Optimum Transonic Wing Design Using Control Theory

1. Introduction While aerodynamic prediction methods based CFD are now well established , and quite accurate and robust, the ultimate need in the design process is to find the optimum shape which maximizes the aerodynamic performance. One way to approach this objective is to view it as a control problem, in which the wing is treated as a device which controls the flow to produce lift with minimum drag, while meeting other requirements such as low structure weight, sufficient fuel volume, and stability and control constrains. Here we apply the theory of optimal control of systems governed by partial differential equations with boundary control, in this case through changing the shape of the boundary. Using this theory, we can find the Frechet derivative (infinitely dimensional gradient) of the cost function with respect to the shape by solving an adjoint problem, and then we can make an improvement by making a modification in a descent direction. For example, the cost function might be the drag coefficient at a fixed lift, or the lift to drag ratio. During the last decade, this method has been intensively developed, and has proved to be very effective for improving wing section shapes for fixed wing planform [3, 4, 8–11, 13, 14]. In the present work a continuous adjoint formulation has been used to derive the adjoint system of equations, in which the adjoint equations are derived directly from the governing equations and then discretized. This approach has the advantage over the discrete adjoint formulation in that the resulting adjoint equations are independent of the form of discretized flow equations. The ad-joint system of equations has a similar form to the governing equations of the flow, and hence the numerical methods developed for the flow equations [1, 2, 5] can be reused for the adjoint equations. Moreover, the gradient can be derived directly from the adjoint solution and the surface motion, independent of the mesh modification.