Asymptotic expansions and fast computation of oscillatory Hilbert transforms

In this paper, we study the asymptotics and fast computation of the one-sided oscillatory Hilbert transforms of the form $$H^{+}(f(t)e^{i\omega t})(x)=-int_{0}^{\infty}e^{i\omega t}\frac{f(t)}{t-x}dt,\qquad \omega>0,\qquad x\geq 0,$$ where the bar indicates the Cauchy principal value and $f$ is a real-valued function with analytic continuation in the first quadrant, except possibly a branch point of algebraic type at the origin. When $x=0$, the integral is interpreted as a Hadamard finite-part integral, provided it is divergent. Asymptotic expansions in inverse powers of $\omega$ are derived for each fixed $x\geq 0$, which clarify the large $\omega$ behavior of this transform. We then present efficient and affordable approaches for numerical evaluation of such oscillatory transforms. Depending on the position of $x$, we classify our discussion into three regimes, namely, $x=\mathcal{O}(1)$ or $x\gg1$, $0<x\ll 1$ and $x=0$. Numerical experiments show that the convergence of the proposed methods greatly improve when the frequency $\omega$ increases. Some extensions to oscillatory Hilbert transforms with Bessel oscillators are briefly discussed as well.Comment: 32 pages, 6 figures, 4 table

New hybrid quadrature schemes for weakly singular kernels applied to isogeometric boundary elements for 3D Stokes flow

This work proposes four novel hybrid quadrature schemes for the efficient and accurate evaluation of weakly singular boundary integrals (1/r kernel) on arbitrary smooth surfaces. Such integrals appear in boundary element analysis for several partial differential equations including the Stokes equation for viscous flow and the Helmholtz equation for acoustics. The proposed quadrature schemes apply a Duffy transform-based quadrature rule to surface elements containing the singularity and classical Gaussian quadrature to the remaining elements. Two of the four schemes additionally consider a special treatment for elements near to the singularity, where refined Gaussian quadrature and a new moment-fitting quadrature rule are used. The hybrid quadrature schemes are systematically studied on flat B-spline patches and on NURBS spheres considering two different sphere discretizations: An exact single-patch sphere with degenerate control points at the poles and an approximate discretization that consist of six patches with regular elements. The efficiency of the quadrature schemes is further demonstrated in boundary element analysis for Stokes flow, where steady problems with rotating and translating curved objects are investigated in convergence studies for both, mesh and quadrature refinement. Much higher convergence rates are observed for the proposed new schemes in comparison to classical schemes

The exponentially convergent trapezoidal rule

It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators

Development of BEM for ceramic composites

It is evident that for proper micromechanical analysis of ceramic composites, one needs to use a numerical method that is capable of idealizing the individual fibers or individual bundles of fibers embedded within a three-dimensional ceramic matrix. The analysis must be able to account for high stress or temperature gradients from diffusion of stress or temperature from the fiber to the ceramic matrix and allow for interaction between the fibers through the ceramic matrix. The analysis must be sophisticated enough to deal with the failure of fibers described by a series of increasingly sophisticated constitutive models. Finally, the analysis must deal with micromechanical modeling of the composite under nonlinear thermal and dynamic loading. This report details progress made towards the development of a boundary element code designed for the micromechanical studies of an advanced ceramic composite. Additional effort has been made in generalizing the implementation to allow the program to be applicable to real problems in the aerospace industry

Exploring Periodic Orbit Expansions and Renormalisation with the Quantum Triangular Billiard

A study of the quantum triangular billiard requires consideration of a boundary value problem for the Green's function of the Laplacian on a trianglar domain. Our main result is a reformulation of this problem in terms of coupled non--singular integral equations. A non--singular formulation, via Fredholm's theory, guarantees uniqueness and provides a mathematically firm foundation for both numerical and analytic studies. We compare and contrast our reformulation, based on the exact solution for the wedge, with the standard singular integral equations using numerical discretisation techniques. We consider in detail the (integrable) equilateral triangle and the Pythagorean 3-4-5 triangle. Our non--singular formulation produces results which are well behaved mathematically. In contrast, while resolving the eigenvalues very well, the standard approach displays various behaviours demonstrating the need for some sort of ``renormalisation''. The non-singular formulation provides a mathematically firm basis for the generation and analysis of periodic orbit expansions. We discuss their convergence paying particular emphasis to the computational effort required in comparision with Einstein--Brillouin--Keller quantisation and the standard discretisation, which is analogous to the method of Bogomolny. We also discuss the generalisation of our technique to smooth, chaotic billiards.Comment: 50 pages LaTeX2e. Uses graphicx, amsmath, amsfonts, psfrag and subfigure. 17 figures. To appear Annals of Physics, southern sprin

Proceedings of the Eleventh UK Conference on Boundary Integral Methods (UKBIM 11), 10-11 July 2017, Nottingham Conference Centre, Nottingham Trent University

This book contains the abstracts and papers presented at the Eleventh UK Conference on Boundary Integral Methods (UKBIM 11), held at Nottingham Trent University in July 2017. The work presented at the conference, and published in this volume, demonstrates the wide range of work that is being carried out in the UK, as well as from further afield