Uncertainty Based Optimization Strategy for the Gappy-POD Multi-Fidelity Method

Abstract In the surrogate model-based optimization of turbine airfoils, often only the prediction values for objective and constraints are employed, without considering uncertainties in the prediction. This is also the case for multi-fidelity optimization strategies based on e.g. the Gappy-POD approach, in which results from analyses of different fidelities are incorporated. However, the consideration of uncertainties in global optimization has the advantage that a balanced coverage of the design space between unexplored regions and regions close to the current optimum takes place. This means that on the one hand regions are covered in which so far only a few sample points are present and thus a high degree of uncertainty exists (global exploration), and on the other hand regions with promising objective and constraint values are investigated (local exploitation). The genuine new contribution in this work is the quantification of the uncertainty of the multi-fidelity Gappy-POD method and an adapted optimization strategy based on it. The uncertainty quantification is based on the error of linear fitting of low-fidelity values to the POD basis and subsequent forward propagation to the high-fidelity values. The uncertainty quantification is validated for random airfoil designs in a design of experiment. Based on this, a global optimization strategy for constrained problems is presented, which is based on the well-known Efficient Global Optimization (EGO) strategy and the Feasible Expected Improvement criterion. This means that Kriging models are created for both the objective and the constraint values depending on the design variables that consider both the predictions and the uncertainties. This approach offers the advantage that existing and widely used programs or libraries can be used for multi-fidelity optimization that support the (single-fidelity) EGO algorithm. Finally, the method is demonstrated for an industrial test case. A comparison between a single-fidelity optimization and a multi-fidelity optimization is made, each with the EGO strategy. A coupling of 2D/3D simulations is used for multi-fidelity analyses. The proposed method achieves faster feasible members in the optimization, resulting in faster turn-around compared to the single-fidelity strategy.