Airfoil design via optimal control theory

In airfoil design, one problem of great interest is to find the target airfoil profile to achieve a given target velocity distribution. It can be formulated as an optimal control problem, with the control being the airfoil profile and the governing equation being the full potential equation in the transonic regime. To discretize the problem, one approach is to employ the finite element method. In the discretized space, a direct relationship between the objective function and the unknown profile co-ordinates can be defined via the finite element basis functions. Moreover, it is advantageous to derive the gradient in the discretized space rather than the continuous space to avoid contamination by discretization errors. In this paper, this approach is studied. In particular, a new formulation is proposed. A novel decomposition of the discrete space for the potential function, the gradient is derived and an efficient algorithm using the quasi-Newton method is described. In generating and adjusting the mesh during iterations, the elliptic mesh generation technique is used.