These are ideas suggested by our industrial sponsors for M.Sc. dissertations. If you are interested in one of these projects you should get in touch directly with the supervisor.
A pdf version of the projects is also available.
Contact: oliver@maths.ox.ac.uk
Background and problem statement
Oil reservoir operations are optimised using mathematical models. Both numerical and analytical techniques are used, depending on the model. During optimisation one evalautes quantities of interest as functions of the input and control parameters.
Approach and techniques
In this project a collection of related models of flow through porous media will be analysed for the purpose of guiding engineers and geoscientists in the selection and use of the best model.
As part of this a reduction in the number of parameters can be made by using the techniques of dimensional analysis, based on a transformation of the model equations. By studying the parameters involved in some actual oil reservoirs, the magnitudes of the dimensionless groups will be found, and recommendations as to which terms in the equations can be ignored will be listed. The method is essentially to supply the tools needed for us to apply the same techniques used for analysing physical, laboratory experiments to numerical experiments.
Objective
The aim is to produce a table of models with their associated dimensionless groups, some exact solutions where these can be found, and the simplified models that follow when some terms are small.
Reference
Contact: Endre.Suli@comlab.ox.ac.uk
The aim of the project is to develop adaptive finite algorithms for dumbbell-type kinetic models for dilute polymers.
A typical model of this kind (such as FENE - Finitely Extensible Nonlinear Elastic model) involves nonlinear coupling of the incompressible Navier-Stokes equations, including the divergence of the non-Newtonian extra-stress tensor as source term, and a high-dimensional (typically (4+1)-dimensional or (6+1)-dimensional) degenerate parabolic equation (Fokker-Planck equation) with an unbounded drift coefficient.
The project will focus on simplified, low-dimensional, versions of this model, with the aim to understand the amount of computational work that needs to be invested into solving the Fokker-Planck equation for the probability density function, associated with the random fluctuation of the polymer molecules in the solvent (an incompressible Newtonian fluid), in order to obtain accurate numerical approximations of macroscopic quantities, such as the velocity and the pressure.
The project will require familiarity with spectral methods, finite element methods and their mathematical analysis (studied in the Hilary Term course "Finite Element Methods for PDEs"), and strong background in fluid mechanics.
The key numerical analysis tools in the project include:
References
Contact: Andy.Wathen@comlab.ox.ac.uk
If one considers z=f(x,y) as a surface then any three points on this surface define a triangle with a lowest (and highest) vertex or edge. By folding over an edge (taking the mirror image of the highest vertex in a side) and expanding or shrinking such a triangle depending on its current orientation, one can define an algorithm for finding a minimum point of the surface. The extension of this idea of a tumbling and stretching simplex in higher dimensions is the basis of the Nelder-Mead (Simplex) algorithm for unconstrained optimization.
Many optimization problems however require minimization of a function of several variables in the presence of linear (hyperplane) or nonlinear constraints.
The well-known Numerical Algorithms Group software library currently contains a simple implementation of the Nelder-Mead algorithm for unconstrained optimization and NAG have an interest in understanding and implementing an extension of this algorithm to incorporate linear and possibly nonlinear constraints. This MSc project would be to investigate how constraints might be applied and how the dynamic behaviour of such a modified algorithm might differ from the unconstrained version.
Skills required: basically accessible to most students, but best for those with an interest in numerical algorithms.
Contact: Andy.Wathen@comlab.ox.ac.uk
There are considerable advantages (as well as some less obvious disadvantages!) in using approximation spaces for 2- and higher dimensional approximation which do not require a geometric mesh or grid structure. One of the most elegant and powerful set of approximating functions are the so-called "Radial Basis Functions". In a radial basis function approximation one selects a number of "centres" xi &isin Rm, i = 1, . . . , n and a single function φ:R &rarr Rn and seeks an approximation of the form p(x) = ∑ni=1 αi φ ( || x-xi ||).
That is one seeks an approximation which is an expansion of functions which vary only radially about the centres. In one dimesion (m = 1) for example choosing φ(r) = r gives the functions | x - xi | which are continuous but not differentiable at xi: any linear combination is therefore a continuous piecewise linear or linear spline function. One can similarly build cubic spline spaces with φ(r) = | r3 | (which is in C2 but not C3 if you think about it!). There are a number of radial functions φ which are used in 2- and 3-dimensions in various different situations.
This MSc project would involve learning about such radial basis function representations for the problems of data interpolation - finding an approximating function that exactly passes through a given set of data points - and to consider how to implement them in software. The particular context would be 2-dimensions for this project and the use of so-called thin plate splines which generalise the idea of a cubic spline from one dimension.
Skills required: basically accessible to most students, but best for those with an interest in numerical algorithms.
Contact: tindallm@maths.ox.ac.uk
With the incidence of obesity on the increase understanding how the body digests fat (lipid metabolism) is an area of growing research. When the liver digests fatty acids it excretes lipoprotein particles comprised of triglyceride and cholesterol. The ratio of triglyceride and cholesterol content of each particle defines the lipoprotein type and has important consequences on the "healthiness" of the particles. For instance, an abundance of low density lipoproteins (LDL) can lead to coronary artery disease.
We have recently formulated a partial differential equation (PDE) model of lipid metabolism in collaboration with the Systems Biology Group at Unilever Colworth Laboratories. Unlike other models in the literature, our PDE approach allows the distribution of lipoprotein particles to be tracked in a two-dimensional cholesterol-triglyceride space. However, the sparseness of the particle distributions within the cholesterol-triglyceride and the effects of implementing certain biological mechanisms raises some interesting computational issues.
This project will primarily focus on implementing current methods for solving hyperbolic PDEs in two dimensions, using standard computational packages as well as coding of specific numerical routines. New methodologies will be developed, where required, to improve computational efficiency of the model solvers. The project will also involve extending the current model to include new biological mechanisms.
Contact: Patrick.McSharry@eng.ox.ac.uk
Background
The assessment of human frailty conventionally involves the observation of individuals constructed to undertake specific tasks and may include physical function measures such as balance, repeated chair stands and short walks. Such tests are frequently interpreted in a categorical way or an assessment made on the time to complete the procedure. Healthy Ageing have been interested in exploring the progressive changes in physical function that may lead ultimately to frailty and are currently engaged in developing and applying new procedures in aged cohorts to explore potential continuous measures of physical function.
The Unilever CR APPT study is collecting triaxial accelerometer data from four body sites (left and right wrists, left and right ankles) on many healthy individuals divided by age decade between twenty and eighty years to establish baseline movement data in the subjects when executing specific tasks from a short physical performance test battery. These tests comprise three separate balance tests (feet side by side, semi-tandem and full tandem stands), a repeated chair stand and a short measured (8 ft) walk. The accelerometers acquire estimates of acceleration (referenced to 'g') from each of the three axis. The objective of the study is to establish baseline data
The exploration of this data for new objective (categorical and continuous) measures of age associated functional change is the starting point for this project.
Project Proposal: Clinical assessment of frailty
The exploration of this data for new objective (categorical and continuous) measures of age associated functional change from continuous accelerometer measurements across four body sites.
Objectives
Contact: porterm@maths.ox.ac.uk
Background and problem statement
The purpose of radar systems, which are ubiquitous for defence purposes, is to detect objects and determine their identities and possible threat levels. One method for collecting intelligence from radars is to simply listen for the signals emitted by other radar systems in the local environment. By identifying which radar systems are in the environment, and what their pattern of behaviour is, it is possible to identify the other vehicles in the area, where they are, and whether or not they pose a threat to you.
A self-organizing map (SOM), or a Kohonnen network, is a static system that is used to classify a set of input vectors (i.e., radar emitter patterns). One can generalize this concept to "plastic SOMs" (PSOMs), which also classify input vectors but learn continuously as new input vectors arrive. (That is, the PSOM is an example of a continuously learning classifier where, instead of there being separate, sequential training and operating phases, these two phases occur simultaneously as part of the same process.) The goal of this project is to use PSOMs for radar emitter classification in non-stationary environments, in which the characteristics of objects or noise can vary with time, new types of objects can arrive, and so on.
Approach and techniques
This project will use ideas from network theory, data mining, statistics, and perhaps time-series analysis as well as simulated "training data" provided from THALES with which to test and develop these ideas. (This data will come from a specific radar and radar mode, and the objective would be to derive radar type and class of intent.) In a PSOM, one represents the data using a dynamically updating graph in which nodes and edges are added and deleted as new data comes in. In this project, appropriate mathematical tools will be used to analyze and improve the present graph-updating procedures, compare them to various graph clustering (community-detection) algorithms, and develop and test quantitative measures of performance based on graph-theoretic concepts.
Aims
The goal of the project is to take PSOM analysis as far as possible in the radar context. For example, the student will attempt to use appropriate mathematical and computational techniques to classify and group detected objects according to class of radar, type, mode of operation, and particular instance of detection. In each case, the reliability of the classification should be as high as possible, so it will be essential to devise and test metrics that measure this reliability.
Contact: allwrigh@maths.ox.ac.uk
Background and Problem
A sonar set consists of an array of hydrophones (underwater microphones) and the associated electronics, signal processing and software. The processing is designed to detect contacts, i.e. approximately plane acoustic waves incident on the array from a definite direction, and to distinguish these contacts from noise, i.e. effectively random pressure fluctuations that may be due to turbulence or to acoustic waves coming from the many other sources of underwater noise (e.g. rain, surface waves, distant shipping etc.). The statistics of this background noise are generally unknown, and time-varying, and therefore have to be estimated from the data gathered by the array at the same time as detecting contacts. The process of estimating the background noise at the same time as listening for contacts from definite directions is called adaptive beamforming. When the array has many hydrophones, it is difficult because (among other things) the estimated noise variance matrix is ill-conditioned. TUS is therefore interested in work on Adaptive Beamforming for large acoustic arrays. One approach of interest is the use of the "Support Vector Machine" (SVM) formalism. This is intended to aid in the optimisation while satisfying the required constraints. (These are the constraints on the sidelobes and beamwidth that characterize how well the array distinguishes waves incident from different directions.) The approach is described in "Robust beamforming with sidelobe control using support vector machines", C.C.Gaudes et al. IEEE Transactions on Signal Processing, Vol.55, pp574-584 (2007) and the aim of the project would be to study the approach mathematically, compare it with other ways of optimizing the beamformer, and compare its performance in the circumstances of interest to TUS.
We are specifically interested in applying adaptive beamforming to two dimensional (e.g. bow and flank) sonar arrays in order to improve detection performance in the presence of strong interference sources and to improve the performance in non-isotropic noise fields. A typical 2-D array of interest has about 1000 hydrophones and, therefore, about 1000 adaptive degrees of freedom are available. If the simplest MVDR adaptive beamformer is used, around 3000-10000 data snap-shots (i.e. time samples) will be required to estimate the O(1000 1000) data covariance matrix accurately enough to achieve a reasonably low level of "sidelobe jitter". It is unlikely that the data statistics are sufficiently stationary for this. Furthermore, uncalibrated amplitude and phase mismatches between the hydrophones will tend to cause further problems with target cancellation.
Planned Approach and Required Techniques
An interesting approach is to apply a large number of desired inequality constraints to the beam pattern in order to reduce the available degrees of freedom. By relaxing the inequality constraints the adaptive beamformer can be reformulated as a support vector machine (SVM). The SVM can be solved using an iterated reweighted least squares (IRWLS) method. We are looking for an efficient method for a solution which evolves gradually with time due to non-stationarity. The method will be coded up in MATLAB and tested on simulated (and possibly real) data.
Aims and Objectives
The aim is to demonstrate that the proposed SVM provides a robust solution when the number of available data snap-shots is inadequate for the standard MVDR approach and to demonstrate that sensor errors have less impact on the energy of the signal of interest. Alternative approaches to this problem could be considered.
Contact: allwrigh@maths.ox.ac.uk
The objective of this project is to identify and quantify the source of a spoking phenomnenon observed in the response of a sonar array under a sonar dome. Some previous work on this can be seen in the OCIAM MSc dissertation of Gareth Jones. Random forcing (due to turbulence) on the shell is observed to cause an apparent plane wave incident on the hydrophones from a particular direction. Studies at Thales have shown that this effect is due to both the curvature of the shell and the asymmetry of how it is coupled to the surrounding steel casing. So the project will involve studying the vibrational response of a curved shell at an asymmetric junction. An interest in waves would be a good background, and knowledge of elastic wave propagation in a shell would have to be acquired as part of the project.
Contact: allwrigh@maths.ox.ac.uk
Background and Problem statement
A towed sonar array is a long flexible array of hydrophones (underwater microphones) towed (by a cable) behind a surface ship or submarine. In order to interpret correctly the acoustic signals received by the array, its position must be known accurately. The position is estimated by combining data from heading sensors along the array with a mathematical model for the shape of the cable and array called DynaTow. However, DynaTow assumes the array is being towed through water at rest. In practice the tow cable sometimes passes through the flow field aft of the submarine, and this has effects on its position. If the flow field is known, there are accepted models for the average force on the cable. The aim of the project is to establish good approximate parametric models for the average flow field in the region aft of the submarine. This can be considered as consisting of the jet from the propulsor (the main drive of the submarine), the wake from the whole submarine, and the vortices from the fins when turning. Trials in which the array was towed from one of the horizontal tail-fins have shown the effects that can occur when the cable passes through the propulsor jet: in this case the deviation of the actual array shape from the shape predicted by DynaTow only occurs when the submarine turns in a way that makes the cable enter the jet region. A previous study by Ed Tarleton (MSc, OCIAM) used the Sclichting model of a turbulent jet, and showed that the velocity field of the jet was sufficient to cause array deflections of the observed order of magnitude. However, TUS would like to take this further and include either the vortices or wake, whichever is likely to have more effect, and to have a better representation of the jet velocity field during turns. TUS has relevant real data against which model enhancements can be compared.
General approach
It is planned that the project will begin with a review of the literature on the flow fields in the jets, wakes and vortices of self-propelled bodies. In particular it is expected that work of Dr Marvin Jones (Southampton) on vortices will be relevant. Although the jet is turbulent, TUS are happy for it to be treated by approximate theories for a slender turbulent jet.
Contact: allwrigh@maths.ox.ac.uk
A Towed Array Sonar array consists of a long length (several hundred metres) of flexible, viscoelastic PVC hose containing highly sensitive acoustic and non-acoustic sensors plus associated electronics and wiring. In use, the array is pulled through the water by a ship or submarine. Under normal conditions the hose extends due to the hydrodynamic drag over the surface. Under more extreme conditions (high speeds and warmer waters) the hose can extend/creep excessively causing it to collapse. The type of collapse observed is generally referred to as "rucking". Visually, rucking is the type of collapse seen when one pushes the sleeve of a sweater or shirt up ones arm.
To date a relatively simple MATHCAD program has been generated to model the conditions that cause rucking to occur - this was based on Euler Buckling Theory, which is now thought to be inadequately representative of the type of collapse seen in practice. In addition to the type of collapse needing to be modelled in a more representative way, the complexity of the modelling will be increased by the tight fitting internal components that constrain the movement of the PVC hose at intervals along its length.
The array contains a central cable that takes the main tension. The drag force is transmitted from the hose to this cable by bulkheads, rigid discs occupying the cross-section of the hose, connected to the central cable and fitting tight against the hose. These bulkheads are regularly spaced along the length. The project is to analyse this, calculating the stress distribution in the hose and either using existing models for this kind of rucking or developing something more appropriate. The work requires interest in solid mechanics but this can be learned during the project if necessary.
If a model can be developed that will more closely predict the conditions leading to the inception of rucking then it may be possible to take measures in the design of the Towed Array Sonar that would extend the operational parameters of the sonar system. Any relaxation in operational limitations imposed by the sonar equipment will always be welcomed by ship or submarine commanders!
Contact: allwrigh@maths.ox.ac.uk
In some sonar sets, acoustic waves travelling through the sea are incident first on a sonar dome (a curved shell, designed to be acoustically transparent) then travel through the still water behind the shell and then are incident on the hydrophones. TUS study the vibration of the shell in response to incident acoustic waves, and generally do this in terms of ray theory and the "tangent plane" approximation. In this, an acoustic ray is considered incident on the shell at a point, and the acoustic field transmitted into the water behind the array is calculated as if the shell were replaced by a flat sheet of the same material tangent to the actual shell at that point. In reality the array is usually a doubly curved shell. TUS have assumed that the errors due to this approximation are small, but would like to quantify the size of the errors in terms of the radii of curvature of the shell and its thickness and the acoustic wavelength. The shell material will be treated as linearly elastic and isotropic. An interest in waves would be a good background, and knowledge of elastic wave propagation in a shell would have to be acquired as part of the project.
Contact: chapman@maths.ox.ac.uk
Background and problem statement
In medical diagnostic tests, including pregnancy testing and tests for typed red blood cells, a small fluid sample is placed at one end of a capillary channel, which has been lined with a dried reagent. If the sample contains the analyte (the substance being tested for) then an agglutination reaction occurs between it and the reagent in the channel, and the agglutinated complexes progressively slow the flow and may even block the channel, partially or completely, so that the flow only reaches the far end very slowly, or not at all. The aim is that this should give a reliable detection of quite low concentrations of analyte in the sample. Platform Diagnostics would like a mathematical model of the process, so that, for known binding forces in the agglutination complexes, the channel size and shape, and the fluid viscosity, can be designed to maximize the reliable detection of low concentrations. A key question is how the flow time depends on channel size, fluid surface tension and viscosity, (a) in the absence of agglutination, and (b) in the presence of agglutination.
Planned approach and the techniques needed
The flow problem will involve lubrication theory for the channel (with different cross-sectional geometries), and capillary statics (Laplace-Young equation) for the driving force. The project will involve a mix of asymptotic analysis and numerical simulation.