Efficacy of the Weak Formulation of Sparse Nonlinear Identification in Predicting Vortex-Induced Vibrations

Abstract

Vortex-induced vibrations (VIV) remain a canonical yet complex manifestation of fluid-structure interactions, where coupled nonlinear dynamics govern the motion of bluff bodies. For several years, we have relied on traditional reduced-order mathematical models derived from empirical and oscillator-based formulations; however, such models often fail to reproduce the quantitative dynamics observed in realistic flow environments. In this study, we explore a data-driven framework that leverages sparse identification of nonlinear dynamics (SINDy) and its weak formulation to uncover the governing equations of a single-degree-of-freedom cylinder undergoing VIV, using both data generated from previously developed reduced-order models and high-fidelity simulation results to assess the interpretation and efficacy of models discovered from a purely data-driven approach, particularly when the underlying dynamics are not fully known. The weak formulation (WSINDy), which replaces numerical differentiation with an integral-based representation, demonstrates marked robustness for aperiodic dynamics in particular. A complementary analysis using proper orthogonal decomposition (POD) is employed to extract the dominant spatio-temporal structures of the flow and to assess whether the temporal evolution of the wake can be represented on a reduced-dimensional manifold. The findings establish that data-driven identification can recover interpretable, quantitatively reliable models of VIV, providing a robust and computationally efficient pathway for modelling fluid-structure interactions directly from data. In particular, WSINDy is shown to be a more robust and interpretable alternative to standard SINDy for discovering VIV equations from aperiodic response dynamics, paving the way for predictive, data-informed design of fluid-structure interaction systems.

Figures (28)

• Figure 1: (a) Comparison between SINDy-predicted and numerically integrated non-dimensional structural amplitude ($y_0=\frac{A_y}{D}$, where $A_y$ is the dimensional amplitude) values obtained from the wake-oscillator model for varying reduced velocity $U_r$; (b) percentage relative error ($\epsilon_{y_0}$) in the SINDy-predicted $y_0$ for different $U_r$ values.
• Figure 2: Time histories of (a) $y_1$ and (b) $q_1$ for $U_r = 5$; and (c) $y_1$ and (d) $q_1$ for $U_r = 15$. A shorter time range is plotted to more clearly highlight the dynamics.
• Figure 3: Comparison of dominant oscillation frequencies of $y(=Y/D)$ from SINDy and numerical integration for (a) $U_r = 5$ and (b) $U_r = 15$.
• Figure 4: CFD results for the transverse displacement of the elastically mounted 1-DoF circular cylinder across reduced velocities $U_r \in [3.0, 5.5]$. (a) Time histories of the dimensionless transverse displacement $y$ as a function of dimensionless time $t^*$, coloured by reduced velocity $U_r$. (b) Three-dimensional representation of the fast Fourier transform (FFT) spectral content of $y$, showing the amplitude spectrum $|y|$ as a function of dimensionless frequency $f^*$ for each $U_r$, illustrating the evolution of the dominant shedding frequency across the reduced-velocity range.
• Figure 5: Comparison of the CFD simulation, SINDy, and WSINDy reconstructions for the 1DOF single-cylinder vortex-induced vibration (VIV) system in aperiodic regime. The time histories of the transverse displacement $y_1$ and the wake variable $q_1$ are shown for (a–b) $U_r = 3$, (c–d) $U_r = 3.5$.