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This paper review the formulation and application of optimization techniques based on control theory for aerodynamic shape design in both inviscid and viscous compressible flow . The theory is applied to a system defined by the partial differential equations of the flow, with the boundary shape acting as the control. The Frechet derivative of the cost function is determined via the solution of an adjoint partial differential equation, and the boundary shape is then modified in a direction of descent. This process is repeated until an optimum solution is approached. Each design cycle requires the numerical solution of both the flow and the adjoint equations, leading to a computational cost roughly equal to the cost of two flow solutions. Representative results are presented for viscous optimization of transonic wing-body combinations and inviscid optimization of complex configurations.

this paper addresses the validity of this design methodology for the problem of high-lift design. Traditionally, high-lift designs have been realized by careful wind tunnel testing. This approach is both expensive and challenging due to the extremely complex nature of the flow interactions that appear. CFD analyses have recently been incorporated to the high-lift design process

Professor Darmofal and the generous funding provided by NASA Langley (grant number NAG1-03035). Secondly, the effort put into Project X by faculty and students (past and present) have made it possible to carry out the computational demonstrations in higher-order DG. In particular, Krzysztof Fidkowski and Todd Oliver are to be acknowledged for their contributions towards the development of the flow solvers and also for providing some of the grids for the test cases demonstrated. Finally, thanks must go to thesis committee members Professors Peraire and Willcox as well as thesis readers Dr. Natalia Alexandrov and Dr. Steven Allmaras for the time they put into reading the thesis and providing the valuable feedbacks. 3 46 Adjoint approach to shape sensitivity 117 6.1 Introduction...............................

This paper discusses the role that computational fluid dynamics plays in the design of aircraft. An overview of the design process is provided, covering some of the typical decisions that a design team addresses within a multi-disciplinary environment. On a very regular basis trade-offs between disciplines have to be made where a set of conflicting requirements exist. Within an aircraft development project, we focus on the aerodynamic design problem and review how this process has been advanced, first with the improving capabilities of traditional computational fluid dynamics analyses, and then with aerodynamic optimizations based on these increasingly accurate methods. The optimization method of the present work is based on the use of the adjoint of the flow equations to compute the gradient of the cost function. Then, we use this gradient to navigate the design space in an efficient manner to find a local minimum. The computational costs of the present method are compared with that of other approaches to aerodynamic optimization. A brief discussion regarding the formulation of a continuous adjoint, as opposed to a discrete one, is also included. Two case studies are provided...

These Lecture Notes review the formulation and application of optimization techniques based on control theory for aerodynamic shape design in both inviscid and viscous compressible flow. The theory is applied to a system defined by the partial differential equations of the flow, with the boundary shape acting as the control. The Frechet derivative of the cost function is determined via the solution of an adjoint partial differential equation, and the boundary shape is then modified in a direction of descent. This process is repeated until an optimum solution is approached. Each design cycle requires the numerical solution of both the flow and the adjoint equations, leading to a computational cost roughly equal to the cost of two flow solutions. Representative results are presented for viscous optimization of transonic wing-body combinations.

This paper provides a survey of shape parameterization techniques for multidisciplinary

The application of proper orthogonal decomposition for incomplete (gappy) data for compressible external aerodynamic problems has been demonstrated successfully in this paper for the first time. Using this approach, it is possible to construct entire aerodynamic flowfields from the knowledge of computed aerodynamic flow data or measured flow data specified on the aerodynamic surface, thereby demonstrating a means to effectively combine experimental and computational data. The sensitivity of flow reconstruction results to available measurements and to experimental error is analyzed. Another new extension of this approach allows one to cast the problem of inverse airfoil design as a gappy data problem. The gappy methodology demonstrates a great simplification for the inverse airfoil design problem and is found to work well on a range of examples, including both subsonic and transonic cases.

An algorithm for efficiently incorporating the effects of mesh sensitivities in a computational design framework is introduced. The method is based on an adjoint approach and eliminates the need for explicit linearizations of the mesh movement scheme with respect to the geometric parameterization variables, an expense that has hindered practical large-scale design optimization using discrete adjoint methods. The effects of the mesh sensitivities can be accounted for through the solution of an adjoint problem equivalent in cost to a single mesh movement computation, followed by an explicit matrix–vector product scaling with the number of design variables and the resolution of the parameterized surface grid. The accuracy of the implementation is established and dramatic computational savings obtained using the new approach are demonstrated using several test cases. Sample design optimizations are also shown. = vector of design variables = cost function = linear elasticity coefficient matrix = Lagrangian function = Lift-to-drag ratio = vector of dependent variables = discretized residual vector = computational mesh = vector of adjoint variables Nomenclature I.

We solve theproblem of determining airfoils that approximate, in a least square sense, given surface pressure distributions in transonic ight regimes. The owis modeled by means of the Euler equations and thesolution procedure is an adjointbased minimization algorithm that makesuseoftheinverse Theodorsen transform in order to parameterize the airfoil. Fast convergence to theoptimal solution is obtained by means of the pseudo-time method. Results are obtained using three di erent pressure distributions for several free stream conditions. The airfoils obtained have given a trailing edge angle. 1