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Title Mathematical image analysis of 4D X-ray microtomography data for crack propagation in aluminium wire bonds in power electronics Algebra and Analysis, Industrial and Applied Mathematics Dr Yves van Gennip Supervised by Dr Yves van Gennip and Dr Pearl Agyakwa In this project we will develop mathematical image processing methods for the analysis of 4D (3d+time) X-ray microtomography data of wire bonds [1]. Wire bonds are an essential but life-limiting component of most power electronic modules, which are critical for energy conversion in applications like renewable energy generation and transport. The key issue we will examine is how we can use mathematical image processing and image analysis techniques to study how defects in wire bonds arise and evolve under operating conditions; this will facilitate more accurate lifetime prediction. [1] Agyakwa et al., “A non-destructive study of crack development during thermal cycling of Al wire bonds using x-ray computed tomography[2] Brune C., “4D imaging in tomography and optical nanoscopy”, PhD thesis, University of Münster (2010)[3] van Gennip, Y., Athavale, P., Gilles, J. and Choksi, R., 2015, "A Regularization Approach to Blind Deblurring and Denoising of QR Barcodes”, IEEE Transactions on Image Processing, 24(9), 2864-2873 3D X-ray microtomography provides non-destructive observations of defect growth, which allows the same wire bond to be evaluated over its lifetime, affording invaluable new insights into a still insufficiently understood process.  Analysis and full exploitation of the useful information contained within these large datasets is nontrivial and requires advanced mathematical techniques [2,3]. A major challenge is the ability to subsample compressible information without losing informative data features, to improve temporal accuracy. Our goal is to construct new mathematical imaging and data analysis methods for evaluating 2D and 3D tomography data for the same wire bond specimen at various stages of wear-out during its lifetime, to better understand the degradation mechanism(s). This will include methods for denoising of data, detection and segmentation of the wires and their defects in the tomography images, and image registration to quantify the wire deformations that occur over time.
Title Crystallisation in polymers Industrial and Applied Mathematics Dr Richard Graham Polymers are very long chain molecules and many of their unique properties depend upon their long chain nature. Like simple fluids many polymer fluids crystallise when cooled. However, the crystallisation process is complicated by the way the constituent chains are connected, leading to many curious and unexplained phenomena. Furthermore, if a polymer fluid is placed under flow, this strongly affects both the ease with which the polymer crystallises and the arrangement of the polymer chains within the resulting crystal. This project will develop and solve models for polymer dynamics and phase transitions using a range of analytical, numerical and stochastic techniques, with the ultimate aim of improving our understanding of polymer crystallisation. The project offers the opportunity to collaborate with a wide range of scientists working in the field, including several world-leading experimental groups.
Title Dynamics of entangled polymers Industrial and Applied Mathematics Dr Richard Graham Polymers are extraordinarily long molecules, made out of chains of simpler molecules. They occur everywhere in our everyday lives, including in the DNA chains that make up our genetics, in many high-tech consumer products and in the simple plastic bag. Often these applications depend crucially on the way that the polymer chains move. This is especially true in concentrated polymer liquids, where the chain dynamics are controlled by how the chains become entangled with each other. A powerful mathematical framework for describing these entangled systems has been under development for some time now, but the ideas have yet to be fully developed, tested and exploited in practical applications. Working on this PhD project will give the opportunity to train in a wide range of mathematical techniques including analytical work, numerical computations and stochastic simulation and to apply these to problems of real practical impact. This lively research field involves mathematicians, scientists and engineers and a keenness to learn from and co-operate with researchers from a range of backgrounds would be a real asset in this project.
Title Instabilities of fronts Industrial and Applied Mathematics Dr Stephen Cox, Dr Paul Matthews Chemical reactions often start at a point and spread through a reactant, much as a fire spreads through combustible material. The advancing zone in which the reactions take place is called a reaction front. In the simplest cases, the reaction front is smooth (flat, cylindrical or spherical), but it may develop irregularities due to instability. Sometimes the instability is so strong that it destroys the front itself; in other cases, it just results in a slight modulation to the front shape. This project involves studying a partial differential equation, the Nikolaevskiy equation, that describes the nonlinear development of the instability of a front. Numerical simulations of the Nikolaevskiy equation show highly complicated, chaotic solutions. This project will involve a mix of numerical simulations and analytical work to understand the behaviour of the Nikolaevskiy equation and of front instabilities in general.
Title Power converters Industrial and Applied Mathematics Dr Stephen Cox, Dr Stephen Creagh In a wide range of applications, it is necessary to convert one electrical power supply to another, of different voltage or frequency. Power converters are devices which achieve this, but they often suffer highly undesirable instabilities, which significantly compromise their operation. The goal of this project is to develop mathematical models for existing power converter technologies and to use these to provide a detailed description of their operation and a thorough understanding of the instability. Through mathematical modelling, it may prove possible to improve existing power converter designs to reduce or eliminate the stability problems! This project will be theoretical in nature, relying largely on analytical and numerical techniques for differential equations, and will involve significant interaction with the Power Electronics Group in the Department of Electrical and Electronic Engineering. S M Cox and J C Clare Nonlinear development of matrix-converter instabilities. Journal of Engineering Mathematics 67 (2010) 241-259. S M Cox and S C Creagh Voltage and current spectra for matrix power converters. SIAM Journal on Applied Mathematics 69 (2009) 1415-1437. S M Cox Voltage and current spectra for a single-phase voltage-source inverter. IMA Journal of Applied Mathematics 74 (2009) 782-805.
Title Class-D audio amplifiers Industrial and Applied Mathematics Dr Stephen Cox, Dr Stephen Creagh The holy grail for an audiophile is distortion-free reproduction of sound by amplifier and loudspeaker. This project concerns the mathematical modelling and analysis of class-D audio amplifiers, which are highly efficient and capable of very low distortion. Designs for such amplifiers have been known for over 50 years, but only much more recently have electronic components been up to the job, making class-D amplifiers a reality. (Class-D amplifiers rely on very high frequency – around 1MHz – sampling of the input signal, and so test their components to the limit.) Unfortunately, while the standard class-D design offers zero distortion, it has poor noise characteristics; when the design is modified by adding negative feedback to reduce the noise, the amplifier distorts. By a further modification to the design it is possible to eliminate (most of) the distortion. This project involves modelling various class-D designs and determining their distortion characteristics, with the aim of reducing the distortion. The project will be largely analytical, applying asymptotic methods and computer algebra to solve the mathematical models. Simulations in matlab or maple will be used to test the predictions of the mathematical models. J Yu, M T Tan, S M Cox and W L Goh Time-domain analysis of intermodulation distortion of closed-loop Class D amplifiers. IEEE Transactions on Power Electronics 27 (2012) 2453-2461. S M Cox, C K Lam and M T Tan A second-order PWM-in/PWM-out class-D audio amplifier. IMA Journal of Applied Mathematics (2011). S M Cox, M T Tan and J Yu A second-order class-D audio amplifier. SIAM Journal on Applied Mathematics 71 (2011) 270-287. S M Cox and B H Candy Class-D audio amplifiers with negative feedback. SIAM Journal on Applied Mathematics 66 (2005) 468-488. S M Cox and S C Creagh Voltage and current spectra for matrix power converters. SIAM Journal on Applied Mathematics 69 (2009) 1415-1437.
Title Mathematical modelling and analysis of composite materials and structures Industrial and Applied Mathematics Dr Konstantinos Soldatos Nottingham has established and maintained, for more than half a century, world-wide research leadership in developing the Continuum Theory of fibre-reinforced materials and structures. Namely, a theoretical mechanics research subject with traditional interests to engineering and, more recently, to biological material applications. The subject covers extensive research areas of mathematical modelling and analysis which are of indissoluble adherence to basic understanding and prediction of the elastic, plastic, visco-elastic or even viscous (fluid-type) behaviour observed during either manufacturing or real life performance of anisotropic, composite materials and structural components.Typical research projects available in this as well as in other relevant research subjects are related with the following interconnected areas:Linearly and non-linearly elastic, static and dynamic analysis of homogeneous and inhomogeneous, anisotropic, solid structures and structural components;Forming flow (process modelling) of fibre-reinforced viscous resins;Plastic behaviour modelling as well as study of failure mechanisms/modes of fibrous-composites;Mass-growth modelling of soft and hard tissue, such as human nail and hair.  Mathematical modelling and analysis of the behaviour of thin walled, anisotropic structures in terms of high-order, linear or non-linear, ordinary and partial differential equations;Development and/or use of various analytical, semi-analytical and/or numerical mathematical methods suitable for solving the sets of differential equations which emerge in the above research areas.The large variety of topics and relevant problems emerging in these subjects of Theoretical Mechanics and Applied Mathematics allow considerable flexibility in the formation of PhD projects. A particular PhD project may accordingly be formed/designed around the strong subjects of knowledge of a potential post-graduate student. The candidate’s relevant co-operation is accordingly desirable and, as such, will be appreciated at the initial, but also at later stages of tentative research collaboration.
Title Dynamics of coupled nonlinear oscillators Industrial and Applied Mathematics Dr Paul Matthews Coupled oscillators arise in many branches of science and technology and also have applications to biological systems. One spectacular example is swarms of fireflies that flash in synchrony. This research field is an expanding area in applied mathematics because of the many applications within physics and biology and because of the variety of behaviour which such systems can exhibit. Recent work on coupled oscillators has revealed some interesting novel results: nonlinear oscillators can synchronise to a common oscillation frequency even if they have different natural frequencies, provided the coupling is above some threshold; the breakdown of synchronisation as the coupling strength decreases involves periodic behaviour and chaos. The project involves extending and improving this work in two ways. First, the oscillator model used in earlier work was simple and idealised; the model will be refined to make it more realistic.Second, most earlier work used a simple linear global coupling so that each oscillator is equally coupled to all of the others. In most practical examples this is not the case and a coupling law over a two- or three-dimensional lattice would be more appropriate, with stronger coupling between nearer pairs of oscillators. The research will be carried out using a combination of numerical and analytical techniques.
Title Dynamo action in convection Industrial and Applied Mathematics Dr Paul Matthews The magnetic fields of the Earth and Sun are maintained by dynamo action. Fluid motions are generated by thermal convection. The kinetic energy of these fluid motions is then converted to magnetic energy, in a manner similar to that of a bicycle dynamo. Dynamo theory studies how this conversion takes place. It is known that in order for a dynamo to work efficiently, the fluid flow must exhibit chaos.This project will investigate dynamo action in convection, using 3D numerical simulation of the equations for the fluid motion and the magnetic field. An existing computer program will be used to study the dynamo problem. A sequence of numerical simulations will be carried out to determinethe conditions under which convection is able to generate a magnetic field,whether rotation is necessary or advantageous for dynamo action,the underlying topological stretching mechanism responsible for the magnetic field generation.The project is also suitable for analytical work, either based on an asymptotic analysis of the equations, or in investigating or proving 'anti-dynamo' theorems.
Title Nonlinear penetrative convection Industrial and Applied Mathematics Dr Paul Matthews The phenomenon of convection, in which heat is transferred by fluid motion, occurs very commonly in nature. Examples include in the Earth's atmosphere, the interior of the Sun, the Earth's liquid outer core, lakes and oceans. The most commonly used mathematical model for convection assumes that a layer of fluid is bounded above and below by boundaries that are maintained at a fixed temperature. This is not a good model for most of the environmental applications, where typically part of the fluid layer is thermally unstable and part is stable. Convection in the unstable layer overshoots and penetrates into the stable layer. This phenomenon, known as 'penetrative convection', has received relatively little investigation. The research project will study penetrative convection in the nonlinear regime. An existing computer program will be adapted to investigate penetrative convection numerically, and analytical work will be carried out using asymptotic methods and methods of bifurcation theory. In particular, the extent of penetration into the stable layer and the possibility of instability to a mean flow will be explored.
Title Coupling between optical components Industrial and Applied Mathematics Dr Stephen Creagh Evanescent coupling between different optical components is a very important process in optical communications. In this effect, light travelling along an optical fibre effectively spills out a little bit into the region of space immediately surrounding the fibre itself and can then leak into and become captured by other, nearby optical components. Among other uses, this mechanism forms a basis for optical switches, which transfer light from one fibre to another, and for wavelength filters, which selectively transmit or redirect light in only certain frequency ranges.This project will investigate the coupling between cylindrical and spherical optical components in two and three dimensions using the geometry of the underlying ray solutions. The aim will be to exploit and generalise approximations which have been developed in the context of quantum waves but which should be equally applicable to the optics problem.
Title Is Periodic Behaviour an Emergent Phenomenon? Industrial and Applied Mathematics Dr Keith Hopcraft Periodic behaviours can be described with great power and economy using the rather simple mathematical machinery associated with wave phenomena. However periodic effects can also be ‘observed’ in collections of discrete objects, be they individuals sending emails, fire-flies signaling to attract mates, synapses firing in the brain or photons emerging from a cavity. The manifestation of periodicity requires both a dynamical process and a ‘medium’ in which it operates and the project will seek to identify the essential properties of the dynamics and the structure of the medium required to do this without invoking the ideas of the continuum, determinism or reversibility.A very simple but surprisingly rich model has been constructed, that involves purely random dynamics acting on a graph, which nevertheless exhibits amorphous, coherent and collapsed states as a single control parameter is changed. The coherent states indeed do exhibit periodic behaviours, and the criteria for this emergence to occur have been identified. Periodicity requires a minimum of three nodes in the graph, for there to be a bias in the direction for flow of information around the network and for the control parameter to exceed a threshold. It also requires the concept of ‘action at a distance’, which is familiar to any field theory. The project will investigate some of the other emergent properties this model possesses, before seeing whether the assumption of a field can be relaxed by considering the self-interaction of a node’s dynamics.
Title Modeling acoustic emission energy propagation Industrial and Applied Mathematics Dr Stephen Creagh, Prof Gregor Tanner This project will develop modelling techniques to predict acoustic emissions radiated from a water jet hitting a target object and subsequently reverberating within the structure. The approach taken will be to exploit dynamical systems and ray-propagation approaches developed in the context of wave and quantum chaos. The source is complex and statistically characterized in this scenario, so ray-tracing techniques must be adapted to predict features such as average intensities and correlations rather than the field amplitude itself.The motivation for this project is provided by manufacturing processes in which such water jets are used to create deformation or attrition on the target object, and will be undertaken in collaboration with Dr Amir Rabani of the School of Mechanical, Materials and Manufacturing Engineering. In the experiments, high-energy fluid jet milling where a multi phase media is used as the source of attrition is an example of such processes. These deformations or attritions are primarily due to the mechanical energy applied to the target object. This mechanical energy propagates within the target object in the shape of high frequency elastic and plastic waves that can be picked up using acoustic emission sensors [1]. Modeling the propagation of the mechanical energy of the propagating waves and their attenuation can provide valuable information about the applied energy to the target object by the source. This information can potentially provide means to monitor/control the deformation or attrition process of the target object. The applications of the acoustic emission energy propagation model can go beyond the manufacturing arena and can be used in condition monitoring to diagnose the causes of deformations and attritions. This provides measures to take fail preventative actions that other methods such as non-destructive evaluation (NDE) methods fail to provide.   References:[1] Rabani, A., Marinescu, I., & Axinte, D. (2012). Acoustic emission energy transfer rate: a method for monitoring abrasive waterjet milling. International journal of machine tools and Manufacture, 61, 80-89.
Title Solitons in higher dimensions Industrial and Applied Mathematics Prof Jonathan Wattis The localisation of energy and its transport is of great physical interest in many applications. The mechanisms by which this occurs have been widely studied in one-dimensional systems; however, in two- and three-dimensional systems a greater variety of waves and wave phenomena can be observed; for example, waves can be localised in one or both directions.This project will start with an analysis of the nonlinear Schrodinger equation (NLS) in higher space dimensions, and with more general nonlinearities (that is, not just $\gamma=1$). Current interest in the Bose-Einstein Condensates which are being investigated in the School of Physics and Astronomy at Nottingham makes this topic particularly timely and relevant.The NLS equation also arises in the study of astrophysical gas clouds, and in the reduction of other nonlinear wave equations using small amplitude asymptotic expansions. For example, the reduction of the equations of motion for atoms in a crystal lattice; this application is particularly intriguing since the lattice structure defines special directions, which numerical simulations show are favoured by travelling waves. Also the motion of a wave through a hexagonal arrangement of atoms will differ from that through a square array of atoms. The project will involve a combination of theoretical and numerical techniques to the study such systems.
Title Modelling the vibro-acoustic response of complex structures Industrial and Applied Mathematics Prof Gregor Tanner The vibro-acoustic response of mechanical structures (cars, airplanes, ...) can in general be well approximated in terms of linear wave equations. Standard numerical solution methods comprise the finite or boundary element method (FEM, BEM) in the low frequency regime and so-called Statistical Energy Analysis (SEA) in the high-frequency limit. Major computational challenges are posed by so-called mid-frequency problems - that is, composite structures where the local wave length may vary by orders of magnitude across the components.Recently, I propsed a set of new methods based on ideas from wave chaos (also known as quantum chaos) theory.  Starting from the phase space flow of the underlying - generally chaotic - ray dynamics, the new method called Dynamical Energy Analysis (DEA) interpolates between SEA and ray tracing containing both these methods as limiting cases. Within the new theory SEA is identified as a low resolution ray tracing algorithm and typical SEA assumptions can be quantified in terms of the properties of the ray dynamics. I have furthermore developed a hybrid SEA/FEM method based on random wave model assumptions for the short-wavelength components. This makes it possible to tackle mid-frequency problems under certain constraints on the geometry of the structure.The PhD project wil deal with extending these techniques towards a DEA/FEM hybrid method as well as  considering FEM formulations of the method. The work will comprise a mix of analytic and numerical skills and will be conducted in close collaboration with our industrial partners inuTech GmbH, Nurenberg, Germany and Jaguar/landrover, Gaydon, UK. Wave chaos in acoustics and elasticity, G. Tanner and N. Soendergaard, J. Phys. A 40, R443 - R509 (2007).Dynamical Energy Analysis - determining wave energy distributions in complex vibro-acoustical structures, G. Tanner, Journal of Sound and Vibration 320, 1023 (2009).
Title Ruin, Disaster, Shame! Industrial and Applied Mathematics Dr Keith Hopcraft Naturally occurring disasters, such as a freak wave that inundates a ship, a bear market that plunges an economy into recession, or those caused by extremes in weather resulting from ‘global warming’, cannot be avoided. But they can be planned for so that their devastating effects can be ameliorated. This project will study the mathematical properties extremal events that are caused by a stochastic process exceeding a threshold. It forms part of a larger programme that will generate data from an optical analogue of extremal events – the generation of caustics, and from analyses of financial and climate data. The project will investigate the extrema produced by a non-gaussian stochastic process that is represented mathematically by the nonlinear filtering of a signal, and will determine such useful quantities as the fluctuations in number of extremal events, and the time of occurrence to the next event. The project will involve modelling of stochastic processes, asymptotic analysis, simulation and data processing. Direct involvement with the experimental programme will also be encouraged. A Case Award supplement may be available for a suitably qualified candidate.
Title Projects in the mechanics of crystals Industrial and Applied Mathematics Dr Gareth Parry The aim is to understand different aspects of plastic behaviour in complex defective crystals. It is not surprising that methods of traditional continuum mechanics play a role in this area of materials science, but it is perhaps unexpected that classical ideas of differential geometry are central to an appreciation of the issues involved. A student of traditional background in either pure or applied mathematics will be guided, first of all, in reading and in other preparatory exercises, in such a way as to strengthen his or her knowledge appropriately. Possible research projects in this area are the following.Locking mechanisms in defective crystals: The traditional view regarding the propagation of defects such as dislocations is that they move under the influence of stress until they encounter some imperfection or other inhomogeneity. The aim of the project is to quantify this idea in the context of a mathematical model of defective crystals.Slip in complex crystals: Existing work details the types of slip that are allowed in a continuum theory of crystals for which the appropriate "state" is given by prescribing three linearly independent vector fields at each point of the region occupied by the crystal. Many crystals do not fit into this scheme, since more than three vector fields are required in the model. The project will extend the existing work to encompass these more complex cases.Geometrical structure of defective crystals: In some cases, the geometrical structure of defective crystals derives from the structure of certain three-dimensional Lie groups. This connection has not been exploited at all in the past, and the project will begin the cross-fertilization of mathematics and mechanics in this context.Constitutive functionals for defective crystals: Specifying relationships between stress and strain, subject to certain invariance requirements, is a classic and well-developed procedure in traditional continuum mechanics. In the presence of defects, the procedure is not so clear cut first of all one has to decide on appropriate measures of changes in geometry (like strain) and then decide if ideas like stored energy and stress are realistic. Finally, symmetry requirements deriving from microscopic considerations have to be derived. The project thus provides an essential prerequisite for any study of the continuum mechanics of this type of crystal.Variational problems: In the classical calculus of variations formulation of the theory of elasticity, the task is to find the infimum of the "stored energy" functional, sometimes accomplished by choosing a function (representing the elastic deformation) which actually provides a minimum of the functional. In defective crystals, the corresponding task is to find the infimum of an appropriate functional by choosing two functions representing (i) the elastic deformation (ii) the rearrangement (or slip) of the crystal. The project will consist of the study of mathematical problems of this type.Thermodynamics of defective crystals: It is accepted in the materials science community that friction and energy dissipation are involved in the slip of one crystal lattice plane over another and that temperature effects are important in the mechanics of crystals allowed to deform by slip. The task is to incorporate modern thermodynamic ideas into a theory of the mechanics of such processes.
Title The frequency of catastrophes Industrial and Applied Mathematics Dr Keith Hopcraft We have recently developed analytical stochastic models that are capable of describing the frequency of discrete events that have (essentially) an arbitrary distribution, including such extreme cases as when the mean does not exist. Such models can be used to investigate the frequency of rare or extremal events, and can be used to quantify the size of fluctuations that are generated by systems that are close to a critical point, where correlations have a dominating role. The current interest on global climate change provides an interesting and important area with which to apply these models. Climate records provide a detailed source of data from which one can deduce extremal events, such as the number of times the temperature or precipitation exceeds the mean during a period and the models then provide the capacity to estimate the future frequency of such occurrences. The work will involve time-series analysis of climate records, stochastic model building and solution of those models using analytical and numerical techniques.
Title Caustics: optical paradigms of complex systems Industrial and Applied Mathematics Dr Keith Hopcraft A complex system is multi-component and heterogeneous in character, the interactions between its component parts leading to collective, correlated and self-organising behaviours. Manifestations of these behaviours are diverse and can range from descriptions of matter near a critical point, through turbulence, to the organising structures that emerge in societies. The interactions which generate these behaviours are always nonlinear and often triggered by the system crossing a threshold, the frequency of crossing this barrier provides an important characteristic of the system under consideration. The pattern of caustics observed on the bottom of a swimming pool is one commonly experienced manifestation of such a threshold phenomenon, the caustics being caused by the stationary points of the water's surface. This illustrates how a continuous fluctuation- i.e. the water's surface, leads to the occurrence of a discrete the number of events — the caustics. The project will investigate the how the number of caustics depends on the properties of the surface and propagation distance (i.e. the depth of the swimming pool). The work will be mainly analytical in nature, involving elements of stochastic model building and their solution, with some simulation. There is a possibility of comparing models with experimental data of light propagation through 'model swimming-pools' and entrained fluids.
Title The discrete random phasor Industrial and Applied Mathematics Dr Keith Hopcraft In 1965 Richard Feynman wrote ‘I think I can safely say that nobody understands quantum mechanics’, and that situation has not changed in the intervening years despite its continued predictive capacity. One of the many paradoxes that the theory presents is wave-particle duality – for example an electric field behaves as a continuous wave disturbance according to Maxwell’s theory, but also presents phenomenology associated with discrete photons at microscopic scale-sizes. In the first instance this project will investigate how a very simple representation of electric field behaviour, a phasor of constant amplitude but random phase, has real and imaginary parts that can be represented by a population of classically interacting particles (photons). The project will proceed by seeking a generalization to this population model with characteristic that can be interpreted as being the addition of two random phasors, each of constant amplitude but independent phase. Such a model leads  to interference effects. No prior knowledge of quantum mechanics is required.
Title Machine learning for first-principles calculation of physical properties. Industrial and Applied Mathematics Dr Richard Graham The physical properties of all substances are determined by the interactions between the molecules that make up the substance. The energy surface corresponding to these interactions can be calculated from first-principles, in theory allowing physical properties to be derived ab-initio from a molecular simulation; that is by theory alone and without the need for any experiments. Recently we have focussed on applying these techniques to model carbon dioxide properties, such as density and phase separation, for applications in Carbon Capture and Storage. However, there is enormous potential to exploit this approach in a huge range of applications. A significant barrier is the computational cost of calculating the energy surface quickly and repeatedly, as a simulation requires. In collaboration with the School of Chemistry we have recently developed a machine-learning technique that, by using a small number of precomputed ab-initio calculations as training data, can efficiently calculate the entire energy surface. This project will involve extending the approach to more complicated molecules and testing its ability to predict macroscopic physical properties.This project will be jointly supervised by Dr Richard Wheatley in the School of Chemistry.
Title Modelling Thermal Effects within Thin-Film Flows Industrial and Applied Mathematics Notice: Undefined index: etzrsh in /maths/www/html/postgraduate/projects/index.php on line 477 Notice: Undefined index: etzrsh in /maths/www/html/postgraduate/projects/index.php on line 477 A number of technologies in aerospace gas turbine transmission systems must maintain appropriate cooling of component surfaces and mitigate contact by the use of thin fluid films. In many cases the operating requirements for these components include high rotation speeds, high pressures and high temperatures. Modern design processes for aeroengine components depend increasingly on high quality modelling tools to guide the creation of new products to obtain a comprehensive understanding of the underlying flow characteristics. This project aims to develop detailed understanding of heat transfer in highly sheared thin-film flows through the creation of sophisticated modelling approaches and numerical tools. Using this capability there will be an opportunity to perform detailed analysis of several engine-relevant configurations. The classical theory of thin-film flow is associated with solutions typically for low fluid speeds (Reynolds equation). For higher speed flow an important physical process, often neglected from many current thin-film flow models, is the generation, transfer and effect of heat within the film and from the surrounding structures. Advanced modelling requires the careful development of fully representative equations and the specification of appropriate boundary conditions. A new model to incorporate non-isothermal effects relevant to a bearing chamber context is provided byThis multi-disciplinary project will be undertaken by a graduate student in mathematics, engineering or related degree with a strong applied mathematics background and with an interest in fluid mechanics, mathematical and numerical methods. • Kay, E. D, Hibberd, S. and Power, H., (2014); A depth-averaged model for non-isothermal thin-film rimming flow. Int Jnl. Heat and Mass Transfer,70, 1003-1015. doi 10.1016/j.ijheatmasstansfer.2013.11.040 The supervision team will include include Prof H Power, Faculty of Engineering and a project partner at Rolls-Royce plc. This project is eligible as an EPSRC Industrial CASE award supported by Rolls –Royce plc that includes an additional stipend and a period of experience working locally at Rolls-Royce plc.
Title On mathematical models for high speed non-isothermal air bearings Industrial and Applied Mathematics Notice: Undefined index: etzrsh in /maths/www/html/postgraduate/projects/index.php on line 477 Notice: Undefined index: etzrsh in /maths/www/html/postgraduate/projects/index.php on line 477 There are a number of technologies that must maintain appropriate cooling of component surfaces and may include resisting contact by the use of thin fluid films. Modern design processes increasingly on high quality numerical modelling tools to guide the creation of new products and identify operating requirements that include high rotation speeds, high pressures and high temperatures. These demanding operating conditions make a comprehensive understanding of the underlying flow characteristics essential. Advanced modelling requires the careful development of representative equations for all thermal effects, coupling with an appropriate (film) Reynolds equation and the specification of appropriate boundary conditions. Further, implementation of appropriate numerical methods and analysis is required in these demanding model systems. High speed gas film bearing bearings (and seals), as proposed for future aero-engines, are designed to work with no contact and very small gaps and applicable to a wide range of industrial applications. Air-riding bearings have inherent dynamic advantages in making use of local structural features to maintain sufficient gap between the rotating parts but these may lead to significant instabilities as a result of the dynamic behaviour of the gas film and potential thermal and mechanical distortions.  Building on recent PhD studies this project aims to develop detailed understanding of heat transfer in highly sheared thin-film flows through the creation of sophisticated and numerical and modelling approaches. BAILEY, N.Y., CLIFFE, K.A., HIBBERD, S. and POWER, H., 2014. Dynamics of a parallel high speed fluid lubricated bearing with Navier slip boundary conditions. IMA Journal of Applied Mathematics, doi:10.1093/immat/hxu053, 1-22.VOSPER, H., CLIFFE, K.A., HIBBERD, S. and POWER, H., 2013. On thin film flow in hydrodynamic bearings with a radial step at finite Reynolds number. Journal of Engineering Mathematics. DOI 10.1007/s10665-013-9627-8, 1-22 The supervision team will include Prof H Power, Faculty of Engineering
Title Classical and quantum Chaos in 3-body Coulomb problems Industrial and Applied Mathematics, Mathematical Physics Prof Gregor Tanner The realisation that the dynamics of 2 particles interacting via central forces is fundamentally different from the dynamics of three particles can be seen as the birth of modern dynamical system theory. The motion of two particles (for example the earth-moon problem neglecting the sun and other planets) is regular and thus easy to predict. This is not the case for three or more particles (especially if the forces between all these particles are of comparable size) and the resulting dynamics is in general chaotic, a fact first spelt out be Poincaré at the end of the 19th century. An important source for chaos in the three-body problem is the possibility of triple collisions, that is, events where all three particles collide simultaneously. Triple collisions form essential singularities in the equation of motions, that is, trajectories can not be smoothly continued through triple collision events. This is related to the fact, that the dynamics at the triple collision point itself takes place on a collision manifold of non-trivial topology.During the project, the student will be introduced to scaling techniques which allow to study the dynamics at the triple collision point. We will in particular consider three-body Coulomb problems, such as two-electron atoms, and study the influence of the triple-collision on the total dynamics of the problem. As a long term goal, we will try to uncover the origin of approximate invariants of the dynamics whose existence is predicted by experimental and numerical quantum spectra of two-electron atoms such as the helium atom. The semiclassical helium atom, G. Tanner and K Richter, www.scholarpedia.org/article/Semiclassical_theory_of_helium_atom
Title Electromagnetic compatibility in complex environments: predicting the propagation of electromagnetic waves using wave-chaos theory Industrial and Applied Mathematics, Mathematical Physics Dr Stephen Creagh, Prof Gregor Tanner The focus of this project is the development of a mathematical framework to understand the propagation of electromagnetic fields within complicated environments – a challenging task especially in the high frequency limit. Modern technology is typically stuffed with electronic componentry. Devices ranging from a mobile phone to a pc to an Airbus A380 will have many internal electronic components operating at high frequencies and therefore radiating electromagnetic waves. If the waves radiated from one component are strong enough, they can interfere with the functioning of another component somewhere else in the unit. The field of Electromagnetic Compatibility (EMC) aims to mitigate these effects by better understanding the emitted radiation.The outcome of the research will help to design electronic devices, which are protected from interference from other EM sources within buildings, pc enclosures or even planes. The innovative idea in the proposed approach rests on combining EM-field propagation with ideas of chaos theory and nonlinear dynamics. In particular, the representation of waves emitted from a complex source is described in terms of their ray-dynamics in phase space using the so-called Wigner distribution function (WDF) formalism.  It allows us to replace the wave propagation problem with one of propagating classical densities within phase space.
Title Wave propagation in complex built-up structures – tackling quasi-periodicity and inhomogeneity Industrial and Applied Mathematics, Mathematical Physics Prof Gregor Tanner, Dr Stephen Creagh Computing the dynamic response of modern aerospace, automotive and civil structures can be a computationally challenging task. Characterising the structural dynamics in terms of waves in a uniform or periodic medium is often an important first step in understanding the principal propagating wave modes. Real mechanical structures are rarely fully periodic or homogeneous – variations in shape or thickness, boundaries and intersections as well as curvature destroy the perfect symmetry. The aim of the project is to extend periodic structure theory to wave propagation in quasi-periodic and inhomogeneous media such as stiffened structures. The modelling of waves can then be recast in terms of Bloch theory, which will be modified by using appropriate energy or flux conservation assumptions. The information about the propagating modes will then be implemented into modern high-frequency wave methods – such as the so-called Dynamical Energy Analysis developed in Nottingham - making it possible to compute the vibrational response of structures with arbitrary complexity at large frequencies.
Title Network performance subject to agent-based dynamical processes Industrial and Applied Mathematics, Statistics and Probability Dr Keith Hopcraft, Dr Simon Preston Networks – systems of interconnected elements – form structures through which information or matter is conveyed from one part of an entity to another, and between autonomous units. The form, function and evolution of such systems are affected by interactions between their constituent parts, and perturbations from an external environment. The challenge in all application areas is to model effectively these interactions which occur on different spatial- and time-scales, and to discover howi)     the micro-dynamics of the components influence the evolutionary structure of the network, andii)    the network is affected by the external environment(s) in which it is embedded.Activity in non-evolving networks is well characterized as having diffusive properties if the network is isolated from the outside world, or ballistic qualities if influenced by the external environment. However, the robustness of these characteristics in evolving networks is not as well understood. The projects will investigate the circumstances in which memory can affect the structural evolution of a network and its consequent ability to function.Agents in a network will be assigned an adaptive profile of goal- and cost-related criteria that govern their response to ambitions and stimuli. An agent then has a memory of its past behaviour and can thereby form a strategy for future actions and reactions. This presents an ability to generate ‘lumpiness’ or granularity in a network’s spatial structure and ‘burstiness’ in its time evolution, and these will affect its ability to react effectively to external shocks to the system. The ability of externally introduced activists to change a network’s structure and function - or agonists to test its resilience to attack - will be investigated using the models. The project will use data of real agent’s behaviour.
Title Fluctuation Driven Network Evolution Industrial and Applied Mathematics, Statistics and Probability Dr Keith Hopcraft, Dr Simon Preston A network’s growth and reorganisation affects its functioning and is contingent upon the relative time-scales of the dynamics that occur on it. Dynamical time-scales that are short compared with those characterizing the network’s evolution enable collectives to form since each element remains connected with others in spite of external or internally generated ‘shocks’ or fluctuations. This can lead to manifestations such as synchronicity or epidemics. When the network topology and dynamics evolve on similar time-scales, a ‘plastic’ state can emerge where form and function become entwined. The interplay between fluctuation, form and function will be investigated with an aim to disentangle the effects of structural change from other dynamics and identify robust characteristics.
Title Exploiting network symmetries for analysis of dynamics on neural networks Mathematical Medicine and Biology, Industrial and Applied Mathematics Dr Rachel Nicks, Prof Stephen Coombes, Dr Paul Matthews Networks of interacting dynamical systems occur in a huge variety of applications including gene regulation networks, food webs, power networks and neural networks where the interacting units can be individual neurons or brain centres. The challenge is to understand how emergent network dynamics results from the interplay between local dynamics (the behaviour of each unit on its own), and the nature and structure of the interactions between the units.Recent work has revealed that real complex networks can exhibit a large number of symmetries. Network symmetries can be used to catalogue the possible patterns of synchrony which could be present in the network dynamics, however which of these exist and are stable depends on the local dynamics and the nature of the interactions between units. Additionally, the more symmetry a network has the more possible patterns of synchrony it may possess. Computational group theory can be used to automate the process of identifying the spatial symmetries of synchrony patterns resulting in a catalogue of possible network cluster states.This project will extend current methods for analysing dynamics on networks of (neural) oscillators through automating the process of determining possible phase relations between oscillators in large networks in addition to spatial symmetries. This will be used to investigate dynamics on coupled networks of simplified (phase-amplitude reduced or piecewise-linear) neuron and neural population models. We will also consider the effect on the network dynamics of introducing delays in the coupling between oscillators which will give a more realistic representation of interactions in real world networks. M Golubitsky and I Stewart (2016) Rigid patterns of synchrony for equilibria and periodic cycles in network dynamics Chaos 26, 094803P Ashwin, S Coombes and R Nicks (2016) Mathematical frameworks for network dynamics in neuroscience. Journal of Mathematical Neuroscience. 6:2.B. D. MacArthur, R. J. Sanchez-Garcia and J.W. Anderson (2008) Symmetry in complex networks, Discrete Applied Mathematics 156 (18), 3525-3531F Sorrentino, L M Pecora, A M Hagerstrom, T E Murphy, and R Roy (2016) Complete characterization of the stability of cluster synchronization in complex dynamical networks. Science Advances. 2, e1501737–e1501737.
Title Topological Resonances on Graphs Mathematical Physics, Industrial and Applied Mathematics Dr Sven Gnutzmann If a light wave in a resonator between two almost perfect mirrors shows resonance if the wavelength is commensurate with the distance between the two mirrors. If this condition is satisfied it will decay much slower than at other wavelengths which are not commensurate. This is one of the simplest mechanisms for a resonace in a wave system. There are other weill known mechanisms that rely on complexity and disorder. It has recently been observed that a netork of wire may have a further mechanism that leads to resonances. This mechanism relies on cycles in the network and leads to various signatures which cannot be explained using other well-known mechanisms for resonances. In this project these signatures will be analysed in detail. GNUTZMANN, S., SMILANSKY, U. and DEREVYANKO, S., 2011. Stationary scattering from a nonlinear network Physical Review A. 83, 033831
Title Quantum Chaos in Combinatorial Graphs Mathematical Physics, Industrial and Applied Mathematics Dr Sven Gnutzmann Graphs consist of V vertices connected by B bonds (or edges). They are used in many branches of science as simple models for complex structures. In mathematics and physics one is strongly interested in the eigenvalues of the V x V connectivity matrix C of a graph. The matrix element C_ij of the latter is defined to be the number of bonds that connect the i'th vertex to the j'th vertex.In this PhD project the statistical properties of the connectivity spectra in (generally large) graph structures will be analysed using methods known from quantum chaos. These methods have only recently been extended to combinatorial graphs (Smilansky, 2007) and allow to represent the density of states and similar spectral functions of a graph as a sum over periodic orbits. The same methods have been applied successfully to metric graphs and quantum systems in the semiclassical regime for more than two decades. Idan Oren, Amit Godel and Uzy Smilansky Trace formulae and spectral statistic for discrete Laplacians on regular graphs (I) J. Phys. A: Math. Theor. 42 (2009) 415101Idan Oren, Amit Godel and Uzy Smilansky Trace formulae and spectral statistic for discrete Laplacians on regular graphs (II) J. Phys. A: Math. Theor. 43 (2010) 225205
Title Supersymmetric field theories on quantum graphs and their application to quantum chaos Mathematical Physics, Industrial and Applied Mathematics Dr Sven Gnutzmann Quantum graphs are a paradigm model for quantum chaos. They consist of a system of wires along which waves can propagate. Many properties of the excitation spectrum and the spatial distribution of standing waves can be mapped exactly onto a supersymmetric field theory on the network. In a mean-field approximation one may derive various universal properties for large quantum graphs. In this project we will focus on deviations from universal behaviour for finite quantum graphs with the field-theoretic approach. GNUTZMANN, S, KEATING, J.P. and PIOTET, F., 2010. Eigenfunction statistics on quantum graphs Annals of Physics. 325(12), 2595-2640GNUTZMANN, S., KEATING, J.P. and PIOTET, F., 2008. Quantum Ergodicity on Graphs PHYSICAL REVIEW LETTERS. VOL 101(NUMB 26), 264102GNUTZMANN, S. and SMILANSKY, U., 2006. Quantum graphs: applications to quantum chaos and universal spectral statistics Advances in Physics. 55(5-6), 527-625GNUTZMANN, S. and ALTLAND, A., 2005. Spectral correlations of individual quantum graphs. Physical Review E. 72, 056215GNUTZMANN, S. and ALTLAND, A., 2004. Universal spectral statistics in quantum graphs. Physical Review Letters. 93(19), 194101
Title Pseudo-orbit expansions in quantum graphs and their application Mathematical Physics, Industrial and Applied Mathematics Dr Sven Gnutzmann Quantum graphs are a paradigm model to understand and analyse the effect of complexity on wave propagation and excitations in a network of wires. They have also been used as a paradigm model to understand topics in quantum and wave chaos where the complexity has a different origin while the mathematical framework is to a large extent analogous.Many properties of the waves that propagate through such a network can be described in terms of trajectories of a point particle that propagates through the network. The ideas is to write a property of interest as a sum over amplitudes (complex numbers) connected to all possible trajectories of the point particle. These sums remain challenging objects for explicit evaluations. Recently a numer of advanced methods for their summation have been introduced. The latter are built on so-called pseudo-orbits. In this project these methods will be develloped further and applied to questions related to quantum chaos and random-matrix theory. Daniel Waltner, Sven Gnutzmann, Gregor Tanner, Klaus Richter, A sub-determinant approach for pseudo-orbit expansions of spectral determinants, arXiv:1209.3131 [nlin.CD] Ram Band, Jonathan M. Harrison, Christopher H. Joyner, Finite pseudo orbit expansions for spectral quantities of quantum graphs, arXiv:1205.4214 [math-ph]
Title The ten-fold way of symmetries in quantum mechanics. An approach using coupled spin operators. Mathematical Physics, Industrial and Applied Mathematics Dr Sven Gnutzmann About 50 years ago Wigner and Dyson proposed a three-fold symmetry classification for quantum mechanical systems -- these symmetry classes consisted of time-reversal invariant systems with integer spin which can be described by real symmetric matrices, time-reversal invariant systems with half-integer spin which can be described by real quaternion matrices, and systems without any time-reversal symmetry which are described by complex hermitian matrices. These three symmetry classes had their immediate application in the three classical Gaussian ensembles of random-matrix theory: the Gaussian orthogonal ensemble GOE, the Gaussian symplectic ensemble GSE, and the Gaussian unitary ensemble GUE. In the 1990's this classification was extended by adding charge conjugation symmetries -- symmetries which relate the positive and negative part of a spectrum and which are described by anti-commutators.The classification was completed by Altland and Zirnbauer who have shown that there are essentially only seven further symmetry classes on top of the Wigner-Dyson classes leading to what is now known as the 'ten-fold way'. All symmetry classes have applications in physics. The new symmetry classes are realised by various cases of the Dirac equation and the Bogoliubov-de Gennes equation. For a long time people have thought of these symmetries only in the context of many-body physics or quantum field theory. However there are simple quantum mechanical realisations of all ten symmetry classes which in terms of two coupled spins where the classification follows from properties of the coupling parameters and of the irreducible SU(2) representations on which the spin operators act. This project will explore these simple representations in the quantum mechanical and semiclassical context. One goal will be to understand the implications of the quantum mechanical symmetries for the corresponding classical dynamics which appears in the semiclassical limit of large spins. M.R. Zirnbauer, Riemannian symmetric superspaces and their origin in random matrix theory, J. Math. Phys. 37, 4986 (1996)A. Altland, M.R. Zirnbauer, Non-standard symmetry classes in mesoscopic normal-/superconducting hybrid structures, Phys. Rev. B 55, 1142 (1997)S. Gnutzmann and B. Seif, Universal spectral statistics in Wigner-Dyson, chiral, and Andreev star graphs. I. Construction and numerical results, Physical Review E 69, 056219 (2004)S. Gnutzmann and B. Seif, Universal spectral statistics in Wigner-Dyson, chiral, and Andreev star graphs. II. Semiclassical approach. Physical Review E 69, 056220 (2004)
Title Nonlinear waves in waveguide networks Mathematical Physics, Industrial and Applied Mathematics Dr Sven Gnutzmann Many wave guides (such as optical fibres) show a Kerr-type effect that leads to nonlinear wave propagation. If th wave guides are coupled at junctions then there is an additional element of complexity due to the non-trivial connectivity of wave guides. In this project the impact of the structure and topology of the network on wave propagation will be studied starting from simple geometries such as a Y-junctions (three waveguides coupled at one junction), a star (many waveguides at one junction), or a lasso (a waveguide that forms a loop and is connected at one point to a second waveguide). Sven Gnutzmann, Uzy Smilansky, and Stanislav Derevyanko, Stationary scattering from a nonlinear network, Phys. Rev. A 83, 033831 (2011)
Title The statistics of nodal sets in wavefunctions Mathematical Physics, Industrial and Applied Mathematics Dr Sven Gnutzmann If a membrane vibrates at one of its resonance frequencies there are certain parts of the membrane that remain still. These are called nodal points and the collection of nodal points forms the nodal set. Building on earlier work this project will look at the statistical properties of the nodal set -- e.g. for 3-dimensional waves the nodal set consists of a coillection of surfaces and one may ask questions about how the area of the nodal set is distributed for an ensemble of membranes or for an ensemble of different resonances of the same membrane. This project will involve a strong numerical component as wavefunctions of irregular membranes need to be found and analysed on the computer. Effective algorithms to find the area of the nodal set, or the number of domain in which the sign does not change (nodal domains) will need to be developed andimplemented. Galya Blum, Sven Gnutzmann, Uzy Smilansky, Nodal domains statistics- a criterion for quantum chaos, Phys. Rev. Lett. 88, 114101 (2002)Alejandro G. Monastra, Uzy Smilansky and Sven Gnutzmann, Avoided intersections of nodal lines, J. Phys. A. 36, 1845-1853 (2003)G. Foltin, S. Gnutzmann and U. Smilansky, The morphology of nodal lines- random waves vs percolation, J. Phys. A 37, 11363 (2004)Yehonatan Elon, Sven Gnutzmann, Christian Joas and Uzy Smilansky, Geometric characterization of nodal domains: the area-to-perimeter ratio, J. Phys. A 40, 2689 (2007) S. Gnutzmann, P. D. Karageorge and U Smilansky, Can one count the shape of a drum?, Phys. Rev. Lett. 97, 090201 (2006)
Title Coherent states, nonhermitian Quantum Mechanics and PT-symmetry Mathematical Physics, Industrial and Applied Mathematics Dr Sven Gnutzmann Heisenberg's uncertainty principle states that momentum and position cannot be sharp at the same time because there is a lower bound for the product of the uncertaincies. Coherent states can be defined as the states that minimize the uncertainty -- in this sense they are as close as quantum mechanics allows to describe a classical point particle. When a quantum system starts in a coherent states it's expectation values follow the classical equations of motion while the shape of the wave function often changes only very slowly. Coherent states are an important tool to understand the corresp[ondence between quantum and classical dynamics.In this project this correspondence will be analysed for a generalized quantum dynamics where the Hamilton operator is not required to be Hermitian. Such dynamics can arise in practice as an effective description for an open quantum system with eitehr decay or gain. Accordingly the energy eigenvalues may have an imaginary part that describes the loss or gain. Recently there have also be suggestions that non-hermitian Hamilton operators could play a fundamental role in quantum mechanics if the Hamilton operator remains symmetric with respect to a combined operatyion of parity P and time reversal T. Such PT-symmetric dynamics have a balance between gain and loss which can lead to real energy eigenvalues. Classical to quantum correspondence for such systems remains an open research topic and this project will aim at getting a clear understanding of the underlying classical dynamics using coherent states as the main tool. S.Gnutzmann, M Kus, Coherent states and the classical limit on irreducible SU3 representations, J. Phys. A 31, 9871 (1998)E.-M. Graefe, M. Höning, H.J. Korsch, Classical limit of non-Hermitian quantum dynamic - a generalized canonical structure, J. Phys A 43, 075306 (2010)E.-M. Graefe, R. Schubert, Complexified coherent states and quantum evolution with non-Hermitian Hamiltonians, J. Phys. A 45, 244033 (2012)C.M. Bender, M. DeKieviet, S.P. Klevansky, PT Quantum Mechanics, Philosophical Transactions of the Royal Society A 371, 20120523 (2013)