Application of domain decomposition methods to problems in topology optimisation

摘要

Determination of the optimal layout of structures can be seen in everyday life, from nature to industry, with research dating back to the eighteenth century. The focus of this thesis involves investigation into the relatively modern field of topology optimisation, where the aim is to determine both the optimal shape and topology of structures. However, the inherent large-scale nature means that even problems defined using a relatively coarse finite element discretisation can be computationally demanding. udThis thesis aims to describe alternative approaches allowing for the practical use of topology optimisation on a large scale. Commonly used solution methods will be compared and scrutinised, with observations used in the application of a novel substructuring domain decomposition method for the subsequent large-scale linear systems. Numerical and analytical investigations involving the governing equations of linear elasticity will lead to the development of three different algorithms for compliance minimisation problems in topology optimisation. Each algorithm will involve an appropriate preconditioning strategy incorporating a matrix representation of a discrete interpolation norm, with numerical results indicating mesh independent performance.