Viscous Aerodynamic Shape Optimization of Wings Including Planform Variables

Abstract

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...

Aero-Structural Wing Planform Optimization

“…Our previous work 9 validates a design methodology for wing planform optimization with simple estimation of wing weight. We find that if the cost function contains both aerodynamic and structural weight dependencies, the trade-off between these two dependencies will lead to a useful design.…”

“…9 The results indicate that the step size almost has no effect on the adjoint gradient, showing an additional advantage of the adjoint gradient over the finite difference gradient. Similar results have been indicated by Kim et.…”

“…If α3 α1 is high enough, the optimum shape will have lower C D and higher C W than another optimum shape with lower α3 α1 value. Leoviriyakit and Jameson 9 propose an intuitive choice of α 1 and α 3 by equating a problem of maximizing range of an aircraft to a problem of minimizing the cost function…”

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.…”