Discontinuous Galerkin discretised level set methods with applications to topology optimisation

This thesis presents research concerning level set methods discretised using discontinuous Galerkin (DG) methods. Whilst the context of this work is level set based topology optimisation, the main outcomes of the research concern advancements which are agnostic of application. The first of these outcomes are the development of two novel DG discretised PDE based level set reinitialisation techniques, the so called Elliptic and Parabolic reinitialisation methods, which are shown through experiment to be robust and satisfy theoretical optimal rates of convergence. A novel Runge-Kutta DG discretisation of a simplified level set evolution equation is presented which is shown through experiment to be high-order accurate for smooth problems (optimal error estimates do not yet exist in the literature based on the knowledge of the author). Narrow band level set methods are investigated, and a novel method for extending the level set function outside of the narrow band, based on the proposed Elliptic Reinitialisation method, is presented. Finally, a novel hp-adaptive scheme is developed for the DG discretised level set method driven by the degree with which the level set function can locally satisfy the Eikonal equation defining the level set reinitialisation problem. These component parts are thus combined to form a proposed DG discretised level set methodology, the efficacy of which is evaluated through the solution of numerous example problems. The thesis is concluded with a brief exploration of the proposed method for a minimum compliance design problem.