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In practical optimization design, an excessive number of design variables have a highly detrimental influence on the efficiency and accuracy of the final design scheme and expose the optimization problem to the curse of dimensionality. Therefore, incorporating only the most essential variables into an optimization design problem facilitates obtaining accurate and cost-efficient solutions. Reported here is an optimization design method based on parameter reduction and active subspaces, and it is used to redistribute the tip load in a transonic fan. Specifically, a coupled design strategy is developed to reduce the number of parameters needed to describe the three-dimensional blade shape, which leads to far fewer design variables being involved in the optimization design. Moreover, active subspaces are used to perform sensitivity analysis and establish low-dimensional surrogate models. After the coupled design, a blade is represented effectively by only three parameters, each of which has a significant influence on the fan performance. Three one-dimensional active subspaces are established for maximum mass flow rate, maximum total pressure ratio, and maximum efficiency, based on which the linear surrogate models are obtained. Next, the chordwise tip blade loading is optimized, after which the rotor efficiency at the design point is increased by 1.1%, while the total pressure ratio remains nearly unchanged. Finally, the flow field is analyzed to understand the mechanism for this performance improvement, and the results show that the optimized blade loading reduces the aerodynamic losses caused by shock-induced flow separation and the interaction between shocks and tip leakage flows.
Ma et al. (Sun,) studied this question.