Parametric Nonlinear Volterra Series via Machine Learning: Transonic Aerodynamics

Abstract

This study introduces an approach for modeling unsteady transonic aerodynamics within a parametric space, using Volterra series to capture aerodynamic responses and machine learning to enable interpolation. The first- and second-order Volterra kernels are derived from indicial aerodynamic responses obtained through computational fluid dynamics, with the second-order kernel calculated as a correction to the dominant linear response. Machine learning algorithms, specifically artificial neural network and Gaussian process regression, are used to interpolate kernel coefficients within a parameter space defined by Mach number and angle of attack. The methodology is applied to two and three dimensional test cases in the transonic regime. Results underscore the benefit of including the second-order kernel to address strong nonlinearity and demonstrate the effectiveness of neural networks. The approach achieves a level of accuracy that appears sufficient for use in conceptual design.

Figures (17)

• Figure 1: Examples of linear (left) and nonlinear (right) synthetic responses under a step input. The squares indicate the exact responses, the continuous lines denote the responses reconstructed with Volterra series.
• Figure 2: Examples of linear (top) and nonlinear (bottom) synthetic responses under a sinusoidal input at reduced frequency $k = 0.3$. Continuous lines indicate the reference Volterra series derived from the exact solutions, while circles and squares denote the reconstructed responses using kernels predicted by FCNN and GPR, respectively.
• Figure 3: Impression of the NACA0012 CFD grid.
• Figure 4: Samples for $M$ and $\alpha_0$ for NACA0012 test case. Red markers denote flight conditions of CT2 and CT2 experiments landon1982naca.
• Figure 5: Harmonic signal reconstruction with Volterra ROM for the AGARD CT2 and CT5 cases landon1982naca. Experimental data are represented by square markers, while continuous and dashed lines denote the reconstructed responses using nonlinear Volterra kernels predicted by FCNN and GPR, respectively.