M.Sc. Projects

# Mainly numerical projects

• ## Discontinuous Galerkin Methods

Discontinuous Galerkin methods are a class of numerical methods that do not require the approximate solutions to lie in the space of continuous functions. Instead, solutions are located in the space of functions with bounded variation. In practice, the spatial domain is discretised into a number of elements and the solution is continuous within each element, but not between elements. The coupling between adjacent elements is achieved by the specification of a flux function. One advantage of such methods is that systems that develop shock structures (for example, non-linear wave equations) where the solution is locally multi-valued can be accurately modelled. Potential projects in this area include:
• Modelling the development and motion of curved shocks in two and three dimensions.
• A comparison between Discontinuous Galerkin and finite volume methods.
• Simulating physical systems modelling by (mainly) hyperbolic equations; for example, gas dynamics, transport of charged particles via the Boltzmann equation, purely advective transport, radiative heat transfer.
• ## Spectral Element Methods

Spectral element methods are related to finite element methods, but the basis functions are chosen to be high-order polynomials, rather than the low-order polynomials used in finite elements. The idea is that spectral (exponential) convergence can, in principle, be achieved by keeping the number of element fixed and increasing the order of the basis functions (p-refinement). In contrast, fixing the order of the basis functions and increasing the number of elements (h-refinement) gives only algebraic convergence, the power depending on the order of the basis function. That said, the cost associated with computing the elemental contributions can increase dramatically with the order of the polynomial, so it not clear which is the better strategy, h or p-refinement. The idea of the project is to investigate the different refinement strategies for a number of fluid-flow problems, including the flow between rotating discs, flow in porous channels and even free-surface flows, if time permits. The project would contain a sizeable programming component in ANSI C++.

Modelling the free surface of a fluid is complicated by the fact that its position is an unknown in the problem. As the free surface deforms the fluid domain changes. This can be a particular problem when solving the system numerically because the fluid domain must be re-meshed at each time step (an expensive procedure). An alternative approach to re-meshing is to represent the position of the free surface by a signed distance function, often known as a level set. The discontinuity in the normal stress (velocity gradient) at the interface is handled by enriching the standard finite element basis with a set of basis functions that are discontinuous across the interface (X-FEM). The idea of the project is to compare X-FEM with level sets against the more traditional moving-mesh methods for the evolution of free surfaces.
• ## Large-scale (numerical) bifurcation detection and tracking

The accurate detection and tracking of bifurcations (points at which the qualitative nature of a dynamical system changes) is one important way of understanding how a system changes under variation of its governing parameters. Although a number of very good tools exist for small and medium-sized systems, there is a lack of easy-to-use tools for large-scale problems. This project will explore the implementation of bifurcation detection routines in C++ that can be applied to general (large) systems. In particular, the parallelisation of the assembly and solution of these systems will be considered. Careful block-factorisation of the augmented systems that describe standard bifurcations can lead to the solution of bifurcation-detection problems being reduced to the repeated solution of similar sets of linear equations. In principle, any pre-conditioning methodology appropriate for the original system of equations can then be applied in the bifurcation-tracking context. It should be possible, therefore, to use iterative methods to track bifurcations efficiently even for very large systems. One particular area of application is in the study of the bifurcation and evolution of periodic solutions of sets of partial differential equations. The necessity of discretising the entire period of the solution leads to very large dimensional problems.

# Biologically motivated projects

• ## Dynamics of fluid in the eye

The vitreous humour that fills the majority of the cavity within the eye is essentially a gel that is firmly attached to the retina. With age, pockets of the vitreous humour can liquify, these liquid pockets can enlarge and might lead to retinal detachment. The aim of this project is to investigate the effect of rapid eye motion of the behaviour of the fluid within these liquid pockets. Initially, the classical problem of spin-up in a spherical geometry will be considered, which has been well-studied in the geophysics literature. Extensions to the case of an elastic, rather than rigid boundary, will examine the stresses that can be transmitted to the retina.
• ## NO transport in the lung

Nitric Oxide (NO) is a simple chemical that has only recently been discovered in the exhaled breath of humans. The concentration of exhaled NO is typically higher in those with asthma, suggesting that it can be used as a non-invasive measure of airway inflammation. At present, the models for the production and transport of NO within the lung are rather simplistic (one-dimensional) and contain a number of gross simplifications. The idea of the project is to critically re-examine the existing models and to examine whether the inclusion of more realistic assumptions (axial diffusion, lateral mixing, etc) can affect the model predictions.
• ## The influence of a collapsing airway on its neighbours

Airways in the lung are prone to a fluid-elastic instability that can cause collapse. One question that arises is whether the collapse of one airway predisposes surrounding airways to collapse leading to a cascade and collapse of an entire section of the lung. In this project a very simple model will be considered in which the airways of the lung are modelled as holes in a two-dimensional (hyper)elastic material. If one of these holes has collapsed (buckled), how is the stability of surrounding holes affected? The answer will presumably depend upon the degree of collapse of the initial hole, the distance and direction(?) of the holes from each other.

# Fluid mechanics problems

• ## Keeping the honey on the knife: free-surface flows on rotating objects

A solid body in a uniform gravitational field is coated in a film of viscous fluid. For fast-enough rotation rates, the film can remain attached to the solid body. Classic studies of this problem consider a uniform solid cylinder with circular cross section and use a thin-film approximation for the fluid motion. Even under these simplifications complex free-surface dynamics have been found. The aim of this project is to review the classic cylindrical studies and to extend the investigation to other cross-sections by numerical simulation and asymptotic methods. One question to be addressed is how the cross-section affects the mass of fluid that can be supported for a given rotation rate.
• ## Falling liquid films

A classic problem in fluid mechanics is the flow of a liquid film along a flat plate, which has an exact analytic solution due to Nusselt. Under different types of forcing the flow is know to be unstable and surface waves can readily develop. The aim of this project is to review the literature on the stability of falling liquid films and to examine the influence of free and fixed obstacles on the stability of the films. In particular, how important are the small regions near the obstacles in which the traditional lubrication (thin film) assumptions break down.
• ## Dynamics of convection

Convective instabilities can be induced by considering the effects of temperature on the behaviour of a viscous fluid. The effects of the temperature on the fluid density affects the gravitational body-force term in the Navier--Stokes equation (Bernard convection). The temperature can also affect the surface tension at a free surface (Marangoni convection). Interplay between the two mechanisms can lead to complex dynamics. The introduction of surfactant into the system gives another means of altering the surface tension and provides a mechanism for oscillatory onset of the convective instability. The aim of this project is to use numerical bifurcation detection and tracking techniques to understand the dynamics of these types of system and the transitions between different types of instability.

# Solid Mechanics problems

• ## The influence of nearby walls on the buckling of an elastic ring

A "free" elastic ring experiences a buckling instability when the transmural (internal - external) pressure exceeds a critical value. If the ring is confined to lie within external walls, of a particular shape, how does the buckling behaviour change? Do the symmetries of the external shape influence the behaviour at all?
• ## Pattern formation in regular elastic lattices

Recently, Tom Mullin and co-workers, discovered that compression of an elastomer containing a periodic array of holes leads to a striking instability: the initially circular holes become ellipses with alternating orientations in either the vertical and horizontal direction. The instability is remarkably robust, but different patterns can arise if the system is sufficiently perturbed. The aim of this project is to explore numerical and analytic models of the system to try to understand the details of the mechanism underlying the instability. What are the minimum ingredients? How does distance between the holes affect the instability? Can we deduce a simple model by considering a few well-separated holes?