Solution of the Unsteady Discrete Adjoint for Three-Dimensional Problems on Dynamically Deforming Unstructured Meshes

The formulation and solution of the adjoint problem for unsteady flow simulations using the Reynolds-averaged Navier-Stokes equations in the presence of dynamically deforming unstructured meshes is demonstrated. A discrete adjoint approach is used, and the full linearization is built up in a systematic and modular fashion. Discrete conservation in the analysis problem is ensured through the geometric conservation law, which is linearized consistently for the adjoint problem. An agglomeration multigrid scheme is used to solve the time-implicit problem at each time step for both the analysis problem, the adjoint problem, and the mesh and mesh adjoint problems. The methodology is demonstrated through a simple time-dependent pitching wing optimization problem.