Robust Airfoil Design in the Context of Multi-objective Optimization

We apply the concept of robustness to multi-objective optimization for finding robust Pareto optimal solutions. The multi-objective optimization and robustness problem is solved by using the $$\varepsilon $$ -constraint method combined with the non-intrusive polynomial chaos approach for uncertainty quantification. The resulting single-objective optimization problems are solved with a deterministic method using algorithmic differentiation for the needed derivatives. The proposed method is applied to an aerodynamic shape optimization problem for minimizing drag and maximizing lift in a steady Euler flow. We consider aleatory uncertainties in flight conditions and in the geometry separately to find robust solutions. In the case of geometrical uncertainties we apply a Karhunen-Loeve expansion to approximate the random field and make use of a dimension-adaptive quadrature based on sparse grid methods for the numerical integration in random space.