Efficient evaluation of highly oscillatory acoustic scattering surface integrals

We consider the approximation of some highly oscillatory weakly singular surface integrals, arising from boundary integral methods with smooth global basis functions for solving problems of high frequency acoustic scattering by three-dimensional convex obstacles, described globally in spherical coordinates. As the frequency of the incident wave increases, the performance of standard quadrature schemes deteriorates. Naive application of asymptotic schemes also fails due to the weak singularity. We propose here a new scheme based on a combination of an asymptotic approach and exact treatment of singularities in an appropriate coordinate system. For the case of a spherical scatterer we demonstrate via error analysis and numerical results that, provided the observation point is sufficiently far from the shadow boundary, a high level of accuracy can be achieved with a minimal computational cost.

Boundary integral methods in high frequency scattering

In this article we review recent progress on the design, analysis and implementation of numerical-asymptotic boundary integral methods for the computation of frequency-domain acoustic scattering in a homogeneous unbounded medium by a bounded obstacle. The main aim of the methods is to allow computation of scattering at arbitrarily high frequency with finite computational resources.

Numerical-asymptotic boundary integral methods in high-frequency acoustic scattering

In this article we describe recent progress on the design, analysis and implementation of hybrid numerical-asymptotic boundary integral methods for boundary value problems for the Helmholtz equation that model time harmonic acoustic wave scattering in domains exterior to impenetrable obstacles. These hybrid methods combine conventional piecewise polynomial approximations with high-frequency asymptotics to build basis functions suitable for representing the oscillatory solutions. They have the potential to solve scattering problems accurately in a computation time that is (almost) independent of frequency and this has been realized for many model problems. The design and analysis of this class of methods requires new results on the analysis and numerical analysis of highly oscillatory boundary integral operators and on the high-frequency asymptotics of scattering problems. The implementation requires the development of appropriate quadrature rules for highly oscillatory integrals. This article contains a historical account of the development of this currently very active field, a detailed account of recent progress and, in addition, a number of original research results on the design, analysis and implementation of these methods.

Top down influence on visuo-tactile interaction modulates neural oscillatory responses

Multisensory integration involves bottom-up as well as top-down processes. We investigated the influences of top-down control on the neural responses to multisensory stimulation using EEG recording and time-frequency analyses. Participants were stimulated at the index or thumb of the left hand, using tactile vibrators mounted on a foam cube. Simultaneously they received a visual distractor from a light emitting diode adjacent to the active vibrator (spatially congruent trial) or adjacent to the inactive vibrator (spatially incongruent trial). The task was to respond to the elevation of the tactile stimulus (upper or lower), while ignoring the simultaneous visual distractor. To manipulate top-down control on this multisensory stimulation, the proportion of spatially congruent (vs. incongruent) trials was changed across blocks. Our results reveal that the behavioral cost of responding to incongruent than congruent trials (i.e., the crossmodal congruency effect) was modulated by the proportion of congruent trials. Most importantly, the EEG gamma band response and the gamma-theta coupling were also affected by this modulation of top-down control, whereas the late theta band response related to the congruency effect was not. These findings suggest that gamma band response is more than a marker of multisensory binding, being also sensitive to the correspondence between expected and actual multisensory stimulation. By contrast, theta band response was affected by congruency but appears to be largely immune to stimulation expectancy.

A high frequency $hp$ boundary element method for scattering by convex polygons

In this paper we propose and analyze a hybrid $hp$ boundary element method for the solution of problems of high frequency acoustic scattering by sound-soft convex polygons, in which the approximation space is enriched with oscillatory basis functions which efficiently capture the high frequency asymptotics of the solution. We demonstrate, both theoretically and via numerical examples, exponential convergence with respect to the order of the polynomials, moreover providing rigorous error estimates for our approximations to the solution and to the far field pattern, in which the dependence on the frequency of all constants is explicit. Importantly, these estimates prove that, to achieve any desired accuracy in the computation of these quantities, it is sufficient to increase the number of degrees of freedom in proportion to the logarithm of the frequency as the frequency increases, in contrast to the at least linear growth required by conventional methods.

Effect of building gap to improve pedestrian comfort levels and ventilation

Rapid rates of urbanization have resulted into increased concerns of urban environment. Amongst them, wind and thermal comfort levels for pedestrians have attracted research interest. In this regards, urban wind environment is seen as a crucial components that can lead to improved thermal comfort levels for pedestrian population. High rise building in modern urban setting causes high levels of turbulence that renders discomfort to pedestrians. Additionally, a higher frequency of high ris e buildings at a particular region acts as a shield against the wind flow to the lower buildings beyond them resulting into higher levels of discomfort to users or residents. Studies conducted on developing wind flow models using Computational Fluid Dynami cs (CFD) simulations have revealed improvement in interval to height ratios can results into improved wind flow within the simulation grid. However, high value and demand for land in urban areas renders expansion to be an impractical solution. Nonetheless, innovative utilization of architectural concepts can be imagined to improve the pedestrian comfort levels through improved wind permeability. This paper assesses the possibility of through-building gaps being a solution to improve pedestrian comfort levels.

A high frequency boundary element method for scattering by a class of nonconvex obstacles

In this paper we propose and analyse a hybrid numerical-asymptotic boundary element method for the solution of problems of high frequency acoustic scattering by a class of sound-soft nonconvex polygons. The approximation space is enriched with carefully chosen oscillatory basis functions; these are selected via a study of the high frequency asymptotic behaviour of the solution. We demonstrate via a rigorous error analysis, supported by numerical examples, that to achieve any desired accuracy it is sufficient for the number of degrees of freedom to grow only in proportion to the logarithm of the frequency as the frequency increases, in contrast to the at least linear growth required by conventional methods. This appears to be the first such numerical analysis result for any problem of scattering by a nonconvex obstacle. Our analysis is based on new frequency-explicit bounds on the normal derivative of the solution on the boundary and on its analytic continuation into the complex plane.

A frequency-independent boundary element method for scattering by two-dimensional screens and apertures

We propose and analyse a hybrid numerical–asymptotic hp boundary element method (BEM) for time-harmonic scattering of an incident plane wave by an arbitrary collinear array of sound-soft two-dimensional screens. Our method uses an approximation space enriched with oscillatory basis functions, chosen to capture the high-frequency asymptotics of the solution. We provide a rigorous frequency-explicit error analysis which proves that the method converges exponentially as the number of degrees of freedom N increases, and that to achieve any desired accuracy it is sufficient to increase N in proportion to the square of the logarithm of the frequency as the frequency increases (standard BEMs require N to increase at least linearly with frequency to retain accuracy). Our numerical results suggest that fixed accuracy can in fact be achieved at arbitrarily high frequencies with a frequency-independent computational cost, when the oscillatory integrals required for implementation are computed using Filon quadrature. We also show how our method can be applied to the complementary ‘breakwater’ problem of propagation through an aperture in an infinite sound-hard screen.

A case study of boundary layer ventilation by convection and coastal processes

It is often assumed that ventilation of the atmospheric boundary layer is weak in the absence of fronts, but is this always true? In this paper we investigate the processes responsible for ventilation of the atmospheric boundary layer during a nonfrontal day that occurred on 9 May 2005 using the UK Met Office Unified Model. Pollution sources are represented by the constant emission of a passive tracer everywhere over land. The ventilation processes observed include shallow convection, turbulent mixing followed by large-scale ascent, a sea breeze circulation and coastal outflow. Vertical distributions of tracer are validated qualitatively with AMPEP (Aircraft Measurement of chemical Processing Export fluxes of Pollutants over the UK) CO aircraft measurements and are shown to agree impressively well. Budget calculations of tracers are performed in order to determine the relative importance of these ventilation processes. Coastal outflow and the sea breeze circulation were found to ventilate 26% of the boundary layer tracer by sunset of which 2% was above 2 km. A combination of coastal outflow, the sea breeze circulation, turbulent mixing and large-scale ascent ventilated 46% of the boundary layer tracer, of which 10% was above 2 km. Finally, coastal outflow, the sea breeze circulation, turbulent mixing, large-scale ascent and shallow convection together ventilated 52% of the tracer into the free troposphere, of which 26% was above 2 km. Hence this study shows that significant ventilation of the boundary layer can occur in the absence of fronts (and thus during high-pressure events). Turbulent mixing and convection processes can double the amount of pollution ventilated from the boundary layer.

A collocation method for high frequency scattering by convex polygons

We consider the problem of scattering of a time-harmonic acoustic incident plane wave by a sound soft convex polygon. For standard boundary or finite element methods, with a piecewise polynomial approximation space, the computational cost required to achieve a prescribed level of accuracy grows linearly with respect to the frequency of the incident wave. Recently Chandler–Wilde and Langdon proposed a novel Galerkin boundary element method for this problem for which, by incorporating the products of plane wave basis functions with piecewise polynomials supported on a graded mesh into the approximation space, they were able to demonstrate that the number of degrees of freedom required to achieve a prescribed level of accuracy grows only logarithmically with respect to the frequency. Here we propose a related collocation method, using the same approximation space, for which we demonstrate via numerical experiments a convergence rate identical to that achieved with the Galerkin scheme, but with a substantially reduced computational cost.

A Galerkin boundary element method for high frequency scattering by convex polygons

In this paper we consider the problem of time-harmonic acoustic scattering in two dimensions by convex polygons. Standard boundary or finite element methods for acoustic scattering problems have a computational cost that grows at least linearly as a function of the frequency of the incident wave. Here we present a novel Galerkin boundary element method, which uses an approximation space consisting of the products of plane waves with piecewise polynomials supported on a graded mesh, with smaller elements closer to the corners of the polygon. We prove that the best approximation from the approximation space requires a number of degrees of freedom to achieve a prescribed level of accuracy that grows only logarithmically as a function of the frequency. Numerical results demonstrate the same logarithmic dependence on the frequency for the Galerkin method solution. Our boundary element method is a discretization of a well-known second kind combined-layer-potential integral equation. We provide a proof that this equation and its adjoint are well-posed and equivalent to the boundary value problem in a Sobolev space setting for general Lipschitz domains.

Detection of Parkinson’s disease tremor-onset with a frequency analysing artificial neural network

Deep Brain Stimulator devices are becoming widely used for therapeutic benefits in movement disorders such as Parkinson's disease. Prolonging the battery life span of such devices could dramatically reduce the risks and accumulative costs associated with surgical replacement. This paper demonstrates how an artificial neural network can be trained using pre-processing frequency analysis of deep brain electrode recordings to detect the onset of tremor in Parkinsonian patients. Implementing this solution into an 'intelligent' neurostimulator device will remove the need for continuous stimulation currently used, and open up the possibility of demand-driven stimulation. Such a methodology could potentially decrease the power consumption of a deep brain pulse generator.

Boundary-layer ventilation by baroclinic lifecycles

Ventilation of the boundary layer has an important effect on local and regional air quality and is a prerequisite for long-range pollution transport. Once in the free troposphere, pollutants can alter the chemical composition of the troposphere and impact on the Earth's radiative forcing. Idealised baroclinic life cycles, LC1 and LC2, have been simulated in a three-dimensional dry hemispheric model in the presence of boundary-layer turbulent fluxes. A passive tracer is added to the simulations to represent pollution emitted at, or near, the surface. A simple conveyor-belt diagnostic is developed to objectively identify regions of the boundary layer that can be ventilated by either warm or cold conveyor belts. Transport of pollutants within and above the boundary layer is examined on synoptic scales. Three different physical mechanisms are found to interact with each other to ventilate pollutants out of the boundary layer. These mechanisms are turbulent mixing within the boundary layer, horizontal advection due to Ekman convergence and divergence within the boundary layer, and advection by the warm conveyor belt. The mass of tracer ventilated by the two life cycles is remarkably similar given the differences in frontal structure, suggesting that the large-scale baroclinicity is an effective constraint on ventilation.

The frequency and dynamics of stratospheric sudden warmings in the 21st century

Changes to stratospheric sudden warmings (SSWs) over the coming century, as predicted by the Geophysical Fluid Dynamics Laboratory (GFDL) chemistry climate model [Atmospheric Model With Transport and Chemistry (AMTRAC)], are investigated in detail. Two sets of integrations, each a three-member ensemble, are analyzed. The first set is driven with observed climate forcings between 1960 and 2004; the second is driven with climate forcings from a coupled model run, including trace gas concentrations representing a midrange estimate of future anthropogenic emissions between 1990 and 2099. A small positive trend in the frequency of SSWs is found. This trend, amounting to 1 event/decade over a century, is statistically significant at the 90% confidence level and is consistent over the two sets of model integrations. Comparison of the model SSW climatology between the late 20th and 21st centuries shows that the increase is largest toward the end of the winter season. In contrast, the dynamical properties are not significantly altered in the coming century, despite the increase in SSW frequency. Owing to the intrinsic complexity of our model, the direct cause of the predicted trend in SSW frequency remains an open question.