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Nonlinear Model Reduction to Random Spectral Submanifolds in Random Vibrations

Zhenwei Xu, Roshan S. Kaundinya, Shobhit Jain, George Haller

TL;DR

The paper tackles the challenge of obtaining reliable statistics for high-dimensional nonlinear systems subjected to bounded random excitations. It extends the deterministic spectral submanifold (SSM) reduction framework to random forcing by proving the existence of random SSMs $\mathcal{W}_{\varepsilon}(E;\boldsymbol{\theta}^{t}(\boldsymbol{\nu}))$ that are $C^{\rho(E)}$-smooth and attract trajectories, with reduced dynamics that perturb the deterministic SSM by a leading-order random term. The authors derive leading-order reduced equations and demonstrate, via four mechanical examples, that Monte Carlo simulations on the low-dimensional random SSMs reproduce full-model statistics with substantial speed-ups. Methodologically, the work integrates a rigorous random-invariant-manifold theory with equation-driven (SSMTool) and data-driven (SSMLearn) reductions, and leverages the bounded-noise assumption to maintain physically realistic forcing. The results offer a practically impactful pathway to efficient stochastic analysis for complex engineering systems, with potential extensions to higher-order corrections and data-driven random SSMs. All claims are framed with the mathematical structure of random invariant manifolds and reduced dynamics, enabling rigorous yet computationally efficient stochastic modeling.

Abstract

Dynamical systems in engineering and physics are often subject to irregular excitations that are best modeled as random. Monte Carlo simulations are routinely performed on such random models to obtain statistics on their long-term response. Such simulations, however, are prohibitively expensive and time consuming for high-dimensional nonlinear systems. Here we propose to decrease this numerical burden significantly by reducing the full system to very low-dimensional, attracting, random invariant manifolds in its phase space and performing the Monte Carlo simulations on that reduced dynamical system. The random spectral submanifolds (SSMs) we construct for this purpose generalize the concept of SSMs from deterministic systems under uniformly bounded random forcing. We illustrate the accuracy and speed of random SSM reduction by computing the SSM-reduced power spectral density of the randomly forced mechanical systems that range from simple oscillator chains to finite-element models of beams and plates.

Nonlinear Model Reduction to Random Spectral Submanifolds in Random Vibrations

TL;DR

The paper tackles the challenge of obtaining reliable statistics for high-dimensional nonlinear systems subjected to bounded random excitations. It extends the deterministic spectral submanifold (SSM) reduction framework to random forcing by proving the existence of random SSMs that are -smooth and attract trajectories, with reduced dynamics that perturb the deterministic SSM by a leading-order random term. The authors derive leading-order reduced equations and demonstrate, via four mechanical examples, that Monte Carlo simulations on the low-dimensional random SSMs reproduce full-model statistics with substantial speed-ups. Methodologically, the work integrates a rigorous random-invariant-manifold theory with equation-driven (SSMTool) and data-driven (SSMLearn) reductions, and leverages the bounded-noise assumption to maintain physically realistic forcing. The results offer a practically impactful pathway to efficient stochastic analysis for complex engineering systems, with potential extensions to higher-order corrections and data-driven random SSMs. All claims are framed with the mathematical structure of random invariant manifolds and reduced dynamics, enabling rigorous yet computationally efficient stochastic modeling.

Abstract

Dynamical systems in engineering and physics are often subject to irregular excitations that are best modeled as random. Monte Carlo simulations are routinely performed on such random models to obtain statistics on their long-term response. Such simulations, however, are prohibitively expensive and time consuming for high-dimensional nonlinear systems. Here we propose to decrease this numerical burden significantly by reducing the full system to very low-dimensional, attracting, random invariant manifolds in its phase space and performing the Monte Carlo simulations on that reduced dynamical system. The random spectral submanifolds (SSMs) we construct for this purpose generalize the concept of SSMs from deterministic systems under uniformly bounded random forcing. We illustrate the accuracy and speed of random SSM reduction by computing the SSM-reduced power spectral density of the randomly forced mechanical systems that range from simple oscillator chains to finite-element models of beams and plates.
Paper Structure (13 sections, 1 theorem, 43 equations, 9 figures, 3 tables)

This paper contains 13 sections, 1 theorem, 43 equations, 9 figures, 3 tables.

Table of Contents

  1. Introduction
  2. SSMs in randomly forced systems
  3. Setup
  4. The unforced system ($\epsilon=0$)
  5. The forced system ($\epsilon>0$)
  6. Examples
  7. Suspension system moving on an irregular road
  8. Earthquake response of a building
  9. Random base excitation of a von Kármán beam
  10. Von Kármán plate subject to a stochastic pressure field
  11. Conclusion
  12. Proof of Theorem 1
  13. Spectral density for linear systems

Key Result

Theorem 1

Assume that Then, for $\epsilon>0$ small enough: (i) The random ODE (eq:random ODE) has a class-$C^{\rho\left(E\right)}$, random inflowing-invariant manifold $\mathcal{W}_{\epsilon}(E;\boldsymbol{\theta}^{t}\left(\boldsymbol{\nu}\right))$, which can locally be written as where $\mathbf{h}_{1}\left(\boldsymbol{\xi};\boldsymbol{\theta}^{t}(\boldsymbol{\nu}),\epsilon\right)$ is measurable in $\bold

Figures (9)

  • Figure 1: The geometry of the slow spectral subspace $E$, the deterministic spectral submanifold $\mathcal{W}_{0}(E)$ and the random invariant manifold $\mathcal{W}_{\epsilon}\left(E;\boldsymbol{\theta}^{t}\left(\boldsymbol{\nu}\right)\right)$.
  • Figure 2: Left: An illustration of the suspension model of quarter-car moving with speed $v$ along an irregular road of length $L$ . Right: A side profile of the model with system parameters shown.
  • Figure 3: PSD $\phi_{\mathbf{x}}^{x_{s}}$ of the displacement of mass $m_{s}$ obtained from the SSM-reduced order model (blue) and for the full order model simulation (yellow) for two regimes of road irregularities. We also plot in red the analytic PSD for the linear system. (a) $\epsilon=1.5$ indicates minor road elevations (b) $\epsilon=2.5$ the road irregularities are amplified.
  • Figure 4: Schematic model of $n$-storey building under seismic excitation.
  • Figure 5: PSD $\phi_{\mathbf{x}}^{u_{10}}$ of the roof displacement of the building, from the SSM-reduced model (blue) and from the full simulation (yellow) for two earthquake magnitudes. We also plot in red the analytic PSD for the linear system. (a) $\epsilon=0.5$ models small ground acceleration intensity. (b) $\epsilon=1$ models large ground acceleration intensity.
  • ...and 4 more figures

Theorems & Definitions (8)

  • Theorem 1
  • proof
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Definition 1