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PhD opportunities

Below is a list of PhD projects available within our research group.  If you are interested in applying or would like to find out more about a project, then please submit an expression of interest form or email the project supervisor.

List of projects:

Hidden dynamics of decision making

Supervisor: Mike Jeffrey

Decisions obviously occur all around us at every moment, not just in the form of human social and economic decisions, but in auto-regulatory ‘choices’ made by genetic and neural mechanisms in living organisms, and increasingly more in electronic decisions in all manner of automated AI and robotic systems. However, decisions are more problematic from a mathematical point of view than is typically realised.

A decision that induces a switch in behaviour or structure introduces a discontinuity. Control engineering over the last century has forced applied mathematicians to seriously tackle the conceptual difficulties of discontinuities caused by decision switches. What has only come to light recently is that when multiple decisions interact in a system, the interaction between their discontinuities breaks down all this careful new mathematics, resulting in uncertainty and unpredictability.

A simple scenario has been posed of two players in a game whose behaviour results in seemingly illogical outcomes, e.g. two investors whose spending decisions appear to be rational, but the value of the company they are investing in becomes wildly unpredictable. You will investigate how widespread a phenomenon this is, how it scales with multiple decision makers, into larger and less ideal networks of ‘players’ like an economic system, a robotic controller, or a living brain.

Contact: Mike Jeffrey (mike.jeffrey@bristol.ac.uk)

Singularities in dynamical systems and pattern formation

Supervisor: Mike Jeffrey

Singularities are incredibly important to all kinds of systems, not just because they create extraordinary points like black holes in space, but because they organise much of the structure and patterning of much more ordinary objects and behaviours around us. Unfortunately, the incredibly general “singularity theory” developed around half a century ago, is too highly technical to put to use in most of the complex high dimensional systems we encounter in everyday engineering and biology. Recently, though, it has been discovered that most singularities of interest can be found by looking for a much simpler object, an “underlying catastrophe”. These reduce sometimes millions of calculations needed to identify a singularity, to just a handful of calculations needed to identify its catastrophe. There are both theoretical and applied questions to explore coming out of this discovery:

  • On the theoretical side, how do the underlying catastrophes fit into established singularity theory? Why do some singularities seem to be ruled out as being likely to occur in practice, because they do not have well-defined underlying catastrophes?
  • On the application side, what can these tell us about the role of singularities in high-dimensional problems that have been inaccessible before now, particularly so-called ‘reaction-diffusion’ problems that have become incredibly important in the last 20 years, largely fulfilling Turing’s ideas on pattern formation, for understanding patterns of growth and structure formation in living and crystalline media.

Contact: Mike Jeffrey (mike.jeffrey@bristol.ac.uk)

Hybrid modelling and nonlinear dynamics in aircraft design

Supervisors: Prof David Barton and Prof Mark Lowenberg

There is a drive towards high-efficiency and high-performance aircraft. This requires lighter and more fuel-efficient structural designs, which are often more flexible than traditional rigid designs. This flexibility can result in disastrous nonlinear behaviour; for example, the destruction of the NASA Helios prototype. This project seeks to develop new approaches to nonlinear behaviour in aircraft and so enable a new generation of low-carbon aircraft.

Within our well-equipped test facilities at the University of Bristol, Prof Mark Lowenberg and colleagues have developed a manoeuvre rig based on a scale model of a Hawk trainer aircraft. This 5 degree-of-freedom model provides an ideal test bed for investigating nonlinear aerodynamic behaviour. To explore the complex nonlinear dynamics, Prof David Barton has developed a range of experimental techniques known as Control-Based Continuation (CBC) that can track the onset of instabilities as system parameters change (e.g., the onset of flutter at a critical airspeed, Lee et al, 2023). As such, CBC can be used to investigate behaviour in the physical system that would have previously been out of reach. The combination of the manoeuvre rig and CBC opens up many possibilities for exploitation.

The data generated from these experiments is ideal for building new models and can be used in combination with scientific machine learning to create hybrid models: high-fidelity models that combine physics-based modelling with data-driven approaches. These models will then enable further design work to either mitigate or take advantage of the nonlinear behaviour in the experiment.

The overall aims are threefold:

  • To generate industrially-relevant insights from the manoeuvre rig.
  • To extend CBC to multi-degree-of-freedom systems, with application to other engineered structures.
  • To develop a hybrid modelling approach for aerospace systems that combines machine learning with physics-based models.

Contact: David Barton (david.barton@bristol.ac.uk)

Online learning for tactile robotics

Supervisors: Prof David Barton and Prof Nathan Lepora

Machine learning (ML) equips robots with the ability to learn from and adapt to new environments, enhancing autonomy and efficiency in complex tasks. However, many ML approaches rely on extensive data for training. This heavy reliance on large volumes of data can hamstring a robot’s ability to function in dynamic real-world conditions—imagine a robotic assistant in a home struggling to recognize new types of objects because it was trained on a limited set of household items. Such data dependency limits the robots’ effectiveness in diverse real-world applications. We seek to overcome this problem in the context of tactile robotics by exploiting generative online learning, enabling robots to learn ‘in the moment’ and adapt swiftly to the unpredictability of real-world tasks.

Online learning is where new data is continuously integrated into a model as a task is being performed. As such, the performance of the robot increases over time. Joint with Prof Nathan Lepora, I have previously demonstrated a generative online learning approach (Learning to live life on the edge) that uses predictions of the tactile sensor output to track the edges of objects. This has been implemented on a robot arm and a quadruped robot.

Significant improvements are needed enable this generative online learning approach to be used on both a broader suite of tasks (not just edge following) and within a higher dimensional space (not just a plane). This will require non-trivial generalisation of the current machine learning approach (a Gaussian Process-Latent Variable Model) and exploration of other computationally efficient generative methods. Insights into the geometry of the tasks of interest are likely to be key to creating an effective approach.

This approach to online learning has the potential to greatly expand the use of tactile sensors and enable them to be used in novel contexts, bringing us closer to building robots that can both touch and feel, and interact safely with the environment around them.

Contact: David Barton (david.barton@bristol.ac.uk)

Surrogate modelling and machine learning for electrical power systems

Supervisors: Prof David Barton and Dr Ian Laird

Designing new devices, particularly in electrical power conversion for renewable energy, is often challenging because of constraints on mass, volume, and cost. Designers must innovate within these boundaries, making trade-offs to meet specifications without compromising performance. This PhD project will employ surrogate modelling and machine learning to improve the efficiency of design processes.

Power electronic device design involves choosing from a limited library of existing parts as well as dealing with behaviors across timescales, from nano-second switching to bulk behaviour over several seconds. Commercial simulation tools, such as Plex, offer accuracy but lack the computational speed needed for quick iterative design. Collaborating with Dr. Ian Laird, who brings extensive experience in power system design optimisation, this project aims to provide the fundamental research developments needed to create rapid design tools.

The project will focus on developing surrogate models using reservoir computing techniques to accelerate simulations for components like modular multi-level converters (MMCs), crucial in renewable energy systems. These models will enable faster design iteration, optimising systems to meet specific application constraints, such as those for offshore wind turbines. Key challenges include modelling the discrete switching of electrical components and the wide range of dynamic timescales. Addressing these challenges is essential for capturing sudden changes and complex interactions, with potential applications extending beyond power systems to fields like synthetic biology and electromagnetic sensing.

The anticipated result is a tool for rapidly designing optimized electrical power systems, streamlining resource use and cost in system deployment. The research has the potential to yield significant publications with broad impacts across multiple disciplines.

Contact: David Barton (david.barton@bristol.ac.uk)

Stochasticity in aircraft icing – ice block formation and shedding.

Supervisor: Ryan Palmer

Aircraft icing is a significant topic in aviation safety today. When aircraft pass through convective cloud systems they are impacted by ice crystals and supercooled water droplets. These impacts can lead to ice layers forming on crucial parts of the aircraft, in particular the engines, measurements devices and the wings and fuselage. Whilst significant improvements have been made throughout the industry in recent years, the issue continues to be a contributory factor in accidents and incidents.

The detailed physics of the icing process is complex. Our understanding has increased to a great extent by numerous research programmes around the world, but there is more to understand. A further challenge in the integration of this research industrial codes. The aeronautical industry requires problems to be solved in a practical timescale, which often requires simplifying assumptions to be made. Aviation has, generally, a very good safety record; however, to further reduce the number of incidents and increase aircraft efficiency, further improved capability in modelling techniques is required. There are two problems of interest:

How to best use probabilistic information within icing analysis

In one sense, aircraft icing is deterministic – it is explained by heat exchange, droplet/crystal transport and ice nucleation/formulation. Yet, in practice, stochasticity arises in experimental data on ice growth and adhesion, and in terms of the mixtures of droplets and ice crystals that impact aircraft. This stochasticity is driven by local variation in physical parameters, variation of aircraft aerodynamics as ice grows, and uncertainty/unknowability of the specific icing conditions (e.g., mix of precipitate, size, and shapes of crystals/droplets). The aim here is to introduce stochasticity into standard icing analysis that captures statistical variability in experiments, reflects different environments and icing conditions, all within the time and computational power constraints that exist in industry.

Ice block formation and shedding

Ice growth is both a function of the precipitate mix, the local conditions (temperature, pressure) as well as the changing aerodynamic flow over the ice block. Once again, this leads to an element of stochasticity as the fluid flow, precipitate and ice block interact.

  1. Can a physical model be implemented that captures each driver, updating as the ice grows?
  2. Can a statistical or probabilistic (machine learning?) model be implemented that capture the variation in ice formation and these processes?
  3. From a model of ice growth, can accurate ice shapes be predicted? and subsequent ice shedding/trajectory be predicted?

The proposed projects are of clear industrial interest in aviation and engineering.

Contact: Ryan Palmer (ryan.palmer@bristol.ac.uk)

Inverse problems in sensory biology

Supervisor: Ryan Palmer

Insects explore the world using a variety of senses – vision, hearing, olfaction (smell), touch, taste and more. Mechanoreception, the detection of stimuli using deflectable hairs, is a fundamental mechanical process that underpins several of these senses across Arthropoda (i.e., insects, arachnids, and crustaceans). These mechanosensory hairs display remarkable morphological diversity, even within a single specimen, and have a wide range of functional properties, enabling an animal to carry out a variety of sensory and non-sensory tasks (e.g., hearing, fluid flow sensing or thermal insulation and pollen grain capture).

The problems of interest here is: given the collective motion of several hairs, can the environmental source be determined?

This is an inverse problem since we need to work out the unknown stimulus and parameters that caused our system of interest to behave in a certain way. For example, each mechanoreceptive hair is innervated such that when deflected by an external force (e.g., electrostatic, aero-acoustic) a neuron fires and transmits information to the animal. However, it is unclear what information an animal can elicit from these seemingly simple receptors and how it relates to their environments. The aim is to explore how, through the multiplicativity of the deflections of several hairs (potentially thousands in reality), insects can ascertain useful information about their environment from the deflection of hairs. Physical, dynamic modelling and machine learning approaches are required. Furthermore, the project need not be limited to deflectable hairs.

The motivation for each problem is grounded in biology where understanding of such systems and sensory processes help to form our broader view of how insects interact with their environments, and potentially inform conservation initiatives, for example.

Contact: Ryan Palmer (ryan.palmer@bristol.ac.uk)

Climate change and species invasion

Supervisor: Dr Helena Stage

As climate change progresses, temperature and humidity variations affect the suitability of our environment to different species. Some species may migrate in search of better climates, and others may find increased areas in which to thrive. These changes pose ecological challenges for species conservation, but they can also pose new health risks when such migrating species are able to transmit diseases to humans or livestock.

The aim of this PhD project is to use European climate data to model the current and future suitability of Northern Europe to invasive mosquitoes, using a multi-scale approach. Directions this project can develop include:

  • the construction of data-based spatial models of climactic suitability;
  • statistical inference of key parameters for species invasion;
  • mathematical modelling and numerical simulations of transport-diffusion processes;
  • modelling intervention scenarios to monitor and mitigate species invasion.

The project would suit a candidate with an interest in interdisciplinary research at the interface of mathematics, ecology, and climate science. A background in mathematics or statistics is essential.

Contact: Helena Stage (Helena.stage@bristol.ac.uk)

Human behaviour in epidemiological models

Supervisors: Dr Leon Danon and Dr Helena Stage

The COVID-19 pandemic has underscored the key role that human behaviour plays in our understanding of disease transmission and how it can be managed or mitigated. We have seen individuals’ social contact patterns change, changing risk perceptions affecting protective behaviours of individuals, and the a large variation in how various sub-populations in a group responded to the same national guidance.

The aim of this PhD project is to develop tools by which human behaviour (and its heterogeneities) can be incorporated into future epidemiological models.

Directions this project can develop include:

  • developing new models of self-organised behavioural change driven by real or perceived risk of infection;
  • using of delay-differential equations to model feedback mechanisms in behavioural responses;
  • mathematical modelling of population heterogeneity in the context of disease transmission to explore varying participation in public health initiatives;
  • developing machine learning approaches to identifying behavioural changes from large data sets.

The project would suit a candidate with an interest in interdisciplinary research at the interface of mathematics, data science, and epidemiology. A background in mathematics, statistics, or data science is essential.

Contact: Helena Stage (Helena.stage@bristol.ac.uk)

Data-driven reduced order modeling of physical systems

Supervisor: Dr Robert Szalai

This project is about identifying reduced order models from data, especially from mechanical systems that have equilibria, are forced periodically or quasi-periodically. Examples include structural vibrations of buildings and uncovering their resiliance to earthquakes, vibration analysis in aerospace structures, robotic arms in order to improve their performance or be able top control them within problem specifications. The project is mathematical/computational in nature and involves machine learning techniques, designing function approximators and training them to a prescribed accuracy.

The successful candidate has strong foundations in applied mathematics, nonlinear dynamics and able to write scientific software to carry out the required calculations.

Background. Many mathematical modeling problems cannot be solved by building models from physical laws. This is because the uncertainty and unknowns about what is involved with the sought after phenomena. Examples, where mathematical modeling becomes challenging includes mechanical friction, electromagnetic hysteresis, social dynamics, and the physiology of living things. To address the problem the models need to based or incorporate data. It is also not desirable to identify a fully detailed model, when we only require information about a specific phenomenon within our system. We also want to make sure that the created model is interpretable and can be used to inform new experiments, design new systems or efficiently predict how our current system behaves.

Reference: https://doi.org/10.1007/s00332-023-09932-y

Contact: Robert Szalai (r.szalai@bristol.ac.uk)

Contact problems in continuum mechanics

Supervisor: Dr Robert Szalai

This project is about resolving the computational difficulties that arise in Finite Element modeling of mechanical structures that include joints, surfaces impacting or slipping on each other. The computational modeling of mechanical contact is a hard task, because after discretising the continuum problem we are left with a large number of non-smooth ordinary differential equations. It is known that non-smooth systems are extremely hard to simulate do to the number switches between the states of the system that increases exponentially with the number of dimensions and the frequency of state switches occurring with the fastest resolved time-scale within the system. The outcome is that the more accurate the computational model is, the harder it is to obtain a solution. The project is about to remedy this inherent issue, by accounting for dynamical properties that all numerical methods throw away, which is the continuum nature of the problem.

The successful candidate has strong knowledge of continuum mathematics, including functional analysis and able to write simple computational code that carries out numerical simulations. Familiarity with continuum mechanics and finite element methods is desirable, but not essential.

Reference: https://doi.org/10.1007/s00332-018-9508-4

Contact: Robert Szalai (r.szalai@bristol.ac.uk)

Looking Inside Aircraft Turbine Blades for Flaws

Supervisor: Professor Anthony Mulholland 

There are many man-made structures that should they fail would be catastrophic. An example is an aircraft engine and in particular the turbine blades within it. The risk is so high that this safety critical component is regularly imaged for the presence of cracks. The materials used to make turbine blades components are becoming increasingly complex and heterogeneous. For example, additive manufacturing and carbon fibre reinforced polymers (CFRP) are increasingly being used. This is making it harder to find these flaws using conventional ultrasound imaging techniques. There are many stakeholders in industry who are interested in this topic. Bristol is part of an international industrial-university consortium that is focused on advancing knowledge in this area. The 50 or so companies include Airbus, Rolls Royce, EDF Energy, IHI Corporation Japan, Petrobras and BP work with their university partners under the auspices of the Research Centre for Non-Destructive Evaluation (RCNDE) (www.rcnde.ac.uk). The technical challenge is to build a mathematical model of ultrasound wave propagation in these materials and use this to (a) deepen our knowledge of how waves travel in these materials and (b) use that knowledge to improve the detection and characterisation of cracks and flaws. The student will use a stochastic differential equations approach to building a model of wave propagation in heterogeneous materials to examine how the material microstructure affects the ability to detect flaws.

Contact: Prof Tony Mulholland (anthony.mulholland@bristol.ac.uk)

Bubbles to Beat Cancer

Supervisor: Professor Anthony Mulholland

Microbubbles, with a size approximately matching that of a red blood cell, have the potential for use in therapeutic applications such as drug delivery and gene delivery. The basic idea is to load an encapsulated microbubble with a drug (such as a cancer treating drug), inject it into the blood stream and use ultrasound to manoeuvre the bubble into the offending tissue and burst it to release its contents in a very localised fashion. This obviously reduces many of side effects of these potent drugs and could lead to more effective treatments. The idea is to manoeuvre the bubble using an array of ultrasound sensors (acoustic tweezers) and then burst it using High Power Focused Ultrasound (HIFU). It is clearly a good idea to develop a mathematical model based version of this scenario so that different designs of the encapsulated bubble and the ultrasound equipment can be assessed and optimised before any expensive and time consuming experiments are performed. This work will be of interest to medical physicists and clinicians working in treating cancer and other diseases using localised drug delivery methods. The challenge is to develop a theoretical model of the dynamics of an encapsulated bubble, its interaction with an external ultrasonic field, its interaction with a nearby wall, and its bursting. The student will develop and study a theoretical model of the dynamics of an encapsulated bubble. They will then look at the dynamics of this bubble when insonified by ultrasound. They will build a model that describes the interaction on a nearby wall the shear stresses that result. They will then develop a model to describe the bursting of an encapsulated bubble.

Contact: Prof Tony Mulholland (anthony.mulholland@bristol.ac.uk)

Waves in Fractal Networks

Supervisor: Professor Anthony Mulholland

Ultrasound sensors are used for detecting dangerous cracks in structures such as aircraft engines, oil, and gas pipelines, and nuclear plants. The imaging resolution of these sensors is dictated to a large extent by the range of frequencies over which they can emit and receive ultrasound waves. At present piezoelectric ultrasound transducers (the most commonly used ones) are made by cutting up a piezoelectric material with a saw and so their design has a regular chess board type pattern. As a resonating system this single length scale leads to the devices only being able to operate over a narrow range of frequencies. With the advent of 3D printing there is an opportunity to have designs that have a range of length scales and hence can operate over a range of frequencies. Given this new freedom what designs should we try ? One direction is to look to mathematics and Fractal shapes since these are characterised by having geometrical features on a range of length scales. Fractals are irregular shapes which recur repeatedly to form objects such as snowflakes, ferns and cauliflowers, making their structure appear more complex than it often actually is. The same concept also lies behind the hearing system of animals including bats, dolphins, cockroaches and moths. If we can send out sound waves that are complicated and have different frequencies, we can work towards simulating what nature does. If there are defects in a nuclear plant or an oil pipeline, we would be able to detect cracks that have a range of sizes and do so at an early stage. This device could not only improve safety but also save a great deal of money, as early detection means inspections don’t have to be carried out as often. There are many stakeholders in industry who are interested in this topic. Bristol is part of an international industrial-university consortium that is focused on advancing knowledge in this area for the non-destructive testing of safety critical structures such as nuclear plants, oil pipelines, and aircraft engines. The 50 or so companies include Airbus, Rolls Royce, EDF Energy, IHI Corporation Japan, Petrobras and BP work with their university partners under the auspices of the Research Centre for Non-Destructive Evaluation (RCNDE) (www.rcnde.ac.uk). To build a mathematical model of a fractal ultrasonic transducer and compare its operating qualities with an equivalent conventional design. The student will build model of ultrasonic wave propagation in a Sierpinski Gasket fractal network. They will then add electrical and mechanical loads to this model to model its reception and transmission frequency response. They will then compare that response to a model of an equivalent conventional design.

Contact: Prof Tony Mulholland (anthony.mulholland@bristol.ac.uk)

Mechanical-biochemical cell polarity modelling

Supervisors: Alan Champneys & Matt Hennessy

Cells are typically created as sphere-like objects but change their shape by creating a sense of polarity, a front and a back. The underlying process by which they do this is thought to contain a number of features, which are initiated by a biochemical process of the reaction and diffusion of small G-proteins, also known as rho-proteins. The spatial organisation of these is thought to be governed by model partial differential equations that capture both fast diffusing inactive proteins and membrane-bound slow diffusing active forms. Considerable research has gone into to understanding the process by which the relevant partial differential equations undergo bifurcation (qualitative change in limit state) as biologically relevant control parameters are varied. That work has been applied in one, two and three-dimensional geometries. The goal of this PhD project is to understand how the differential growth caused by these bifurcated patterns can influence cell shape, and how cell shape feeds back to the biochemical processes. The work will involve mathematical modelling, dynamical systems theory and continuum mechanics. It is highly likely to be carried out in collaboration with experimental and modelling groups in different Universities in the UK and internationally.

Contact: Alan Champneys (a.r.champneys@bristol.ac.uk)

Mathematical modelling of judder, squeak and rattle.

Supervisors: Alan Champneys & Robert Szalai

We have considerable expertise in modelling strange behaviour such as the so-called Painleve paradox which explains the strange woodpecker-like motion that occurs when chalk is pushed rather than pulled across a blackboard. This kind of modelling has also been used to explain how rats whiskers stick and slip in order to detect texture and how musical instruments such as the violin generate sound. In this project we shall be interested in two particular problems, the onset of squeak in brakes and other frictional interfaces and the judder experienced in robotic arms. This latter problem is of specific interest to the nuclear industry who are interested in the faithful control of motions of long-reach robotic arms in future nuclear fusion reactors. This PhD project shall explore these questions from both a fundamental point of view using mathematical analysis of piecewise-smooth system, and at the level of mathematical modelling with industry. There is the possibility to work with industrial partners such as the UK Atomic Energy Authority.

Contact: Alan Champneys (a.r.champneys@bristol.ac.uk)

Mathematical modelling to provide optimal solutions to cardiovascular waiting lists

Supervisors: Alan Champneys and Ryan Palmer

Cardiovascular disease (CVD) affects over 7 million people and accounts for 27% of all deaths in the UK. CVD has been the largest non-COVID-19 cause of excess mortality in England since March 2020. This project builds on work convened by the Newton Gateway to Mathematics. Clinicians and mathematicians co-designed a simple mathematical model using routinely collected open access data to provide insights and solutions to the challenge of prolonged waiting lists in UK CVD Care. This nascent systems-dynamics model needs further development, optimisation, mathematical analysis and validation.

The over-arching aim of the project is to further develop, optimise and validate a systems-level, operations research model of the care pathway for patients waiting for referral, diagnosis and treatment of CV disease. Specific objectives are to:

  • Optimise the nascent systems dynamic model. Incorporate additional data streams and parameters. Use a variety of mathematically rigorous optimisation methods to understand the fitting of the model to data and the amount of uncertainty in future predictions.
  • Analyse the model using ideas from nonlinear dynamical systems and bifurcation theory. Develop more general theories of bifurcation analysis of systems-dynamics models
  • Validate the model using routinely collected, open access NHS data (e.g. NHS Digital) Validate the effects of both COVID-19 and non-COVID-19 impacts on the model and start to work on a bespoke software solutions alongside clinicians and healthcare planners.

Contact: Alan Champneys (a.r.champneys@bristol.ac.uk)

Shrinkage and swelling during 3D printing

Supervisor: Matthew Hennessy

3D printing has led to a manufacturing revolution by enabling complex objects to be rapidly created at a low cost. Many 3D printers work by using light to convert liquid into solid in a layer-by-layer fashion. That is, a layer of liquid is placed on top of an existing solid structure and then exposed to light to create a new solid layer. Mathematical modelling plays a critical role in 3D printing by predicting the experimental conditions, such as the light intensity and exposure time, that are needed to grow a given shape.

During 3D printing, the solid structure can shrink or swell. On one hand, shrinkage and swelling are detrimental because they lead to large distortions. 3D printers must take this distortion into account in order to produce desired shapes. On the other hand, shrinkage and swelling can lead to bending and mechanical instabilities such as wrinkling. These instabilities can be harnessed to self-assemble complex shapes that are difficult to produce using conventional 3D printers.

The overall goal of this project is to develop a new suite of models for 3D printing that account for shrinkage and swelling. These models will be used to create a software package that takes as input the 3D shape to be produced and outputs the required experimental conditions. Specific objectives include:

  • Building discrete and continuum models of 3D printing that account for the mechanics of shrinkage and swelling
  • Developing code (Python, Julia, or Matlab) to solve the models and simulate 3D printing
  • Using machine learning to predict the experimental conditions needed to produce a given 3D shape

This project will be carried out in collaboration with researchers at Imperial College London.

Contact: Matthew Hennessy (matthew.hennessy@bristol.ac.uk)

Mathematical modelling of magnetically responsive gels

Supervisors: Matthew Hennessy and Stuart Thomson

Magnetically responsive gels are soft solids that change their size and shape in response to a magnetic field. The ability to create such materials has driven the development of new technologies for soft robotics, biomedicine, and drug delivery. However, a key challenge of using magnetically responsive gels in applications is that their complex behaviour is difficult to predict and not well understood. The complexities arise from the wide range of physics that come into play, which include elasticity, magnetism, and fluid transport.

The objective of this project is to develop experimentally validated mathematical models of magnetically responsive gels.  The long-term goal is to use these models to design "programmable" gels that will morph into specific shapes when subject to an external a magnetic field. Specific objectives of this project include:

  • developing mathematical models of magnetically responsive gels based on continuum mechanics;
  • validating the models against experimental data;
  • deriving simplified models for thin gels;
  • numerically solving the models using the finite element method;
  • using machine learning to determine how to design gels that morph into specific shapes.

This project will be carried out in collaboration with researchers in the Department of Mechanical and Manufacturing Engineering at the University of Cyprus.

Contact: Matthew Hennessy (matthew.hennessy@bristol.ac.uk)

Optimisation of hydrogel-based drug-delivery systems

Supervisor: Matthew Hennessy

Hydrogel-based drug-delivery systems allow for the targetted and controlled release of pharmaceutical agents. Targetted delivery means that drug molecules are delivered to specific sites within the body. Controlled delivery means that drugs are released at a specific rate. Both targetted and controlled delivery can reduce the effects of drug toxicity. A common problem with hydrogel-based drug delivery systems is the "burst effect", where a large amount of drug molecules are suddenly released. Another issue is that hydrogels can degrade, which leads to a premature release of drugs. The objective of this project is to use mathematical modelling to optimise the performance of hydrogel-based drug-delivery systems and overcome these issues.

Some examples of questions that can be explored in this project include:

  • Can machine learning be used to reduce the burst effect by optimising the properties (e.g. size, stiffness, shape) of the gel?
  • What impact does hydrogel degradation have on drug delivery?
  • How does an external fluid flow affect the release of drug molecules?

Contact: Matthew Hennessy (matthew.hennessy@bristol.ac.uk)

Phase separation in soft, porous solids

Supervisors: Matthew Hennessy and Matteo Taffetani

Phase separation describes the process of a homogeneous mixture spontaneously separating into its constituent components.  A common example of phase separation occurs in oil-water mixtures.  At high temperatures, oil and water form a homogeneous mixture, but at low temperatures, oil will separate from the water to form small droplets.  Phase separation can also occur in porous, fluid-filled solids, where it gives rise to a rich variety of patterns that defy explanation in terms of traditional theories.  The patterns that arise from phase separation have important implications for the operation of lithium-ion batteries, cell biology, and the onset of neurodegenerative diseases, and therefore need to be understood.  The objective of this project is to develop the theoretical foundations of phase separation in soft, porous solids by:

  • Deriving mathematical models using continuum mechanics that capture the physics of phase separation (e.g. elasticity, fluid transport)
  • Carrying out numerical simulations and mathematical analyses (e.g. stability analysis)
  • Determining how stimuli, such as an electric field or an imposed deformation, can trigger, alter, or suppress phase separation

Contact: Matthew Hennessy (matthew.hennessy@bristol.ac.uk)

Mathematical modelling of golf-ball impact dynamics

Supervisors: Stuart Thomson, Alan Champneys (Bristol); Mark Grattan (Royal & Ancient Rules Ltd)

The game of golf is enjoyed throughout the world. Outside North America, the rules of golf are owned and maintained by the Royal & Ancient (R&A), based in St Andrews in Scotland. They maintain a large technical facility for testing whether golf balls and clubs are consistent with the rules of golf. Part of this research requires accurate modelling of the bounce and roll of golf balls. Recent collaboration with the R&A has led to an improved understanding of golf ball bounce using theories from piecewise smooth dynamical systems. What makes golf somewhat different from other sports such a tennis and football, is that the ball is relatively rigid, and it is the compliance of the surface (the turf) rather than the ball that determines the nature of the bounce. Another complication is that golf balls can carry significant amounts of spin, so we need to understand the mechanical response of the turf to large normal deformations as well as high tangential velocities.

The difficulty in any accurate model of bounce and roll though is to understand how turf behaves. Previous modelling assumed that the ball is only in contact at a single point, and hence the turf is replaced by equivalent point stiffness and damping functions. This is too simplistic since the ball forms a contact set with the turf, so one needs to consider a continuum theory.

The first goal of this project will be to understand and validate exactly what properties of the surface devices such as the “Truform Turf Thumper” and the “Clegg Hammer” measure and how that relates to the properties of the turf that determine the bounce of the ball. The second goal is to derive a mathematical model of the turf deformation and resulting golf-ball trajectory, informed by extant theories of soil mechanics and granular media under large dynamic loads.

The project will be carried out in collaboration with R&A Rules Ltd, and there is the possibility of conducting experimental measurements on realistic turf at their facility near St Andrews.

Contact: Stuart Thomson (stuart.thomson@bristol.ac.uk)

Real-time monitoring of human hormonal-response

Supervisors: Stuart Thomson, Alberto Gambaruto, Alan Champneys, Stafford Lightman

U-RHYTHM is an emerging technology, developed at the University of Bristol, for non-invasive monitoring of body-chemistry, measuring daily fluctuations in the levels of multiple hormones, metabolites and drugs present in the human body. The technology has been tested in over 600 patients in the United Kingdom, Norway, Sweden, Greece, Switzerland and the Netherlands.

The aim of this PhD project is to address some of the scientific challenges that need to be overcome in order for U-RHYTHM to realize its full potential. Depending on interest and area of expertise, some specific objectives include:

  • developing and simulating mathematical models of the flow of biological fluids through porous micro-dialysis probes to investigate and extend the family of molecules that can be measured by U-RHYTHM;
  • using machine-learning techniques to interpret complex time-series measurements obtained from real patient data (for example, comparing the time series of hormones with other physiological measurements such as body temperature or heartrate);
  • experimental analysis of clinically important molecules using physiochemical methods including neutron/synchrotron X-ray scattering

Broadly speaking, this project would suit a candidate with an interest in interdisciplinary collaboration between chemistry, mathematics, engineering and clinical medicine. Successful outcomes of the project will enable the use of U-RHYTHM for endocrinology, metabolism, neuroscience, rheumatology, inflammation, and cardiovascular health.   

Contact: Stuart Thomson (stuart.thomson@bristol.ac.uk)

Modelling organoid and organ-on-chip dynamics

Supervisors: Martin Homer and Lucia Marucci

Complex biological systems involve and coordinate several mechanisms across scales; their understanding often requires a system-level characterisation. We are interested in the emerging dynamics of organoids, i.e., miniaturized in vitro versions of organs grown in a dish, and of organs-on-chip (microfluidic cell culture devices recapitulating in vivo organs).The use of mathematical models that allow to recapitulate multiscale processes such as intracellular cellular dynamics, cell division, biomechanical interactions and the cellular environment can be instrumental to predict and understand organoids and organs-on-chip behaviour and morphology, save experimental time and get quantitative insights.

Possible projects in this area:

  • 3D modelling of organoids
  • Modelling organ-on-chip dynamics, integrating fluid dynamics with a mathematical representation of the organoid

Contact: Lucia Marucci (lucia.marucci@bristol.ac.uk)

Cybergenetics control of living cells

Supervisors: Dr Lucia Marucci and Dr Ludovic Renson (Imperial College)

Cybergenetics technologies have recently emerged at the interface of control engineering and synthetic biology to steer target phenotypes in living cells and organisms. The key idea is to use tools and design principles from control theory, as they are known to enable robust process regulation in naturally occurring (e.g., regulation of blood pressure in humans) and man-made (e.g., automobile engine in cruise control) systems. We have developed microfluidics/microscopy platforms for external feedback control: the controlled processes are within cells, while the controller is implemented externally. Real-time measurements of control outputs (e.g., fluorescent protein reporters) in living cells inform control algorithms, which calculate online time-varying inducer molecule/light stimuli (control inputs) to minimize the difference (control error) between the output and the control reference; inputs are provided by actuators (e.g., pumps).

We now want to extend the repertoire of feedback control algorithms to use, and the applications of these technologies.

Possible projects in this area:

  • Experimental control-based continuation to track non-linear dynamics of synthetic gene networks
  • Implementation of novel control algorithms (machine learning-based) to steer mammalian cell dynamics

Contact: Lucia Marucci (lucia.marucci@bristol.ac.uk)

Whole-cell genome design

Supervisors: Dr Lucia Marucci, Prof Claire Grierson, and Dr Zahraa Abdallah

Whole-cell models are large, complex models that simulate the entire life cycle of a cell. Their capacity for producing large and diverse quantities of data means that they can give a more detailed insight into the interactions between different cellular processes than in vivo data, and are cheaper and easier than performing lab experiments. Their uses vary from genome design to drug testing, and although they will never be as accurate as living cells, they are powerful tools that can inform experiment design and be used in tandem with lab work. Our group has used various computational methods with the two existing whole-cell models (of the bacteria M. genitalium and E. coli) for applications such as finding minimal genomes and metabolic flux analysis, where our long-term goals are to understand how we can design genomes to produce desired phenotypes.  We are also applying machine learning and AI to predict cell phenotypes from whole-cell model outputs.

Possible projects in this area:

  • Time-series analysis of whole-cell model outputs using machine learning
  • Embedding whole-cell models within synthetic biology design-build-test-learn cycles
  • Automatic whole-cell model calibration using machine learning/AI

Contact: Lucia Marucci (lucia.marucci@bristol.ac.uk)

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