A fully discrete adjoint method for optimization of flow problems on deforming domains with time-periodicity constraints

The central contribution of this work is the derivation of the adjoint equations and the corresponding adjoint method for fully discrete, time-periodically constrained partial differential equations. These adjoint equations constitute a linear, two-point boundary value problem that is provably solvable. The periodic adjoint method is used to compute gradients of quantities of interest along the manifold of time-periodic solutions of the discrete partial differential equation, which is verified against a second-order finite difference approximation. These gradients are then used in a gradient-based optimization framework to determine the energetically optimal flapping motion of a 2D airfoil in compressible, viscous flow over a single cycle, such that the time-averaged thrust is identically zero. In less than 20 optimization iterations, the flapping energy was reduced nearly an order of magnitude and the thrust constraint satisfied to 5 digits of accuracy.