An Efficient Method for Aerodynamic Shape Optimization

We present simultaneous pseudo-timestepping as an efficient method for aerodynamic shape optimization. In this method, instead of solving the necessary optimality conditions by iterative techniques, pseudo-time embedded nonstationary system is integrated in time until a steady state is reached. The main advantages of this method are that it requires no additional globalization techniques and that a preconditioner can be used for convergence acceleration which stems from the reduced SQP method. The important issue of this method is the trade-off between the accuracy of the forward and adjoint solver and its impact on the computational cost to approach an optimum solution is addressed. The method is applied to a test case of drag reduction for an RAE2822 airfoil, keeping it’s thickness constant. The optimum overall cost of computation that is achieved in this method is less than 4 times that of the forward simulation run. Nomenclature (x, y) ∈ R :cartesian coordinates H :total enthalpy (ξ, η) ∈ [0, 1] :generalized coordinates M :Mach number Ω :flow field domain )∞ :values at free stream ∂Ω :flow field boundary γ :ratio of specific heats ~n := ( nx ny ) :unit outward normal Cref :chord length α :angle of attack CD :drag coefficient ρ :density I :cost unction u :x-component of velocity w :vector of state variables v :y-component of velocity q :vector of design variables p :pressure λ :vector of adjoint variables E :total energy J :Jacobian Cp :pressure coefficient B :reduced Hessian