The Unsteady Continuous Adjoint Method Assisted by the Proper Generalized Decomposition Method

In adjoint-based optimization for unsteady flows, the adjoint PDEs must be integrated backwards in time and, thus, the primal field solution should be available at each and every time-step. There are several ways to overcome the storage of the entire unsteady flow field which becomes prohibitive in large scale simulations. The most widely used technique is checkpointing that provides the adjoint solver with the exact primal field by storing the computed primal solution at a small number of time-steps and recomputing it for all other time-steps. Alternatively, approximations to the primal solution time-series can be built and used. One of them relies upon the use of the Proper Generalized Decomposition (PGD), as a means to approximate the time-series of the primal solution for use during the unsteady adjoint solver and this is where this paper is focusing on. The original contribution of this paper it that, apart from the standard PGD method, an incremental variant, running simultaneously with the time integration of unsteady primal equation(s) is proposed and tested. For the purpose of demonstration, three optimization problems based on different physical problems (unsteady heat conduction and unsteady flows around stationary and pitching isolated airfoils) are worked out by implementing the continuous adjoint method to both of them. The proposed incremental PGD technique is generic and can be used in any problem, to support either continuous or discrete unsteady adjoint.