Aerodynamic shape optimization using stabilized finite element method

Abstract

An algorithm for aerodynamic shape optimization using a gradient-based approach is presented. A stabilized finite element method is utilized to solve the equations governing the fluid flow and adjoint variables. The effectiveness of the method in finding nontrivial designs that satisfy the posed objectives and constraints is demonstrated. It is shown that the initial guess plays an important role in the optimal solution that is realized. Shapes of bumps that are associated with a specified volume, lift and drag are determined and presented. The method is applied to design of airfoils in unsteady flows. The objective function is based on time-averaged force coefficients. The effect of enriching the design space is studied. It is shown that this being a local method, where the search direction is based on gradient of the objective function, the gradual enrichment of design space leads to superior performance. The idea is demonstrated via spontaneous appearance of corrugations on an airfoil surface during the optimization, for maximum lift, by gradually increasing the number of control points. The method is extended to three dimensions. Its application is demonstrated via optimization of the planform of a finite wing for maximum lift-to-drag ratio. A bi-parametric tensorial NURBS (non-uniform rational bi-cubic spline) surface is interpolated on a 3D control net to represent a wing surface. It is shown that for low Reynolds number, it is possible to design a planform that is more efficient than an ellipse. Unlike the elliptic planform, the optimal wing computed by the present method, is associated with a short winglet-like structure at the wing-tip and the maximum chord length at around mid-span.