Shape optimisation and robust solvers for incompressible flow

When designing a new car or a plane, engineers need to solve the Navier-Stokes equations to understand how air flows around the object. Based on experience and intuition, they modify the design slightly, then solve the equations again, and inspect the changes. This process is iterated many times until a final design that minimises or maximises some quantity of interest, such as drag or lift, is found. The goal of shape optimisation is to automate this type of process. In this thesis we address several issues related to shape optimisation. Focussing on the case when the shape is discretised using a mesh and when PDE constraints are solved using the finite element method, we describe a reformulation of the shape derivative as the derivative of the pushforward from the reference element. This viewpoint allows for automated calculation of shape derivatives in finite element software. When shape optimisation is performed by deforming an initial mesh, the choice of deformation is important. We propose a new Hilbert space structure on the space of deformations that results in high mesh quality of the deformed domains. We then focus on the solution of a particular PDE constraint given by the steady, incompressible Navier–Stokes equations that govern laminar flow. The solution of these equations becomes challenging for large Reynolds number. We develop augmented Lagrangian based preconditioners that exhibit robust performance as the Reynolds number is increased. The effectiveness and scalability of the developed solvers is demonstrated for a range of test problems.