Data-driven surrogate model for aerodynamic design using separable shape tensor method

In the context of increasing dimensionality of design variables and the complexity of constraints, the efficacy of Surrogate-Based Optimization (SBO) is limited. The traditional linear and nonlinear dimensionality reduction algorithms are mainly to decompose the mathematical matrix composed of design variables or objective functions in various forms, the smoothness of the design space cannot be guaranteed in the process, and additional constraint functions need to be added in the optimization, which increases the calculation cost. This study presents a new parameterization method to improve both problems of SBO. The new parameterization is addressed by decoupling affine transformations (dilation, rotation, shearing, and translation) within the Grassmannian submanifold, which enables a separate representation of the physical information of the airfoil in a high-dimensional space. Building upon this, Principal Geodesic Analysis (PGA) is employed to achieve geometric control, compress the design space, reduce the number of design variables, reduce the dimensions of design variables and enhance predictive performance during the surrogate optimization process. For comparison, a dimensionality reduction space is defined using 95% of the energy, and RAE 2822 for transonic conditions are used as demonstrations. This method significantly enhances the optimization efficiency of the surrogate model while effectively enabling geometric constraints. In three-dimensional problems, it enables simultaneous design of planar shapes for various components of the aircraft and high-order perturbation deformations. Optimization was applied to the ONERA M6 wing, achieving a lift-drag ratio of 18.09, representing a 27.25% improvement compared to the baseline configuration. In comparison to conventional surrogate model optimization methods, which only achieved a 17.97% improvement, this approach demonstrates its superiority.