# Waves and Continuum Mechanics

# Sample PhD Projects in Waves and Solid Mechanics

## General Interests:

Continuum mechanics; composite media; fracture mechanics; periodic structures; scientific computing and image algorithms

### 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 Daniel Colquitt

### 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.

### Professor Ke Chen

### Scientific computing and numerical analysis

The purpose of numerical analysis is to analyse existing algorithms to achieve insights, to use the understanding for fundamental improvements and optimization of algorithms, and ultimately to serve the scientific computing needs of other sciences and engineering. In this research field, a project can be designed to focus on one of these scientific challenges

- Solution of structured matrix problems by exploring the structure to design fast algorithms with optimal complexity to provide computing capabilities beyond existing computer power at any level;
- Study of new non-linear system solvers arising from discretized partial differential equations. Here we aim to tackle non-linearity operator spitting methods and homotopy continuation ideas to enable the convergence of iterative methods;
- Study of new non-linear optimization solvers arising from minimization problems. Here we aim to tackle non-convexity by convexification ideas. There is much scope to go in constructing reliable and fast algorithms for finding a global minimizer.
- Design and analysis of high order discretization schemes for partial differential equations using adaptivity and sparse grid ideas.

### Development of novel methods for inverse problems

Most equations from various applications are so called forward problems. Inverse problems are new research topics where one knows part of the solution but there are missing or inaccessible data to find out. Examples are wide: given known observations (e.g. of sound or pollution or radar signals or outside measurements), how to work out the source information. The commonly used models are variational, involving minimization of some geometries of functions and practical solution of partial differential solutions.

### Modelling of high resolution imaging problems

Using advanced mathematics of continuous functions and equations, beyond matrices and vectors, one can restore high resolution images from given low resolution ones. This capability gives information science a major boost in intelligence tasks. A related problems is to use different image modalities to obtain new information that are not visible in current technology. One challenge is to fuse infra-red images with digital ones and another for fusing ultra-sound with CT images. The key components are a study of functional mappings and advanced data fitting for functions.

### Dr Gayane Piliposyan

### Spin and elasto-spin surface waves in ferromagnetic media

In the past few years, there has been a rapidly growing interest in the properties of spin waves (or magnons) in ordered magnetic materials. These are excitations that characterize the dynamical behavior of the magnetization variables in ferromagnets, ferrimagnets and antiferromagnets. The aim of this project is to develop a mathematical model and investigate the propagation of surface spin and elasto-spin waves in piecewise homogeneous ferromagnetic media consisting of different homogeneous parts with different ferromagnetic properties. .

### 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

### Hybrid asymptotic-numerical approach for the problem of initiation of excitation waves

Excitation waves play important role in biological signalling. Mathematical theory describing propagation of such waves is well developed. The classification of initial conditions that give rise to such waves, i.e. the problem of initiation, is understood much less. It is difficult and involves non-stationary solutions to nonlinear partial differential equations with no internal symmetries. Some analytical progress in this direction has been made recently by the supervisor's group for the Zeldovich-Frank-Kamenetsky equation (also known as Huxley equation or Nagumoe equation in cardiac physiology and Schloegl model in chemical kinetics). In the proposed project, we shall extend this approach to a more realistic situation when the critical nucleus and the ignition mode for a given excitable model have to be found numerically.

### Analytical and numerical study of the chaotic behaviour of scroll waves

Scroll waves of excitation are three-dimensional analogues of two-dimensional spiral waves. They are thought to underlie dangerous cardiac arrhythmias in the hearts of larger animals, including humans..Due to the extra spatial dimension, scroll waves have more degrees of freedom and more possibilities to go chaotic. Ventricular fibrillation, which is the most lethal of all arrhythmias, is a fully developed chaos of cardiac excitation. The project will use dynamical systems methods to study ways of transition to chaos as the thickness of model cardiac muscle grows, thus providing a continuous transition from two to three spatial dimensions.

### Iterative methods of calculation of response functions of spiral waves

Spiral waves are a form of self-organization observed in distributed active media, such as some catalytic chemical reactions or heart muscle. A peculiar feature of spiral waves is "wave-particle duality": being waves, they behave like particles when drifting in response to generic small perturbations. This allows simplified asymptotic description of their drift in terms of ordinary differential equations of motion. The success of this asymptotic approach depends on knowledge of so called response functions. A numerical method for calculating the response functions has been developed recently by the supervisors' group, and its workability has been successfully demonstrated on a number of concrete models; however it has its limitations. The subject of the present project will be development and investigation of alternative methods of calculation of response functions.

### 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 elastic 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.

[3] I. Thompson. Wave diffraction by a rigid strip in a plate modelled by Mindlin theory. In progress; to be submitted in early 2020.

### 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.

### 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.

### Links

Research Centre in Mathematics and Modelling