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.

A cell by cell anisotropic adaptive mesh ALE method

A cell by cell anisotropic adaptive mesh Arbitrary Lagrangian Eulerian (ALE) method for the solution of the Euler equations is described. An efficient approach to equipotential mesh relaxation on anisotropically refined meshes is developed. Results for two test problems are presented.

Modelling the hydraulics of the Carlisle 2005 flood event

The performance of a 2D numerical model of flood hydraulics is tested for a major event in Carlisle, UK, in 2005. This event is associated with a unique data set, with GPS surveyed wrack lines and flood extent surveyed 3 weeks after the flood. The Simple Finite Volume (SFV) model is used to solve the 2D Saint-Venant equations over an unstructured mesh of 30000 elements representing channel and floodplain, and allowing detailed hydraulics of flow around bridge piers and other influential features to be represented. The SFV model is also used to corroborate flows recorded for the event at two gauging stations. Calibration of Manning's n is performed with a two stage strategy, with channel values determined by calibration of the gauging station models, and floodplain values determined by optimising the fit between model results and observed water levels and flood extent for the 2005 event. RMS error for the calibrated model compared with surveyed water levels is ~±0.4m, the same order of magnitude as the estimated error in the survey data. The study demonstrates the ability of unstructured mesh hydraulic models to represent important hydraulic processes across a range of scales, with potential applications to flood risk management.

Voronoi, Delaunay, and Block-Structured Mesh Refinement for Solution of the Shallow-Water Equations on the Sphere

Alternative meshes of the sphere and adaptive mesh refinement could be immensely beneficial for weather and climate forecasts, but it is not clear how mesh refinement should be achieved. A finite-volume model that solves the shallow-water equations on any mesh of the surface of the sphere is presented. The accuracy and cost effectiveness of four quasi-uniform meshes of the sphere are compared: a cubed sphere, reduced latitude–longitude, hexagonal–icosahedral, and triangular–icosahedral. On some standard shallow-water tests, the hexagonal–icosahedral mesh performs best and the reduced latitude–longitude mesh performs well only when the flow is aligned with the mesh. The inclusion of a refined mesh over a disc-shaped region is achieved using either gradual Delaunay, gradual Voronoi, or abrupt 2:1 block-structured refinement. These refined regions can actually degrade global accuracy, presumably because of changes in wave dispersion where the mesh is highly nonuniform. However, using gradual refinement to resolve a mountain in an otherwise coarse mesh can improve accuracy for the same cost. The model prognostic variables are height and momentum collocated at cell centers, and (to remove grid-scale oscillations of the A grid) the mass flux between cells is advanced from the old momentum using the momentum equation. Quadratic and upwind biased cubic differencing methods are used as explicit corrections to a fast implicit solution that uses linear differencing.

Predicting mesh density for adaptive modelling of the global atmosphere

The shallow water equations are solved using a mesh of polygons on the sphere, which adapts infrequently to the predicted future solution. Infrequent mesh adaptation reduces the cost of adaptation and load-balancing and will thus allow for more accurate mapping on adaptation. We simulate the growth of a barotropically unstable jet adapting the mesh every 12 h. Using an adaptation criterion based largely on the gradient of the vorticity leads to a mesh with around 20 per cent of the cells of a uniform mesh that gives equivalent results. This is a similar proportion to previous studies of the same test case with mesh adaptation every 1–20 min. The prediction of the mesh density involves solving the shallow water equations on a coarse mesh in advance of the locally refined mesh in order to estimate where features requiring higher resolution will grow, decay or move to. The adaptation criterion consists of two parts: that resolved on the coarse mesh, and that which is not resolved and so is passively advected on the coarse mesh. This combination leads to a balance between resolving features controlled by the large-scale dynamics and maintaining fine-scale features.

Distribution and mobility of trace elements in soils and vegetation around the mining and smelting areas of Tharsis, Riotinto and Huelva, Iberian Pyrite Belt, SW Spain

Trace elements may present an environmental hazard in the vicinity of mining and smelting activities. However, the factors controlling their distribution and transfer within the soil and vegetation systems are not always well defined. Total concentrations of up to 15,195 mg center dot kg (-1) As, 6,690 mg center dot kg(-1) Cu, 24,820 mg center dot kg(-1) Pb and 9,810 mg center dot kg(-1) Zn in soils, and 62 mg center dot kg(-1) As, 1,765 mg center dot kg(-1) Cu, 280 mg center dot kg(-1) Pb and 3,460 mg center dot kg (-1) Zn in vegetation were measured. However, unusually for smelters and mines of a similar size, the elevated trace element concentrations in soils were found to be restricted to the immediate vicinity of the mines and smelters (maximum 2-3 km). Parent material, prevailing wind direction, and soil physical and chemical characteristics were found to correlate poorly with the restricted trace element distributions in soils. Hypotheses are given for this unusual distribution: (1) the contaminated soils were removed by erosion or (2) mines and smelters released large heavy particles that could not have been transported long distances. Analyses of the accumulation of trace elements in vegetation (median ratios: As 0.06, Cu 0.19, Pb 0.54 and Zn 1.07) and the percentage of total trace elements being DTPA extractable in soils (median percentages: As 0.06%, Cu 15%, Pb 7% and Zn 4%) indicated higher relative trace element mobility in soils with low total concentrations than in soils with elevated concentrations.

Factors affecting distribution and mobility of trace elements (Cu, Pb, Zn) in a perennial grapevine (Vitis vinifera L.) in the Champagne region of France

Soil and Vitis vinifera L (coarse and fine roots, leaves, berries) concentration and geochemical partitioning of Cu, Pb and Zn were determined in a contaminated calcareous Champagne plot to assess their mobility and transfer. Accumulation ratios in roots remained low (0.1-0.4 for Cu and Zn, <0.05 for Pb). Differences between elements resulted from vegetation uptake strategy and soil partitioning. Copper, significantly associated with the oxidisable fraction (27.8%), and Zn with the acid soluble fraction (33.3%), could be mobilised by rhizosphere acidification and oxidisation, unlike Pb, essentially contained in the reducible fraction (72.4%). Roots should not be considered as a whole since the more reactive fine roots showed higher accumulation ratios than coarse ones. More sensitive response of fine roots, lack of correlation between chemical extraction results and vegetation concentrations, and very limited translocation to aerial parts showed that fine root concentrations should be used when assessing bioavailability. (C) 2008 Elsevier Ltd. All rights reserved.

A wavenumber independent boundary element method for an acoustic scattering problem

In this paper we consider the impedance boundary value problem for the Helmholtz equation in a half-plane with piecewise constant boundary data, a problem which models, for example, outdoor sound propagation over inhomogeneous. at terrain. To achieve good approximation at high frequencies with a relatively low number of degrees of freedom, we propose a novel Galerkin boundary element method, using a graded mesh with smaller elements adjacent to discontinuities in impedance and a special set of basis functions so that, on each element, the approximation space contains polynomials ( of degree.) multiplied by traces of plane waves on the boundary. We prove stability and convergence and show that the error in computing the total acoustic field is O( N-(v+1) log(1/2) N), where the number of degrees of freedom is proportional to N logN. This error estimate is independent of the wavenumber, and thus the number of degrees of freedom required to achieve a prescribed level of accuracy does not increase as the wavenumber tends to infinity.