Shape optimization of a breakwater

In this paper, we optimize the shape of a breakwater which protects a harbour basin from incoming waves. More specifically, our objective is reducing the harbour resonance due to long-range ocean waves. We consider the complex-valued Helmholtz equation as our model state equation and minimize the average wave height in the harbour basin with the shape of the breakwater as optimization variable. The geometry is described by the level set method, i.e. the domain is given as the subzero level set of a function. In contrast to many publications we use the volume expression of the shape derivative, which lends itself naturally to a level set update via a transport equation. The model problem features intrinsic geometric constraints which we treat in the form of forbidden regions. We guarantee feasibility of the iterates by projecting the gradient onto a suitable admissible set.