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

•Derived adjoint equations for time-periodically constrained fully discrete PDE.•Introduced shooting method to solve periodic adjoint equations (2pt BVP).•Introduced adjoint method for computing gradients on manifold of periodic solutions.•PDE optimization method introduced for optimizing on manifold of periodic solutions•Energetically optimal flapping motion for foil in compressible, viscous flow found. A variety of shooting methods for computing fully discrete time-periodic solutions of partial differential equations, including Newton–Krylov and optimization-based methods, are discussed and used to determine the periodic, compressible, viscous flow around a 2D flapping airfoil. The Newton–Krylov method uses matrix-free GMRES to solve the linear systems of equations that arise in the nonlinear iterations, with matrix-vector products computed via the linearized sensitivity evolution equations. The adjoint method is used to compute gradients for the gradient-based optimization shooting methods. The Newton–Krylov method is shown to exhibit superior convergence to the optimal solution for these fluid problems, and fully leverages quality starting data. 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.