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 combined BIE-FE method for the Stokes equations

A numerical algorithm for the biharmonic equation in domains with piecewise smooth boundaries is presented. It is intended for problems describing the Stokes flow in the situations where one has corners or cusps formed by parts of the domain boundary and, due to the nature of the boundary conditions on these parts of the boundary, these regions have a global effect on the shape of the whole domain and hence have to be resolved with sufficient accuracy. The algorithm combines the boundary integral equation method for the main part of the flow domain and the finite-element method which is used to resolve the corner/cusp regions. Two parts of the solution are matched along a numerical ‘internal interface’ or, as a variant, two interfaces, and they are determined simultaneously by inverting a combined matrix in the course of iterations. The algorithm is illustrated by considering the flow configuration of ‘curtain coating’, a flow where a sheet of liquid impinges onto a moving solid substrate, which is particularly sensitive to what happens in the corner region formed, physically, by the free surface and the solid boundary. The ‘moving contact line problem’ is addressed in the framework of an earlier developed interface formation model which treats the dynamic contact angle as part of the solution, as opposed to it being a prescribed function of the contact line speed, as in the so-called ‘slip models’. Keywords: Dynamic contact angle; finite elements; free surface flows; hybrid numerical technique; Stokes equations.

Periodic Reordering

For many networks in nature, science and technology, it is possible to order the nodes so that most links are short-range, connecting near-neighbours, and relatively few long-range links, or shortcuts, are present. Given a network as a set of observed links (interactions), the task of finding an ordering of the nodes that reveals such a range-dependent structure is closely related to some sparse matrix reordering problems arising in scientific computation. The spectral, or Fiedler vector, approach for sparse matrix reordering has successfully been applied to biological data sets, revealing useful structures and subpatterns. In this work we argue that a periodic analogue of the standard reordering task is also highly relevant. Here, rather than encouraging nonzeros only to lie close to the diagonal of a suitably ordered adjacency matrix, we also allow them to inhabit the off-diagonal corners. Indeed, for the classic small-world model of Watts & Strogatz (1998, Collective dynamics of ‘small-world’ networks. Nature, 393, 440–442) this type of periodic structure is inherent. We therefore devise and test a new spectral algorithm for periodic reordering. By generalizing the range-dependent random graph class of Grindrod (2002, Range-dependent random graphs and their application to modeling large small-world proteome datasets. Phys. Rev. E, 66, 066702-1–066702-7) to the periodic case, we can also construct a computable likelihood ratio that suggests whether a given network is inherently linear or periodic. Tests on synthetic data show that the new algorithm can detect periodic structure, even in the presence of noise. Further experiments on real biological data sets then show that some networks are better regarded as periodic than linear. Hence, we find both qualitative (reordered networks plots) and quantitative (likelihood ratios) evidence of periodicity in biological networks.

Visualizing knowledge in project-based work

This article considers how visual practices are used to manage knowledge in project-based work. It compares project-based work in a capital goods manufacturer and an architectural firm. Visual representations are used extensively in both cases, but the nature of visual practice differs significantly between the two. The research explores the kinds of knowledge that are (and aren't) developed and made visible in strategizing and planning activities. For example, whereas the emphasis of project-based work in the former firm is on exploitation of knowledge and it visualizes its project context largely in commercial and processual terms, the emphasis in the latter is on exploration and it uses a wide range of visual materials to understand physical interdependencies across the project boundary. We contend particular kinds of visual tools can help project teams step between exploration and exploitation within a project, and articulate the types of representations, foci of attention and patterns of interaction involved. The findings suggest that business managers can make more deliberate choices about how knowledge is made visible, and can change visual practice to align the project with exploring and exploiting opportunities. It raises the question: What don't you see within your organization? The work contributes to academic debates about managing through projects, strategising and organizing, while the focus on visual representation disrupts the tacit-codified dichotomy in the broad debate on knowledge and learning, and highlights the craft skills central to strategizing and organizing.

Visual practices and the objects of design

Recent interest in material objects - the things of everyday interaction - has led to articulations of their role in the literature on organizational knowledge and learning. What is missing is a sense of how the use of these 'things' is patterned across both industrial settings and time. This research addresses this gap with a particular emphasis on visual materials. Practices are analysed in two contrasting design settings: a capital goods manufacturer and an architectural firm. Materials are observed to be treated both as frozen, and hence unavailable for change; and as fluid, open and dynamic. In each setting temporal patterns of unfreezing and refreezing are associated with the different types of materials used. The research suggests that these differing patterns or rhythms of visual practice are important in the evolution of knowledge and in structuring social relations for delivery. Hence, to improve their performance practitioners should not only consider the types of media they use, but also reflect on the pace and style of their interactions.

Numerical study of 2D heat transfer in a scraped surface heat exchanger

A numerical study of fluid mechanics and heat transfer in a scraped surface heat exchanger with non-Newtonian power law fluids is undertaken. Numerical results are generated for 2D steady-state conditions using finite element methods. The effect of blade design and material properties, and especially the independent effects of shear thinning and heat thinning on the flow and heat transfer, are studied. The results show that the gaps at the root of the blades, where the blades are connected to the inner cylinder, remove the stagnation points, reduce the net force on the blades and shift the location of the central stagnation point. The shear thinning property of the fluid reduces the local viscous dissipation close to the singularity corners, i.e. near the tip of the blades, and as a result the local fluid temperature is regulated. The heat thinning effect is greatest for Newtonian fluids where the viscous dissipation and the local temperature are highest at the tip of the blades. Where comparison is possible, very good agreement is found between the numerical results and the available data. Aspects of scraped surface heat exchanger design are assessed in the light of the results. (C) 2003 Elsevier Ltd. All rights reserved.

An Exponentially Convergent Nonpolynomial Finite Element Method for Time-Harmonic Scattering from Polygons

In recent years nonpolynomial finite element methods have received increasing attention for the efficient solution of wave problems. As with their close cousin the method of particular solutions, high efficiency comes from using solutions to the Helmholtz equation as basis functions. We present and analyze such a method for the scattering of two-dimensional scalar waves from a polygonal domain that achieves exponential convergence purely by increasing the number of basis functions in each element. Key ingredients are the use of basis functions that capture the singularities at corners and the representation of the scattered field towards infinity by a combination of fundamental solutions. The solution is obtained by minimizing a least-squares functional, which we discretize in such a way that a matrix least-squares problem is obtained. We give computable exponential bounds on the rate of convergence of the least-squares functional that are in very good agreement with the observed numerical convergence. Challenging numerical examples, including a nonconvex polygon with several corner singularities, and a cavity domain, are solved to around 10 digits of accuracy with a few seconds of CPU time. The examples are implemented concisely with MPSpack, a MATLAB toolbox for wave computations with nonpolynomial basis functions, developed by the authors. A code example is included.

A high frequency boundary element method for scattering by convex polygons with impedance boundary conditions

We consider scattering of a time harmonic incident plane wave by a convex polygon with piecewise constant impedance boundary conditions. Standard finite or boundary element methods require the number of degrees of freedom to grow at least linearly with respect to the frequency of the incident wave in order to maintain accuracy. Extending earlier work by Chandler-Wilde and Langdon for the sound soft problem, we propose a novel Galerkin boundary element method, with the 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. Theoretical analysis and numerical results suggest that the number of degrees of freedom required to achieve a prescribed level of accuracy grows only logarithmically with respect to the frequency of the incident wave.

High frequency scattering by convex curvilinear polygons

We consider the scattering of a time-harmonic acoustic incident plane wave by a sound soft convex curvilinear polygon with Lipschitz boundary. For standard boundary or finite element methods, with a piecewise polynomial approximation space, the number of degrees of freedom required to achieve a prescribed level of accuracy grows at least linearly with respect to the frequency of the incident wave. Here we propose a novel Galerkin boundary element method with a hybrid approximation space, consisting of the products of plane wave basis functions with piecewise polynomials supported on several overlapping meshes; a uniform mesh on illuminated sides, and graded meshes refined towards the corners of the polygon on illuminated and shadow sides. Numerical experiments suggest that the number of degrees of freedom required to achieve a prescribed level of accuracy need only grow logarithmically as the frequency of the incident wave increases.

Surface based hypothesis verification in intensity images using geometric and appearance data

In this paper we discuss current work concerning Appearance-based and CAD-based vision; two opposing vision strategies. CAD-based vision is geometry based, reliant on having complete object centred models. Appearance-based vision builds view dependent models from training images. Existing CAD-based vision systems that work with intensity images have all used one and zero dimensional features, for example lines, arcs, points and corners. We describe a system we have developed for combining these two strategies. Geometric models are extracted from a commercial CAD library of industry standard parts. Surface appearance characteristics are then learnt automatically by observing actual object instances. This information is combined with geometric information and is used in hypothesis evaluation. This augmented description improves the systems robustness to texture, specularities and other artifacts which are hard to model with geometry alone, whilst maintaining the advantages of a geometric description.

A very rare hydrogen-bridged hexanuclear CuII complex containing a triangular Cu3O core capped by an unusual triply coordinated perchlorate anion

An unusual hexanuclear Cu-II complex, [{[Cu(NHDEPO)](3)(mu(3)-O)(O3ClO)}(2)(mu-H)]center dot 7ClO(4)center dot 4H(2)O (1) was prepared starting from Cu(ClO4)(2)center dot 6H(2)O and the oxime-based Schiff base ligand NHDEPO (= 3-[3-(diethylamino)propylimino]butan-2-one oxime). Structural characterization of the complex reveals that it consists of two triangular Cu3O units, the copper ions being at the corners of an equilateral triangle, separated by an O center dot center dot center dot O distance of 2,447(5) angstrom, held together solely by a proton. In each triangle, the copper atoms are in square-pyramid environments. The equatorial plane consists of the bridging oxygen of the central OH-(O2-) group together with three atoms (N, N, O) of the Schiff base. All Unusual triply coordinated perchlorate ion (mu(3)-kappa O:kappa O':kappa O '') interacts in axial position with the three copper ions, Variable-temperature (2-300 K) magnetic susceptibility measurements show that complex 1 is antiferromagnetically Coupled (J = -148 cm(1-)). The EPR data at low temperature clearly indicates the presence of spin frustration phenomenon in the complex.