You may contact a Proposer directly about a specific project or contact the Postgraduate Admissions Secretary with general enquiries.

Title Foundations of adaptive finite element methods for PDEs
Group(s) Scientific Computation, Algebra and Analysis
Proposer(s) Dr Kris van der Zee
Description

Foundations of adaptive finite element methods for PDEs
(Or- Why do adaptive methods work so well?) 

Adaptive finite element methods allow the computation of solutions to partial differential equations (PDEs) in the most optimal manner that is possible. In particular, these methods require the least amount of degrees-of-freedom to obtain a solution up to a desired accuracy! In recent years a theory has emerged that explains this behaviour. It relies on classical a posteriori error estimation, Banach contraction, and nonlinear approximation theory. Unfortunately, the theory so far applies only to specific model problems.

Challenges for students:
* How can the theory be extended to, for example, nonsymmetric problems, nonlinear problems, or time-dependent problems?
* What about nonstandard discretisation techniques such as, discontinuous Galerkin, isogeometric analysis, or virtual element methods? 

Depending on the interest of the student, several of these issues (or others) can be addressed.
Also, the student is encouraged to suggest a second supervisor, possibly from another group! 

Relevant Publications
  • J.M. Cascon, C. Kreuzer, R.H. Nochetto, and K.G. Siebert, Quasi-optimal convergence rate for an adaptive finite element method, SIAM J. Numer. Anal. 26 (2008), pp. 2524-2550
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Title Partitioned-domain concurrent multiscale modelling
Group(s) Scientific Computation, Mathematical Medicine and Biology
Proposer(s) Dr Kris van der Zee
Description

Partitioned-domain concurrent multiscale modelling
(Or- How does one get cheap, but accurate, models?)

Multiscale modeling is an active area of research in all scientific disciplines. The main aim is to address problems involving phenomena at disparate length and/or time scales that span several orders of magnitude! An important multiscale-modeling type is known as partitioned-domain concurrent modelling. This type addresses problems that require a fine-scale model in only a small part of the domain, while a coarse model is employed in the remainder of the domain. By doing this, significant computational savings are obtained compared to a full fine-scale model. Unfortunately, it is far from trivial to develop a working multiscale model for a particular problem.

Challenges for students:
* How can one couple, e.g., discrete (particle) systems with continuum (PDE) models?
* Or a fine-scale PDE with a coarse-scale PDE?
* How can one decide on the size and location of the fine-scale domain?
* Is it possible to proof the numerically observed efficiency of concurrent multiscale models? 
* Can the multiscale methodology be applied to biological growth phenomena (e.g., tumours) where one couples cell-based (agent-based) models with continuum PDE models?

Depending on the interest of the student, several of these issues (or others) can be addressed.
Also, the student is encouraged to suggest a second supervisor, possibly from another group!

Relevant Publications
  • J.T. Oden, S. Prudhomme, A. Romkes, and P.T. Bauman, Multiscale modeling of physical phenomena: Adaptive control of models, SIAM J. Sci. Comput. 28 (2006), pp. 2359-2389
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Title Phase-field modelling of evolving interfaces
Group(s) Scientific Computation, Mathematical Medicine and Biology
Proposer(s) Dr Kris van der Zee
Description

Phase-field modelling of evolving interfaces
(Or – How does one effectively model and simulate interfacial phenomena?)

Evolving interfaces are ubiquitous in nature, think of the melting of the polar ice caps, the separation of oil and water, or the growth of cancerous tumours. Two mathematical descriptions exist to model evolving interfaces: those with sharp-interface descriptions, such as parametric and level-set methods, and those with diffuse-interface descriptions, commonly referred to as phase-field models.

Challenges for students:
* Can one develop a phase-field model for a particular interfacial phenomenon?
* What are the foundational laws underpinning phase-field models?
* What is the connection between sharp-interface models and phase-field models?
* Can one design stable time-stepping schemes for phase-field models?
* Or efficient adaptive spatial discretisation methods?

Depending on the interest of the student, one of these issues (or others) can be addressed.
Also, the student is encouraged to suggest a second supervisor, possibly from another group! 

Relevant Publications
  • H. GOMEZ, K.G. VAN DER ZEE, Computational Phase-Field Modeling, in Encyclopedia of Computational Mechanics, Second Edition, E. Stein, R. de Borst and T.J.R. Hughes, eds., to appear
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Title Numerical methods for stochastic partial differential equations
Group(s) Scientific Computation, Statistics and Probability
Proposer(s) Prof Michael Tretyakov
Description

Numerics for stochastic partial differential equations (SPDEs) is one of the central topics in modern numerical analysis. It is motivated both by applications and theoretical study. SPDEs essentially originated from the filtering theory and now they are also widely used in modelling spatially distributed systems from physics, chemistry, biology and finance acting in the presence of fluctuations. The primary objectives of this project include construction, analysis and testing of new numerical methods for SPDEs.

 

Relevant Publications
Other information

Web-page http://www.maths.nott.ac.uk/personal/pmzmt

Title Property Prediction of Composite Components Prior to Production
Group(s) Scientific Computation, Statistics and Probability
Proposer(s) Prof Michael Tretyakov, Prof Frank Ball
Description

Property Prediction of Composite Components Prior to Production

Supervisors: Dr Frank Gommer1*, Prof Michael Tretyakov2*, Prof Frank Ball2 , Dr Louise P. Brown1

University of Nottingham, University Park, Nottingham NG7 2RD, UK

1 Polymer Composites Group, Faculty of Engineering

2 School of Mathematical Sciences

* Contact: F.Gommer@nottingham.ac.uk or Michael.Tretyakov@nottingham.ac.uk

 

This is an exciting opportunity for a postgraduate student to join a vibrant inter-disciplinary team and to work in the modern area of Uncertainty Quantification.

Fibre reinforced composites are increasingly used in the transport industry to decrease the structural weight of a vehicle and thus increase its fuel efficiency. The importance of the UK composite sector is reflected in the current growth rate of 17% pa for high performance composite components and the expected gross value of £2 billion in 2015 [1]. However, due to the large number of production steps and the necessary saturation of the fibre preform with a resin matrix, a significant amount of waste is produced, which may range between 2% and 20% of the production volume [2]. A major cause of rejecting parts is variability in the reinforcement, such as varying yarn spacing and yarn path waviness, which can significantly influence subsequent properties. For example, these variabilities can affect resin flow and may cause dry spots or reduce mechanical properties. This PhD project will enable the successful candidate to work at the forefront of material science, combining engineering standards, applied mathematics and statistics, with a potential of making an impact on the way of manufacturing composite parts in the future.

This proposed doctoral study aims to demonstrate that properties of light-weight fibre reinforced plastics can be predicted in real time before a part is actually manufactured. Data gained from images taken of each layer of a composite during the stacking process are used to determine local geometries and variabilities, within and in-between individual layers [3]. For example, based on the measured textile geometries it will be possible to predict the resin flow within a preform during a liquid composite moulding (LCM) process considering individual variabilities before injection. These specific flow predictions will allow adjustments of the process parameters during the impregnation process to ensure full saturation of the entire preform with a liquid resin matrix. This will be especially useful when a number of inlet and outlet ports are present such as in the case of complex or large parts. The formation of dry spots will be avoided, which will reduce immediate wastage. For these predictions, faster solutions than currently available are necessary. To find such solutions, appropriate advanced statistical techniques and stochastic modelling for quantifying uncertainties in composites production will be developed in the course of the PhD project.

In addition, the developed techniques will also allow virtual testing of a finished component with its specific inherent reinforcement variability. This will make it feasible to customise predictions for every fabricated component. In combination with continuous health monitoring of a structure, it may be possible to estimate the influence of loading conditions, load cycles and damage evaluation. This will also make it possible to predict an individual life expectancy of a part in service. These data can then be used to determine customised inspection intervals for each component.

We require an enthusiastic graduate with a 1st class degree in Mathematics or Engineering, preferably of the MMath/MSc level, with good programming skills and willing to work as a part of an interdisciplinary team. A candidate with a solid background in statistics will have an advantage.

References

[1] CompositesUK. www.compositesuk.co.uk/Information/FAQs/UKMarketValues.aspx.

[2] A. C. Long, Design and Manufacture of Textile Composites: Woodhead Publ, 2005.

[3] F. Gommer, L. P. Brown, and R. Brooks, “Quantification of meso-scale variability and geometrical reconstruction of a textile”, submitted to Compos Part A-Appl S, 2015.

Relevant Publications
Other information

This project is supported by  EPSRC DTG Centre in Complex Systems and Processes, see elligibility and how to apply at http://www.nottingham.ac.uk/complex-systems/index.aspx

Title Statistical analysis of fibre variability in composites manufacture
Group(s) Statistics and Probability, Scientific Computation
Proposer(s) Prof Frank Ball, Prof Michael Tretyakov
Description

Multidisciplinary collaborations are a critical feature of material science research enabling integration of data collection with computational and/or mathematical modelling. This PhD study provides an exciting opportunity for an individual to participate in a project spanning research into composite manufacturing, stochastic modelling, statistical analysis and scientific computing. The project is integrated into the EPSRC Centre for Innovative Manufacturing in Composites, which isled by the University of Nottingham and delivers a co-ordinated programme of research in composites manufacturing.

This project focuses on the development of a manufacturing route for composite materials capable of producing complex components in a single process chain based on advancements in the knowledge, measurement and prediction of uncertainty in processing. The outcome of this work will enable a step change in the capabilities of composite manufacturing technologies to be made, overcoming limitations related to part thickness, component robustness and manufacturability as part of a single process chain, whilst yielding significant developments in mathematics and statistics with generic application in the fields of stochastic modelling and inverse problems.

The specific aims of this project are: (i) statistical analysis of fibber placements based on textile and composite material data sets; (ii) statistical analysis and stochastic modelling of permeability of textiles and composites; (iii) efficient sampling techniques of stochastic permeability. A student will obtain an excellent grasp of various statistical and stochastic techniques (e.g., spatial statistical methods, use of random fields, Monte Carlo methods), how to apply them, how to work with real data and how to do related modelling and simulation. This knowledge and especially experience are transferable to other applications of statistics and probability.

The PhD programme contains a training element, the exact nature of which will be mutually agreed by the student and their supervisors.

We require an enthusiastic graduate with a 1st class honours in Mathematics (in exceptional circumstances a 2(i) class degree can be considered), preferably at the MMath/MSc level, with good programming skills and williness to work as a part of an interdisciplinary team. A candidate with a solid background in statistics and stochastic processes will have an advantage.

The studentship is available for a period of three and a half years from September/October 2015 and provides a stipend and full payment of Home/EU Tuition Fees. Students must meet the EPSRC eligibility criteria.

Informal enquiries should be addressed to Prof. Michael Tretyakov, email: michael.tretyakov@nottingham.ac.uk.

To apply, please access: https://my.nottingham.ac.uk/pgapps/welcome/. Please ensure you quote ref: SCI/1262x1. This studentship is open until filled. Early application is strongly encouraged.

Relevant Publications
Other information