Local-in-time adjoint-based method for design optimization of unsteady flows

We develop a new local-in-time adjoint-based method for minimization of ow matching functionals subject to the 2-D unsteady compressible Euler equations. The new method-ology is aimed at circumventing the memory requirements associated with conventional time-dependent adjoint-based optimization methods that require the ow solutions to be available at all time levels. These storage requirements quickly become prohibitive for large-scale applications. The key idea of the local-in-time method is to divide the entire time interval into several subintervals and approximate the solution of the global-in-time adjoint equations and the global sensitivity derivative as the combination of the corre-sponding local quantities computed on each subinterval. Since each subinterval contains relatively few time steps, the storage cost of the local-in-time method is much lower than that of the global adjoint formulation, thus making the time-dependent optimization feasi-ble for practical applications. The paper presents a detailed comparison of the local- and global-in-time adjoint-based methods for design optimization of unsteady subsonic and su-personic ows around a bump. Our numerical results show that the local-in-time method converges to the optimal solution obtained with the global counterpart, while drastically reducing the memory cost as compared to the global-in-time adjoint formulation. I. Discrete Design Optimization Problem We consider time-dependent optimization problems that include design optimization of unsteady com-pressible ows governed by the 2-D Euler equations. The governing equations are discretized by using a node-centered nite-volume scheme,1 where solution values are stored at the mesh nodes. Inviscid uxes at cell interfaces are computed using the upwind scheme of Roe.2 The discretized Euler equations including the boundary conditions can be written in the following form