Shape optimization of a breakwater

Abstract

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.

Self-consistent gradient flow for shape optimization

“…The choice of the concrete function space influences the resulting descent method, since it can encode information about desired smoothing and even things like geometric constraints. We refer to [ 15 ] for a comparison of various spaces. In the following, we always choose , as this space performs well in practice.…”

Geometric constraints in descent methods for shape optimisation

“…Thus, if we choose some Hilbert space H for the speed fields, we can find the Riesz representative of the shape derivative and use it as speed field. See [1] for a general functional-analytic treatment of various spaces in this context, and [8] for a numerical comparison of a few selected spaces. For our purposes, the choice H = H 1 (D) seems to perform quite well in practice.…”

“…To do so, there are, again, two possible strategies we want to discuss: The most natural way is to project the speed field in the same Hilbert space H that is also used for the computation of the shape gradient with (3). This is done in [8], based, in particular, on Theorem 3 there. Projection in H ensures that the constraints are incorporated into the descent method in a consistent way.…”

“…It is straight-forward to modify our results to constraints of the form B ⊂ Ω instead. See [8] for a practical problem that gives rise to both types of constraints in a natural way. Section 5 of [8] also briefly discusses methods to deal with the constraints, as does Section 2.13 of [7].…”

“…See [8] for a practical problem that gives rise to both types of constraints in a natural way. Section 5 of [8] also briefly discusses methods to deal with the constraints, as does Section 2.13 of [7]. In this work, we want to expand this discussion and give more details, particularly about the underlying theoretical considerations.…”