Numerical solution of steady free-surface flows by the adjoint optimal shape design method

Numerical solution of flows that are partially bounded by a freely moving boundary is of great importance in practical applications such as ship hydrodynamics. Free-boundary problems can be reformulated into optimal shape design problems, which can in principle be solved efficiently by the adjoint method. In this work we investigate the suitability of the adjoint shape optimization method for solving steady free-surface flows. The asymptotic convergence behaviour of the method is determined for free-surface flows in 2D and 3D. It is shown that the convergence behaviour depends sensitively on the occurrence of critical modes. The convergence behaviour is moreover shown to be mesh-width independent, provided that proper preconditioning is applied. Numerical results are presented for 2D flow over an obstacle in a channel. The observed convergence behaviour is indeed mesh-width independent and conform the derived asymptotic estimates.