Unsteady Discrete Adjoint Formulation for Two-Dimensional Flow Problems with Deforming Meshes

Abstract

A method to apply the discrete adjoint for computing sensitivity derivatives in two-dimensional unsteady flow problems is presented. The approach is to first develop a forward or tangent linearization of the nonlinear flow problem in which each individual component building up the complete flow solution is differentiated against the design variables using the chain rule. The reverse or adjoint linearization is then constructed by transposing and reversing the order of multiplication of the forward problem. The developed algorithm is very general in that it applies directly to the arbitrary Lagrangian-Eulerian form of the governing equations and includes the effect of deforming meshes in unsteady flows. It is shown that an unsteady adjoint formulation is essentially a single backward integration in time and that the cost of constructing the final sensitivity vector is close to that of solving the unsteady flow problem integrated forward in time. It is also shown that the unsteady adjoint formulation can be applied to time-integration schemes of different orders of accuracy with minimal changes to the base formulation. The developed technique is then applied to three optimization examples, the first in which the shape of a pitching airfoil is morphed to match a target time-dependent load profile, the second in which the shape is optimized to match a target time-dependent pressure profile, and the last in which the time-dependent drag profile is minimized without any loss in lift.