There exists substantial applications motivating the study of nonlinear longitudinal wave propagation in layered (or laminated) elastic waveguides, in particular within areas related to non-destructive testing, where there is a demand to understand, reinforce, and improve deformation properties of such structures. It has been shown [76] that long longitudinal waves in such structures can be accurately modelled by coupled regularised Boussinesq (cRB) equations, provided the bonding between layers is sufficiently soft. The work in this thesis firstly examines the initial-value problem (IVP) for the system of cRB equations in [76] on the infinite line, for localised or sufficiently rapidly decaying initial conditions. Using asymptotic multiple-scales expansions, a nonsecular weakly nonlinear solution of the IVP is constructed, up to the accuracy of the problem formulation. The asymptotic theory is supported with numerical simulations of the cRB equations. The weakly nonlinear solution for the equivalent IVP for a single regularised Boussinesq equation is then constructed; constituting an extension of the classical d'Alembert's formula for the leading order wave equation. The initial conditions are also extended to allow one to separately specify an O(1) and O(ε) part. Large classes of solutions are derived and several particular examples are explicitly analysed with numerical simulations. The weakly nonlinear solution is then improved by considering the IVP for a single regularised Boussinesq-type equation, in order to further develop the higher order terms in the solution. More specifically, it enables one to now correctly specify the higher order term's time dependence. Numerical simulations of the IVP are compared with several examples to justify the improvement of the solution. Finally an asymptotic procedure is developed to describe the class of radiating solitary wave solutions which exist as solutions to cRB equations under particular regimes of the parameters. The validity of the analytical solution is examined with numerical simulations of the cRB equations. Numerical simulations throughout this work are derived and implemented via developments of several finite difference schemes and pseudo-spectral methods, explained in detail in the appendices.

The mean variance framework (MV) developed by Markowitz in his groundbreaking paper offers a quantitative and rational approach to portfolio selection. It is well known to market practitioners however that the MV optimal portfolios tend to perform subpar. One of the issues of the MV portfolios is that they require the inverse of a large covariance matrix, which is often ill-conditioned. In this thesis, we develop a new approach to circumvent these issues. We propose an optimisation approach akin to least squares linear regression and compare the performance with an establish method, covariance shrinkage. When tested on a set of 30 futures contracts, we find that the models yield promising results albeit somewhat lower than that of the benchmark. / Mean variance ramverket (MV) framtaget av Markowitz i sin banbrytande artikel möjliggör en kvantitativ och rationell metod för portföljallokering. Det är däremot ett väletablerat faktum bland marknadsaktörer att Markowitz-optimala portföljer tenderar att prestera relativt dåligt. Ett av tillkortakommandena av ramverket är den ofta problemtyngda inverteringen av, den ofta stora, kovariansmatrisen som är illa konditionerad. I denna uppsats föreslår vi en ny metod för att kringgå detta problem. Vi föreslår en optimeringsmetodologi mycket lik minsta kvadratmetoden i linjär regression. Denna metod utvärderas sedan mot en vedertagen metod, kovarianskrympning. När vi utvärderar vår modell på 30 stycken terminskontrakt ser vi lovande resultat men finner en Sharpekvot något lägre än referensportföljens.

Le but principal de cette thèse est de développer un modèle numérique pour modéliser les phénomènesde bifurcation et du branchement des fissures. Pour réaliser cet objectif, il était indispensablede posséder un modèle permettant un couplage fort entre le modèle de Plasticité cristalline etcelui de l'Endommagement régularisé. Dans un premier temps, quelques outils de post-traitement ont été développés pour analyser les systèmes de glissement actifs. Ces outils ont été utilisés surdes simulations d'éprouvettes réelles, et comparés à des résultats expérimentaux. Par ces comparaisons, l'application du modèle de Plasticité cristalline aux superalliages monocristallins a été validée. Ce modèle a ensuite été couplé avec le modèle d'endommagement régularisé. Le couplage a été réalisé dans les deux sens, c'est-à-dire que l'évolution de la plasticité a une influence sur l'endommagement et vice-versa. Le nouveau modèle peut être implémenté simplement, avec la méthode traditionnelle des Éléments Finis. Des expériences étudiant la propagation de fissure sous des chargements de types différents ont été simulées à l'aide de ce nouveau modèle :éprouvettes CT,fissuration en Mode II et rupture en fluage. Une méthode pour l'identification des paramètres matériaux a également été proposée. / The main goal of this dissertation was to develop a model to simulate the processes of crack bifurcation and crack branching in anisotropic materials. To achieve this goal, a thorough coupling of crystal plasticity and regularised damage models was deemed necessary. Firstly, post-processing tools were developed to better analyse the results obtained from standard Crystal Plasticity simulations. These were then compared with experiments, thereby validating the use of Crystal Plasticity models for Nickel base single crystal superalloys. The validated Crystalplasticity model was then coupled with a regularised microdamage model such that the evolution of plasticity influenced damage and vice versa. The newly developed model allows for the simulation of cracks using the standard Finite Element Approach. Experiments studying crack propagation under different types of loads were simulated using the newly developed model, including CT, shear andcreep specimens. A methodology was also proposed for the identification of the newly introduced material parameters.