The thesis starts with a short introduction to smooth dynamical systems and Hamiltonian dynamical systems. The aim of the introductory chapter is to collect basic results and concepts used in the thesis to make it self–contained. The second chapter of the thesis deals with the interaction of two charges moving in R2 in a magnetic field B. This problem can be formulated as a Hamiltonian system with four degrees of freedom. Assuming that the magnetic field is uniform and the interaction potential has rotational symmetry we reduce this Hamiltonian system to one with two degrees of freedom; for certain values of the conserved quantities and choices of parameters, we obtain an integrable system. Furthermore, when the interaction potential is of Coulomb type, we prove that, for suitable regime of parameters, there are invariant subsets on which this system contains a suspension of a subshift of finite type. This implies non–integrability for this system with a Coulomb type interaction. Explicit knowledge of the reconstruction map and a dynamical analysis of the reduced Hamiltonian systems are the tools we use in order to give a description for the various types of dynamical behaviours in this system: from periodic to quasiperiodic and chaotic orbits, from bounded to unbounded motion. In the third chapter of the thesis we study the interaction of two charges moving in R3 in a magnetic field B. This problem can also be formulated as a Hamiltonian system, but one with six degrees of freedom. We keep the assumption that the magnetic field is uniform and the interaction potential has rotational symmetry and reduce this Hamiltonian system to one with three degrees of freedom; for certain values of the conserved quantities and choices of parameters, we obtain a system with two degrees of freedom. Furthermore, when the interaction potential is chosen to be Coulomb we prove the existence of an invariant submanifold where the system can be reduced by a further degree of freedom. The reductions simplify the analysis of some properties of this system: we use the reconstruction map to obtain a classification for the dynamics in terms of boundedness of the motion and the existence of collisions. Moreover, we study the scattering map associated with this system in the limit of widely separated trajectories. In this limit we prove that the norms of the gyroradii of the particles are conserved during an interaction and that the interaction of the two particles is responsible for a rotation of the guiding centres around a fixed centre in the case of two charges whose sum is not zero and a drift of the guiding centres in the case of two charges whose sum is zero.

This thesis focuses on the role played by fluctuations in both thermal optimisation techniques and diffusion limited aggregation. The key idea is that by tuning the level of input noise asymptotic results may be attained more rapidly. Stochastic optimisation problems are considered, where the function being optimised cannot be known exactly and may only be estimated. The noise in the estimates is used as the analogue of thermal fluctuations in simulated annealing. This analogy is made exact by use of an acceptance function, and stochastic annealing is seen to be the generalisation of simulated annealing to stochastic optimisation problems. The probabilistic travelling salesman problem (PTSP) is used as a test-bed for stochastic annealing, and significant new results are found. A good characterisation is found for the PTSP and scaling arguments are shown to be accurate for determining the expected length of the pruned and a priori tours, specifically E(Lpruned)=βpruned(p)√np. An oil field project, as a complex commercial problem, is considered and stochastic annealing is seen to make a large improvement in the expected return of the project. Noise reduction in diffusion limited aggregation (DLA) is known to be crucial to our understanding. A generalisation of noise reduction off-lattice is introduced, and noise reduction is shown to be a central parameter controlling the growth of DLA. In 2 dimensions, all quantities appear to be influenced by the slowest correction to scaling, with exponent ~ 0.3. In 3 dimensions, some quantities are not affected by the slowest correction to scaling, exponent ~ 0.2. The renormalisation of DLA is considered, and the noise reduction at the fixed point is measured. The noise, given as the relative variability in extremal cluster radius is found to be ε*2D ≃ 0.003 and ε*3D ≃ 0.006 in 2 and 3 dimensions, respectively.

There are many branches of engineering science that require solution of fluid flow problems. Some of these examples are aerodynamics of aircraft and vehicles, hydrodynamics of ships, electrical and electronic engineering and many others. Some of these flows may involve complex geometrical shapes which are usually modelled using the unstructured mesh discretisation techniques. There are well established methods that are used in such simulations. The aim of this project is to investigate the staggered positioning of variables on an unstructured based context and hence compare it to well known methods such as the cell-centred approach. A two dimensional unstructured staggered mesh discretisation method for the solution of fluid flow and heat transfer problems has been developed. This method stores and solves the vector variables at the cell faces and other scalar variables are stored at the cell centres. The very well known pressure based scheme SIMPLE is employed for pressure and velocity coupling. Three different approaches on unstructured staggered meshes are proposed. The first method solves for normal velocity component and interpolates the tangential velocity component, the second method solves for normal and tangential velocity components whereas the third method also solves for normal and tangential velocity components but uses a different upwind scheme for convection. The discretisation on unstructured staggered mesh methods is validated for a variety of fluid flow and heat transfer problems and comparisons are made between unstructured staggered mesh methods, the cell-centred approach and benchmark solutions. The first and third unstructured staggered mesh methods are shown to perform well and give comparable results to benchmark solutions. The third unstructured staggered mesh method does not always work.

The research work reported herein addresses the problem of mathematical modelling of fluid flow and combustion in internal combustion engines. In particular, the investigation of three topics that constitute prime sources of uncertainty, in current numerical models, namely turbulence modelling, inaccuracies in the solution procedure specific to moving grids, and combustion modelling. Two and three-dimensional computations of the in-cylinder turbulent flow in a diesel engine are described first, with emphasis on the modifications made to the standard k- model of turbulence to account for rapid compression/expansion, and on the k-W model also used in the computations. It is concluded that the standard k- model may lead to poor predictions when used for internal combustion engine simulations, and that the modified model leads to more reasonable length-scale distributions, improving significantly the overall agreement of velocity predictions with experiment. It is also demonstrated that the k-W model provides better turbulence predictions than the unmodified k- model for the cases considered. The moving boundary within a reciprocating engine poses the problem that as it moves toward the cylinder head it compresses the computational grid cells, creating large aspect ratios that can adversely affect the numerical accuracy and convergence. A conservative scheme has therefore been devised that allows for the removal or addition of grid cells during the simulation, so as to maintain reasonable aspect ratios. It is concluded that with the proposed scheme convergence is obtained within fewer iterations, computational cost is therefore reduced, and that the results are generally in better agreement with experimental data. The third part of this study investigates and compares the performance of the two most commonly used combustion models (the eddy-break-up and the Arrhenius models) and proposes a new formulation of a flame-front model. Calculations have been performed for a one-dimensional test case and for a representative spark-ignition engine in order to determine the grid and time step requirements for numerical accuracy, the sensitivity of results to empirical input and the physical realism of the predictions by comparison with experimental data. It has been found for the cases considered that neither the eddy-break-up nor the Arrhenius models are appropriate for predicting engine combustion. The Arrhenius model does not represent well the combustion process for the cases considered. The eddy-break-up model is not capable of predicting the observed flame front, and the empirical constants in the model require extensive tuning to obtain predictions that match experiments. The flame-front model however, in spite of many simplifications, produces much more realistic flame-front propagation and the empirical input of the model, i.e. the flame speed, can in principle be obtained by other means other than ad-hoc tuning. It is concluded that the flame-front model requires refinement, but for the cases considered, it provides the basis of a very promising combustion model for predicting premixed combustion in engines.

Thesis (M. Phil.)--University of Hong Kong, 2008. / Also available in print.

In this dissertation we consider some boundary value and stability problems for electro-active soft rubberlike materials which withstand finite deformations elastically. In the beginning we consider in detail the problem of finite deformation of a pressurized electroelastic circular cylindrical tube with closed ends with compliant electrodes at its curved boundaries. Expressions for the dependence of the pressure and reduced axial load on the deformation and a potential difference between the electrodes, or uniform surface charge distributions, are obtained in respect of a general isotropic electroelastic energy function. To illustrate the behaviour of the tube specific forms of energy functions accounting for different mechanical properties coupled with a deformation independent quadratic dependence on the electric field are used for numerical purposes, for a given potential difference and separately for a given charge distribution. Numerical dependences of the non-dimensional pressure and reduced axial load on the deformation are obtained for the considered energy functions. Results are then given for the thin-walled approximation as a limiting case of a thick-walled cylindrical tube without restriction on the energy function. The theory provides a general basis for the detailed analysis of the electroelastic response of tubular dielectric elastomer actuators, which is illustrated for a fixed axial load in the absence of internal pressure and fixed internal pressure in the absence of an applied axial load. Using the theory of small incremental electroelastic deformations superimposed on an electroelastic finitely deformed body, we then look for solutions of underlying configurations which are different from perfect cylindrical shape of the tube. First, we consider prismatic bifurcations. We obtain the solutions which show that for neo-Hookean electroelastic material prismatic modes of bifurcation become possible under inflation. This result is different from the pure mechanical case considered previously in Haughton and Ogden (1979), because in Haughton and Ogden (1979) prismatic bifurcation modes were found only for an externally pressurised tube. Second, we consider axisymmetric bifurcations, and we obtain results for neo-Hookean and Mooney-Rivlin electroelastic energy functions. Our solutions show that in the presence of an electric field the electroelastic tube become more unstable: axisymmetric bifurcations become possible at lower values of circumferential stretches as compared with the values of circumferential stretches found for analogous problems solved for electromechanically indifferent materials, or equivalently, when electric field is not present. Within similar lines we consider the bifurcation of a thick-walled electroelastic spherical shell with compliant electrodes at its curved boundaries under internal and external pressure. The solutions obtained for neo-Hookean electroelastic energy function show that in some cases axisymmetric modes of bifurcation become possible under inflation in the presence of electric field.

The problem of aerodynamic noise is considered following the Computational Aeroacoustics approach which is based on direct numerical simulation of the sound field. In the region of sound generation, the unsteady airflow is computed separately from the sound using Computational Fluid Dynamics (CFD) codes. Overlapping this region and extending further away is the acoustic domain where the linearised Euler equations governing the sound propagation in moving medium are solved numerically. After considering a finite volume technique of improved accuracy, preference is given to an optimised higher order finite difference scheme which is validated against analytical solutions of the governing equations. A coupling technique of two different CFD codes with the acoustic solver is demonstrated to capture the mechanism of sound generation by vortices hitting solid objects in the flow. Sub-grid turbulence and its effect 011sound generation has not been considered in this thesis. The contribution made to the knowledge of Computational Aeroacoustics can be summarised in the following: 1) Extending the order of accuracy of the staggered leap-frog method for the linearised Euler equations in both finite volume and finite difference formulations; 2) Heuristically determined optimal coefficients for the staggered dispersion relation preserving scheme; 3) A solution procedure for the linearised Euler equations involving mirroring at solid boundaries which combines the flexibility of the finite volume method with the higher accuracy of the finite difference schemes; 4) A method for identifying the sound sources in the CFD solution at solid walls and an expansion technique for sound sources inside the flow; 5) Better understanding of the three-level structure of the motions in air: mean flow, flow perturbations, and acoustic waves. It can be used, together with detailed simulation results, in the search for ways of reducing the aerodynamic noise generated by propellers, jets, wind turbines, tunnel exits, and wind-streamed buildings.

In this thesis we examine the properties of Cosmic Strings in the theory of General Relativity. We begin by considering static Cosmic Strings in flat space-time. We derive the field equations for the Cosmic String and show that the solution depends upon a single scaling parameter a which is constructed from the physical constants. Using this result we construct 1-parameter families of solutions which depend on an auxiliary parameter e and which describe the thin-string limit of a Cosmic String. By interpreting these solutions as elements of the simplified Colombeau algebra we may interpret the relativistic energy density Too of the thin string as an element of the Colombeau algebra with delta-function mass-per-unit-length. Furthermore, for a critically coupled Cosmic String the energymomentum tensor in the thin-string limit may be given a distributional interpretation. We also solve the string equations numerically for various values of a. This is done by compactifying the space-time to include infinity as part of the numerical grid and then using a relaxation method to suppress exponentially growing un-physical solutions. In curved space-time we derive the equations for the scalar and vector fields which are now coupled to the geometric variables through Einstein's equations. We again examine the thin-string limit in the Colombeau algebra by considering a 1-parameter family of solutions. W'e derive an expression for the deficit angle in terms of the distributional energy-momentum tensor of the thin string. We use this result to investigate the gravitational lensing properties of the string and relate this to the deficit angle. In the special case of a cone we find the scattering angle is equal to the deficit angle. We also solve the coupled equations numerically using techniques similar to those used in flat space-time. The second part of the thesis involves the dynamics of Cosmic Strings. Einstein's equations then lead to wave equations for both the matter and metric variables. However, the space-time is not asymptotically flat and this leads to problems in applying the appropriate boundary conditions. By using a Geroch transformation it is possible to reformulate the equations in terms of geometrical variables defined on an asymptotically flat (2+l)-dimensional space-time. Three exact vacuum solutions describing gravitational radiation due to Weber-Wheeler, Xanthopoulos and Piran et al. are used to excite the string which is found to oscillate with frequencies which are proportional to the masses of the scalar and vector fields of the string. This is in agreement with the exact results obtained using the linearised equations of the thin dynamic string. The behaviour of the dynamic string is studied by solving the equations numerically using an implicit fully characteristic scheme. The use of the Geroch transformation allows us to compactify the space-time and include null infinity as part of the numerical grid. This enables us to use the correct boundary conditions at infinity and hence suppress un-physical divergent solutions. The code is tested by comparing the results with exact solutions, by checking that it agrees with the static code and by undertaking a time dependent convergence test. The code is found to be accurate, stable and exhibit clear second order convergence.

The purpose of this thesis is to give a fully gauge-natural formulation of gravitation theory, which turns out to be essential for a correct geometrical formulation of the coupling between gravity and spinor fields. In Chapter 1 we recall the necessary background material from differential geometry and introduce the fundamental notion of a gauge-natural bundle. Chapter 2 is devoted to expounding the general theory of Lie derivatives, its specialization to the gauge-natural context and, in particular, to spinor structures. In Chapter 3 we describe the geometric approach to the calculus of variations and the theory of conserved quantities. Then, in Chapter 4 we give our gauge-natural formulation of the Einstein (-Cartan) -Dirac theory and, on applying the formalism developed in the previous chapter, derive a new gravitational superpotential, which exhibits an unexpected freedom of a functorial origin. Finally, in Chapter 5 we complete the picture by presenting the Hamiltonian counterpart of the Lagrangian formalism developed in Chapter 3, and proposing a multisymplectic derivation of bi-instantaneous dynamics. Appendices supplement the core of the thesis by providing the reader with useful background information, which would nevertheless disrupt the main development of the work. Appendix A is devoted to a concise account of categories and functors. In Appendix B we review some fundamental notions on vector fields and flows, and prove a simple, but useful, proposition. In Appendix C we collect the basic results that we need on Lie groups, Lie algebras and Lie group actions on manifolds. Finally, Appendix D consists of a short introduction to Clifford algebras and spinors.

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