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This thesis investigates the solution of equations of the form G(Lambda,chi)=0, G: R chi X > X, where X is a real Banach space. Such equations are often called non-linear eigenvalue problems. If (Lambda,chi) is a solution for which Gx(Lambda,chi) is invertible, there are well-known existence and uniqueness results for solutions near (Lambda,chi), which are easily made constructive. However, in this thesis, we are interested in solutions (Lambda,chi) for which Gx(Lambda,chi) is not invertible, and specifically in the case when Gx(Lambda,chi) has only a l-dimensional null-space. Our approach is to apply the Newton-Kantorovich theorem, first to determine these so-called singular points, and second to compute nearby solutions. In the former case we modify the equations to avoid singular systems, and in the latter case we obtain accurate starting values which compensate for the near-singularity of Gx(Lambda,chi). Hence the advantages of a quadratically convergent method are retained. Chapter 1 contains a brief, general introduction to non-linear eigenvalue problems in which we distinguish between the two important types of singular point, turning points and bifurcation points. In Chapter 2 we consider the common case G-(Lambda,O) = O for arbitrary Lambda. Thus (Lambda,o) is a solution, the singular points (Lambda,O) can be computed by standard methods, and the major difficulty is to determine solutions (Lambda,chi), near (Lambda,O), with non-zero chi-component. Turning points are the subject of Chapter 3, where we show how to compute both the points themselves and the solutions beyond them. In Chapter 4 we consider the problem of bifurcation points in a similar way and, finally, the stability of such points under small perturbations is discussed in Chapter 5.

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Model reduction by balanced truncation for bounded real and positive real input-stateoutput systems, known as bounded real balanced truncation and positive real balanced truncation respectively, is addressed. Results for finite-dimensional systems were established in the mid to late 1980s and we consider two extensions of this work. Firstly, using a more behavioral framework we consider the notion of a finite-dimensional dissipative system, of which bounded real and positive real input-state-output systems are particular instances. Specifically, we work in a framework where we make no a priori distinction between inputs and outputs. We derive model reduction by dissipative balanced truncation, where a gap metric error bound is obtained, and demonstrate that the aforementioned bounded real and positive real balanced truncation can be seen as special cases. In the second part we generalise bounded real and positive real balanced truncation to classes of bounded real and positive real systems respectively that have non-rational transfer functions, so called infinite-dimensional systems. Here we work in the context of well-posed linear systems. We derive approximate transfer functions, which we prove are rational and preserve the relevant dissipativity property. We also obtain error bounds for the difference of the original transfer function and its reduced order transfer function, in the H-infinity norm and gap metric for the bounded real and positive real cases respectively. This extension to bounded real and positive real balanced truncation requires new results for Lyapunov balanced truncation in the infinite dimensional case, which we also describe. We conclude by highlighting possible future research.

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The specific contribution of this thesis is to present a detailed FE modelling approach to simulate the stiffness behaviour 3D fibre reinforced composites and facilitate progression to failure using commercially available software and existing computational capabilities. The composite constituents including the fibre, interface and resin regions are modelled with Tetrahedral, Pyramid and Hexahedral finite elements respectively. An analysis detailing the how geometric details such as inter tow spacing contribute to a models predictive performance is formed with increasing resin thicknesses reducing the elastic modulus. It is demonstrated that an elliptical tow shape is preferable for modelling 2D woven composites such as the plain weave and a racetrack tow shape is preferable for modelling 3D woven composites, with a 10% error between predicted and manufactured modulus values. The implications of modelling on the micro level are established, with the tow sub division into filaments shown to reduce model stiffness. Further geometrical considerations along with numerical factors such as the trade off between reducing the element number and increasing the accuracy of the geometry representation are also detailed. A large catalogue of 3D fibre reinforced composite structures exists but their behaviour under in-plane loading conditions is not fully understood. As the manufacturing technology associated with production of 3D woven structures becomes more sophisticated, the types of 3D reinforcing architectures that can be produced will increase. The need therefore exists for the capability to predict the mechanical performance of fibre reinforced composite materials.

This research presents a new method, the decoupled overlapping grids method, for the numerical modelling of transient pressure and rate properties of oil wells. The method is implemented in two stages: a global stage solved in the entire domain with a point or line source well approximation, and a local (post-process) stage solved in the near-well region with the well modelled explicitly and boundary data interpolated from the global stage results. We have carried out simulation studies in two- and three- dimensions to investigate the accuracy of the method. For homogeneous case studies in 2D, we have demonstrated the convergence rate of the maximum error in the quantities of interest of the global and local stage computations by numerical and theoretical means. We also proposed a guideline for the selection of the relative mesh sizes of the local and global simulations based on error trends. Comparison to other methods in the literature showed better performance of the decoupled overlapping grids method in all cases. We carried out further investigations for heterogeneous case studies in 2D and partially-penetrating wells in 3D which show that the error trends observed for the 2D homogeneous case deteriorate only slightly, and that a high level of accuracy is achieved. Overall the results in this thesis demonstrate the potential of the method of decoupled overlapping grids to accurately model transient wellbore properties for arbitrary well con gurations and reservoir heterogeneity, and the gain in computational e ciency achieved from the method.