Adjoint-based optimization for flapping wings

Adjoint-based methods show great potential in flow control and optimization of complex problems with high- or infinite-dimensional control space. It is attractive to solve an adjoint problem to understand the complex effects from multiple control parameters to a few performance indicators of the flight of birds or insects. However, the traditional approach to formulate the adjoint problem becomes either impossible or too complex when arbitrary moving boundary (e.g. flapping wings) and its perturbation is considered. Here, we use non-cylindrical calculus to define the perturbation. So that, a simple adjoint system can be derived directly in the inertial coordinate. The approach is first applied to the optimization of cylinder oscillation and later to flapping wings.