Transformation techniques in optimal design problems with application to harbour shapes
In this thesis the previously unsolved problem of finding the shape of a harbour which minimises the wave height inside is considered. This is a problem of optimal shape design in which the wave height variation is given by a partial differential equation defined over the harbour region and the optimisation is performed with respect to the shape of the boundary of this same region. A brief review of methods that have been used in the past to solve similar problems is given. It is noted that here, in contrast to most previous approaches which have been iterative, the solution of the partial differential equation and the domain optimisation proceed simultaneously. A simple two-dimensional model of a harbour, with part of the boundary unknown, is formulated in Cartesian co-ordinates. A co-ordinate transformation, in terms of an unknown parameter, is selected which maps all admissible harbour regions onto a semi-circular domain with unit radius. Using the method of Lagrange the shape optimisation problem is then formulated as a nonlinear variational problem over a known, fixed, domain. The equivalence of the transformed and original problems is demonstrated by noting that both lead to the same set of necessary conditions for a solution. The finite element method is applied to the optimisation problem, leading to a set of algebraic equations which are solved using an implementation of the Marquardt Algorithm. The convergence of this method, under certain conditions, is demonstrated. Results for the optimal harbour shape from these calculations are presented. Improvements in the smoothness of the solution with increasing mesh resolution and higher order finite elements are noted. In addition, all solutions show similar general features. Hence it is concluded that the problem has been successfully solved.