A new theoretical error estimate of the method of fundamental solutions applied to reduced wave problems in the exterior region of a disk

AbstractIn this paper, we present a mathematical study of the method of fundamental solutions (MFS) applied to reduced wave problems with Dirichlet boundary conditions in the exterior domain of a disk. A theorem in this paper shows that the MFS with N source points in equi-distantly equally phased arrangement with assignment parameter q(0<q<1), which characterizes the position of the source points and the collocation points, gives an approximate solution with error of O(qN) if the Fourier coefficients of the boundary data decay exponentially. This error estimate is an extension of the results of the previous studies. Numerical examples make good agreements with the results of the theoretical study

A software perspective on infinite elements for wave diffraction and wave forces on marine risers

This thesis describes work on the problem of the scattering of water waves by fixed objects. The method used to solve this problem is that of finite and infinite elements. In particular the development of a new wave infinite element is described. Various aspects of the wave scattering problem are considered, but always from the perspective of the numerical methods, the algorithms and the computer implementations used. These deal not only with the modelling of the wave equations, but also the pre and post processing of the finite element algorithms. This encompasses the generation of suitable finite element meshes, in an accurate and economical way, and the presentation of the results, particularly as accurate contour plots of the wave surface. The first two chapters gives a brief introduction to water waves, and a summary of the basic concepts of finite and infinite elements. In the third chapter the new infinite element for waves, which is a development of an earlier infinite element, is described in detail, including the new mapping, the necessary shape functions and the integration of the element matrix. The earlier infinite element was restricted to the exterior of circular problems. For scattering objects of large aspect ratio this led to meshes with many finite elements, which performed no useful function, and which were computationally wasteful. The mapping in the new infinite element allows the mesh of infinite elements to be tailored to the shape of the diffracting body, without any observed loss of accuracy. It is therefore much more flexible and computationally efficient, because the infinite elements no longer need to be placed radially. The next three chapters, concentrate on the computer science aspects of the implementation of the finite and infinite elements dealing with the linked list data structures for storage of the element information, the special purpose mesh generation programs, which make it possible to analyse a large range of practical scattering problems and the plotting programs for the display of the results. The chief work in chapter six is the implementation of the Akin and Grey accurate predictor-corrector contour plotting algorithm, with colour fill. The advantage of an accurate contour plotting algorithm is that any discontinuities in the contours represent discontinuities in the results, rather than plotting deficiencies. Chapter seven shows results which validate the new infinite element, particularly on the problem of waves diffracted by an ellipse. In the remaining chapters eight to eleven, the emphasis is on a practical problem of the wave forces on groups of risers, which are the tubes which carry hydrocarbons from the sea-bed to the working areas of offshore platforms. The aim was to see if the forces on a group of risers were different from the sum of the forces on the individual risers, calculated on the assumption that the risers do not modify the wave field. The conclusion is that more detailed studies may well bring financial benefits to the companies operating offshore installations

Trapped Modes of the Helmholtz Equation

In the framework of the classical theory of linearised water waves in unbounded domains, trapped modes consist of non-propagating, localised oscillation modes of finite energy occurring at some well-defined frequency and which, in the absence of dissipation, persist in time even in the absence of external forcing. Jones (1953) proved the existence of trapped modes for problems governed by the Helmholtz equation in semi-infinite domains. Trapped modes have been studied in quantum mechanics, elasticity and acoustics and are known, depending on the context, as bound states, acoustic resonances, Rayleigh-Bloch waves, sloshing modes and motion trapped modes. We consider trapped modes in two dimensional infinite waveguides with either Neumann or Dirichlet boundary conditions. Such problems arise when considering obstacles in acoustic waveguides or bound states in quantum wires for example. The mathematical model is a boundary value problem for the Helmholtz equation. Under the usual assumptions of potential theory, the solution is written in terms of a boundary integral equation. We develop a Boundary Element Method (BEM) program which we use to obtain approximate numerical solutions. We extend existing results by identifying additional trapped modes for geometries already studied and investigate new structures. We also carry out a detailed investigation of trapped modes, using the planewave spectrum representation developed for various characteristic problems from the classical theories of radiation, diffraction and propagation. We use simple planewaves travelling in diverse directions to build a more elaborate solution, which satisfies certain conditions required for a trapped mode. Our approach is fairly flexible so that the general procedure is independent of the shape of the trapping obstacle and could be adapted to other geometries. We apply this method to the case of a disc on the centreline of an infinite Dirichlet acoustic waveguide and obtain a simple mathematical approximation of a trapped mode, which satisfies a set of criteria characteristic of trapped modes. Asymptotically, the solution obtained is similar to a nearly trapped mode, which is a perturbation of a genuine trapped mode

Finite element and boundary element modelling of a physiotherapy transducer and its near-field

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