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A discrete adjoint method is developed and demonstrated for aerodynamic design optimization on unstructured grids. The governing equations are the three-dimensional Reynolds-averaged Navier-Stokes equations coupled with a one-equation turbulence model. A discussion of the numerical implementation of the flow and adjoint equations is presented. Both compressible and incompressible solvers are differentiated and the accuracy of the sensitivity derivatives is verified by comparing with gradients obtained using finite differences. Several simplifying approximations to the complete linearization of the residual are also presented, and the resulting accuracy of the derivatives is examined. Demonstration optimizations for both compressible and incompressible flows are given. *Graduate Student, Aerospace and Ocean Engineering Department, Student Member AIAA. †Senior Research Scientist, Aerodynamic and Acoustic Methods Branch, Fluid Mechanics and Acoustics Division, Assoc. Fellow, AIAA. Copyright © 1998 by Eric J. Nielsen. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. Introduction As computational power has continued to advance in recent years, researchers have been able to extend the use of computational tools to increasingly more complex problems. Computational fluid dynamics (CFD) has been exploited as an analysis tool for some time, and is currently receiving attention as a design optimization tool. Early attempts in CFD-based design problems made use of finite-difference calculations to obtain sensitivity information. This technique can be used to obtain the derivatives of all the flow quantities with respect to each design variable and can be easily retrofitted to existing flow solvers. One problem with this approach is the computational time required. To obtain the design sensitivities for a system involving design parameters using a central-difference approach requires well-converged solutions of flow analysis problems. For complex cases with many design variables, this requirement may become prohibitive. Another problem often encountered with the finite-difference approach is the sensitivity of the derivatives to the choice of the step size. It is desirable to have a small step size so that the truncation error is minimal, while at the same time, avoid exceedingly small step sizes which could yield large cancellation errors. To mitigate the difficulties associated with the choice of step size used in finite differences, direct differentiation can be employed.14, 19, 25 In this approach, the sensitivity derivatives of all the variables in the flow field are obtained but the solution of a large linear system of equations for each design variable is required. Therefore, for problems involving many design variables, obtaining the sensitivity derivatives can be expensive. In recent years, adjoint formulations have grown in popularity, and are rapidly being developed for use in aerodynamic design sensitivity computations (see e.g. Refs. 2, 4, 9, 10, 15, 18). The adjoint approach has the advantage of being able to compute cost function gradients at an expense independent of the number of design parameters. This feature makes adjoint methods extremely attractive for problems involving a large number of design variables. n 2n In the current work, a discrete adjoint approach is used in an unstructured-grid framework to compute design sensitivities for problems based on the Navier-Stokes equations. A one-equation turbulence model is used which is fully differentiated and coupled into the solution of the adjoint equations so that the resulting sensitivity derivatives are consistent with those obtained using finite differences. In addition to a compressible solver, an incompressible formulation is also differentiated in order to accommodate a wide range of applications. The accuracy of the resulting derivatives is established and sample calculations are shown for twoand threedimensional cases using both implementations. Conclusions and suggestions for future research are also given.