We have investigated aerodynamic design problems via control theory for viscous and inviscid compressible flows. The present procedure solves an analysis problem to evaluate the cost function and an adjoint problem to evaluate its gradient, then uses these information in an optimization algorithm to get optimal shapes. Discrete adjoint approach was used because of its simplicity and flexibility.; Study of nozzle design problem using quasi one dimensional Euler equations revealed the effect of cost function and grid sensitivity. In the presence of shock waves, spikes in the gradient are observed. Two methods are presented to fix this problem.; The smoothness of the cost function is an important factor to get a high quality optimized shape. The superiority of mass flux cost function compared to pressure cost function is demonstrated using Euler and Cauchy/Riemann equations.; Approximate gradients based on physical and hierarchical models are formulated. For inviscid flow, analysis problem based on Euler equations and adjoint problem based on Cauchy/Riemann equations are combined to get an approximate gradient. In a design cycle a non-monotone behavior in cost function is observed, but it can be removed by building entropy variation into the cost function. For viscous flow, analysis problem based on Navier-Stokes equations and adjoint problem based on Cauchy/Riemann and Euler equations are combined together. In order to circumvent ill-posedness, an equivalent flow field is introduced. These two combinations give satisfactory results for inverse design problem with and without shock waves. Hence, approximate gradients offer useful and efficient engineering design strategies for industrial applications.
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