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++.
Level set methods for free-surface flow
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?