Contribution to Problems T4 and T6 Finite Element GMRES and Conjugate Gradient Solvers

A GMRES solver based on the estimation of the gradient of the objective function gradient algorithm are used for the reconstruction and optimization of airfoils represented by cubic splines and operating at subsonic and transonic conditions. It is shown that the number of iterations has been multiplied by three between the subsonic and the transonic case. For the optimization problem, the drag objective function is penalized to take into account the lift constraint. Two descent methods, namely a fixed step gradient and a conjugate gradient algorithm are investigated. Results are compared with a GMRES optimizer for the solution of the optimality conditions using the calculus of variations with adjoint equations. These methods have been developped in close cooperation with O. Pironneau and A. Vossinis at INRIA Rocquencourt.