This thesis is concerned with the estimation of the effects of an external boundary on some dynamical system of continuum mechanics which have previously been analysed for infinite media. The problems considered are:- (i) The forced torsional oscillations of a rigid spheroidal inclusion in a bounded axisymmetric elastic solid. (ii) The diffraction of torsional stress waves travelling along the axis of an infinite elastic cylinder by a) a fixed rigid inclusion and b) a penny shaped crack. (iii) The diffraction of harmonic sound waves in a circular tube by a) a soft spheroidal obstacle and b) a rigid disc. (iv) The steady swirling flow of an inviscid fluid past a rigid spheroidal body in a coaxial tube (v) The forced torsional oscillations of a) a rigid sphere and b) a rigid disc in an axisymmetric container of viscous fluid. These are particular examples of a general class of boundary value problems for the reduced wave equations which may be formulated by means of Green's theorem and appropriate Green's functions as Fredholm integral equations of the first kind. By perturbing on the static solution low frequency" approximations for quantities of physical interest exhibiting explicitly the first order effects of the external boundaries are obtained. The advantage of this procedure is that it may be used in a large variety of situations where the geometry of the problem prohibits the use of exact processes such as the method of separation of variables. Integral equation formulations may also facilitate the use of a direct boundary perturbation on the infinite medium solution; this technique is used here in a few particular examples.

The diffraction of high frequency torsion waves by disc-shaped obstacles, situated in solids which are homogeneous, isotropic and of infinite extent, are considered in this thesis. In a high frequency limit these problems are formulated as Fredholm integral equations of the second kind. The thesis is divided into two chapters:- Chapter I: diffraction of high frequency torsion waves by a penny-shaped crack. Explicit asymptotic expressions are obtained for the dynamic stress intensity factors and the scattering coefficients. These results predict an oscillatory behaviour of the stress intensity factors at high frequencies. Chapter II: diffraction of high frequency torsion waves by a rigid disc. Explicit asymptotic expressions are obtained for the torque resisting the motion of the disc, and for the scattering coefficients. In both chapters extensive, numerical results are given.

In this thesis a variety of mixed boundary-value problems, taken from the fields of elastodynamics, magnetohydrodynamics and acoustics are investigated. The work is divided into two main parts. In each part a different method is employed to solve the integral equations governing the particular problems under consideration. Numerical results are obtained and whenever possible comparisons made with results taken from other works. Part I deals with the solution of a class of Fredholm integral equations of the second kind by means of a kernel approximation technique. Starting from a typical mixed boundary-value problem an analysis is presented in such a way as to indicate the theoretical basis of the method. The exact kernel of the integral equation is replaced by an approximate kernel, derived from a Pade approximation, which permits the solution of the equation in closed form. Numerical comparisons between exact and approximate quantities are presented. The method is applied to transient elasto-dynamic problems of the Reissner-Sagoci and Boussinesq types and to the case of a disc rotating in a conducting fluid. Finally an extension of the approximation leading to improved accuracy is discussed. The far field pattern produced by a vibrating circular piston set in an infinite, plane, non-rigid baffle is investigated in the second part of the work. An approximation to the solution of the problem, depending upon the solution of an integral equation which arises in the problem of diffraction of a normally incidented plane wave by a rigid disc, is found by following a complicated asymptotic analysis. Because of the ranges of certain parameters three sets of solutions are found. These are combined to form composite graphs of the radiation pattern function for various values of the baffle impedence.

No description available.

This thesis presents aspects of noncommutative spectral geometry as an approach to formulate a model of gravity and particle physics, while addressing open issues associated with this approach. We propose a novel de nition of the bosonic spectral action using zeta function regularisation, in order to address the issues of renormalisability, ultraviolet completeness and spectral dimensions. We compare the zeta spectral action with the usual (cuto based) spectral action and discuss its purely spectral origin, predictive power, stressing the importance of the issue of the three dimensionful fundamental constants, namely the cosmological constant, the Higgs vacuum expectation value, and the gravitational constant. We emphasise the fundamental role of the neutrino Majorana mass term for the structure of the bosonic action. We subsequently show that the regularised zeta spectral action gives a stable linearised gravitational theory despite being a 4th-order derivative theory. Afterwards, we explore the notion of Lorentzian noncommutative geometry, where the bosonic action is not well-de ned. However, in such a case, the dynamics of fermions is still well-de ned. We have shown that one could give a geometrical meaning to the energy-momentum dispersion relation of fermions.

Atomic scale phenomena concurring to atomic bond ruptures at a crack tip determine the chemomechanical properties of oxide materials, thus a better understanding of them is instrumental in addressing engineering issues related to the brittleness of oxides. In a fracturing material, the macroscopic stress field couples with the chemical reactions occurring at the crack tip in a bidirectional interplay requiring a concurrent multiscale (QM/MM) computational approach. Due to long range electrostatic interactions, the dynamics of chemically accurate description of the neighbourhood of a breaking bond requires ab initio calculations on several hundred atoms on a timescale of picoseconds. First, to make these prohibitively demanding calculations tractable I developed an ensemble parallel QM/MM computational strategy comprising a novel graph partitioning method for optimal load balancing that is able to efficiently parallellise the workload over hundreds of thousands of cores on supercomputing facilities. Secondly, I present a computational study of crack propagation in two–dimensional silica systems that have recently been experimentally synthesized, which provide ideal and physically observed structures that are key to the understanding of atomic scale phenomena in fracture events in oxides. The atomic structure, either crystalline or amorphous, and the emerging set of free energy barriers to crack advance are the basis to understand the fundamental difference in the crack dynamics observed at a larger scale. Finally, I explored different pathways to make efficient use of the information produced by ab initio calculations by studying machine learning methods capable of predicting local physical observables as a function of the local atomic environment. This includes a machine learning–augmented method to obtain free energy barriers of hybrid accuracy that only require DFT calculations on just a small fraction of the sampled atomic configurations.

No description available.

The work presented in this thesis concerns the application of methods based on the one-electron Green's integral operator to certain quantum mechanical wave equations associated in particular with atomic and molecular systems. Estimates of eigenvalues and wavefunctions for such systems are computed by means of variational functionals based on this operator. In the case of molecular systems, the major difficulty is that of evaluating the resulting integrals. In the early part of the thesis, this problem is discussed in some detail with particular reference to the H+2 molecular ion.

In this thesis, the matrix free Characteristic Based Split (CBS) scheme based on an artificial compressibility (AC) method and the semi-implicit CBS scheme are presented for laminar and turbulent incompressible flows. Numerical simulations of steady and unsteady state incompressible flow problems have been carried out on structured and unstructured meshes of linear triangular and tetrahedral elements. The standard Galerkin method was used for spatial discretization of the governing equations in their semi-discrete CBS form. Four different Reynolds average Navier-Stokes (RANS) turbulence models have been studied in detail. They are the one-equation linear kappa-l model of Wolfshtein, the one-equation Spalart-Allmaras model, the two-equation linear kappa-epsilon model with two different low Reynolds number treatments (Lam-Bremhorst damping functions and Fan- Lakshminarayana-Barnett damping functions), and the two-equation nonlinear near-wall kappa-epsilon model with Kimura-Hosoda's parameters. The results of standard steady flow in a channel, inside a lid-driven cavity, over a backward facing step, around a stationary sphere and through an upper human airway are adequately predicted. In addition to steady state flow problems, unsteady Reynolds-averaged Navier-Stokes (URANS) model was employed to solve vortex shedding behind a circular cylinder using a dual-time stepping technique. The two- and three-dimensional results presented show that both the CBS-AC matrix free procedure and semi-implicit CBS formulation are accurate and efficient.

We present a new generalised eigenfunction of the reduced two-particle, mixed-charge, hyperbolic Ruijsenaars-Schneider (or, relativistic A_1-Calogero-Moser) Hamiltonian. The asymptotics of this function displays transmission and reflection in a way that generalizes the familiar non-relativistic picture. Using this function we construct integral transforms diagonalizing the Hamiltonian (an analytic difference operator, or A\DeltaO). When the three parametric dependences of the Hamiltonian are restricted to a certain polytope, these transforms can be used for a functional-analytic Hilbert space theory with all the desired quantum mechanical features (self-adjointness, spectrum, S-matrix etc.). As a final consideration we see how such a theory can be constructed in a different way for a special choice of the coupling parameter, with accompanying special features.