Metric-Based Mathematical Derivation of Aerofoil Design Variables

Derivation of fundamental geometric aerofoil design variables is considered. An efficient parameterization and deformation scheme is important in an aerodynamic shape parameterization framework to allow flexible deformation of the surface and volume mesh with full design space coverage attainable. To allow full design space coverage a method has been developed to allow the derivation of aerofoil geometric design variables, which is done by the mathematical decomposition of a training library. The resulting geometric modes are independent of parameterization scheme, surface and volume mesh, and flow solver, thus are generally applicable. Furthermore, a benchmark performance measure, the aerofoil technology factor, has been incorporated into the scheme, to allow intelligent, metric-based selection of the training library, to ensure high performance aerofoil modes are extracted. Three criteria for the selection of the training library are considered: high performance; varied performance; random selection. Results are presented for several geometric shape recovery problems, and these show that having a wide variety of aerofoil profiles in the training data produces modes that have greater design space coverage and are able to reconstruct target aerofoils to a greater accuracy than only having a high performance training library. I. Background and Introduction Numerical simulation methods are now used routinely in industrial design, and increasing computer power has resulted in their integration into the optimization process being widely accepted. Integrating an effective geometry control method with an aerodynamic model and a numerical optimization scheme can result in an aerodynamic shape optimization approach, which is invaluable during design. The quality and applicability of results gained by numerical optimization are inherently dependent on the fidelity of the analysis tool used, and so developing a modular approach means that the complexity and cost of each module required at various stages of the design process can be changed, in terms of aerodynamic or fluid dynamic fidelity, number of surface shape design degrees of freedom, and the complexity of the optimization scheme. Computational fluid dynamics (CFD) is at the forefront of aerodynamic analysis capabilities, and application of numerical optimization algorithms with compressible CFD has been used in numerous optimizations of two-dimensional aerofoil sections, 1,2 three-dimensional aircraft, 3–5 and three-dimensional aeroelastic aircraft, 6 and there has been a recent increase in activity in the rotor area. 7–11 The authors have also presented work in this area, having developed a modularised, generic optimization tool, that is flow-solver and mesh type independent, and applicable to any aerodynamic problem. 12–14 The key aspect of a flexible optimization and design process is an effective geometry parameterization and surface control technique. The effectiveness of these techniques can be measured in terms of being i) flexible enough to allow sufficient design space investigation, ii) robust enough to be applicable to any geometry or design surface, and iii) efficient enough to cover the design space with a small number of design parameters. The latter is a metric which relates to the computational power required to perform numerical aerodynamic shape optimization, particularly if using gradient-based optimisers which require the computation of the sensitivity of the objective function (which if done using finite-difference is proportional to the number of