Development and application of TLNS3D-adjoint - A practical tool for aerodynamic shape optimization

An efficient, versatile, robust, and easy-to-use adjoint code for aerodynamic shape optimization has been developed. Derivation of the adjoint of the Euler equations, its discretization and implementation in the optimization process are presented. The primary goal of this effort was to develop capability to efficiently perform supersonic/transonic multipoint optimization of a complete High Speed Civil Transport (HSCT) configuration for minimum thrust at specified lift and trim conditions. The results of optimizing an HSCT wing/body configuration at supersonic and transonic cruise points are shown. Also presented are results of a supersonic/transonic multipoint thrust minimization for trimmed flight where, in addition to shape design variables, deflections of multiple control surfaces are also included as design parameters. The optimizations were performed in the AEROdynamic SHape Optimization (AEROSHOP) environment using the NPSOL optimizer. The performance characteristics of the adjoint code, namely, the rate of convergence, accuracy of sensitivities, and computational overhead at supersonic as well as transonic Mach numbers are also discussed. Introduction Designing air vehicles that are economically viable and environmentally friendly is a challenge facing aerospace engineers today. While it is generally recognized that a multidisciplinary design optimization is the key to meeting this challenge, aerodynamic performance improvement alone can significantly contribute to reaching this goal. During the preliminary design phase of an aircraft development, the aerodynamic shape is defined to the point where detailed performance estimates can be made and guaranteed to potential customers. Sometimes the * Principal Engineer. Senior Member AIAA. f Senior Engineer. Senior Member AIAA. * Technical Fellow. Associate Fellow AIAA. Copyright © 2001 by The Boeing Company. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. aerodynamic shape is frozen at the end of this design phase, making further improvements to aerodynamic performance difficult. Traditionally, the aerodynamic shape is defined using a combination of analytical tools and wind-tunnel testing. The analytical tools are often based on linear methods with corrections applied for nonlinear effects. In this approach a sequence of design iterations, each requiring many months of analysis and wind-tunnel testing, are required to finalize the design. During the last two decades, improvements in computer technology, along with progress in Computational Fluid Dynamics (CFD) have substantially reduced the cost and time required to perform nonlinear aerodynamic analysis of complete air vehicles. In the last decade, advances in sensitivity analysis methods, particularly the adjoint method, have significantly decreased the cost and time necessary to optimize aerodynamic shapes using CFD. Making shape optimization practical to use in an industrial design environment can eliminate multiple design iterations and therefore shorten the overall design cycle time and cut cost considerably. It could also become a necessary tool for designing air vehicles that meet the challenging requirements of the market and the environment. Under the NASA/Industry High-Speed Research (HSR) Phase II program (1995-99), several parallel efforts were supported for developing and demonstrating practical aerodynamic shape optimization methods. This paper presents one such technology, which was developed and successfully used for shape optimization of HSCT configurations. In aerodynamic shape optimization, the outer mold line (OML) of the air vehicle is changed, using a set of design variables, such that a selected cost function (liftto-drag ratio for example) reaches a desired optimum. Among the different optimization methods, the gradient-based method is widely used for shape optimization. In this method, an optimizer uses the sensitivity of the cost function to changes in design variables to search for a better design. Here, the key to finding an improved design is computing the sensitivities accurately. In order for this method to be 1 American Institute of Aeronautics and Astronautics (c)2001 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization. practical in the preliminary design phase of an air vehicle development, sensitivities of hundreds of design variables have to be computed efficiently. Among the methods for computing sensitivities, the finitedifference method is the easiest one to implement. However, the accuracy of the sensitivities depends on proper choice of the step-size and level of convergence of the function analysis. Since the number of function analyses required to compute sensitivities is proportional to the number of design variables, this method is also computationally expensive and often impractical when hundreds of design variables are used. Alternatively, the need to choose a step-size, for each design variable, can be eliminated by using the sensitivity equations method. Here, a system of sensitivity equations is solved for each design variable. The sensitivity equations form a linear system. However, for large three-dimensional problems, the cost of solving them could become as high as that of a function evaluation. Although this method provides accurate sensitivities, the computing cost is still proportional to the number of design variables. A method that offers both accuracy and efficiency is the adjoint method. Here, the governing equations of the flow (state equations) along with its adjoint (co-state equations) are solved once, to obtain the sensitivities of all the design variables. The adjoint equations also form a linear system. Again, like the sensitivity equations, the cost of solving these equations could become as high as that of a function evaluation, especially for large three-dimensional problems. However, since only one adjoint solution is required per cost function, the cost of computing sensitivities is independent of the number of design variables. Pironneau' derived an adjoint method for the minimum-drag problem in Stokes flow and for incompressible Navier-Stokes equations. Jameson developed an adjoint method for wing design using Euler equations. The derivations of adjoint equations for potential, Euler, and Navier-Stokes equations are well documented in literature'''. In this paper first, the derivation of the adjoint equations for the Euler equations and its boundary conditions for aerodynamic problems is presented. Next, its discretization and implementation in the optimization framework is discussed. Finally, the performance characteristics of the adjoint code along with selected optimization results are shown. The Adjoint Method Consider an optimization problem, where the objective is to find the shape of a wing such that its drag is a minimum at a specified lift, subject to the governing equations of the flow. Let the wing be perturbed as follows: