AIAA 2003–3955 A continuous adjoint method for unstructured grids

Adjoint based shape optimization methods have proven to be computationally efficient for aerodynamic problems. The majority of the studies on adjoint methods have used structured grids to discretize the computational domain. Due to the potential advantages of unstructured grids for complex configurations, in this study we have developed and validated a continuous adjoint formulation for unstructured grids. The hurdles posed in the computation of the gradient for unstructured grids are resolved by using a reduced gradient formulation. Methods to impose thickness constraints on unstructured grids are also discussed. Results for two and three dimensional simulations of airfoils and wings in transonic flow are used to validate the design procedure. Finally, the design procedure is applied to redesign the shape of a transonic business jet configuration, and we were able to reduce the drag of the aircraft from 235 to 216 counts resulting in a shock free wing. Introduction With the availability of high performance computing platforms and robust numerical methods to simulate fluid flows, it is possible to shift attention to automated design procedures which combine CFD with optimization techniques to determine optimum aerodynamic designs. The feasibility of this is by now well established, 1–6 and it is actually possible to calculate optimum three dimensional transonic wing shapes in a few hours, accounting for viscous effects with the flow modeled by the Reynolds averaged Navier Stokes (RANS) equations. By enforcing constraints on the thickness and span-load distribution one can make sure that there is no penalty in structure weight or fuel volume. Larger scale shape changes such as planform variations can also be accommodated. 7 It then becomes necessary to include a structural weight model to enable a proper compromise between minimum drag and low structure weight to be determined. Aerodynamic shape optimization has been successfully performed for a variety of complex configurations using multi-block structured meshes. 8, 9 Meshes of this type can be relatively easily deformed to accommodate shape variations required in the redesign. However, it is both extremely time-consuming and expensive in human costs to generate such meshes. Consequently we believe it is essential to develop shape optimization methods which use unstructured meshes for the flow simulation. Typically, in gradient-based optimization techniques , a control function to be optimized (the wing shape, for example) is parameterized with a set of design variables and a suitable cost function to be min