  An important step in shape optimization with partial differential equation
constraints is to adapt the geometry during each optimization iteration. Common
strategies are to employ mesh-deformation or re-meshing, where one or the other
typically lacks robustness or is computationally expensive. This paper proposes
a different approach, in which the computational domain is represented by
multiple, independent meshes. A Nitsche based finite element method is used to
weakly enforce continuity over the non-matching mesh interfaces. The
optimization is preformed using an iterative gradient method, in which the
shape-sensitivities are obtained by employing the Hadamard formulas and the
adjoint approach. An optimize-then-discretize approach is chosen due to its
independence of the FEM framework. Since the individual meshes may be moved
freely, re-meshing or mesh deformations can be entirely avoided in cases where
the geometry changes consists of rigid motions or scaling. By this free
movement, we obtain robust and computational cheap mesh adaptation for
optimization problems even for large domain changes. For general geometry
changes, the method can be combined with mesh-deformation or re-meshing
techniques to reduce the amount of deformation required. We demonstrate the
capabilities of the method on several examples, including the optimal placement
of heat emitting wires in a cable to minimize the chance of overheating, the
drag minimization in Stokes flow, and the orientation of 25 objects in a Stokes
flow.
