Optimised Mathematical Models for Multi-Physical Systems
The rhythmic beating of tiny whip-like filaments attached to swimming micro-organisms, the crashing of ocean waves onto offshore oil platforms and the manufacturing of self cleaning products are all examples of phenomena which involve many different interacting natural forces. These forces act on the interfaces where fluid and solid or elastic materials meet whether it involves very small scales such as found in cells within the human body or the very large scales involved in man made structures. It has become clear that such multi-physics, multi-phase and multi-scale systems play an important role in the overall health of our technological civilisation especially in the face of environmental concerns.
One way to further understand the physical basis of such problems is through the construction of a mathematical model of such multi-physical systems. While gathering data through experiments is important, some experiments may be too difficult or expensive to carry out. On the other hand computer simulations of a properly constructed mathematical model can be both economic and avoid experimental difficulties while still capturing the vital characteristics of the problem. One of the most remarkable aspects of mathematics and science is that it is possible to accurately model such complex physical phenomena and, that the same physical principles may be applied in the physical, engineering and natural sciences.
Currently, the various approaches to the solution of these complex but vital multi-physics problems remain somewhat ad hoc since they fail to answer two fundamental questions:
· Firstly, how can one simultaneously incorporate within the computer model the action of physical forces
which act on very small scales with those acting on very large ones?
· Secondly, what is the best way to implement the various physical variables in the model so that the
results make physical and mathematical sense?
My own research has been involved with the computer solution of mathematical models involving the flow of multiple interacting fluids, for example, the splash of a raindrop. The model is designed to be used for any kind of physical problem which involves multi-phase systems and can also incorporate multiple physical forces, which may arise from very different scales within the problem, such as gravity or the surface forces that hold together a spherical water droplet for example. The solution on a computer requires special care for which so-called numerical methods have been developed so that the machine is able to carry out the solution of the mathematical problem in a systematic way by following an algorithm. My own research makes use of two ways of doing this, one maintains the value of physical variables, e.g. velocity and pressure, on a fixed grid and the other allows a set of particles, e.g. representing the density, to move freely throughout the grid. The grid itself is useful for accurately capturing model aspects involving the diffusion of physical information whereas the particles can best capture how the interface between the two fluids can be deformed or disrupted. While this work has successfully modelled such multi-phase, multi-physics phenomena more research is needed to establish this approach as one of the best available.
Usually such multi-scale forces are modelled by incorporating the fine scale effects within the higher scales but this induces a loss of information due to the coarsening of the fine scale information and no longer properly represents the fine scale behaviour. One of the advantages of my own hybrid grid-particle approach is that it can incorporate the effects of the unresolvable fine scales within the resolvable coarser scale through the sub-grid resolution available in the particle component of the method. However, a detailed understanding of this approach is still required including how best to implement it and how the grid and particle components communicate with each other. It is the aim of my research to find the best or optimal way to do this through an in depth study of such multi-physics models followed by the optimisation of this process in a precise way while maintaining model accuracy and efficiency.
Such multi-physical problems abound in every area of engineering and science whether it involves chemical reactions taking place on the interface between suspended droplets of chemical reactants in industry or the bubbling of argon gas in molten metals to slow the solidification process. A fuller understanding of such problems can lead to the development of new computer software which can better simulate such problems without a need to carry out dangerous experiments. Simulations can then contribute to the development of new ways to develop commercial products or processes for use in industry. (see proposal_1.pdf)
One way to further understand the physical basis of such problems is through the construction of a mathematical model of such multi-physical systems. While gathering data through experiments is important, some experiments may be too difficult or expensive to carry out. On the other hand computer simulations of a properly constructed mathematical model can be both economic and avoid experimental difficulties while still capturing the vital characteristics of the problem. One of the most remarkable aspects of mathematics and science is that it is possible to accurately model such complex physical phenomena and, that the same physical principles may be applied in the physical, engineering and natural sciences.
Currently, the various approaches to the solution of these complex but vital multi-physics problems remain somewhat ad hoc since they fail to answer two fundamental questions:
· Firstly, how can one simultaneously incorporate within the computer model the action of physical forces
which act on very small scales with those acting on very large ones?
· Secondly, what is the best way to implement the various physical variables in the model so that the
results make physical and mathematical sense?
My own research has been involved with the computer solution of mathematical models involving the flow of multiple interacting fluids, for example, the splash of a raindrop. The model is designed to be used for any kind of physical problem which involves multi-phase systems and can also incorporate multiple physical forces, which may arise from very different scales within the problem, such as gravity or the surface forces that hold together a spherical water droplet for example. The solution on a computer requires special care for which so-called numerical methods have been developed so that the machine is able to carry out the solution of the mathematical problem in a systematic way by following an algorithm. My own research makes use of two ways of doing this, one maintains the value of physical variables, e.g. velocity and pressure, on a fixed grid and the other allows a set of particles, e.g. representing the density, to move freely throughout the grid. The grid itself is useful for accurately capturing model aspects involving the diffusion of physical information whereas the particles can best capture how the interface between the two fluids can be deformed or disrupted. While this work has successfully modelled such multi-phase, multi-physics phenomena more research is needed to establish this approach as one of the best available.
Usually such multi-scale forces are modelled by incorporating the fine scale effects within the higher scales but this induces a loss of information due to the coarsening of the fine scale information and no longer properly represents the fine scale behaviour. One of the advantages of my own hybrid grid-particle approach is that it can incorporate the effects of the unresolvable fine scales within the resolvable coarser scale through the sub-grid resolution available in the particle component of the method. However, a detailed understanding of this approach is still required including how best to implement it and how the grid and particle components communicate with each other. It is the aim of my research to find the best or optimal way to do this through an in depth study of such multi-physics models followed by the optimisation of this process in a precise way while maintaining model accuracy and efficiency.
Such multi-physical problems abound in every area of engineering and science whether it involves chemical reactions taking place on the interface between suspended droplets of chemical reactants in industry or the bubbling of argon gas in molten metals to slow the solidification process. A fuller understanding of such problems can lead to the development of new computer software which can better simulate such problems without a need to carry out dangerous experiments. Simulations can then contribute to the development of new ways to develop commercial products or processes for use in industry. (see proposal_1.pdf)
proposal_1.pdf | |
File Size: | 44 kb |
File Type: |