If you are interested in joining the group in any capacity, you are encouraged to to reach out to Bryn via email [bryn.davies@warwick.ac.uk]. Informal enquiries of any form are always welcome. It's generally advisable to reach out as early as possible to discuss possible projects and the various options for funding them.
Potential PhD students from overseas (not with "home" fee status) should reach out particularly early, so that we can look into arranging suitable funding.
Current Warwick students are always welcome to join the group for URSS projects, 4th year R-projects and MSc projects. We have a handful of students doing each type of project every year.
Some specific open projects are listed below, however this list will never be exhaustive and we may have other opportunities that are better suited to you, so please get in touch.
We will soon have opportunities for postdocs to join the group to work on the Leverhulme Trust Research Leadership Award entitled “Unlocking a New Generation of Imperfection-Resilient Metamaterials”.
Background: The performance of advanced materials such as metamaterials is often highly sensitive to imperfections, significantly hampering high-volume manufacturing and limiting operational lifespan. We will develop new mathematical tools to enable the design of a new generation of imperfection-resilient metamaterials that function despite defects or damage. By driving a discipline-wide shift towards approaching resilience as a design objective, we will reduce the hurdle between theoretical breakthroughs and real-world applications.
Objectives: 1. Predict when metamaterials fail: develop new mathematical tools to predict the magnitude of imperfection that causes a given function to break down (by exploiting topological characterisations of spectra); 2. Understand how metamaterials fail: develop new tools for characterising the function of imperfect systems (by developing methods for PDEs with periodic coefficients); 3. Design and fabricate imperfection-resilient metamaterials: use our new tools to develop imperfection-resilient designs (by developing numerical optimisation strategies and working with experimental collaborators to validate predictions).
If you’re excited by waves, metamaterials, analysis of PDEs, design optimisation and interdisciplinary collaboration, and want to work at the interface of first-principles mathematics and real-world applications, then we're keen to hear from you.
Funding: These positions are funded by the Leverhulme Trust through my Research Leadership Award on developing new imperfection-resilient metamaterials. I expect to advertise the next set of positions in spring 2027 (but feel free to reach out earlier).
Background: Periodic small-scale structures are the backbone of metamaterials and other wave control devices. However, manufacturing imperfections or operational damage can alter their intended function. Many of the standard tools we use to model and design these systems rely on periodicity and struggle to predict how they will behave once imperfections are introduced.
Objectives: This project will develop a range of analytic and numerical methods to model and design imperfect metamaterials. Depending on the student's interests, this could include
perturbative methods for weakly random media [JSV];
developing variants of large deviation theory to characterise when imperfections qualitatively alter wave phenomena [PNAS];
applying localisation landscape techniques to predict the creation of localised modes [PNAS];
generalising radiative transfer methods to describe propagation in perturbed periodic media [WM];
characterising the statistics of eigenmodes, particularly the creation of localised modes [CiMP].
This would be well suited for a student with an interest in combining analysis of PDEs, probability theory and numerical simulation. We will study problems with significant engineering applications and collaborate with a range of industrial and experimental colleagues.
Funding: This project is funded by the Leverhulme Trust through my Research Leadership Award on developing new imperfection-resilient metamaterials. It will be integrated into the Mathematics Centre for Doctoral Training. Candidates will need to apply through the CDT and mention this project (it's also worth reaching out to Bryn via email to let him know that you are applying).
Background: Although we typically rely on linear models to design metamaterials, harnessing nonlinear phenomena can unlock significant benefits. This includes the freedom to alter the frequency composition of a wave, exploit amplitude-dependent responses and create directionally dependent (e.g. diode-like) behaviour. However, many of the standard tools for metamaterials (e.g. asymptotic homogenisation, transfer matrices, Floquet-Bloch analysis) exploit linearity of the underlying equations.
Objectives: This project will develop a propagation operator framework that generalises the transfer matrix method to nonlinear wave regimes. We will use concepts from dynamical systems theory (such as attractors and Lyapunov exponents) to develop suitable notions of band gaps and use asymptotic approximation to develop homogenisation strategies. Depending on the student's interests, we could generalise the approach to nonlinear systems including nonlinear acoustics, vibro-elastic systems and nonlinear optics. We will use our methods to unlock the benefits of nonlinearity for applications including
exploiting amplitude-dependent transmission gaps to build sensors;
designing diode-like non-reciprocal phenomena [APL];
reducing energy reflection to improve wave energy harvesters [NJP];
exploiting hysteresis to give materials the ability to remember, count and compute [PRL].
This would be well suited for a student with an interest in combining mathematical modelling, dynamical systems and numerical simulation. We will study problems with significant engineering applications and collaborate with a range of industrial and experimental colleagues.
Funding: This project is funded by the Leverhulme Trust through my Research Leadership Award on developing new imperfection-resilient metamaterials. It will be integrated into the Mathematics Centre for Doctoral Training. Candidates will need to apply through the CDT and mention this project (it's also worth reaching out to Bryn via email to let him know that you are applying).
Background: A latent symmetry (sometimes known as a "hidden" symmetry) is the phenomenon whereby systems are not symmetric in any classical sense (i.e. they don't have any reflectional or rotational symmetries) but they still exhibit properties of symmetric systems. These ideas originate in graph theory [PhysA] and have been extended to wave systems [PRL]. This project will extend these ideas to PDE models.
Objectives: Find an appropriate formulation of latent symmetry for continuous PDE models. Use this to advance applications to imaging, sensing and communication problems. Work with external collaborators to develop experimental realisations.
Collaborators: Malte Röntgen and Wenlong Gao (Eastern Institute of Technology, Ningbo), experts in the physics of latent symmetries.
Funding: This project does not have specific funding attached to it, however it would be a suitable project for funding through the Mathematics Centre for Doctoral Training or the Warwick-EIT Collaborative PhD Programme.
Background: Metamaterials are formed by combining many small, locally resonant elements in a repeating pattern. This gives materials with exotic properties such as negative effective masses, negative refractive indices or negative Poisson's ratios. The repeating patterns are often either periodic (in which case Floquet-Bloch theory can be applied) or random (in which case ensemble averages can be used). We have been studying metamaterials based on a class of quasiperiodic patterns, which are non-periodic but have long-range order. Our work so far has focused on developing methods to estimate spectra [PRSA] and using these to identify and develop novel applications [PRL].
Objectives: Continue the search for novel applications of quasiperiodic metamaterials, particularly to multi-dimensional systems. Search for examples of super band gaps [PRSA] in multi-dimensional systems and study how to construct supercells optimally. Develop new mathematical methods to facilitate design, such as superspace homogenisation techniques [MMAS]. Use these tools to unlock new applications, e.g. by leveraging exotic spectra to achieve broadband effects [PRL] by exploiting isotropy [AMT].
Collaborators: Various collaborators, both in the UK and overseas, will support the development of experimental implementations of our new designs and concepts.
Funding: This project does not have specific funding attached to it, however it would be a suitable project for funding through the Mathematics Centre for Doctoral Training.