Modelling non-linear aeroelastic loads in long-span bridges with extreme learning machines

Gledson Rodrigo Tondo; Samir Chawdhury; Sergio Andres Castro Giraldo; Guido Morgenthal

cs.CESep 27, 2026

Modelling non-linear aeroelastic loads in long-span bridges with extreme learning machines

Authors: Gledson Rodrigo Tondo, Samir Chawdhury, Sergio Andres Castro Giraldo, Guido Morgenthal

Organizations: Bauhaus-Universität Weimar Weimar, Germany

Abstract

Accurate modelling of aerodynamic loads is essential for predicting instabilities and ensuring the safety of long-span bridges. A methodology is introduced for modelling aerodynamic self-excited forces in bridge-deck cross-sections using extreme learning machines (ELMs). ELMs, as single-layer feedforward neural networks, offer efficient training and accurate predictions. Forced-oscillation datasets from computational fluid dynamics (CFD) or wind-tunnel experiments are used for training, enabling systematic data selection to capture non-linear aerodynamic behaviour often missed by semi-analytical approaches. Once trained, the model predicts self-excited loads for any arbitrary motion composed by frequencies and amplitudes within the training domain. Comparisons with analytical, semi-analytical, and CFD results show superior accuracy in capturing non-linear force components and close agreement for aerodynamic loads and flutter wind speeds. Training required about 1.1% of the time of a conventional neural network, and coupled flutter analysis runs in seconds, providing orders-of-magnitude speed-ups over CFD. These results indicate that ELM-based frameworks are accurate, practical, and efficient alternatives for modelling self-excited loads, particularly when preliminary CFD or wind-tunnel data are available. The presented approach offers a reliable data-driven technique for aeroelastic load modelling in long-span bridges.

Figures & tables

Figure 1: Left: 2D coordinate system for the degrees of freedom ( $h$ and $\alpha$ ) and the corresponding aerodynamic loads ( $L$ and $M$ ), for a cross-section of width $B$ subjected to a flow field with mean wind speed $U$ . Right: the general framework to obtain aerodynamic derivatives from aerodynamic analyses, where a harmonic motion is imposed to the system, and the loads are tracked either in computational fluid dynamics (CFD) or wind tunnel (WT) tests.

Figure 2: Framework of the ELM models proposed in this work. The input layer (blue box) accepts the motion ( $h$ or $\alpha$ ) and its time derivatives, represented by $u$ , $v$ and $a$ . The hidden layer (green box) contains a user-defined number of neurons $N_{n}$ , used to calculate the output force $F$ (red box). In this approach, four ELM models are required to cover all combinations of input motion types and aerodynamic forces ( $C_{i,j}$ for $i\in\{L,M\}$ and $j\in\{h,\alpha\}$ ).

Sec	$Re$	$N_{c}$	$N_{tp}$	$v_{r,\mathrm{min}}$	$v_{r,\mathrm{max}}$	$\hat{h}$	$\hat{\alpha}$	$m_{h}$	$m_{\alpha}$	$f_{h}$	$f_{\alpha}$	$\zeta$
	[-]	[-]	[-]	[-]	[-]	[m]	[deg]	[t/m]	[tm 2 /m]	[Hz]	[Hz]	[%]
FP	-	1	25	2	16	1.0	1.0	22.74	2470	0.100	0.278	1.0
GB	$10^{5}$	10	250	2	16	5.0	12.0	22.74	2470	0.100	0.278	0.5

Table 1: Training properties and structural parameters used for the flat plate (FP, Sec. 3 ) and the Great Belt East bridge (GB, Sec. 4 ) cases: Reynolds number

Re

, number of training oscillation cycles

N_{c}

, number of training points per cycle

N_{tp}

, minimum and maximum reduced velocities

v_{r,min}

and

v_{r,max}

, maximum vertical and torsional amplitudes of motion

\hat{h}

and

\hat{\alpha}

, structural vertical mass

m_{h}

and torsional mass

m_{\alpha}

, bending frequency

f_{h}

and torsional frequency

f_{\alpha}

and damping ratio

\zeta

Figure 3: Analytical flat plate case: prediction of normalized lift and moment forces in vertical (left, $C_{L,h}$ and $C_{M,h}$ ) and rotational (right, $C_{L,\alpha}$ and $C_{M,\alpha}$ ) directions due to a harmonic input motion, for a reduced velocity $v_{r}=5$ .

Figure 4: Analytical flat plate case: comparison of aerodynamic derivatives obtained analytically and from the ELM model, estimated via least squares fit to Scanlan’s formulation (see Eqs. ( 8 , 9 )).

Figure 5: Analytical flat plate case: a pseudo-random motion is applied to the structural model in both vertical (left) and rotational (right) degrees of freedom (top). The aerodynamic lift (centre) and moment (bottom) predicted by the trained ELM models are compared with the analytical solutions for both degrees of freedom.

Figure 6: Analytical flat plate case: comparison metrics related to root mean squared error $\mathcal{M}_{\mathrm{rms}}$ , correlation $\mathcal{M}_{\mathrm{c}}$ , coefficient of determination $\mathcal{M}_{R^{2}}$ , wavelet $\mathcal{M}_{\mathrm{W}}$ , peak amplitude $\mathcal{M}_{\mathrm{peak}}$ , phase angle $\mathcal{M}_{\phi}$ and magnitude $\mathcal{M}_{\mathrm{m}}$ . The results for both lift $C_{L}$ and moment $C_{M}$ forces are shown for the cases of vertical displacements (left) and rotations (right).

Figure 7: Great Belt East Bridge schematic. Left: a side view of the bridge, showing the main and side spans. Right: the section dimensions, wind speed direction, coordinate system and aerodynamic forces.

Figure 8: Great Belt East Bridge: forced vibration simulations ( $v_{r}=2$ and $\hat{\alpha}=5$ deg) from CFD analysis used as training data, in vertical (left) and rotational (right) directions. Vortex particle map (top) at $tU/B=9.66$ , and the respective normalised velocity fields (bottom).

Figure 9: Great Belt East Bridge: forced oscillation CFD simulations for an amplitude of $\hat{\alpha}=10$ deg and a reduced velocity $v_{r}=16$ . Results for the normalized lift (top) and moment (centre-top) due to a vertical harmonic motion and lift (centre-bottom) and moment (bottom) due to a rotational harmonic motion are compared to the ELM predictions and the Scanlan linear fit. Comparisons are given in the time-domain normalized forces (left) and the corresponding power spectral density (right).

Figure 10: Comparison metrics for the forced oscillation case of amplitude of $\hat{\alpha}=10$ deg and a reduced velocity $v_{r}=16$ (c.f. Figure 9 ). The ELM model predictions are compared to Scanlan’s linear fit, taking the CFD forces as a basis. Top: metrics for $C_{L,h}$ (left) and $C_{M,h}$ (right). Bottom: metrics for $C_{L,\alpha}$ (left) and $C_{M,\alpha}$ (right).

Figure 11: Great Belt East Bridge: aerodynamic derivatives obtained from harmonic oscillation analysis computed both from CFD simulations and using the trained ELM models.

Figure 12: Great Belt East Bridge: comparison of aerodynamic forces predicted by the ELM with the ones derived from a CFD analysis of flutter instability, showing limit cycle oscillation at $U=72.5$ m/s. Top: the vertical (left) and rotational (right) structural responses. Bottom: the lift (left) and moment (right) forces, linearly superimposed from each degree of freedom in the case of ELM predictions.

Figure 13: Great Belt East Bridge: normalised Morlet wavelet analysis (with a central frequency $f_{c}=4$ Hz) of the lift forces from a flutter simulation (see Figure 12 ), computed from CFD (left) and predicted by the ELM model (right).

Figure 14: Great Belt East Bridge: flutter analysis in time domain performed using a coupled ELM-structural model. Results are shown for the vertical displacements (left) and rotations (right), and indicate the onset of flutter at $U=71.6$ m/s, with higher wind speeds showing divergent response behaviour.

Figure 15: Great Belt East Bridge: ELM flutter limit wind speeds (left) and the corresponding oscillation frequency of the coupled motion (right) in comparison with the linear quasi-steady (LQS), quasi-steady (QS), corrected quasi-steady (CQS), modified quasi-steady (MQS), linear unsteady (LU) and hybrid nonlinear (HNL) semi-analytical models, as well as CFD and wind tunnel results [ 33 , 36 ] .

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May 21, 2026, cs.LGJ/K move · Enter open · S save

Aerodynamic force reconstruction using physics-informed Gaussian processes

Gledson Rodrigo Tondo, Igor Kavrakov, Guido Morgenthal

Bauhaus-Universität Weimar Weimar, Germany · University of Cambridge Cambridge, United Kingdom

Aerodynamics Gaussian Process Force

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Accurate modeling of aerodynamic loads is essential for understanding and predicting the responses of complex structural systems. However, these models often rely on simplifications of the true physical forces, introducing assumptions that can limit their accuracy. Validating such models becomes particularly challenging in the presence of noisy or incomplete data. To address this, we introduce a probabilistic physics-informed machine learning approach designed to reconstruct the underlying aerodynamic loads from noisy measurements of structural dynamic responses. The model avoids overfitting, eliminates the need for regularization schemes, and allows for the use of heterogeneous and multi-fidelity data during the training process. The efficacy of the approach is demonstrated through the reconstruction of aerodynamic loads on the Great Belt East Bridge, simulated under a linear unsteady assumption. Results show a strong agreement between true and predicted loads, particularly related to root mean squared errors, magnitude, phase angle and peak values of the signals. The method for load reconstructing holds broad applicability, such as modeling validation, future load estimation, and structural damage prognosis.

Abstract

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