Asymptotic and numerical methods for high-frequency scattering problems

This thesis is concerned with the development, analysis and implementation of efficient and accurate numerical methods for solving high-frequency acoustic scattering problems. Classical boundary or finite element methods that are based on approximating the solution by polynomials can be effective for small and moderate frequencies. However, as the frequency increases, the solution to the scattering problem becomes more oscillatory and classical numerical methods cope very badly with high oscillation. For example, for two-dimensional scattering problems, classical numerical methods require their number of degrees of freedom to grow at least linearly with frequency to capture the oscillatory behaviour of the solution accurately. Therefore, at large frequencies, classical numerical methods become essentially numerically intractable. In order to overcome the limitations of classical methods, one can seek to incorporate the known asymptotic behaviour of the solution in the numerical method. This involves using asymptotic theory to determine the oscillatory part of the solution and then using classical numerical methods to approximate the slowly varying remainder. Such methods are often referred to as hybrid numerical-asymptotic methods. Determining the high frequency asymptotics of acoustic scattering problems is a classic problem in applied mathematics, with methods such as geometrical optics or the geometrical theory of diffraction providing asymptotic expansions of the solutions. Considerable amount of research has been directed towards both constructing these asymptotic expansions and proving error bounds for truncated asymptotic series of the solution, notably by Buslaev [23], Morawetz and Ludwig [78], and Melrose and Taylor [75], among others. Often, the oscillatory component of the solution can be determined explicitly from these asymptotic expansions. This can then be used in designing ecient hybrid methods. Furthermore, from the asymptotic expansions, frequency-dependent bounds on the slowly-varying remainder and its derivatives can be obtained (in some cases these follow directly from classical results, in other cases some additional work is required). The frequency-dependent bounds are the key results used in the frequency-explicit numerical error analysis of the approximation of the slowly-varying remainder. This thesis presents a rigorous justification of one of the key result using only elementary techniques. Hybrid numerical-asymptotic methods have been shown in theory to be substantially more efficient than classical numerical methods alone. For example, [40] presented a hybrid numerical-asymptotic method in the context of boundary integral equations (BIEs) for solving the problem of high-frequency scattering by smooth, convex obstacles in two dimensions. It was proved in [40] that in order to maintain the accuracy as the frequency increases, the hybrid BIE method requires the number of degrees of freedom to grow slightly faster than k1=9, where k is a parameter proportional to the frequency. This is a substantial improvement from the classical boundary integral methods that require O(k) number of degrees of freedom to achieve the same accuracy for this problem. Despite this slow growth in the number of degrees of freedom, hybrid numerical-asymptotic methods lead to stiffness matrices with entries that are highly-oscillatory singular integrals that can not be computed exactly. Thus, without efficient and accurate numerical treatment of these integrals, the hybrid numerical-asymptotic methods, regardless of their attractive theoretical accuracy, can not be efficiently implemented in practice. In order to resolve this difficulty, this thesis develops a methodology for approximating the integrals arising from hybrid methods in the context of BIEs. The integrals are transformed under a change of variables into integrals amenable to Filon-type quadratures. Filon-type quadratures are designed to cope well with high oscillations in the integrands. Then, graded meshes are used to capture the singularities accurately. Along with k-explicit error bounds for the integration methods, this thesis derives k-explicit error bounds for the hybrid BIE methods that incorporate the error of the inexact approximation of the entries of the stiffness matrix. The error bounds suggest that, with an appropriate choice of parameters of Filon quadrature and mesh grading, the overall error of the hybrid method does not deteriorate due to inexact approximation of the stiffness matrix, therefore preserving its attractive theoretical convergence properties.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

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

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 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. Our numerical results suggest that fi�xed accuracy can be achieved at arbitrarily high frequencies with a frequency-independent computational cost. Our analysis does not capture this observed behaviour completely, but 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 boundary element methods require $N$ to increase at least linearly with frequency to retain accuracy). 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

High-frequency asymptotic compression of dense BEM matrices for general geometries without ray tracing

Wave propagation and scattering problems in acoustics are often solved with boundary element methods. They lead to a discretization matrix that is typically dense and large: its size and condition number grow with increasing frequency. Yet, high frequency scattering problems are intrinsically local in nature, which is well represented by highly localized rays bouncing around. Asymptotic methods can be used to reduce the size of the linear system, even making it frequency independent, by explicitly extracting the oscillatory properties from the solution using ray tracing or analogous techniques. However, ray tracing becomes expensive or even intractable in the presence of (multiple) scattering obstacles with complicated geometries. In this paper, we start from the same discretization that constructs the fully resolved large and dense matrix, and achieve asymptotic compression by explicitly localizing the Green's function instead. This results in a large but sparse matrix, with a faster associated matrix-vector product and, as numerical experiments indicate, a much improved condition number. Though an appropriate localisation of the Green's function also depends on asymptotic information unavailable for general geometries, we can construct it adaptively in a frequency sweep from small to large frequencies in a way which automatically takes into account a general incident wave. We show that the approach is robust with respect to non-convex, multiple and even near-trapping domains, though the compression rate is clearly lower in the latter case. Furthermore, in spite of its asymptotic nature, the method is robust with respect to low-order discretizations such as piecewise constants, linears or cubics, commonly used in applications. On the other hand, we do not decrease the total number of degrees of freedom compared to a conventional classical discretization. The combination of the ...Comment: 24 pages, 13 figure

On the eigenmodes of periodic orbits for multiple scattering problems in 2D

Wave propagation and acoustic scattering problems require vast computational resources to be solved accurately at high frequencies. Asymptotic methods can make this cost potentially frequency independent by explicitly extracting the oscillatory properties of the solution. However, the high-frequency wave pattern becomes very complicated in the presence of multiple scattering obstacles. We consider a boundary integral equation formulation of the Helmholtz equation in two dimensions involving several obstacles, for which ray tracing schemes have been previously proposed. The existing analysis of ray tracing schemes focuses on periodic orbits between a subset of the obstacles. One observes that the densities on each of the obstacles converge to an equilibrium after a few iterations. In this paper we present an asymptotic approximation of the phases of those densities in equilibrium, in the form of a Taylor series. The densities represent a full cycle of reflections in a periodic orbit. We initially exploit symmetry in the case of two circular scatterers, but also provide an explicit algorithm for an arbitrary number of general 2D obstacles. The coefficients, as well as the time to compute them, are independent of the wavenumber and of the incident wave. The results may be used to accelerate ray tracing schemes after a small number of initial iterations.Comment: 24 pages, 9 figures and the implementation is available on https://github.com/popsomer/asyBEM/release