Geometry Optimization in Three-Dimensional Unsteady Flow Problems using the Discrete Adjoint

The unsteady discrete adjoint equations are utilized to construct objective function gradients for use in geometry optimization in large scale three-dimensional unsteady ow problems. An existing massively parallel unstructured Reynolds-averaged Navier-Stokes equations solver forms the basis for the framework used to solve both the primal and dual (adjoint) problems. The framework is also capable of handling dynamic mesh deformation using the linear elasticity model. Turbulence modeling options using the one-equation Spalart-Allmaras model and the two-equation k-! model are available. The adjoint equations are formulated such that all aspects of the primal problem are exactly linearized and no assumptions such as freezing certain components are made. This leads to discretely exact gradients which can be used for optimization purposes. The adjoint-based gradients are veri ed against those computed by the complex-step method and agreement to machine precision is observed. An agglomeration multigrid solution strategy is employed to accelerate the convergence of the primal problem while GMRES preconditioned by linear agglomeration multigrid is used to solve the dual problem. The method is applied to minimizing the time-integrated torque coe cient of the HART2 rotor in hover while constraining the time-integrated thrust coe cient to the baseline value.