Aero-Structural Wing Planform Optimization

Abstract

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...

Efficient Aerodynamic Shape Optimization

“…By varying these it is possible to calculate the Pareto front of designs which have the least weight for a given drag coefficient, or the least drag coefficient for a given weight. The relative importance of these can be estimated from the Breguet range equation; Figure 5 shows the Pareto front obtained from a study of the Boeing 747 wing, 36 in which the flow was modeled by the Euler equations. The wing planform and section were varied simultaneously, with the planform defined by six parameters; sweepback, span, the chord at three span stations, and wing thickness.…”

Advances in Aerodynamic Shape Optimization

“…By varying these it is possible to calculate the Pareto front of designs which have the least weight for a given drag coefficient, or the least drag coefficient for a given weight. The relative importance of these can be estimated from the Breguet range equation; [17], in which the flow was modeled by the Euler equations. The wing planform and section were varied simultaneously, with the planform defined by six parameters; sweepback, span, the chord at three span stations, and wing thickness.…”

Multipoint Wing Planform Optimization via Control Theory

“…6,7,8,9,10 In the steady flow the transient term of equation (6) drops out and the the adjoint problem can be formulated by combining the variations of equations (7) and (6) using the Lagrange multiplier ψ as…”

“…During the last decade this method has been extensively developed to improve wing section shapes 1,2,3,4,5 and wing planforms. 6,7,8,9,10,11 Our previous works 6,7,8,9,10 reported design methodology for wing planform optimization at a cruise condition. The main objective was to reduce drag of the airplane at constant lift, using the wing structural weight as a constraint to prevent un-realistic designs.…”