This paper describes the formulation of optimization techniques based on control theory for aerodynamic shape design in inviscid compressible flow modelled by the Euler equations. The design methodology has been extended to include wing planform optimization. A model for the structure weight has been included in the design cost function to provide a meaningful design. A practical method to combine the structural weight into the design cost function has been studied. Results of optimizing a wing-fuselage of a commercial transport aircraft show a sucessful trade of planform design, leading to meaingful designs. The results also support the necessity of including the structure weight in the cost function. INTRODUCTIONW HILE aerodynamic prediction methods based on CFD are now well established, and quite accurate and robust, the ultimate need in the design process is to find the optimum shape which maximizes the aerodynamic performance. One way to approach this objective is to view it as a control problem, in which the wing is treated as a device which controls the flow to produce lift with minimum drag, while meeting other requirements such as low structure weight, sufficient fuel volume, and stability and control constrains. Here we apply the theory of optimal control of systems governed by partial differential equations with boundary control, in this case through changing the shape of the boundary. Using this theory, we can find the Frechet derivative (infinitely dimensional gradient) of the cost function with respect to the shape by solving an adjoint problem, and then we can make an improvement by making a modification in a descent direction. For example, the cost function might be the drag coefficient at a fixed lift, or the lift to drag ratio. During the last decade, this method has been intensively developed, and has proved to be very effective for improving wing section shapes for fixed wing planform. 1,2,7,8,[11][12][13][14] In this work we report on recent improvements in the adjoint method, and also consider its extension to planform design. It is well known that the induced drag varies inversely with the square of the span. Hence the induced drag can be reduced by increasing the span. Moreover, shock drag in transonic flow might be reduced by increasing sweep back or increasing the chord to reduce the thickness to cord ratio. Consequencely an optimization which considers only the pressure drag would lead to a wing with excessive span and sweep back. In order to produce a meaningful optimization problem, we therefore include a simple structure weight model based on the span, sweep back, and taper. MATHEMATICAL FORMULATIONIn this work the equations of steady flowwhere w is the solution vector, and f i (w) are the flux vectors along the x i axis are applied in a fixed computational domain, with coordinates ξ i , so thatwhere S ij are the coefficients of the Jacobian matrix of the transformation. Then geometry changes are represented by changes δS ij in the metric coefficients. Suppose one wishes to minimize...

This paper focuses on wing optimization via control theory using a multi-point design method. Based on the design methodology previously developed for wing section and planform optimization at a specific flight condition, it searches for a single wing shape that performs well over a range of flight conditions. A new cost function is defined as the weighted sum of cost functions from a range of important flight conditions. Results of multi-point optimization of a long range transport aircraft show that improvement at each flight condition is not as large as the result from single-point optimization at one of the design points. However improvement in performance measures such as drag divergence Mach number and the lift-to-drag ratio over a range of Mach numbers is significantly greater.

This paper focuses on wing optimization via control theory using a multi-point design method. Based on the design methodology previously developed for wing section and planform optimization at a specific flight condition, it searches for a single wing shape that performs well over a range of flight conditions. Our previous experience with multipoint design without a detailed FE structural model, showed improvements in performance measures such as drag divergence Mach number and the lift-to-drag ratio over a range of Mach numbers. In the current work, the flow solution is modified to allow for shape deformation under load. We achieve this by coupling SYN107 to FEAP (Robert Taylor, University of California at Berkeley). The resulting aero-elastic simulation is then used to determine the optimal airfoil section and wing planform definition. In the multi-point design the actual shape will now be different at the different design points. With the coupled aero-structural analysis we hope to determine the best jig shape for the multipoint design.

During the last decade, aerodynamic shape optimization methods based on control theory have been intensively developed. The methods have proved to be very effective for improving wing section shapes for fixed wing-planforms. Building on this success, extension of the control theory approach to variable planforms has yielded further improvement. This paper describes the formulation of optimization techniques based on control theory for aerodynamic shape design in inviscid compressible flow modeled by the Euler equations. The design methodology has been expanded to include wing planform optimization. It extends the previous work on wing planform optimization based on simple wing weight estimation. A more realistic model for the structural weight, which is sensitive to both planform variations and wing loading, is included in the design cost function to provide a meaningful design. A practical method to combine the structural weight into the design cost function is studied. An extension of a single to a multiple objective cost function is also considered. Results of optimizing a wing-fuselage of a commercial transport aircraft show a successful trade-off between the aerodynamic and structural cost functions, leading to meaningful wing planform designs. The results also support the necessity of including the structural weight in the cost function. Furthermore, by varying the weighting constant in the cost function, an optimal set called "Pareto front" can be captured, broadening the design range of optimal shapes. IntroductionW HILE aerodynamic prediction methods based on CFD are now well established, quite accurate, and robust, the ultimate need in the design process is to find the optimum shape which maximizes the aerodynamic performance. One way to approach this objective is to view it as a control problem, in which the wing is treated as a device which controls the flow to produce lift with minimum drag, while meeting other requirements such as low structural weight, sufficient fuel volume, and stability and control constraints. In this paper, we apply the theory of optimal control of systems governed by partial differential equations with boundary control, in this case through changing the shape of the boundary. Using this theory, we can find the Frechet derivative (infinitely dimensional gradient) of the cost function with respect to the shape by solving an adjoint problem, and then we can make an improvement by making a modification in a descent direction. For example, the cost function might be the drag coefficient at a fixed lift, or the lift to drag ratio. During the last decade, this method has been intensively developed, and has proved to be very effective for improving wing section shapes for fixed wing planforms. Wing planform modification can yield large improvement in wing performance but can also affect wing weight and stability and control issues. It is well known that the induced drag varies inversely with the square of the span. Hence the induced drag can be reduced by increasing the span. Moreov...

This paper describes the formulation of optimization techniques based on control theory for aerodynamic shape design in inviscid compressible flow modelled by the Euler equations. The design methodology has been extended to include wing planform optimization. A model for the structure weight has been included in the design cost function to provide a meaningful design. A practical method to combine the structural weight into the design cost function has been studied. Results of optimizing a wing-fuselage of a commercial transport aircraft show a sucessful trade of planform design, leading to meaingful designs. The results also support the necessity of including the structure weight in the cost function. INTRODUCTIONW HILE aerodynamic prediction methods based on CFD are now well established, and quite accurate and robust, the ultimate need in the design process is to find the optimum shape which maximizes the aerodynamic performance. One way to approach this objective is to view it as a control problem, in which the wing is treated as a device which controls the flow to produce lift with minimum drag, while meeting other requirements such as low structure weight, sufficient fuel volume, and stability and control constrains. Here we apply the theory of optimal control of systems governed by partial differential equations with boundary control, in this case through changing the shape of the boundary. Using this theory, we can find the Frechet derivative (infinitely dimensional gradient) of the cost function with respect to the shape by solving an adjoint problem, and then we can make an improvement by making a modification in a descent direction. For example, the cost function might be the drag coefficient at a fixed lift, or the lift to drag ratio. During the last decade, this method has been intensively developed, and has proved to be very effective for improving wing section shapes for fixed wing planform. 1,2,7,8,[11][12][13][14] In this work we report on recent improvements in the adjoint method, and also consider its extension to planform design. It is well known that the induced drag varies inversely with the square of the span. Hence the induced drag can be reduced by increasing the span. Moreover, shock drag in transonic flow might be reduced by increasing sweep back or increasing the chord to reduce the thickness to cord ratio. Consequencely an optimization which considers only the pressure drag would lead to a wing with excessive span and sweep back. In order to produce a meaningful optimization problem, we therefore include a simple structure weight model based on the span, sweep back, and taper. MATHEMATICAL FORMULATIONIn this work the equations of steady flowwhere w is the solution vector, and f i (w) are the flux vectors along the x i axis are applied in a fixed computational domain, with coordinates ξ i , so thatwhere S ij are the coefficients of the Jacobian matrix of the transformation. Then geometry changes are represented by changes δS ij in the metric coefficients. Suppose one wishes to minimize...