Computational methods are developed to simulate interactions of nonlinear waves with single- and multi-structures through the finite element method based on second order and fully nonlinear theories. The three dimensional (3D) mesh with prism elements is generated through an extension of a two dimensional (2D) unstructured mesh. The potential and velocity in the fluid field are obtained by solving finite element matrix equations at each time step using the conjugate gradient method with SSOR preconditioner. The combined Sommerfeld-Orlanski radiation condition and the damping zone method is used to minimize wave reflection. The regridding and smoothing techniques are employed to improve the stability of the solution and the accuracy of the result. The method is first used to simulate interactions of waves and an array of cylinders in the time domain based on the second order theory. It is shown that the interference between the cylinders has magnificent influence on the phase and amplitude of the waves and forces. The fully nonlinear problem is tackled first by considering the two dimensional problem, which allows the developed method to be properly tested and validated. Simulation is made for a body in a tank and a wedge-shaped body in oscillation. Comparison is made with results obtained from other methods. In the next application, the 3D interactions between single- and multi-cylinders with or without flare and waves generated by a wave maker in a rectangular tank are investigated based on the fully nonlinear theory. The effect of the flare on waves and hydrodynamic forces is analysed, and the mutual interference of multiple cylinders is also studied. The method is also employed to solve the 3D fully nonlinear radiation problems by single- and multi-cylinders undergoing oscillation in the open sea. The result is compared with those obtained from the linear and second order theories. It is concluded that the developed numerical approaches based on the finite element method can be used to effectively simulate interactions of waves and single- and multi- cylinders with or without flare, which can provide some valuable information to design of offshore structures.

Quantitative imaging complements structural imaging by providing quantitative estimations of subsurface material properties as opposed to the sizes, shapes and positions of scatterers available from structural methods. The ability to reconstruct material properties from a series of wave measurements is extremely valuable in a range of applications as it potentially allows diagnostic technology with superior sensitivity and selectivity. Breast cancer, for example, is stiffer and hence of higher sound velocity than the surrounding tissue, so reconstructing velocity from ultrasonic measurements could allow cancer detection. Using this concept, breast ultrasound tomography has the potential to significantly improve the cost, safety and reliability of breast cancer screening and diagnosis over mammography, the gold-standard. Key to unlocking this potential is the availability of an accurate, fast, robust and high-resolution algorithm to reconstruct wave velocity. This thesis introduces HARBUT, the Hybrid Algorithm for Robust Breast Ultrasound Tomography, a new imaging approach combining the complementary strengths of low resolution bent-ray tomography and high resolution diffraction tomography. HARBUT's theoretical foundation is explained and applied to simulated and experimental, in-vivo, breast ultrasound tomography data, confirming that it generates a step change in image quality over existing techniques, revealing lesions that would not be visible on a mammogram. This thesis also shows how, by combining data from many slices, the out-of-plane resolution can be significantly improved compared to treating each slice independently. HARBUT is applied to alternative problems including guided wave tomography, which aims to quantify the remaining wall thickness of a potentially corroded, inaccessible plate-like structure. Thickness estimates within 1mm for a 10mm nominal thickness plate were demonstrated for both simulated and experimental data. The thesis finally investigates HARBUT's performance with limited view configurations, and introduces VISCIT, the Virtual Image Space Component Iterative Technique, which accounts for the missing data, significantly improving the reconstructed image.

The problem of wave scattering by sea-ice of varying thickness and non-zero draught floating on water of finite depth is considered. To do so, the common idealisation of the ice as a thin-elastic plate, which is static in all but its flexural response, is adopted. Furthermore, the assumptions of linear and time harmonic motio,ns are made. The physical situation is initially formulated as a boundary-value problem but is subsequently reformulated as a variational principle. Here, the geometry is unconstrained and the ice covering may be either complete or partial. Additionally, the bed profile is permitted to undulate. The solution method proceeds via application of the Rayleigh-Ritz method in conjunction with the variational principle. This restricts the vertical component of the velocity potential that represents the fluid motion to a finite-dimensional subspace and the stationary point. of the variational principle over this finite space is sought. As the dimension of the subspace is increased, a sequence of approximations is generated, which can be made arbitrarily close to the full linear solution.

Optical singularities are points in complex scalar and vector fields where a property of the field becomes undefined (singular). In complex scalar fields these are phase singularities and in vector fields they are polarisation singularities. In the former the phase of the field is singular and in the latter it is the polarisation ellipse axes. In three dimensions these singularities are lines and natural light fields are threaded by these lines. The interference between three, four and five waves is investigated and inequalities are given which establish the topology of the singularity lines in fields composed of four plane waves. Beyond several waves, numerical simulations are used, supported by experiments, to establish that optical singularties in speckle fields have the fractal properties of a Brownian random walk. Approximately 73% of singularity lines percolate random optical fields, the remainder forming closed loops. The statistical results are found to be similar to those of vortices in random discrete lattice models of cosmic strings, implying that the statistics of singularities in random optical fields exhibit universal behavior. It is also established that a random superposition of plane-waves, such as optical speckle, form singularities which not only map out fractal lines, but create topological features within them. These topological features are rare and include vortex loops which are threaded by infinitely long lines and pairs of loops that form links. Such structures should be not only limited to optical fields but will be present in all systems that can be modeled as random wave superpositions such as those found in cosmic strings and Bose-Einstein condensates. Also reported are results from experiments that generated compact vortex knots and links in real Gaussian beams. These results were achieved through the use of algebraic knot theory and random search optimisation algorithms. Finally, polarisation singularity densities are measured experimentally which confirm analytic predictions.

The theme of the research is on the use of the hypersingular boundary element method for the modelling of surface waves. Surface waves in solids are known to be partially reflected & transmitted and mode converted into body waves at stress discontinuities, which suggests that a formulation continuous in stress and strain might prove beneficial for modelling purposes. Such continuity can be achieved with a subparametric approach where the geometry is approximated using linear elements and the field variables, displacement and traction, are approximated using cubic Hermitian and linear shape functions respectively. The higher order polynomial for approximating displacement is intended to be a more accurate representation of the physics relating to surface wave phenomena, especially at corners, and thus, is expected to capture this behaviour with greater accuracy than the standard isoparametric approach. The subparametric approach affords itself to continuity in stress and strain by imposing a smoothness in the elements, which is not available to the isoparametric approach. As the attention is focused primarily on the modelling of surface waves on the boundary of a medium rather than the interior, the boundary element method lends itself appropriately to this end.A 2D semi analytical integration scheme is employed to evaluate the integrals appearing in the hypersingular boundary integral formulation. The integration scheme is designed to reduce the errors incurred when integrals with singular integrands are evaluated numerically. The scheme involves the application of Taylor expansions to formulate the integrals into two parts. One part is regular and is evaluated numerically and the other part is singular but sufficiently simple to be evaluated analytically. The scheme makes use of the aforementioned subparametric approach and is applied to linear elements for the use in steady state elastodynamic boundary element method problems. The steady state problem is used as it is a simplified problem and is sufficient to permit the investigation of surface vibration at a constant motion. The 2D semi analytical integration scheme presented can be naturally extended to 3D.A particular focus and novelty of the work is the application of different limiting approaches to determine the free terms common to boundary integral methods. The accurate numerical solution of hypersingular boundary integral equations necessitates the precise evaluation of free terms, which are required to counter discontinuous and often unbounded behaviour of hypersingular integrals at a boundary. The common approach for the evaluation of free terms involves integration over a portion of a circular/spherical shaped surface centred at a singularity and allowing the radius of the circle/sphere to tend to zero. This approach is revisited in order to ascertain whether incorrect results are possible as a consequence of shape dependency, which is a recognised issue for hypersingular integrals.Two alternative methods, which are shape invariant, are proposed and investigated for the determination of free terms. The first approach, the point limiting method, involves moving a singularity towards a shrinking integration domain at a faster rate than the domain shrinks. Issues surrounding the choice of approach, shrinkage rates and path dependency are examined. A related and second approach, the boundary limiting method, involves moving an invariant, but shrinking, boundary toward the singularity, again at a faster rate than the shrinkage of the domain. The latter method can be viewed as a vanishing exclusion zone approach but the actual boundary shape is used for the boundary of the exclusion zone. Both these methods are shown to provide consistent answers and can be shown to be directly related to the result obtained by moving a singularity towards a boundary, that is, by comparison with the direct method. Unlike the circular/spherical approach the two methods involve integration over the actual boundary shape and consequently shape dependency is not an issue. A particular highlight of the point limiting approach is the ability to obtain free terms in mixed formulation, which is not available to the circular/spherical approach.There are three numerical problems considered in this research. The first problem considers the longitudinal vibration of a square plate. This is a problem for which a known analytical solution exists and is used to verify the equation formulation and integration scheme adopted for the isoparametric and subparametric formulations. Both formulations are as accurate as each other and produce results that are in keeping with the analytical solution, thus instilling confidence in their predictions.The second problem considers the simulation of surface waves on a square plate. Various boundaries of a square plate have displacement conditions imposed on them as a result of surface wave propagation. The results indicate that the surface wave behaviour is not captured. However, the analytical solution does not make any consideration for the effects from corners; the analytical solution is for a Rayleigh wave propagating upon a planar surface. It does not take into account the wave phenomena encountered at corners. Therefore, these results cannot be used to validate the predictions obtained on the boundary of the problem considered. The purpose of this problem is to illustrate the impact of corners on the surface wave propagation. Sensitivity studies are conducted to illustrate the effect of corners on the computed solution at the boundary.The final problem considers the simulation of surface waves on a circular plate. Various portions of the boundary of the circular plate have displacement conditions imposed on them as a result of surface wave propagation on curved surfaces. The results indicate that the isoparametric and subparametric predictions are similar to one another. However, both displacement profiles predict the presence of other waves. Given the multi faceted nature of the mesh, the computed solution is picking up mode conversion and partial reflection & transmission of surface waves. In reality, this is not expected as the surface of the boundary is smooth. However, due to the discretisation there are many corners in this problem.