Dr Daniel Colquitt
Underwater acoustics
Sound is the primary method used for a wide variety of underwater activities, including communication, sonar, navigation, exploration, and monitoring. These activities span a broad spectrum of domains from marine biology and climatology to defence, navigation, and transport. The earliest recorded use of sonar by humans was in the late 15th century when da Vinci used a partially submerged tube to listen to passing ships. However, we still lack a comprehensive understanding of the complex processes that are involved in the propagation of sound underwater and the mechanisms associated with these phenomena are rarely studied and remain poorly understood, particularly from a mathematical and physical perspective. This project will study a number of critical dynamic phenomena, which affect the propagation of sound waves through ocean environments, including internal waves, multi-scale structural thermal and temporal variations and fluctuations, scattering by non-smooth interfaces and boundaries (e.g. semi-submerged structures, seabed, surface), currents, eddies, and fronts.
Multi-scale mechanical structures for the control of elastic waves
This project involves the mathematical modelling and design of mechanical structures capable of controlling the propagation of surface and bulk waves in elastic solids. The project will employ both numerical and asymptotic analysis to study a combination of continuous and discrete structures in one-, two-, and three-dimensions. The research programme includes scattering, homogenisation, and spectral problems for finite and infinite systems and has a broad range of applications including, filtering of waves, lensing, and cloaking.
Defect states in discrete elastic systems
This project involves the analysis of finite and semi-infinite defects in elastic lattice systems. These defects may be dislocations or variations in inertial properties and, for finite defects, will have an associated spectrum of eigenstates. The focus of the research programme is on the analysis of these eigenstates and the fields in the vicinity of the defect sites; algorithms will be developed to study the solutions in various asymptotic regimes. The research programme will involve both analytical and numerical models. The project may also include the study of edge and interfacial waves in mechanical lattices.
Dr Stewart Haslinger
Scattering of ultrasound by rough defects for industrial application
Non-destructive evaluation (NDE) is a collection of analysis techniques used to assess the properties of a material, component or structure, without introducing damage. It is a highly valuable tool in the energy, power and aerospace engineering sectors since it is capable of inspecting safety-critical systems whilst saving both time and money.
This project will focus on mathematical modelling of ultrasonic testing (UT) inspections where transducers are used to send and receive elastic waves (shear and longitudinal bulk waves) that propagate through a structure being investigated. The scattering of waves by inclusions and defects within a material is used to diagnose the health and lifespan of a component.
This project will use a combination of analytical (inverse problems, asymptotic analysis, stochastic methods), numerical simulation and machine learning to develop mathematical models for predicting the scattering amplitudes from very rough and branched cracks.
Realisation of active cloaking in structured elastic solids
In recent years, there has been a proliferation of theoretical and experimental models for cloaking systems (making objects ``appear” invisible to incoming waves) in the physical sciences, particularly for electromagnetic and acoustic waves. The validation and application of elastic wave cloaking, which has potential application in seismic engineering, has proved more difficult to achieve.
This project will begin with a mathematical model for an elastic plate containing a periodic system of pinned points that may be cloaked using a surrounding array of tuned point sources (active sources). The model will be adapted by introducing finite-sized sources and objects to be cloaked, using a combination of asymptotic and numerical analysis and inverse methods. The goal of the research is to design and validate new models for engineering applications, for instance a system of tuned actuators to protect sensitive components from extreme vibrations.
Professor Alexander B. Movchan
Mathematical models of solids with imperfect interfaces
This project involves analysis of equations of solid mechanics in multi-phase media separated by thin and soft layers. The effective contact conditions involve discontinuities of displacement (and possibly traction) components. The objective of this work is to develop analytical and numerical models describing the fields around imperfect interfaces. This study will require asymptotic theory for singularly perturbed boundary value problems and analysis of singular integral equations. The range of applications involve models (including dynamics) of thin anisotropic plates, shells and models of delamination cracks on imperfect interfaces.
Asymptotic analysis of crack-defect interaction in three dimensions
The project deals with asymptotic models of 3D cracks propagating in elastic domains containing small defects (micro-cracks or small inclusions). A defect is characterized by its dipole tensor. A singular perturbation algorithm is to be developed to describe a deflection of the crack front due to interaction with a small defect (or an array of defects).
Models of periodic structures with defects
This project offers analysis of problems of conductivity, electro-magnetism and elasticity in periodic structures with defects. One of the model formulations includes an array of circular inclusions. The unperturbed structure is supposed to be doubly periodic; in the ''damaged'' structure one or several inclusions have been removed. The objective is to develop an analogue of the Rayleigh method describing this damaged structure. The second group of problems involves lattice structures with defects (dislocations)
and requires homogenization and numerical analysis to describe the gradient fields around the defects.
Asymptotics for eigenvalue problems posed in multi-structures
For some configurations of multi-structures, explicit asymptotic formulae for the first few eigenfrequencies can be derived. It is also interesting to introduce some defect within the structure and analyse how the eigenfrequencies and their order change. The formulae can also be supported by numerical calculations (finite element computations).
Modelling of waves in multi-scale metamaterials
This topic includes analysis of hyperbolic partial differential equations on a multi-scale network. Physical interpretation is in waves mechanics, dispersion, dynamic anisotropy, and exponentially localised wave forms. Asymptotic analysis of a singularly perturbed system with high anisotropy also leads to modelling of so-called ''invisibility cloaks'', which reduce scattering of an incident wave in a solid with finite size obstacles.
Mathematical modelling of fracture in adhesive joints
Vibrations of a weakly nondegenerate 1D-3D multi-structure
Professor Natalia V Movchan
Asymptotic analysis of fracture in composite materials
The project involves asymptotic and numerical analysis of singular integral equations describing cracks in composite media. The integral equation formulation is set for the displacement field on the crack faces. The main attention is given to singular perturbations of domains occupied by cracks and to singular integral equations with a small parameter near the integral term, modelling cracks in fibre-reinforced composites.
Mathematical modelling of wave propagation in phononic crystals
The aim of this project is to develop a qualitatively new class of mathematical models describing phononic band gap structures with inertial structural interfaces. This will involve a combination of continuum and lattice structures and will cover both infinite periodic structures and layered structures. It is also planned to study the effect of disorder and models of defects within periodic structures, and to generalise our earlier models of filters and polarisers of elastic waves developed for stacks of layers to the more general cases of cylindrical and spherical layered structures.
Spectral problems related to sizing and location of defects in elastic structures
The project involves the mathematical study of spectral problems of elasticity for solids with small defects (such as cavities, inclusions and cracks), and it is based on the asymptotic theory of singular perturbations of elliptic operators. The purpose of this study is to develop mathematical techniques to detect, locate and size defects in elastic structures.
Mathematical models of dynamic structural interfaces
The project deals with periodic composite structures with inertial interfaces associated with localised eigenstates of certain spectral problems. Asymptotic algorithms are to be developed to estimate the frequencies corresponding to localised eigensolutions and to study propagation of elastic waves in structures of this type.
Dr Gayane Piliposyan
Electro-elastic waves in piezoelectric cubic crystals
This project is concerned with the mathematical formulation of electro-elastic wave propagation in piezoelectric cubic crystals from an instantaneous impulse type point source. The behaviour of electro-elastic waves will be investigated. In particular, the geometrical form of wave fronts and conditions of existence of cusps and lacunas (undisturbed regions) in wave fields will be considered. The methods of plane waves, integral transforms and Cagniard and Smirnov-Sobolev functional-invariant solutions will be applied.
Dr Ozgur Selsil
Active cloaking of waves in thin plates
Making an object invisible, by manipulating the waves around them, has been a well-explored area of research in the last 20 years. The work in this field may be divided into two main streams: interior and exterior cloaks. In the former, the cloaking device surrounds the object to be cloaked, and in the latter, the cloaking region lies outside the cloaking device. Interior cloaking methods include transformation based cloaking and active cloaking. To achieve active cloaking a discrete number of monopole sources may surround the object and by carefully choosing their intensities propagating components of the scattered wave may be eliminated. Alternatively, a small number of multipole cloaking devices may be used to create a region of `stillness' and an object may be placed in this region with additional attention on the boundary conditions. The proposed project relies on our previous work on active cloaking and aims to combine these two novel methods.
Discrete chiral elastic structures
Waves in elastic lattice systems exhibit dispersion. Lattice structures can be engineered to achieve desired dynamic responses, particularly concerning wave dispersion and localisation. The bandgap phenomenon, which occurs when a non-uniform distribution of mass along a mass-spring system creates destructive interference and prevents the propagation of certain wave frequencies, has been utilised in the design of filters and waveguides. Recent studies on curved two-dimensional mass-spring systems (lattices) have established connections to one-dimensional chains. In addition, geometrically chiral three-dimensional mass-spring systems have been investigated and again connections with one-dimensional chains have been explored. Interesting effects have been noticed with the presence of physical chirality in these systems, by attaching gyroscopic spinners to some of its components, including the uncovering of a mechanical analogue of electromagnetic induction for waves in chiral elastic structures. The proposed project is concerned with creating discrete chiral three-dimensional systems which possess exciting properties and applications.
Dr Ian Thompson
Diffraction problems in elasticity
Modelling wave propagation in elastic media is important for non-destructive evaluation of materials. Waves are transmitted into the material and the scattering pattern is used to detect cracks and other defects. To apply this method, one must first understand how waves interact with these defects. Modelling this process is a challenging mathematical problem, which requires sophisticated complex analysis, asymptotic methods and (in all but the simplest cases) numerical methods. Many diffraction problems in acoustics can be solved using a standard procedure [1], but often the problems that arise in elasticity involve coupled systems of Wiener-Hopf equations, and there is no known general method for solving these [2]. However, a very effective and direct numerical method has recently been introduced and used to solve a coupled Wiener-Hopf system [3]. The objective of this project is to apply the same method to other coupled systems, such as those arising from parallel defects and cracks in plates modelled using Mindlin theory. Programming experience would be an advantage to students applying for this project; willingness to develop scientific programming skills is an absolute necessity.
References
[1] B. Noble. Methods Based on the Wiener-Hopf Technique. Chelsea, 1988.
[2] J. B. Lawrie and I. D. Abrahams, 2007. A brief historical perspective of the Wiener-Hopf technique. Journal of Engineering Mathematics 59, 351-358. Doi: 10.1007/s10665-007-9195-x
[3] I. Thompson, 2020. Wave diffraction by a rigid strip in a plate modelled by Mindlin theory. Proceedings of the Royal Society A 476: 20200648. Doi: 10.1098/rspa.2020.0648
Wave propagation through periodic structures with defects
Recent years have seen a rapid expansion in the research resources devoted to the study of wave interactions with periodic structures that have defects. Defects can be used to trap waves, or to guide propagation along particular paths. A method based on Fourier analysis was used to solve scattering problems involving defects in linear arrays of cylinders in [1]. The aim of this project is to further develop this idea, and use it to model wave interactions with different types of defects in two- and three-dimensional lattices. Acoustic waves will be considered in the first instance; extensions to the more difficult cases of electromagnetism and elasticity are possible. Successful completion of this research will require complex analysis, asymptotics and numerical computations. Programming experience would be an advantage to students applying for this project; willingness to develop scientific programming skills is an absolute necessity.
Reference
[1] I. Thompson and C. M. Linton, 2008. An interaction theory for scattering by defects in arrays. SIAM Journal on Applied Mathematics, 68 (6): 1783–1806. Doi: 10.1137/070703144
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