Initial Implementation of an Adjoint CFD Code for Aeroshell Shape Optimization

Application of computational fluid dynamics to the optimization of aeroshell shapes usually entails high computational cost. Many converged solutions are required to generate gradients and optimize a shape with respect to very few design variables. The benefits of high-fidelity aerodynamic analysis can be reaped early in the design cycle with less computational cost if the traditional direct optimization problem is transformed to an indirect optimization, using optimal control theory. The indirect gradient formulation decouples the effects on the objective function of the design variables and the flow solution. Meaning, all derivatives used to compute the gradient can be generated from a single converged flow solution. Involved in the computation of the gradient is the solution of an adjoint system of PDEs. An incremental approach is developed for the implementation of an adjoint equation solver. The phased approach begins using inexact and computationally costly finite difference derivative calculations. Results are presented for a transonic airfoil and a supersonic wedge to demonstrate that the finite difference gradient is reasonably accurate, providing a meaningful validation as exact numerical derivatives are substituted later in the development cycle. Finally, a roadmap is presented for future implementation of indirect optimization capability for the Euler/Navier-Stokes CFD code, NASCART-GT. Nomenclature AD = automatic differentiation α = vector of design variables C d = drag coefficient CFD = computational fluid dynamics C l = lift coefficient E = energy state γ = ratio of specific heats GMRES = generalized minimal residual J = cost function M = computational mesh vector m = number of input variables MDO = multidisciplinary optimization N = number of design variables n = number of output variables υ = vector of adjoint variables NASCART-GT = Numerical Aerodynamic Simulation via Cartesian Grid Techniques p = pressure PDEs = partial differential equations q = vector of flow state variables R = vector of discretized residuals ρ = density state T = wetted surface triangulation vector u x , u y = Cartesian velocity components x, y = Cartesian coordinates