Derivatives for Time-Spectral Computational Fluid Dynamics Using an Automatic Differentiation Adjoint

In computational fluid dynamics, for problems with periodic flow solutions, the computational cost of spectral methods is significantly lower than that of full, unsteady computations. As is the case for regular steady-flow problems, there are various interesting periodic problems, such as those involving helicopter rotor blades, wind turbines, or oscillating wings, that can be analyzed with spectral methods. When conducting gradient-based numerical optimization for these types of problems, efficient sensitivity analysis is essential. The authors developed an accurate and efficient sensitivity analysis for time-spectral computational fluid dynamics. By combining the cost advantage of the spectral-solution methodology with an efficient gradient computation, the total cost of optimizing periodic unsteady problems can be significantly reduced. The efficient gradient computation takes the form of an automatic differentiation discrete adjoint method, which combines the efficiency of an adjoint method with the accuracy and rapid implementation of automatic differentiation. To demonstrate the method, the authors computed sensitivities for an oscillating ONERA M6 wing. The sensitivities are shown to be accurate to 8-12 digits, and the computational cost of the adjoint computations is shown to scale well up to problems of more than 41 million state variables.