GA-hardness of aerodynamic optimization problems: Analysis and proposed cures

A study about convergence of Genetic Algorithms (GAs) applied to aerodynamic optimization problems for transonic flows of dilute and dense gases is presented. Specific attention is devoted to working fluids of the Bethe - Zel'dovich - Thompson (BZT) type, which exhibit non classical dynamic behaviors in the transonic/supersonic regime, such as the disintegration of compression shocks. A reference, single-objective optimization problem, namely, wave drag minimization for a non-lifting transonic flow past a symmetric airfoil is considered. Several optimizations runs are performed for perfect and BZT gases at different flow conditions using a GA coupled with a flow solver. For each case, GA-hardness, i.e. the capability of converging more or less easily toward the global optimum for a given problem, is measured by means of statistical tools. For GA-hard problems, reduced convergence rate and high sensitivity to the choice of the starting population are observed. Results show that GA-hardness is greater for flow problems characterized by very weak shocks, and is strongly affected by numerical inaccuracies in the evaluation of the objective function. Then, some possible cures to GA-hardness are proposed and numerically verified. An efficient objective-function evaluation procedure based on Richardson extrapolation is proposed, which drastically reduces GA-hardness with a very moderate increase (and sometimes a slight decrease) in computational cost of optimization runs. Finally, an application of the proposed strategy to a multi-objective optimization problem is provided, clearly demonstrating the advantages deriving by the use of the proposed technique.