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Data-Assisted Non-Intrusive Model Reduction for Forced Nonlinear Finite Elements Models

Mattia Cenedese, Jacopo Marconi, George Haller, Shobhit Jain

TL;DR

This paper develops a data-driven, nonintrusive ROM framework based on Spectral Submanifolds to predict forced responses of nonlinear FE models from unforced simulations. By leveraging the linear FE information and learning the nonlinear SSM parametrization and reduced dynamics, the method yields accurate forced-response curves and backbone/damping characteristics, even in internal resonance scenarios. The approach is validated on a spectrum of FE models—from beams and shells to a MEMS device with over a million DOFs—demonstrating substantial computational savings (offline data generation) while preserving predictive accuracy (NMTE typically within a few percent). The work advances nonlinear model reduction for large-scale structures, enabling fast, reliable forced-response analysis under periodic and quasi-periodic forcing.

Abstract

Spectral submanifolds (SSMs) have emerged as accurate and predictive model reduction tools for dynamical systems defined either by equations or data sets. While finite-elements (FE) models belong to the equation-based class of problems, their implementations in commercial solvers do not generally provide information on the nonlinearities required for the analytical construction of SSMs. Here, we overcome this limitation by developing a data-driven construction of SSM-reduced models from a small number of unforced FE simulations. We then use these models to predict the forced response of the FE model without performing any costly forced simulation. This approach yields accurate forced response predictions even in the presence of internal resonances or quasi-periodic forcing, as we illustrate on several FE models. Our examples range from simple structures, such as beams and shells, to more complex geometries, such as a micro-resonator model containing more than a million degrees of freedom. In the latter case, our algorithm predicts accurate forced response curves in a small fraction of the time it takes to verify just a few points on those curves by simulating the full forced-response.

Data-Assisted Non-Intrusive Model Reduction for Forced Nonlinear Finite Elements Models

TL;DR

This paper develops a data-driven, nonintrusive ROM framework based on Spectral Submanifolds to predict forced responses of nonlinear FE models from unforced simulations. By leveraging the linear FE information and learning the nonlinear SSM parametrization and reduced dynamics, the method yields accurate forced-response curves and backbone/damping characteristics, even in internal resonance scenarios. The approach is validated on a spectrum of FE models—from beams and shells to a MEMS device with over a million DOFs—demonstrating substantial computational savings (offline data generation) while preserving predictive accuracy (NMTE typically within a few percent). The work advances nonlinear model reduction for large-scale structures, enabling fast, reliable forced-response analysis under periodic and quasi-periodic forcing.

Abstract

Spectral submanifolds (SSMs) have emerged as accurate and predictive model reduction tools for dynamical systems defined either by equations or data sets. While finite-elements (FE) models belong to the equation-based class of problems, their implementations in commercial solvers do not generally provide information on the nonlinearities required for the analytical construction of SSMs. Here, we overcome this limitation by developing a data-driven construction of SSM-reduced models from a small number of unforced FE simulations. We then use these models to predict the forced response of the FE model without performing any costly forced simulation. This approach yields accurate forced response predictions even in the presence of internal resonances or quasi-periodic forcing, as we illustrate on several FE models. Our examples range from simple structures, such as beams and shells, to more complex geometries, such as a micro-resonator model containing more than a million degrees of freedom. In the latter case, our algorithm predicts accurate forced response curves in a small fraction of the time it takes to verify just a few points on those curves by simulating the full forced-response.
Paper Structure (17 sections, 1 theorem, 48 equations, 10 figures, 1 table)

This paper contains 17 sections, 1 theorem, 48 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Setup
  3. Spectral submanifolds and their properties
  4. Learning SSMs from data
  5. Setting up graph-style approaches
  6. Autonomous SSM geometry and reduced dynamics
  7. Including external forcing in the reduced-order model
  8. Examples
  9. Von Kármán beam
  10. Prismatic beam in 1:3 internal resonance
  11. Von Kármán shells with and without 1:2 internal resonance
  12. Von Kármán plate in 1:1 internal resonance
  13. High-dimensional FE model of a MEMS device
  14. Conclusion
  15. Non-modal graph style parametrizations
  16. ...and 2 more sections

Key Result

Proposition B.1

If the rank of $\mathbf{Y}_{\mathrm{nl}}$ is equal to $n_{2:M}$, the optimal solution in Eq. (eq:regressparamat) is unique and always such that $\mathbf{W}_0\mathbf{v}_{\mathrm{nl}}(\mathbf{y})\equiv \mathbf{0}$.

Figures (10)

  • Figure 1: Illustration of parametrization of an autonomous invariant manifold $\mathcal{W}_0(E^{2m})$ (in blue, with a trajectory on it) using the tangent space at the origin, being the spectral subspace $E^{2m}$ (in orange, with the projected trajectory on it).
  • Figure 2: Plot (a) shows the finite element discretization of the beam, including the equilibrium position and the static deflection when subject to midpoint loading. The force displacement relation of such static loading is instead shown in plot (b), distinguishing the linear (black dashed line) and nonlinear case (blue line), and plotting the midpoint displacement. Plot (c) shows the test trajectory from the numerical simulation of the full model (blue curve) and its prediction (red line) from the SSMLearn reduced-order model. This reduced-order model predicts the backbone curves shown in plot (d,e), in terms of damping and frequency, where the frequency is compared to that extracted by processing the training trajectory with the method of Peak Finding and Fitting, Jin2020. Plot (f) shows the SSM in the physical space along with the training trajectory, where $q$ and $q_a$ are the transverse and longitudinal displacements shown in plot (a), respectively. Plots (g,h) show forced responses in terms of amplitude and phase of $q$ computed via SSMTool (green) and SSMLearn (dark red) for two forcing amplitude values.
  • Figure 3: Plot (a) shows the forced response curve, whose green line depict the frequency to which plot (b) refers, where a section of the three dimensional forced SSM is shown, along with the periodic orbits (red and blue dots), the stable (blue line) and unstable (red line) manifolds of the saddle (red point), and two trajectories (grey lines) coverging to the two attractors. Plots (c,d) show two periodically forced trajectories and their predictions converging to the same attractor from different initial conditions, as shown in plot (a). Plots (e,g,i) show three quasi-periodically forced trajectories with initial condition being the origin, whose forcing is shown in plots (f,h,j), respectively.
  • Figure 4: Plot (a) shows the evaluation of the internal force field nonlinearities under varying imposed modal displacement field, for the first (left) and second mode (right). Plot (c) shows the decaying trajectories in the two modal coordinates obtained from the full-order model (in blue) and their prediction (dark red) from the data-driven reduced-order model and plot (b) shows the spectrogram trajectory in blue on the top left plot of (c). Plots (d,e) show the amplitudes of the first two modal coordinates of forced responses computed via SSMTool and SSMLearn for two forcing amplitude values near the lowest eigenfrequency.
  • Figure 5: Plot (a) shows the finite element discretization of the shallow-arc reference position and the two probe points whose transverse displacements are denoted $q_A$ and $q_B$. Plots (b,c) show the displacement field for the first two conservative mode shapes of configuration (i) without internal resonance, while plots (d,e) show these shapes for configuration (ii) with $1:2$ internal resonance. The plots also show the natural frequencies in Hz and the damping ratios.
  • ...and 5 more figures

Theorems & Definitions (2)

  • Proposition B.1
  • proof