Gradient-based Pareto front approximation applied to turbomachinery shape optimization

Multi-objective optimization has been rising in popularity, especially within an industrial environment, where several cost functions often need to be considered during the design phase. Traditional gradient-free approaches, such as evolutionary algorithms, can be employed to compute a front of equally suitable solutions a designer can choose from, with a high computational cost though, particularly in high-dimensional design spaces. In this paper, a gradient-based algorithm is developed for efficiently tracing the Pareto front in bi-objective aerodynamic shape optimization problems, where an adjoint method is used for the computation of the objective functions’ gradients with respect to the design variables. After obtaining a starting point on the front, a prediction–correction approach is employed to compute new Pareto points. Satisfying the Karush–Kuhn–Tucker conditions provides a prediction for the next point, which is, then, corrected by solving a minimum distance problem. The prediction step, though, requires the costly computation of the Hessian matrix. This is avoided here using the BFGS (Nocedal and Wright in Numerical optimization, Springer, New York, 2006) technique. The proposed method is first demonstrated in a less expensive lift/drag optimization of an isolated airfoil and then applied to the bi-objective optimization of a 3D compressor stator. The extension of the proposed method to cases with more than two objectives is straightforward, on condition that an algorithm is found to coordinate the way the Pareto front is swept.