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Opportunities for PhD Researchscicomp@leeds |
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Research within the Scientific Computation group focuses on analysis and implementation of fast, efficient and reliable numerical algorithms for computational simulation of physical problems. Much of our work is collaborative, both with industrial and academic partners, and we have particular interests in the application of our work to real-world problems in other subject areas.
There are many potential projects for PhD research within the group. On this page we outline some of these projects with a contact name for further information. This list is by no means definitive and you should feel free to e-mail any of us expressing your own interests.
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Contact: Peter Jimack (pkj@comp.leeds.ac.uk)
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This project will seek to extend the state-of-the-art capabilities described above for phase field simulations to more general multiphase problems. In particular the work will focus on the flow of two immiscible fluids and the flow of such fluids within a porous medium. Rather than using a phase-field model, this project will model the interface using the so-called level set method. This approach introduces a non-physical level-set variable whose zero-contour is used to represent the interface between the fluids. The need to have a highly refined mesh around this moving interface is still extremely important, and for many problems adaptive implicit time-stepping will provide the most efficient numerical scheme so long as a suitable non-linear algebraic solver can be found. As for the phase-field solver described above, the use of multigrid methods will provide just such a solver. Hence this computational approach offers the ability to solve many important two-phaseflow problems with an accuracy and efficiency that is not currently possible. Test problems that will be used to demonstrate the advances made will include: the displacement of oil by brine in a porous medium, as used in oil extraction processes; and the displacement of bone marrow by non-Newtonian cements, as used in certain medical interventions for osteoporotic bone weakness.
Contact: Peter Jimack (pkj@comp.leeds.ac.uk)
Collaboration with Dr Oliver Harlen, Applied Mathematics
It is well-understood that temperature variations occur during most polymer-processing experiments and that the rheology of the polymer is affected by the temperature change. Isothermal, finite-element numerical models developed at Leeds, based on quantitative molecular models of polymers in flow, have had great success in simulating industrially-important flows. However, these ignore the temperature variations that are apparent in the corresponding physical experiments.
This work would extend the finite-element models to include thermal effects and investigate the effect on both the polymer rheology and the overall flow field.
Contact: Mark Walkley (markw@comp.leeds.ac.uk)
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Boussinesq equations model nearshore, shallow-water waves and allow for both dispersive and nonlinear wave effects. As such they are ideally suited for modelling the flow in and around harbours and other coastal structures. Numerical models based on finite element methods offer the geometric flexibility to model the complex geometries and the accuracy required to capture the physical effects.
Previous work has developed a low-order finite element model for this problem. This project would investigate higher-order extensions to this work. It is probable that parallel computing techniques will be required to solve industrial-scale problems.
Contact: Mark Walkley (markw@comp.leeds.ac.uk)
New computational algorithms are continually being developed for the simulation of fluid flow, whether they are to be used in modelling the effects of storms, predicting climate change or examining the flow of blood around the body. Many very robust algorithms already exist but although they readily produce plausible results for the most complex of situations it soon becomes apparent that they often misinterpret the underlying physical processes. This means that while rapidly increasing computer power can allow far more detailed modelling, the additional detail does not necessarily provide a more accurate representation of the flow.
Fluctuation distribution algorithms have been deliberately designed to include genuinely multidimensional physical processes into the computational representation of the problem and they have been demonstrated to be significantly more accurate than the most popular schemes in current use. The reason why they are not more widely used is that they are less robust, especially for time-varying problems, and not as flexible.
This project aims to address these issues by developing more robust generalisations of these methods to systems of nonlinear partial differential equations, paying particular attention to the approximation of time-dependent problems, higher order derivative terms and boundary conditions. It will make use of a new, more adaptable, approach which allows for a discontinuous representation of the underlying flow variables.
Contact: Matthew Hubbard (meh@comp.leeds.ac.uk)
One of the most popular approaches to approximating parabolic and elliptic partial differential equations is the finite element method. It is also widely applied to hyperbolic systems of equations, but with less success due to its tendency to create oscillations in the approximation where they shouldn't exist. This has been partially addressed for steady state problems by designing schemes in which the flow of information across the computational mesh as the algorithm iterates towards the steady state follows the direction of the flow dictated by the mathematical model (a process known as upwinding).
Unfortunately, in order for the finite element approach to retain its accuracy when it is used to model time-dependent problems, it is necessary to invert an additional matrix (the consistent mass matrix) which arises from the discretisation of the temporal derivative. This process reintroduces the unwanted oscillations in to the approximate solution. The aim of this project is to design modifications to the mass matrix which would allow the accuracy of the finite element approach to be retained while avoiding the introduction of spurious oscillations when it is inverted. This would significantly widen the range of applications to which the finite element approach could reliably be applied.
Contact: Matthew Hubbard (meh@comp.leeds.ac.uk)
Many physical processes have important inherent equilibria, typically representing the states that the system will be attracted towards in the absence of external influences. For example, water in a pond will tend to settle down to a state in which it is not moving and has a flat surface. These equilibria are built in to mathematical models designed to describe the system, but often they appear as a balance between two or more separate terms. When these terms involve derivatives, as they often do, it is not always obvious how a computational approximation to the mathematical model should be designed to retain such equilibria.
Returning to the example of still, flat water in a pond, many commonly used mathematical models of this situation involve separate terms for two competing processes: in the absence of any variation in bed topography the net movement is towards shallower water, whereas when the bed is sloping there will also be a tendency for the water to move downhill, i.e. towards deeper water. The equilibrium is maintained by balancing the corresponding terms in the mathematical model, but many widely-used computational methods ignore this, discretise the two terms independently (using techniques which are perfectly sensible choices when the individual terms are considered in isolation), and introduce dramatic disturbances into the supposedly still, flat water.
This issue is particularly problematic when flux-based methods, such as finite volumes or discontinuous Galerkin, are being used as part of the approximation. This is due to the replacement of discrete forms of integrals over local volumes by discrete forms of integrals around the surfaces of those volumes. The aim of this project will be to design a general framework for the approximation of balance terms within standard numerical methods for systems of partial differential equations. Many techniques have been proposed for use on uniform one-dimensional computational meshes, but this research will focus on achieving this goal on irregular, multidimensional meshes using approximation methods of arbitrary order.
Contact: Matthew Hubbard (meh@comp.leeds.ac.uk)
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Many physical and chemical processes, typified by those related to fluid flow, can be modelled mathematically using partial differential equations. These can usually only be solved in the simplest of situations, but solutions in far more complex cases can be approximated using numerical and computational techniques. Traditional approaches to providing these computational simulations have typically modelled the evolution of the system by approximating the equations on a uniform mesh of points covering a domain with a fixed boundary. However, many situations (consider the spreading of a droplet, for example), naturally suggest a domain which evolves with the flow, while the main focus of interest in others (say the movement of a shock wave up and down an aeroplane wing) is in following the motion of a sharp internal feature. For accuracy and efficiency a computational method should not only approximate the partial differential equations appropriately, but also move the computational mesh in a manner which follows such features.
Recent research has developed a finite element approach to the adaptive approximation of time-dependent physical problems involving moving boundaries or interfaces. It has been deliberately designed to preserve inherent properties (such as conservation principles and invariances) of the underlying partial differential equations and hence of the system the mathematics is intended to represent. Extremely promising results have been obtained for a wide range of problems in one and two space dimensions, but the applicability of the approach is still limited (as are all moving mesh methods) by the potential for the computational mesh to ``tangle''. The aim of this project will be to develop an alternative approach, derived within the same framework, which does not require a computational mesh in the same sense, but simply a distribution of nodes at which information about the solution is stored. This would avoid the issue of mesh tangling and greatly improve the robustness of the method when modelling problems involving complex, interacting features.
Contact: Matthew Hubbard (meh@comp.leeds.ac.uk)
In the past few years typical High Performance architectures have changed through the availability of multi-core processors. Many machines, from local clusters to world-leading Blue Gene machines have many computational units (cores and/or processors) per node with gigabytes of memory shared between them.
The most typical parallel strategy is to use MPI in order to connect multiple distributed memory nodes together. Within a node each processing unit is usually imagined to be as separate as those on a different node, hence memory may be replicated.
In this project we are interested in how best to construct matrix solvers and unstructured meshes on such systems without having this redundancy, through the combination of distributed MPI and shared memory techniques.
Contact: Chris Goodyer (ceg@comp.leeds.ac.uk)
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Elastohydrodynamic lubrication (EHL) concerns the behaviour of a non-Newtonian lubricating fluid which acts to keep two elastically-deformable contacts apart by working against an applied load that is trying to force the contacting elements together. This scenario occurs in a wide variety of important engineering situations and so gaining a thorough understanding of EHL can lead to significant improvements in the efficiency of components such as those occuring in engines and other machinery. The importance of this is obvious in today's society where improved engine efficiency can lead to energy savings and has major positive environmental implications.
The aim of this project is to investigate the direct use of nonlinear solution strategies, such as Newton's method, for this problem. To achieve maximum efficiency preconditioned inner-iterations are required for each step of the nonlinear solver. These must also achieve an optimal efficiency if the overall algorithm is to be competitive with existing solution strategies.
This work builds on a long history of research in the group which has developed computational algorithms and tools that are used in industry as well as academia.
Contact: Mark Walkley (markw@comp.leeds.ac.uk)
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Elastohydrodynamic lubrication (EHL) concerns the behaviour of a non-Newtonian lubricating fluid which acts to keep two elastically-deformable contacts apart by working against an applied load that is trying to force the contacting elements together. This scenario occurs in a wide variety of important engineering situations and so gaining a thorough understanding of EHL can lead to significant improvements in the efficiency of components such as those occuring in engines and other machinery. The importance of this is obvious in today's society
Contact: Peter Jimack (pkj@comp.leeds.ac.uk)
Elastohydrodynamic lubrication (EHL) concerns the behaviour of a non-Newtonian lubricating fluid which acts to keep two elastically-deformable contacts apart by working against an applied load that is trying to force the contacting elements together. This scenario occurs in a wide variety of important engineering situations and so gaining a thorough understanding of EHL can lead to significant improvements in the efficiency of components such as those occuring in engines and other machinery. The importance of this is obvious in today's society where improved engine efficiency can lead to energy savings and has major positive environmental implications. Traditional techniques for the computational solution of EHL problems tend to have focused primarily on the use of low order discretization schemes, using large numbers of degrees of freedom when high accuracy is required. Recently however, work at Leeds has demonstrated the potential advantages of high order discontinuous Galerkin (DG) finite element (FE) methods for this class of problem. In order for these advantages to be exploited however it will be essential to combine higher order approximations in space with equivalent orders in time. It will also be necessary to make use of adaptivity to ensure that, at any given time, the degrees of freedom in the spatial discretization are positioned where they will be of most value to the approximation. This project will seek to combine our work at Leeds on high order DG for EHL problems with recent advances in so-called HP-adaptivity for DG. This type of adaptivity provides the capability of altering the degree of the polynomial approximations locally as well as changing the sizes of the finite elements in a local manner too. This provides the maximum flexibility in terms of selecting the approximation space and is likely to lead to significant advances in EHL modelling.
Contact: Peter Jimack (pkj@comp.leeds.ac.uk)
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Elastohydrodynamic lubrication (EHL) concerns the behaviour of a non-Newtonian lubricating fluid which acts to keep two elastically-deformable contacts apart by working against an applied load that is trying to force the contacting elements together. This scenario occurs in a wide variety of important engineering situations and so gaining a thorough understanding of EHL can lead to significant improvements in the efficiency of components such as those occuring in engines and other machinery. The importance of this is obvious in today's society where improved engine efficiency can lead to energy savings and has major positive environmental implications.
A recently completed PhD was into investigating the use of adjoint error estimation techniques in one dimension. This project would look at the extension to two dimensions.
Adjoint techniques are applied here in order to ascertain the best areas of the domain for refinement. This is done based on a Quantity of Interest, typically friction for our cases. This means that the overall solution may have some areas seemingly under-resolved, however if any changes in the solution from there don't affect the QoI then adding extra work to resolve these further is wasted.
As part of this project it could be possible to also investigate adaptation of the film thickness calculation, probably through use of the differential deflection method. Another possible direction could be investigating alternative parallel strategies than previously applied.
Contact: Chris Goodyer (ceg@comp.leeds.ac.uk)
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The outer layer of mammalian epidermis provides much of the body's barrier function. This means that the top 40 micrometres of skin stop most chemicals that the body comes in contact with from being absorbed into the bloodstream. This layer is built up of large almost impermeable keratinsed cells call corneocytes. They are surrounded by lipid through which the diffusion takes place. This means that there is a very complex geometry forming the tortuous path into the body See http://www.comp.leeds.ac.uk/ceg/skinperiodichex.html for example pictures.
In this project it is proposed to look at how global error estimation techniques can be applied to such cases. Given a quantity of interest, such as the rate of chemical entering the body, or the lag time (a measure of the time to steady state), it would be useful to be able to take appropriate timestep sizes in order to get these transient output quantities as accurate as possible.
Important work here would be in the development of the adaptation and adjoint algorithms for these cases, and consideration of the computing storage requirements for performing back-solves. Work on both 2-d and 3-d codes (using SPRINT and maybe PETSc solvers respectively) would be envisaged.
Contact: Chris Goodyer (ceg@comp.leeds.ac.uk)
A wide variety of mathematical models have been developed in order to predict tumour growth, as controlled by the supply of nutrients from surrounding tissue. In the early stages of growth, the small size of the tumour means that its constituent cells can be treated individually, with processes such as subdivision and movement considered in terms of discrete random events with clearly defined consequences. However, as the tumour develops, models of this type can become unwieldy and continuum approximations, which consider the cell activity in some smoothly averaged sense, become more common.
Continuum models take the form of systems of partial differential equations which must be considered throughout a domain which changes in size and shape as tumour spreads. It is possible to analyse these models mathematically, but useful results are typically restricted to one space dimension or idealised multidimensional situations, e.g. radially symmetric cases. The underlying mathematical models actually allow far more complex behaviour, but this can only be investigated through the application of computational simulation techniques.
This project would consist of applying state-of-the-art computational algorithms for the approximation of partial differential equations on time-dependent domains to model the growth of tumours in more realistic situations. These methods would be used to track both the surface of the tumour and interior interfaces between regions occupied by different types of cell. The results would be used to inform the development of new mathematical models for tumour growth and provide deeper insight in to the underlying biological processes.
Contact: Matthew Hubbard (meh@comp.leeds.ac.uk)
Tumour growth is dependent on the supply of oxygen and nutrients from the surrounding tissue, which governs the rate at which the constituent cells can multiply. Without these resources being replenished, the size to which the tumour can grow is severely limited. In order to combat this, the tumours induce the growth of blood vessels in the neighbouring tissue through the secretion of special chemicals, known as growth factors, which have the effect of initiating the growth of blood vessels from nearby capillaries, a process known as angiogenesis, and attracting them towards the tumour. Once the tumour has its own blood supply it has a continual supply of oxygen and nutrients, a means of disposing of waste products and a ready-made pathway along which cancerous cells can spread throughout the body.
The process of developing realistic mathematical models of angiogenesis is still in its early stages. The complexity of the basic processes and the geometry of the problem have meant that mathematical analysis of the models has generally proved intractable in any situation which is complex enough to be meaningful to the biomedical scientists studying the process from the experimental point of view. As a result, computational modelling techniques are providing a vital tool for understanding the mechanisms which drive angiogenesis, since they can be used to approximate mathematical models which incorporate enough of the underlying biochemistry to provide a useful complement to the work of the experimentalists.
One family of computational models of angiogenesis uses reinforced random walks to predict the paths of the tips of the new blood vessels. However, a more natural approach, in the context of the continuum models typically used to represent the distribution of nutrients within the surrounding tissue, would be to use an equivalent stochastic differential equation to simulate the random process. This project would involve building a computational model of this type and using it to investigate the process of tumour angiogenesis.
Contact: Matthew Hubbard (meh@comp.leeds.ac.uk)
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The heart is essentially and electromechanical pump. It is the rapid propagation of electrical activation through cardiac tissue which initiates its contraction, but its electrical behaviour is influenced by the resulting mechanical activity and is very difficult to predict. Research in to these processes routinely uses computational models to simulate the cardiac electrical activity, but the mechanical properties of cardiac tissue are nonlinear and anisotropic, and simplifying assumptions often have to be made to make coupling of electrical and mechanical behaviour in these problems feasible.
The aim of this project will be to develop and apply a computational model of electromechanical activity, described by a system of stiff, nonlinear, coupled ordinary and partial differential equations, in which the heart's shape is modelled by a deforming boundary. Emphasis will be placed on the modelling of ventricular fibrillation, a lethal cardiac arrhythmia that claims up to 60000 lives in the UK each year and is initiated by a disturbance in the normal pattern of electrical activation which results in immediate mechanical failure. The extent to which cardiac mechanics affects the initiation and maintenance of ventricular fibrillation is unknown and computational models provide a valuable complement to animal tissue experiments which will help to address these open questions.
Contact: Matthew Hubbard (meh@comp.leeds.ac.uk)
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last updated Spring 2008 |