  We consider the method of mappings for performing shape optimization for
unsteady fluid-structure interaction (FSI) problems. In this work, we focus on
the numerical implementation. We model the optimization problem such that it
takes several theoretical results into account, such as regularity requirements
on the transformations and a differential geometrical point of view on the
manifold of shapes. Moreover, we discretize the problem such that we can
compute exact discrete gradients. This allows for the use of general purpose
optimization solvers. We focus on an FSI benchmark problem to validate our
numerical implementation. The method is used to optimize parts of the outer
boundary and the interface. The numerical simulations build on FEniCS,
dolfin-adjoint and IPOPT. Moreover, as an additional theoretical result, we
show that for a linear special case the adjoint attains the same structure as
the forward problem but reverses the temporal flow of information.
