Moving mesh methods
Reference no.: JAM1
Supervisor: Dr John Mackenzie
Date advertised: 6th February 2007
Please contact Dr John Mackenzie for further information.
The efficient numerical solution of partial differential equations requires the use of solution adaptive techniques.
This is currently a very active area of research with a wide range of applications including computational fluid dynamics, computational finance, and problems in materials and life sciences.
Central to the use of most methods is a computational grid or mesh that covers the spatial domain on which the problem is posed.
Traditionally, solution adaptivity is achieved by locally subdividing the grid in areas where the solution requires greater resolution and possibly coarsening the mesh where the numerical solution is deemed to be overly resolved.
An alternative to this approach is to move mesh points towards areas where additional resolution is required leading to so-called moving grid methods.
For time dependent problems this approach has been shown to be quite successful at solving a wide range of problems.
For movies go to http://www.maths.strath.ac.uk/~caas61/movies.html
There are a number of outstanding issues related to moving mesh methods that would form the basis of a PhD project. Two examples are given below.
Analysis of moving mesh methods. At present there is a great need for mathematical analysis of the stability and accuracy of moving mesh techniques.
Recent work has established some results on the stability of some moving mesh methods.
However much remains to be done and there are a large number of interesting problems such as identifying situations where mesh movement is clearly advantageous, and also when other forms of adaptivity should be performed instead.
Ideally these decisions would be based on reliable a priori and a posteriori error estimates. This project would suit someone who has an interest in doing some analysis which can be applied to a practically relevant area.
Partial differential equations defined on surfaces are seen in many practical areas such as polymer rheological analysis, modeling of the heart, ocean modeling and global climate simulation.
Moving mesh methods are natural techniques for solving problems with localised moving singularities.
This project would consider the generation of adaptive moving meshes on surfaces.
Furthermore, it is envisaged to investigate situations when the surface also moves in time e.g the heart contracting and then expanding.
This area of research is relatively untapped and there would be scope for the project again could be more or less computationally oriented depending on the interest of the student.
Domain growth in liquid crystals
Reference no.: JAM/NJM1
Supervisors:
Date advertised: 6th February 2007
Please contact Dr John Mackenzie for further information.
Liquid crystal displays play an increasingly important role in today's world.
There is now a need for power efficient, high-definition small screens for hand-held devices such as mobile phones, personal organisers and internet access units and for large screen, space saving computer VDUs and televisions.
The main issues of importance to LCD manufacturers are, the ability to produce high speed switching to allow video images to be displayed, increased optical performance i.e better contrast and viewing angle, and power consumption, to extend the life-time of batteries in portable devices.
With this in mind, research is being undertaken in the Maths Department to understand fast switching processes and how defects affect switching.
One problem is that switching often occurs by domain growth and no one knows why the domains are "boat-shaped", how we can affect the shape of the domains, what governs the speed of growth of the domains and how zig-zag defects affect domain growth.
This project aims to model this system as a set of differential equations in order to answer these questions. The theory of liquid crystals that has been developed in the Maths Department will be used as well as state-of-the-art numerical techniques developed by the Numerical Analysis group.
This work involves numerically solving nonlinear ordinary and partial differential equations governing the fluid flow, molecular orientation together with Maxwell's equations governing the electric field throughout the LCD and the behaviour of light through the display.
This PhD project will provide a postgraduate student with extensive training in model building, problem solving, fluid dynamics and a number of numerical techniques.
Adaptive Meshless Methods Using Moving Nodes
Reference no.: OD/JAM1
Supervisors:
Date advertised: 6th February 2010
Please contact Dr Oleg Davydov for further information.
This project will look at the use of moving node techniques to enhance the accuracy of meshless methods to solve numerically partial differential equations (PDEs).
Meshless methods are a new class of techniques that have the potential to revolutionise computational science and engineering.
The main attractive property of these methods is that they do not need a computational mesh but use a distribution of nodes within the spatial domain.
Mesh generation is often seen as the most time consuming aspect of the simulation of real-world problems. However, for most interesting problems the nodes in a meshless method need to be placed intelligently to avoid under-sampling of important physical features and for time-dependent problems these nodes also need to move.
Here, theoretical a priori and a posteriori error bounds should be used as guidance. In particular, methods based on radial basis functions are very promising because these functions provide an effective tool for the approximation on arbitrary nodes without the need to generate a mesh.
The interest in meshless methods is rapidly growing internationally among both engineers and numerical analysts.
Dr John Mackenzie has considerable experience in the analysis and development of moving mesh methods to solve PDEs. He has also recently supervised four PhD students in this area of research.
For time dependent problems moving mesh methods have been shown to be successful at solving a wide range of problems including melting and solidification problems, the simulation of liquid crystals, pattern formation in mathematical biology and the evaluation of options in mathematical finance.
For some movies go to http://www.maths.strath.ac.uk/~caas61/movies.html
Dr Oleg Davydov has worked on the approximation properties of radial basis functions and in the analysis of finite element methods.
His recent work includes the development and implementation of efficient algorithms for scattered data fitting based on local approximation with radial basis functions (see http://www.maths.strath.ac.uk/~aas04108/scat_data.html), as well as theoretical error bounds for the interpolation with radial basis functions.
This project would suit a student with an interest in the computational solution of PDEs and numerical approximation.
There would be plenty of scope within the project for the student to concentrate on analysis and/or computational aspects.
For the latter, confidence with MATLAB programming is expected.
It is envisaged that the developed algorithms will be applied to a realistic problem in the area of mathematical finance, mathematical biology or computational mechanics.