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.

No description available.

The 4-manifold S1 x S3, when endowed with the structure of a certain complex Hopf surface, is an example of a principal elliptic fibration. We use this structure to study the moduli spaces of anti-self-dual connections (instantons) on SU(2) bundles over S1 x S3. Chapter 1 is introductory. We define Buchdahl's notion of stability and outline the correspondence between instantons and stable holomorphic SL(2,C) bundles over S1 x S3. In Chapter 2 we study holomorphic line and SL(2, C) bundles over a general principal elliptic surface using an extension of the ‘graph’ invariant introduced by Braam and Hurtubise. We prove some auxiliary results needed in later chapters and introduce a stratification of the moduli space. In Chapter 3 we construct elements of one of the strata using the ‘Serre construction’ of algebraic geometry and deduce a structure result for the charge 1 case. Chapter 4 applies the results of the previous chapters in the construction of monopoles on the solid torus with a hyperbolic metric. We recover easily a result of Braam and Hurtubise. In Chapter 5 we adapt a construction of Friedman to describe a method of construction for elements of the remaining strata of the moduli spaces over the Hopf surface. In the charge 1 case we again determine the diffeomorphism type of the stratum completely. Combined with the results of Chapter 3 we deduce the natural action of S1 x S3 on the charge 1 moduli space is free. In Chapter 6 we study the charge 1 instanton moduli spaces over secondary Hopf surfaces diffeomorphic to the product of S1 and a Lens space. Chapter 7 considers twistorial methods and their application in the construction of explicit solutions. We define an invariant of an instanton, the spectral surface, which is a 2-dimensional analogue of Hitchin’s spectral curve. We use it to deduce that methods of Atiyah and Ward fail to generate a full family of charge 1 solutions. Finally we show how the spectral surface can be used in a sheaf theoretic construction of the ‘missing’ solutions.

The present thesis considers the space of connections modulo based gauge equivalence on a principal SU(2) bundle over a closed simply-connected smooth four-dimensional manifold M. Up to homotopy equivalence, this is the space of basepoint-preserving maps from M to BSU('2), the classifying space of SU(2). It depends only on the homotopy type of M which is characterized by the intersection form. The Z/pZ-homology of the mapping space for p a prime not equal to 3 is computed and given in terms of the data associated to the intersection form. For the prime 3, partial results are obtained. The main method is to consider a fibration associated to a CW decomposition of M and to show that the corresponding Eilenberg- Moore spectral sequence collapses. These results generalize from manifolds to spaces homotopy equivalent to a bouquet of 2-spheres with a single 4-cell attached. For the possible homotopy types the space of connections modulo gauge equivalence ran attain, a classification is obtained in the following sense. The homotopy type of this space is uniquely determined by the rank, type and signature modulo eight of the intersection form. On the other hand, the homotopy type determines the rank, type and signature modulo four of the intersection form. Both results together give a complete classification for the case of spin manifolds. The homotopy types of the spaces of connections modulo gauge equivalence over two spin manifolds agree if and only if the intersection forms are of the same rank. These results use a classification of unimodular bilinear forms over the ring Z/4Z. In a final part, a map is constructed from the labelled configuration spaces of points in the manifold to the mapping space. This map is shown to be asymptotically surjective in homology with Z/2Z-coefficients. For homology with general coefficients, classes are constructed which are not approximated by this map.

Monodromy fields on I3 are a family of lattice fields in two dimensions which are a natural generalisation of the two dimensional Ising field occurring in the C*-algebra approach to Statistical Mechanics. A criterion for the critical limit one point correlation of the monodromy field tra(M) at a 6 l3, Um(#.(M)). is deduced for matrices M € GL(p, C) having non-negative eigenvalues. Using this criterion a non-identity 2x2 matrix is found with a finite critical limit one point correlation. The general set of p x p matrices with finite critical limit one point correlations is also considered and a conjecture for the critical limit n point correlations postulated. The boson-fermion correspondence for the representation of the CAR algebra over L3(Sl, C) defined by the (t,B) KMS state with chemical potential p is considered and the non-bijectivity shown. Using an alternative formulation the correlations are recalculated leading to a determinant identity reminiscent of Saego’s Theorem.

This work attempts to utilise perturbation theory to derive discrete mappings which describe the dynamical behaviour of a continuous, and a discrete, chaotic system. The first three chapters introduce some background to the theory of chaotic behaviour In discrete and continuous systems. Chapter 4 considers the dynamical behaviour of Duffings equation. Perturbation theory is applied to Hamiltonian solutions of the system, and a 1-D mapping is derived which models the bifurcation of the system to chaos. Chapter 5 introduces a 2-D chaotic difference map. The qualitative dynamics of the system are investigated and a form of perturbation theory is applied to a parameterised version of the map. The perturbative solutions are shown to exhibit dynamical behaviour very like the original system.

The objective of this research is to develop a design framework for virtual prototyping of electronic packaging. This framework couples computational mechanics and fluid dynamics, based on finite volume method with integrated finite element routines, with numerical optimisation and statistical methods. This integrated approach is intended as a modelling tool for calculating optimal design solutions for electronic packaging and component assembly with a focus on die reliability and the thermal management. The motivation is to introduce numerical optimisation theory as an approach for a fast, systematic and automated design approach for wide range microelectronics applications. The proposed methodology will also benefit from multi-physics numerical analysis to predict complex behaviour of electronic packages, systems and processes subject to different operational or environmental conditions. This thesis demonstrates multi-physics modelling (i.e. integrated solutions for fluid flow, heat transfer and stress) coupled with gradient/non-gradient based numerical optimisation techniques and associated statistical methods. An explanation and comparison of the two approaches to numerical optimisation — (1) Response Surface Methodology (RSM) based on Design of Experiments (DoE) and (2) direct gradient based and non-gradient methods - are given. Both the advantages and limitations of these virtual design strategies, with respect to their integration with multi-physics modelling, are discussed and demonstrated. This integrated multiphysics/optimisation design approach is demonstrated on a variety of problems from the area of microelectronics design and packaging. The thesis demonstrates this for three industrial examples. These are: The software packages used to develop the design tool and to undertake the outlined studies are PHYSICA and VisualDOC. PHYSICA is a multiphysics finite volume based simulation tool with integrated modules for finite element solid mechanics analysis. The software framework is detailed in Chapter 2, Section 2.4 and further in Chapter 4. The VisualDOC tool offers a collection of numerical optimisation routines and modules for statistical analysis (Design of Experiments) and approximate Response Surface modelling. VisualDOC framework is discussed in Chapter 4, Section 4.8.