Analytical and numerical techniques for wave scattering

In this thesis, we study the mathematical solution of wave scattering problems which describe the behaviour of waves incident on obstacles and are highly relevant to a raft of applications in the aerospace industry. The techniques considered in the present work can be broadly classed into two categories: analytically based methods which use special transforms and functions to provide a near-complete mathematical description of the scattering process, and numerical techniques which select an approximate solution from a general finite-dimensional space of possible candidates. The first part of this thesis addresses an analytical approach to the scattering of acoustic and vortical waves on an infinite periodic arrangement of finite-length flat blades in parallel mean flow. This geometry serves as an unwrapped model of the fan components in turbo-machinery. Our contributions include a novel semi-analytical solution based on the Wiener–Hopf technique that extends previous work by lifting the restriction that adjacent blades overlap, and a comprehensive study of the composition of the outgoing energy flux for acoustic wave scattering on this array of blades. These results provide an insight into the importance of energy conversion between the unsteady vorticity shed from the trailing edges of the cascade blades and the acoustic field. Furthermore, we show that the balance of incoming and outgoing energy fluxes of the unsteady field provides a convenient tool for understanding several interesting scattering symmetries on this geometry. In the second part of the thesis, we focus on numerical techniques based on the boundary integral method which allows us to write the governing equations for zero mean flow in the form of Fredholm integral equations. We study the solution of these integral equations using collocation methods for two-dimensional scatterers with smooth and Lipschitz boundaries. Our contributions are as follows: Firstly, we explore the extent to which least-squares oversampling can improve collocation. We provide rigorous analysis that proves guaranteed convergence for small amounts of oversampling and shows that superlinear oversampling can ensure faster asymptotic convergence rates of the method. Secondly, we examine the computation of the entries in the discrete linear system representing the continuous integral equation in collocation methods for hybrid numerical-asymptotic basis spaces on simple geometric shapes in the context of high-frequency wave scattering. This requires the computation of singular highly-oscillatory integrals and we develop efficient numerical methods that can compute these integrals at frequency-independent cost. Finally, we provide a general result that allows the construction of recurrences for the efficient computation of quadrature moments in a broad class of Filon quadrature methods, and we show how this framework can also be used to accelerate certain Levin quadrature methods.Supported by EPSRC grant EP/L016516/

Numerical integration of Schr\"odinger maps via the Hasimoto transform

We introduce a numerical approach to computing the Schr\"odinger map (SM) based on the Hasimoto transform which relates the SM flow to a cubic nonlinear Schr\"odinger (NLS) equation. In exploiting this nonlinear transform we are able to introduce the first fully explicit unconditionally stable symmetric integrators for the SM equation. Our approach consists of two parts: an integration of the NLS equation followed by the numerical evaluation of the Hasimoto transform. Motivated by the desire to study rough solutions to the SM equation, we also introduce a new symmetric low-regularity integrator for the NLS equation. This is combined with our novel fast low-regularity Hasimoto (FLowRH) transform, based on a tailored analysis of the resonance structures in the Magnus expansion and a fast realisation based on block-Toeplitz partitions, to yield an efficient low-regularity integrator for the SM equation. This scheme in particular allows us to obtain approximations to the SM in a more general regime (i.e. under lower regularity assumptions) than previously proposed methods. The favorable properties of our methods are exhibited both in theoretical convergence analysis and in numerical experiments

Symmetric resonance based integrators and forest formulae

We introduce a unified framework of symmetric resonance based schemes which preserve central symmetries of the underlying PDE. We extend the resonance decorated trees approach introduced in arXiv:2005.01649 to a richer framework by exploring novel ways of iterating Duhamel's formula, capturing the dominant parts while interpolating the lower parts of the resonances in a symmetric manner. This gives a general class of new numerical schemes with more degrees of freedom than the original scheme from arXiv:2005.01649. To encapsulate the central structures we develop new forest formulae that contain the previous class of schemes and derive conditions on their coefficients in order to obtain symmetric schemes. These forest formulae echo the one used in Quantum Field Theory for renormalising Feynman diagrams and the one used for the renormalisation of singular SPDEs via the theory of Regularity Structures. These new algebraic tools not only provide a nice parametrisation of the previous resonance based integrators but also allow us to find new symmetric schemes with remarkable structure preservation properties even at very low regularity.Comment: 71 page

Feature-assisted interactive geometry reconstruction in 3D point clouds using incremental region growing

Reconstructing geometric shapes from point clouds is a common task that is often accomplished by experts manually modeling geometries in CAD-capable software. State-of-the-art workflows based on fully automatic geometry extraction are limited by point cloud density and memory constraints, and require pre- and post-processing by the user. In this work, we present a framework for interactive, user-driven, feature-assisted geometry reconstruction from arbitrarily sized point clouds. Based on seeded region-growing point cloud segmentation, the user interactively extracts planar pieces of geometry and utilizes contextual suggestions to point out plane surfaces, normal and tangential directions, and edges and corners. We implement a set of feature-assisted tools for high-precision modeling tasks in architecture and urban surveying scenarios, enabling instant-feedback interactive point cloud manipulation on large-scale data collected from real-world building interiors and facades. We evaluate our results through systematic measurement of the reconstruction accuracy, and interviews with domain experts who deploy our framework in a commercial setting and give both structured and subjective feedback.Comment: 13 pages, submitted to Computers & Graphics Journa

Learning the Sampling Pattern for MRI.

The discovery of the theory of compressed sensing brought the realisation that many inverse problems can be solved even when measurements are "incomplete". This is particularly interesting in magnetic resonance imaging (MRI), where long acquisition times can limit its use. In this work, we consider the problem of learning a sparse sampling pattern that can be used to optimally balance acquisition time versus quality of the reconstructed image. We use a supervised learning approach, making the assumption that our training data is representative enough of new data acquisitions. We demonstrate that this is indeed the case, even if the training data consists of just 7 training pairs of measurements and ground-truth images; with a training set of brain images of size 192 by 192, for instance, one of the learned patterns samples only 35% of k-space, however results in reconstructions with mean SSIM 0.914 on a test set of similar images. The proposed framework is general enough to learn arbitrary sampling patterns, including common patterns such as Cartesian, spiral and radial sampling

Bridging the gap: symplecticity and low regularity on the example of the KdV equation

Recent years have seen an increasing amount of research devoted to the development of so-called resonance-based methods for dispersive nonlinear partial differential equations. In many situations, this new class of methods allows for approximations in a much more general setting (e.g. for rough data) than, for instance, classical splitting or exponential integrator methods. However, they lack one important property: the preservation of geometric structures. This is particularly drastic in the case of the Korteweg--de Vries (KdV) equation which is a fundamental model in the broad field of dispersive equations that is completely integrable, possessing infinitely many conserved quantities, an important property which we wish to capture -- at least up to some degree -- also on the discrete level. A revolutionary step in this direction was set by the theory of geometric numerical integration resulting in the development of a wide range of structure-preserving algorithms for Hamiltonian systems. However, in general, these methods rely heavily on highly regular solutions. State-of-the-art low-regularity integrators, on the other hand, poorly preserve the geometric structure of the underlying PDE. This work makes a first step towards bridging the gap between low regularity and structure preservation. We introduce a novel symplectic (in the Hamiltonian picture) resonance-based method on the example of the KdV equation that allows for low-regularity approximations to the solution while preserving the underlying geometric structure of the continuous problem on the discrete level

Characterization of a Capacitive Sensor for Particulate Matter

We characterize a novel micro-sensor with pairs of interdigitated combs of microelectrodes designed to detect particles in air. We evaluate the sensor’s response to 1 µm Polystyrene Latex (PSL) particles experimentally and crosscheck the results with simulations. Experiment and simulation show good consistency. Based on the promising results we propose a redesign of the capacitive particle sensor with respect to PM2.5

Simplified Damage Assessment Tool for Rails and Crossings Based on Standard Wear and RCF Models

A numerical tool is proposed to simultaneously assess various damage mechanisms that are driven by contact loading. The tool transfers loads to the contact-patch level using three contact parameters: the maximum contact pressure (pmax), the creepage (c) and the contact length (2a). The local wear and RCF predictions are implemented based on existing models from the literature. The load input can originate from numerical vehicle–track simulations or manual input of the user. The assessment tool is applied for a finite element analysis of a fixed manganese crossing nose to prove its validity. The algorithm is implemented via an automated Python code, which, on the one hand enables damage prediction for track components based on standard damage models. On the other hand, knowledge of novel local contact damage models can be transferred to the scale of track components