Study of Kyrlov methods for topology optimization

Abstract

Topology optimization (TOPOPT) focuses on optimizing the mass distribution of a given design space whilst maintaining certain structural properties. In aerospace engineering, the structural design process is often steered towards the minimization of weight. However, this reduction in mass cannot be achieved at the cost of the structural integrity. Due to the nature of these objectives, the approach to design problems can benefit greatly from the implementation of topology optimization techniques. Within the optimization process, many complex linear systems appear. One of the main current drawbacks of TOPOPT is precisely related to the computational cost of solving these systems. Traditionally, direct methods are used, but these become too slow when the complexity of the system increases. This final thesis is centered around the alternative to direct methods, which are iterative methods. These have the potential of reducing computation times significantly in topology optimization problems. The primary objective of the thesis involves the development of an iterative algorithm for solving sequences of linear systems, and subsequent exhaustive comparisons with direct methods, in terms of computational cost. The particular iterative methods explored are known as Krylov methods. Moreover, different techniques related to the recycling of information from previous systems are employed. The project places significant emphasis on deeply understanding the theory behind these existing mathematical techniques to tailor them into the framework of TOPOPT, a field where these methods are emerging, and provide numerical comparative results in representative examples. To successfully achieve the set goals, an initial thorough research involving the mathematical theory behind iterative and Krylov methods has been conducted. A detailed description of the complete algorithm has been written to facilitate the posterior programming process. A clean object-oriented approach has been carried out for easier maintenance of code. The results underscore the significant decrease in computational cost when using the developed iterative solvers as opposed to direct solvers. Additionally, the comparisons between different possible techniques within the iterative approach demonstrate their effectiveness. The subsequent course of action involves the polishing of the code for greater time reductions, and the exploring of preconditioning techniques to complement the current algorithm.