The inverse problem being addressed is the construction of an aerodynamic airfoil shape. The objective of the present paper is: (i) to formulate the inverse problem such that the mathematical problem is well-posed both theoretically and numerically; (ii) to formulate the adjoint problem such that an efficient solution procedure can be constructed; (iii) to demonstrate how the solution procedure works. The inverse problem is posed as a minimization problem of an objective functional. The minimum of the functional corresponds to the attainment of a target velocity distribution by the aerodynamic airfoil shape. The design variables being considered are geometric parameters of the airfoil and an appropriately defined set of target velocity parameters which are introduced to assure the well-posedness of the problem. The minimization problem is solved by an optimization algorithm. Adjoint method is employed for an efficient computation of the objective functional gradient with respect to the geometric parameters. Numerical results demonstrate that the present inverse problem formulation is well-posed. The significance of the target velocity parameters in obtaining well-posedness is explained in terms of the Lighthill constraints.
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