Structure-preserving reduced-order models of thin panels in supersonic and hypersonic flow

Isogai's case A, a pitch-plunge aerofoil whose elastic axis lies half a chord ahead of the leading edge, has a folded transonic flutter boundary in inviscid flow: near the bottom of the transonic dip several flutter speeds coexist at one Mach number. We map this boundary, and the family obtained by moving the elastic axis, with coupled unsteady Euler simulations, a Gaussian-process surrogate of the damping coefficient trained by batch active learning, and split conformal prediction on independent random simulations, which turns the surrogate into stability calls with a distribution-free error rate. Three failure modes of time-domain damping labels that leave no trace in the residuals are documented, and the half-degree mean incidence left by the common one-degree start is shown to raise the flutter speed at the dip by 2.8%, more than any numerical setting. With 80 simulations the learned case A boundary labels 64 of 65 independent runs correctly, and calibrated calls are wrong in at most 1.6% of cases. Over Mach number, elastic axis and flutter speed index, 193 simulations and 40 calibration runs show that the fold exists only for elastic axes more than about 0.4 chord ahead of the leading edge; it closes between 0.39 and 0.33 chord ahead, on two meshes. Chordwise energy budgets, closed against the solver loads and the growth rate within a few per cent, explain the boundary. The lower branch is crossed through the reduced frequency at unchanged shock phase. On the upper branch the shock is pinned at the trailing edge, and moving the elastic axis aft brings the pivot of the flutter mode towards the section, so that the damping of the supersonic surface falls faster than the residual shock input and the fold closes.

This paper investigates the feasibility of using residual learning to improve unsteady aerodynamic load prediction for aeroelastic applications. The machine learning technique selected for the study is the long short-term memory (LSTM) neural network, which is used for its suitability for sequential data with aerodynamic memory effects. The approach is investigated for the NLR 7301 airfoil benchmark using high-fidelity CFD lift data for prescribed pitch and plunge motions in the transonic flow regime in the presence of shock motion. An analytical unsteady aerodynamic model based on the Wagner function is used as a physics-based baseline, and the neural network is trained to learn the difference between the CFD lift coefficient and the Wagner prediction. The residual model is compared with a direct neural-network model trained to predict the CFD lift coefficient. The comparison includes feature and normalization studies, external benchmark cases, and leave-one-out and leave-family-out generalization tests across a range of sinusoidal and non-sinusoidal motions. The residual model performs best when its inputs align with the Wagner formulation variables, generally giving lower error and more consistent performance across training runs, though the direct model remains more accurate for some high-frequency cases. The residual model also generalizes better in the leave-one-out and leave-family-out tests, with a smaller increase in error than the direct model when entire motion families are withheld from training. Overall, the results indicate that residual learning shows promise as a modular approach for augmenting classical low-order aerodynamic theories, especially when the physics baseline removes a structured part of the aerodynamic response and leaves a lower-variance correction for the neural network to learn.

Assigning reduced coordinates to states near an attracting limit cycle requires the correct invariant-fibre geometry. The classical first-order phase-isostable chart obtained from adjoint Floquet modes projects along the strong-stable quotient fibre, whereas a metric-orthogonal complement of the retained slow bundle generally does not. We prove locally that a chart satisfying the linearised semiconjugacy relation leaves an O(delta^2) invariance residual, while projection along a non-invariant complement generically leaves an O(delta) term. For a nonlinear aeroelastic limit cycle, the metric-normal and strong-stable directions differ by 48.5 to 71.7 degrees, and metric-normal perturbations contain first-order retained phase and slow-amplitude components. Replacing the metric normal by the strong-stable fibre changes the measured residual scaling from delta^1.01 to delta^1.87 without fitted parameters. We then test learned higher-order corrections whose linearisation is pinned to the adjoint-Floquet chart, whose symmetry is exact, and whose reduced flow is fixed. Although they reduce the registered fixed-normalisation latent residual, post-hoc amplitude recalibration and adjoint-Floquet-targeted future consistency move or reverse the ranking. Because the learned maps already share the baseline's first-order gauge and the future target is supplied by the baseline chart, these diagnostics establish neither an independent positive nor negative higher-order result. Correct first-order Floquet geometry is therefore necessary in this benchmark, while the additional predictive value of the learned correction remains unidentified by the available representation-dependent diagnostics.