Aerodynamic shape optimization based on proper orthogonal decomposition reparameterization under small training sets

Abstract

Complex shape aerodynamic optimization frequently encounters the problem of dimensionality curse. Extracting geometric features from the original design space proves effective in this issue by reconstructing shapes with lower-dimension. While deep learning methods possess strong capabilities in dimensionality reduction, they suffer from longer training times and higher computational resource requirements. However, Proper Orthogonal Decomposition (POD) presents a more practical algorithm that can be effectively utilized for complex high-dimensional shapes. In this study, we propose a combination of the POD-based method and Gaussian Process Regression with an automatic kernel construction algorithm (AKC-GPR) to optimize complex shapes in a lower-dimensional reparameterization space using small training sets. This approach aims to explore the potential for reducing optimization cost. By reducing the design parameters from 56 to 18 with an error rate of 1.28 %, we establish surrogate models for optimization using AKC-GPR with progressively decreasing training sets. For samples below 100, the Mean Absolute Percentage Errors (MAPEs) remain below 2 %, while the Mean Relative Percentage Errors (MRPEs) approach 10 %. It emphasizes the proficiency of POD in geometric dimensionality reduction. The optimized shapes exhibit smoother contour, resulting in a 10 % improvement in lift and a relatively modest reduction in drag of less than 3 %. The lift-to-drag ratio experiences an increase of over 14 %. The shapes optimized by various training sets possess minimal differences in geometric shapes and aerodynamic characteristics. This confirms the AKC-GPR algorithm's high-precision fitting capability in effectively handling complex aerodynamic coefficients of multi-parameter shapes during small-sample shape optimization.