Thin-walled structures are often used as kinetic energy absorbers in vehicular systems and infrastructure designs. In such applications, high specific energy absorption is usually desirable, because it is beneficial for weight reduction. The dimpling cold-roll metal forming process introduces dimpled geometry and increases the strength of sheet metal. This thesis aims to investigate the energy absorption characteristics of the dimpled thin-walled structures. A finite element (FE) modelling analysis was performed using ANSYS Explicit Dynamics solver, to predict the response of dimpled structures to dynamic and quasi-static loads. A series of experimental tests were conducted and the FE method was validated through comparing the numerical and experimental results. To understand the response of the dimpled structural components to axial crushing loads, numerical simulations were performed. A parametric study on a key cold-roll forming parameter “forming depth” was carried out to evaluate its effects on the dimpled geometry and material properties. Through the parametric study, manufacturing parameters for the cold-roll forming process were suggested to improve yield strength and energy absorption performance of dimpled steel components. It was shown that the specific energy absorption can be increased by up to 16% after optimizing the forming depth. To take the most advantage of the dimpled geometry, multi-layer dimpled thin-walled columns were analysed. The interlocking mechanism of dimpled plates were investigated and an empirical model was proposed to describe the interaction between dimpled plates. It was shown that a considerable amount of energy can be absorbed through the interaction between dimpled walls. The behaviour of dimpled columns under lateral impact loads was also investigated. It was revealed that the introduced dimpled geometry contributes to reducing the peak impact force without sacrificing the energy absorption capacity. However, this is only valid when at least one end of the dimpled thin-walled column is fully restrained.

This thesis aims to analyse a finite element method applied to an adjusted Cahn-Hilliard equation that has been used for digital image inpainting applications. We consider both the standard model with a smooth double well potential and an alternative where an obstacle potential has been used. Existence and uniqueness results are derived for both formulations by adapting techniques existing in literature for other problems. For each formulation we then propose approximations, by discretising first in space and then in time, and we derive error bounds between the weak solution of the original formulation and the solution of the discrete approximations in terms of the discretisation parameters. We then propose and implement a practical numerical scheme for both models and investigate their use in applications, alongside some other models from literature. We investigate various real digital image examples and compare the resulting inpaintings for these competing models, considering their suitability for real-world applications.