Dynamic-mode-decomposition-based gradient prediction for adjoint-based aerodynamic shape optimization

Accurate and efficient gradient computation is the key to aerodynamic shape optimization. In this paper, dynamic mode decomposition (DMD) is employed to analyze the dynamic characteristics of the early pseudo-time marching of adjoint equations and to predict the gradient. Besides the first-order zero-frequency mode, other zero-frequency modes also contribute to the pseudo iterations of the adjoint equations in the early iterations. Hence, different from existing methods, all zero-frequency modes are retained to reconstruct adjoint fields for gradient prediction. Moreover, to further improve the modeling accuracy, an improved DMD (IDMD) is proposed by omitting the initial snapshots in early iterations. The effect of pseudo-time step on modeling accuracy is also studied. By solving the adjoint equations of the transonic and subsonic flows, the accuracy of the proposed method is verified. Results indicate that the proposed method still works despite the conventional solution process diverges. Through aerodynamic shape optimization examples of transonic flow over an airfoil, the number of adjoint pseudo-time steps is remarkably reduced by 83%, which indicates the proposed IDMD-based gradient prediction method has great potential for improving the efficiency of aerodynamic shape optimization.