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| EPSRC Reference: |
EP/D058791/1 |
| Title: |
Scale Invariant Moving Mesh Finite Elements for Multidimensional Nonlinear Partial Differential Equations |
| Principal Investigator: |
Dr M Hubbard |
| Other Investigators: |
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| Researcher Co-investigator: |
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| Project Partner: |
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| Department: |
Sch of Computing |
| Organisation: |
University of Leeds |
| Scheme: |
Mathematical Sciences Small Grants |
| Starts: |
01 December 2005 |
Ends: |
30 November 2008 |
Value (£): |
5,284
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| EPSRC Research Topic Classifications: |
| Non-linear Systems Mathematics |
Numerical Analysis |
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| EPSRC Industrial Sector Classifications: |
| No relevance to Underpinning Sectors |
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| Related Grants: |
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| Panel History: |
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Summary |
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 these processes by approximating them on a uniform mesh of points on domains with fixed boundaries. However, many situations (consider a spreading droplet, for example) naturally suggest a domain which evolves with the flow, while others (say the movement of a shock wave up and down an aeroplane wing) have their main focus of interest in following the motion of a sharp internal feature. For accuracy and efficiency a computational method should dictate the movement of the mesh accordingly.
This project aims to enhance and analyse a new computational method developed recently by the applicants to incorporate these ideals, which has already shown the ability to model complex situations. Furthermore, the method has been designed so that the approximation preserves inherent properties (such as conservation principles and invariances) of the mathematical model being used to predict the fluid flow. The long term aim is a method which can reliably and accurately predict a range of fluid flows involving moving boundaries or internal features, from the spreading of droplets, through waves breaking on a beach, to chemical explosions.
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| Final Report Summary |
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 these processes by approximating them on a uniform mesh of points on domains with fixed boundaries. However, many situations (consider a spreading droplet, for example) naturally suggest a domain which evolves with the flow, while others (say the movement of a shock wave up and down an aeroplane wing) have their main focus of interest in following the motion of a sharp internal feature. For accuracy and efficiency a computational method should dictate the movement of the mesh accordingly.
This research built on an algorithm developed by the applicants which had previously been demonstrated to accurately predict boundary movement as well the internal evolution of the solution in a range of simple situations. The major outcomes of this project were (1) the application of the algorithm to problems involving a change of phase (from solid to liquid, for example) and the tracking of an internal, moving interface between the phases, and (2) the development of a technique for applying boundary conditions within the computational algorithm while retaining an underlying conservation principle (conservation of mass, for example). The algorithm has also been applied successfully to a range of other problems, including blow-up for reaction-diffusion equations, the swelling of grains, tidal bores using the shallow water equations, and waiting times for nonlinear diffusion.
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| Further Information: |
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| Organisation Website: |
http://www.leeds.ac.uk |
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