Studies of the Continuous and Discrete Adjoint Approaches to Viscous Automatic Aerodynamic Shape Optimization

This paper compares the continuous and discrete viscous adjoint-based automatic aerodynamic optimization. The objective is to study the complexity of the discretization of the adjoint equation for both the continuous and discrete approach, the accuracy of the resulting estimate of the gradient, and its impact on the computational cost to approach an optimum solution. First, this paper presents complete formulations and discretizations of the NavierStokes equations, the continuous viscous adjoint equation and its counterpart the discrete viscous adjoint equation. The differences between the continuous and discrete boundary conditions are also explored. Second, the accuracy of the sensitivity derivatives obtained from continuous and discrete adjoint-based equations are compared to complexstep gradients. Third, the adjoint equations and its corresponding boundary conditions are formulated to quantify the influence of geometry modifications on the pressure distribution at an arbitrary remote location within the domain of interest. Finally, applications are presented for inverse, pressure and skin friction drag minimization, and sonic boom minimization problems.