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Deep Learning of Vortex Induced Vibrations
Maziar Raissi, Zhicheng Wang, Michael S. Triantafyllou, George Em Karniadakis
TL;DR
The paper addresses difficult inference of lift and drag in vortex-induced fluid–structure interactions from sparse velocity or concentration observations, without pressure measurements. It uses coupled deep neural networks constrained by the incompressible Navier–Stokes and structural-motion equations. The framework reconstructs velocity, pressure, structural motion, and parameters accurately, while its demonstrated scope avoids chaotic and turbulent regimes.
Problem
The central problem is predicting lift and drag from limited, scattered velocity data or dye visualizations when pressure is unavailable, a difficult inverse data-assimilation task for standard CFD.
Method
Coupled deep neural networks encode the incompressible Navier–Stokes equations and structural dynamic equation in the loss function while learning fields and motion from scattered space-time data.
Results
The framework accurately infers structural parameters, pressure, velocity, structural motion, lift, and drag from velocity-and-motion data or concentration-field data.
Takeaways & Limitations
The approach supports velocity and pressure quantification from flow snapshots in small subdomains and can be used for flow control and system identification.
Takeaways & Limitations
The demonstrated framework avoids regimes where the Navier–Stokes equations become chaotic and turbulent because optimization of physics-informed residual losses may become difficult.
Abstract
from arXiv · showhide
Vortex induced vibrations of bluff bodies occur when the vortex shedding frequency is close to the natural frequency of the structure. Of interest is the prediction of the lift and drag forces on the structure given some limited and scattered information on the velocity field. This is an inverse problem that is not straightforward to solve using standard computational fluid dynamics (CFD) methods, especially since no information is provided for the pressure. An even greater challenge is to infer the lift and drag forces given some dye or smoke visualizations of the flow field. Here we employ deep neural networks that are extended to encode the incompressible Navier-Stokes equations coupled with the structure's dynamic motion equation. In the first case, given scattered data in space-time on the velocity field and the structure's motion, we use four coupled deep neural networks to infer very accurately the structural parameters, the entire time-dependent pressure field (with no prior training data), and reconstruct the velocity vector field and the structure's dynamic motion. In the second case, given scattered data in space-time on a concentration field only, we use five coupled deep neural networks to infer very accurately the vector velocity field and all other quantities of interest as before. This new paradigm of inference in fluid mechanics for coupled multi-physics problems enables velocity and pressure quantification from flow snapshots in small subdomains and can be exploited for flow control applications and also for system identification.
1. Introduction
The paper targets inverse inference of lift and drag in vortex-induced fluid–structure interactions from limited flow information, extending physics-informed deep learning to coupled multiphysics systems. Its neural networks encode governing equations and use scattered space-time data to reconstruct unmeasured fields and structural behavior.
- Motivation: Vortex-induced vibrations arise when bluff-body vortex shedding approaches the structure’s natural frequency, producing potentially damaging periodic motion.For an elastically mounted cylinder, motion is driven by lift and drag and can reach approximately 1D crossflow and 0.1D streamwise amplitudes.
- Challenges: Inverse coupled CFD is difficult because simulations face geometric, Reynolds-number, grid, and unavailable-structural-parameter constraints.The inverse problem can also be computationally prohibitive and ill-posed, while pressure information may be absent.
- Approach: The proposed approach extends physics-informed deep learning to coupled fluid–structure systems using neural networks for velocity, pressure, and structural motion.The governing equations are included in the loss function as regularization mechanisms, allowing noisy scattered spatio-temporal experimental inputs.
- Contribution: The work demonstrates feasibility and accuracy for inference and data assimilation while extending physics-informed learning toward coupled multiphysics problems.It builds on earlier studies using structured prior information for data-efficient, physics-informed learning machines.
- Approach: Physics-informed neural networks use automatic differentiation and physical constraints to address nonlinear problems without prior linearization or local time-stepping assumptions.They constrain learned functions using symmetries, invariances, and conservation principles from governing physical laws.
- Problem: The paper seeks lift and drag prediction from limited, scattered velocity measurements or dye and smoke visualizations of the flow.This is framed as a data assimilation problem for fluid–structure interactions and vortex-induced vibrations.
2. Problem Setup and Solution Methodology
The paper formulates VIV inference as a coupled fluid–structure inverse problem and uses neural networks constrained by structural and incompressible Navier–Stokes equations. It considers scattered velocity measurements and passive-scalar observations for reconstructing flow, pressure, motion, and forces.
- Problem setup: VIV is modeled for a circular cylinder as incompressible flow coupled to a mass-spring-damper structure moving in the crossflow direction.The displacement is η, while damping b and stiffness k may be unknown parameters.
- Pedagogical formulation: The structural equation is embedded in a neural-network loss, allowing displacement data to jointly estimate damping and stiffness through automatic differentiation.The loss combines displacement regression with the residual of the governing differential equation, and gradients train both network and structural parameters.
- Velocity measurements: The velocity-measurement problem targets full space-time velocity and pressure reconstruction from scattered data collected in small subdomains.Such data may be noisy, spatially and temporally scattered, and limited to a few thousand points or fewer.
- Velocity measurements: For scattered velocity measurements, one network outputs u, v, p, and η from t, x, and y, while residuals enforce momentum and incompressibility equations.The loss combines velocity and displacement data with the Navier–Stokes residuals; inferred pressure and velocity gradients yield lift and drag forces.
- Flow visualizations: For flow visualizations, the network additionally outputs passive-scalar concentration c and enforces its advection-diffusion equation alongside the Navier–Stokes and structural equations.Concentration observations are used to infer hidden velocity and pressure fields, after which lift and drag are computed from pressure and velocity gradients.
3. Results2
The study evaluates physics-informed neural networks for reconstructing fluid and structural quantities in vortex-induced vibrations from sparse measurements. Across velocity and concentration inputs, the framework recovers hidden pressure, forces, and structural parameters with reported accuracy, while network architecture remains an empirical design choice.
- Experimental setup: DNS generated 280 snapshots over the VIV domain, with Reynolds number Re = 100 and 4 million scattered space-time data points used in the velocity-measurement case.The simulations used a deforming-boundary spectral-element method and computed reference lift and drag forces.
- 3.1. A Pedagogical Example: 0.45% and 0.02% relative errors were obtained for damping b = 0.08438281 and stiffness k = 2.2015007 in the displacement-and-lift pedagogical example.The network also produced almost perfect reconstructions of the observed displacement and lift-force time series from N = 111 observations.
- 3.2. Inferring Lift and Drag Forces from Scattered Velocity Measurements: 10−3-order accuracy was reported for velocity reconstruction and pressure inference without any pressure observations in the velocity-measurement case.The predicted velocity and pressure fields were used to compute drag and lift forces, which were compared with exact forces.
- 3.2. Inferring Lift and Drag Forces from Scattered Velocity Measurements: 0.48% and 0.37% relative errors were obtained for damping b = 0.0844064 and stiffness k = 2.1938791 after estimating forces from velocity and pressure fields.The figure reports relative L2 errors for predicted velocity and pressure fields; weaker predictions occur at the initial and final time instants because training data are absent outside the sampled interval.
4. Discussion and Concluding Remarks
The work demonstrates physics-informed deep learning for coupled fluid–structure inference, while identifying computational and theoretical boundaries that motivate further study.
- Discussion and Concluding Remarks: The framework infers pressure, structural parameters, lift, and drag from scattered velocity, displacement, or passive-scalar data in coupled VIV problems.It uses governing equations as loss-function regularizers and couples multiple neural-network outputs for the fluid and structure.
- Discussion and Concluding Remarks: The authors state that efficiency, uniqueness, and accuracy questions for this modeling paradigm cannot yet be answered rigorously.They present these as theoretical and practical questions for future investigation.
- Discussion and Concluding Remarks: Conventional methods can require high-dimensional optimization, repeated solver runs, and grid regeneration when geometry is unknown.These requirements can make conventional approaches almost impractical for the corresponding inverse problems.
- Discussion and Concluding Remarks: The study avoids chaotic and turbulent Navier–Stokes regimes, so turbulent velocity and pressure inference remains a future extension.The stated setting assumes Newtonian, incompressible flow governed by the Navier–Stokes equations.