An aerodynamic design optimization method is presented that generates an airfoil, producing a specified surface pressure distribution at a transonic speed. The design procedure is based on the coupled Euler and boundary-layer technology to include the rotational viscous physics which characterizes transonic flows. A leastsquare optimization technique is used to minimize pressure discrepancies between the target and designed airfoils. The method is demonstrated with several examples at transonic speeds. The design optimization process converges quickly, that makes the method attractive for practical engineering applications. I. Introduction I N recent years, computational fluid dynamics (CFD) has become a valuable engineering tool in the aircraft industry. CFD plays a complementary role, not a replacement, to experiments in practical design communities. Rubbert1 showed some good examples of the use of CFD and experiment, in combination, for transonic design. A major strength of CFD is the ability to produce detailed insights into complex flow phenomena. The process of decomposition and parameterization can help identify the cause of weak aerodynamic performance, and the microscopic understanding of the flow can lead to improved design. Continuing advances in computer hardware and simulation techniques provide an unprecedented opportunity for CFD. Now simulations of more complete configurations with more complex physics can be performed at an affordable cost. Accuracy and reliability of the computation have been continuously improved. The use of high-level flow models and large-size refined grids enables one to analyze flows with complicated structures and various length scales. Compared to the remarkable advances in analysis capability, however, relatively few advances have been made in design technology. Conventional design practices, therefore, often depend on analysis methods through iterative cut-and-try approaches. A unique advantage of CFD is the capability of inverse design. Inverse design directly determines the airfoil geometry that produces the pressure distribution specified by a designer. Many existing inverse design methods are based on the potential flow assumption due to its simplicity. Volpe and Melnik2 employed an inverse design method using the nonlinear full potential formulation. Bauer and colleagues3 used the hodograph method that solves the full potential equation in the hodograph plane where the equations are linear. The potential flow model, however, cannot properly represent transonic features such as embedded shock waves and shock-boundarylayer interactions. An accurate analytic capability is a prerequisite for a successful design, because the quality of the design depends on the quality of the method used to predict the flowfield. Several inverse design methods were demonstrated using the Euler formulations by Giles and Drela,4 and Mani.5 Instead of achieving the prescribed pressure distribution, some design methods use a constrained optimization process