment: 24 pages, 7 figure
Long- and short-time asymptotics of the first-passage time of the Ornstein-Uhlenbeck and other mean-reverting processes
The first-passage problem of the Ornstein-Uhlenbeck process to a boundary is
a long-standing problem with no known closed-form solution except in specific
cases. Taking this as a starting-point, and extending to a general
mean-reverting process, we investigate the long- and short-time asymptotics
using a combination of Hopf-Cole and Laplace transform techniques. As a result
we are able to give a single formula that is correct in both limits, as well as
being exact in certain special cases. We demonstrate the results using a
variety of other models
The homogenisation of Maxwell's equations with applications to photonic crystals and localised waveforms on metafilms
An asymptotic theory is developed to generate equations that model the global
behaviour of electromagnetic waves in periodic photonic structures when the
wavelength is not necessarily long relative to the periodic cell dimensions;
potentially highly-oscillatory short-scale detail is encapsulated through
integrated quantities.
The theory we develop is then applied to two topical examples, the first
being the case of aligned dielectric cylinders, which has great importance in
the modelling of photonic crystal fibres. We then consider the propagation of
waves in a structured metafilm, here chosen to be a planar array of dielectric
spheres. At certain frequencies strongly directional dynamic anisotropy is
observed, and the asymptotic theory is shown to capture the effect, giving
highly accurate qualitative and quantitative results as well as providing
interpretation for the underlying change from elliptic to hyperbolic behaviour
Analytical approximation to the multidimensional Fokker--Planck equation with steady state
The Fokker--Planck equation is a key ingredient of many models in physics,
and related subjects, and arises in a diverse array of settings. Analytical
solutions are limited to special cases, and resorting to numerical simulation
is often the only route available; in high dimensions, or for parametric
studies, this can become unwieldy. Using asymptotic techniques, that draw upon
the known Ornstein--Uhlenbeck (OU) case, we consider a mean-reverting system
and obtain its representation as a product of terms, representing short-term,
long-term, and medium-term behaviour. A further reduction yields a simple
explicit formula, both intuitive in terms of its physical origin and fast to
evaluate. We illustrate a breadth of cases, some of which are `far' from the OU
model, such as double-well potentials, and even then, perhaps surprisingly, the
approximation still gives very good results when compared with numerical
simulations. Both one- and two-dimensional examples are considered.Comment: Updated version as publishe
Spectral algorithms for reaction-diffusion equations
A collection of codes (in MATLAB & Fortran 77), and examples, for solving reaction-diffusion equations in one and two space dimensions is presented. In areas of the mathematical community spectral methods are used to remove the stiffness associated with the diffusive terms in a reaction-diffusion model allowing explicit high order timestepping to be used. This is particularly valuable for two (and higher) space dimension problems. Our aim here is to provide codes, together with examples, to allow practioners to easily utilize, understand and implement these ideas; we incorporate recent theoretical advances such as exponential time differencing methods and provide timings and error comparisons with other more standard approaches. The examples are chosen from the literature to illustrate points and queries that naturally arise
Asymptotic approximations for Bloch waves and topological mode steering in a planar array of Neumann scatterers
We study the canonical problem of wave scattering by periodic arrays, either of infinite or finite extent, of Neumann scatterers in the plane; the characteristic lengthscale of the scatterers is considered small relative to the lattice period. We utilise the method of matched asymptotic expansions, together with Fourier series representations, to create an efficient and accurate numerical approach for finding the dispersion curves associated with Floquet-Bloch waves through an infinite array of scatterers. The approach also lends itself to direct scattering problems for finite arrays and we illustrate the flexibility of these asymptotic representations on some topical examples from topological wave physics
Geometrically navigating topological platonic modes around gentle and sharp bends
Predictive theory to geometrically engineer devices and materials in continuum systems to have desired topological-like effects is developed here by bridging the gap between quantum and continuum mechanical descriptions. A structured elastic plate, a bosoniclike system in the language of quantum mechanics, is shown to exhibit topological valley modes despite the system having no direct physical connection to quantum effects. We emphasize a predictive, first-principles, approach, the strength of which is demonstrated by the ability to design well-defined broadband edge states, resistant to backscatter, using geometric differences; the mechanism underlying energy transfer around gentle and sharp corners is described. Using perturbation methods and group theory, several distinct cases of symmetry-induced Dirac cones, which when gapped yield nontrivial band gaps, are identified and classified. The propagative behavior of the edge states around gentle or sharp bends depends strongly upon the symmetry class of the bulk media and we illustrate this via numerical simulations
Fluid-loaded metasurfaces
We consider wave propagation along fluid-loaded structures which take the
form of an elastic plate augmented by an array of resonators forming a
metasurface, that is, a surface structured with sub-wavelength resonators. Such
surfaces have had considerable recent success for the control of wave
propagation in electromagnetism and acoustics, by combining the vision of
sub-wavelength wave manipulation, with the design, fabrication and size
advantages associated with surface excitation. We explore one aspect of recent
interest in this field: graded metasurfaces, but within the context of
fluid-loaded structures.
Graded metasurfaces allow for selective spatial frequency separation and are
often referred to as exhibiting rainbow trapping. Experiments, and theory, have
been developed for acoustic, electromagnetic, and even elastic, rainbow devices
but this has not been approached for fluid-loaded structures that support
surface waves coupled with the acoustic field in a bulk fluid. This surface
wave, coupled with the fluid, can be used to create an additional effect by
designing a metasurface to mode convert from surface to bulk waves. We
demonstrate that sub-wavelength control is possible and that one can create
both rainbow trapping and mode conversion phenomena for a fluid-loaded elastic
plate model.Comment: 13 pages, 10 figure
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