Source-linked AI summary
Modelling non-linear aeroelastic loads in long-span bridges with extreme learning machines
Gledson Rodrigo Tondo, Samir Chawdhury, Sergio Andres Castro Giraldo, Guido Morgenthal
TL;DR
Long-span bridge safety depends on accurately modelling self-excited aerodynamic forces, but existing approaches have limitations in data sensitivity, computational cost, and nonlinear representation. The paper trains ELMs on forced-oscillation data and reports improved nonlinear-force accuracy and reliable aerodynamic-load and flutter predictions.
Problem
Self-excited forces can trigger flutter, while linear aerodynamic-derivative models are sensitive to identification data and may vary in estimating forces near instability.
Source: §1 · p. 2Method
The framework trains extreme learning machines on forced-oscillation data from CFD or wind-tunnel tests to model nonlinear aerodynamic loads.
Source: §1 · p. 2, §5 · p. 14Results
ELMs improve accuracy in capturing aerodynamic nonlinearities over semi-analytical models and provide reliable load and flutter predictions, including onset wind speed and coupled frequency content.
Source: §5 · p. 14Takeaways & Limitations
The authors conclude that ELMs offer a fast, accurate alternative for aerodynamic evaluation that can be integrated into early bridge analysis and design.
Source: §5 · p. 14The trained ELM models did not reproduce the limit-cycle oscillation observed in CFD simulations.
Source: §4.2 · p. 10
Abstract
from arXiv · showhide
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.
Notation
The notation covers ELM training and network quantities, bridge-motion and aerodynamic variables, and flutter and training parameters.
Notation
- ELM notation distinguishes the training data, outputs, hidden values, weights, activation function, SVD matrices, regularization parameter, and aerodynamic forces.It also names lift and moment coefficients and vertical displacement, velocity, and acceleration.Source: Notation · p. 1
- Bridge-aerodynamics symbols include angular motion, air density, wind speed, section width, reduced velocity, oscillation frequency, and non-dimensional frequency.The notation also includes Scanlan derivatives, Reynolds number, structural masses and natural frequencies, damping ratio, and time.Source: Notation · p. 1
- Flutter and model-size notation covers critical flutter speed and frequency, training-point count, input dimensions, and neuron count.The notation also defines the number of oscillation cycles and training points per cycle.Source: Notation · p. 2
1 Introduction
The introduction motivates accurate self-excited-force modelling for bridge safety, reviews limitations in existing approaches, and presents ELMs as a data-driven alternative.
1 Introduction
- Self-excited forces can lead to flutter and pose structural risks, making their prediction important for long-span bridge safety under changing wind conditions.The passage links concern about higher wind speeds to extreme weather patterns.Source: §1 · p. 2
- Linear aerodynamic-derivative models are sensitive to identification data, while nonlinear and data-driven alternatives can be computationally intensive or depend on modelling assumptions.The introduction also notes that physics-informed approaches require a physical description and are not straightforward to implement.Source: §1 · p. 2
- The proposed ELM framework uses forced-oscillation CFD or wind-tunnel data to retain nonlinear load behaviour while reducing training and prediction costs.The introduction says these datasets can be available from preliminary testing and used to predict coupled structural–aerodynamic responses.Source: §1 · p. 2
2 ELM models for aerodynamic self-excited forces
The section proposes ELMs as a data-driven bridge between efficient but assumption-dependent models and costly CFD or wind-tunnel analysis, trained on forced-oscillation data.
2.1 Problem statement
- The framework targets the gap between efficient, assumption-dependent semi-analytical models and more precise but computationally intensive CFD or resource-intensive wind-tunnel methods.It aims to use CFD or wind-tunnel training data to capture nonlinear and fluid-memory effects, reproduce their outputs efficiently, and support coupled-response analysis from simplified forced-vibration data.Source: §2.1 · p. 3
2.2 Extreme learning machine models
- ELMs randomly initialize and freeze hidden-layer parameters, then analytically fit output weights by least squares rather than iteratively optimizing network weights.Tikhonov regularization penalizes large weights to address overfitting, especially when the neuron count is large.Source: §2.2 · pp. 3–4
- For new inputs, the ELM computes hidden values and applies the fitted output weights to predict outputs.Training fits the output weights to targets through a generalized inverse; regularized parameter values can be selected by cross-validation or L-curves.Source: §2.2 · p. 4
2.3 ELM application to self-excited forces in bridge aerodynamics
- The bridge-aerodynamics models map vertical or rotational motion and its time derivatives to non-dimensional aerodynamic-force values, using four ELMs for motion–force combinations.Training directly on CFD data, with extension to wind-tunnel data, transfers the input-motion relation to complex aerodynamic data.Source: §2.3 · p. 5, Figure 2 · p. 5
2.4 Design of experiments
- Forced-vibration datasets use sinusoidal heave or pitch at selected amplitudes and reduced velocities, with sampling and cycle counts adapted to data noise and nonlinear force content.The study uses more cycles and training points for noisy, nonlinear CFD data; its training ranges are selected to cover expected motion and frequency conditions.Source: §2.4 · pp. 5–6, Table 1 · p. 6
- Cross-validation selects regularization and neuron count by testing candidate settings across data partitions, minimizing prediction error while preferring smaller networks.Evaluation metrics compare both global and local signal properties against analytical or CFD results.Source: §2.4 · p. 6
3 Analytical flat plate benchmark case
The flat-plate benchmark tests ELM predictions against analytical forces, including aerodynamic derivatives and forces under pseudo-random motion. The model agrees well with the analytical results, with some metric differences between vertical and rotational motion.
3.1 Model training
- The analytical training set uses noise-free harmonic forces at reduced velocities from 2 to 16 in steps of 2, with one cycle, 210 neurons, and no regularization.The example uses vertical amplitude 1.0 m and rotational amplitude 1.0 deg.Source: §3.1 · p. 7, Figure 3 · p. 7
3.2 Predictions of aerodynamic derivatives and random motion forces
- ELM aerodynamic derivatives agree well with the analytical flat-plate solution, with minor discrepancies at high reduced velocities as forces approach a quasi-steady state.The derivatives are obtained by fitting ELM-predicted forces to Scanlan’s formulation.Source: §3.2 · p. 7, Figure 4 · p. 8
- Using only harmonic-motion training data, the ELM predicts analytical lift and moment forces well for pseudo-random vertical and rotational inputs.The analytical forces for pseudo-random input are obtained by convolving the motion with Wagner’s indicial function.Source: §3.2 · p. 7, Figure 5 · p. 8
- Comparison metrics are generally close to unity, although vertical-motion predictions are generally more accurate than rotational predictions, particularly for magnitude, phase, and coefficient of determination.RMSE, correlation, wavelet, and peak-amplitude metrics show strong agreement with the analytical forces.Source: §3.2 · p. 7, Figure 6 · p. 9
4 Application in bridge aerodynamics
The Great Belt East Bridge application trains ELMs on forced-oscillation CFD data and tests their predictions of nonlinear forces, aerodynamic derivatives, and flutter. ELM estimates the flutter limit speed with 0.8% error against CFD and runs much faster than CFD, but it does not reproduce the CFD limit-cycle oscillation.
4.1 Model training
- Bridge-deck training uses forced harmonic CFD data across reduced velocities 2–16 and rotational amplitudes 1–12 deg, with 2,500 training points per velocity-amplitude combination.The trained model uses 1,890 neurons and a Tikhonov regularization parameter of 3 × 10^-7.Source: §4.1 · p. 9, Figure 7 · p. 9, Figure 8 · p. 10
4.2 Prediction of aerodynamic derivatives and flutter instability
- ELM-derived aerodynamic derivatives align well with CFD-derived results and can be estimated at any reduced velocity within the training range without additional CFD simulations.The ELM-generated forced motions are fit to Scanlan’s model by least squares.Source: §4.2 · p. 10
4.1 Model training
- At 10 deg amplitude and reduced velocity 16, ELM captures nonlinear force harmonics that the linear Scanlan model misses, and improves magnitude, peak-amplitude, coefficient-of-determination, and wavelet metrics against CFD.Scanlan remains a reasonable approximation for RMSE, correlation, and phase angle, which are primarily influenced by the dominant frequency.Source: §4.1 · pp. 9–10, Figure 9 · p. 11, Figure 10 · p. 12
4.2 Prediction of aerodynamic derivatives and flutter instability
- The ELM flutter-limit-speed error is 0.8% against CFD; only the hybrid nonlinear model performs better, while LQS, QS, CQS, and MQS errors exceed 5% and LU error is 2.2%.Against wind-tunnel results, LU, HNL, CFD, and ELM are all in the range considered critical for flutter onset.Source: §4.2 · p. 11
4.3 Computational performance
- ELM training takes 7.6 seconds, compared with 665.2 seconds for a single-layer feedforward network at the same neuron count and comparable RMS-error target.Gaussian process regression takes over 104 seconds to train.Source: §4.3 · pp. 11–12
4.2 Prediction of aerodynamic derivatives and flutter instability
- At U = 72.5 m/s, CFD flutter exhibits a limit-cycle oscillation; ELM predicts high-amplitude self-excited forces and coupled-flutter frequency content, slightly overestimating lift and underestimating moment.Time-domain ELM analysis estimates flutter onset at U = 71.6 m/s; ELM simulations decay below onset and diverge above it. The ELM models do not reproduce the CFD limit-cycle oscillation.Source: §4.2 · p. 10
4.3 Computational performance
- ELM flutter simulations average 6.2 seconds, or 0.0081% of the 7.7 × 10^4 seconds required by CFD on the same hardware.The paper identifies rapid scenario exploration as valuable in early bridge design and structural optimization.Source: §4.3 · p. 13
5 Conclusion
The paper presents an ELM framework for modelling aerodynamic self-excited forces from forced-oscillation data. It reports accurate flutter and load predictions, efficient analysis, and future extensions to additional motions and load types.
5 Conclusion
- The framework uses extreme learning machines, a class of single-layer feedforward neural networks, to model aerodynamic self-excited forces and instabilities from forced-oscillation datasets.Training data can be selected systematically from CFD simulations or wind-tunnel experiments.Source: §5 · pp. 13–14
- The Great Belt East Bridge coupled ELM-structural analysis indicates flutter onset at 71.6 m/s, with aerodynamic-load and coupled-frequency predictions agreeing closely with CFD or wind-tunnel results.Figure 15 compares ELM flutter limits and coupled-motion frequencies with semi-analytical models, CFD, and wind-tunnel results.Source: Figure 14 · p. 14, §5 · p. 14, Figure 15 · p. 14
- ELMs achieve accuracy comparable to similar-sized standard neural networks with a fraction of their training time, and predict loads much faster than CFD or wind-tunnel experiments.This enables fast aerodynamic analysis of structures.Source: §5 · p. 14
- The authors identify 3D structures, along-wind motion, drag, and buffeting under gusty turbulent winds as possible future extensions of the framework.The current conclusion presents these as directions for further research.Source: §5 · pp. 14–15