A time parallelised adjoint-based optimization strategy applied to incompressible flow configurations

Adjoint-based methods are widely used in various areas of fluid mechanics as a cost effective way of evaluating gradient information in order to perform sensitivity analysis, control, data assimilation, etc. When applied to unsteady configurations, however, the calculation of the direct-adjoint loop requires the use of checkpointing algorithms, causing the optimisation procedure to become time consuming and in some cases even infeasible. One common approach to overcome this problem is using parallelisation. While parallelisation in space has been widely used to reduce the cost of CFD calculations, time parallelisation is less explored. This is to a great extend due to unpredictable convergence rate of existing time parallel methods when applied to highly nonlinear and unsteady complex flow regimes. The linear nature of the adjoint equations, however, make them suitable for such implementations. In this study, we introduce a parallel in time algorithm designed to speed up the integration of the adjoint equations and ultimately the optimisation procedure. The code used to perform the calculations is a two-dimensional incompressible Navier-Stokes solver with immersed boundaries capabilities. Various control strategies are considered from drag reduction around a shedding cylinder to reducing pressure loss across a blade. In all cases, the parallel in time strategy allows the extraction of the gradient at a fraction of a time compared to the time used in the direct adjoint loop.