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 opti-mization. The objective is to study the complex-ity of the discretization of the adjoint equation for both the continuous and discrete approach, the ac-curacy of the resulting estimate of the gradient, and its impact on the computational cost to approach an optimum solution. First, this paper presents com-plete formulations and discretizations of the Navier-Stokes equations, the continuous viscous adjoint equation and its counterpart the discrete viscous adjoint equation. The differences between the con-tinuous 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 complex-step gradients. Third, the adjoint equations and its corresponding boundary conditions are formu-lated to quantify the influence of geometry modi-fications on the pressure distribution at an arbitrary remote location within the domain of interest. Fi-nally, applications are presented for inverse, pressure and skin friction drag minimization, and sonic boom minimization problems