Seminar in Numerical Analysis: Edwin Mai (Universität der Bundeswehr München)
With an increasing range of applications, Shape Optimisation problems receive more and more interest in the engineering community, while solving such problems is still a demanding task. In this talk the example of a Stokes channel flow with the objective to reduce the energy dissipation is considered, on which an optimise-then-discretize approach shall be applied. Starting with a gradient descent method, based on the analytical shape derivative and the adjoint approach, an initial optimisation procedure is discussed and differences in the shape derivative representation and their numerical implications are highlighted. Subsequently a possible way to derive shape hessian information in a so-called tangent-on-reverse method, i.e. combining the adjoint and sensitivity approach, is introduced. The shape hessian is utilised in a reduced SQP method for the equally constrained channel flow problem comprising of the objective, PDE and additional geometric constraints. In contrary to a one-shot approach the reduced approach requires the state and adjoint equations to be solved exactly for each optimisation step. Finally, some features of the numerical implementation using the finite element software package FEniCS and the obtained results are presented to show superiority of using hessian information.
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