Data-free Non-intrusive Model Reduction for Nonlinear Finite Element Models via Spectral Submanifolds

TL;DR

The paper tackles non-intrusive, high-order model reduction for nonlinear FE systems by developing a STEP-based procedure to compute spectral submanifolds (SSMs) and their reduced dynamics using only black-box nonlinearities. The approach supports cubic nonlinearities, velocity-dependent forces, and asymmetric damping/stiffness, and is implemented in SSMTool2 with interfaces to commercial FE software like COMSOL. It demonstrates accurate ROM predictions for forced responses and bifurcations across challenging, large-scale FE examples, including MEMS with over a million DOFs, while achieving significant memory and time savings over intrusive methods. This work bridges rigorous SSM theory with practical FE practice, enabling data-free, physics-based reduction directly inside generic FE pipelines and offering a path toward broader deployment in engineering analyses and design optimization.

Abstract

The theory of spectral submanifolds (SSMs) has emerged as a powerful tool for constructing rigorous, low-dimensional reduced-order models (ROMs) of high-dimensional nonlinear mechanical systems. A direct computation of SSMs requires explicit knowledge of nonlinear coefficients in the equations of motion, which limits their applicability to generic finite-element (FE) solvers. Here, we propose a non-intrusive algorithm for the computation of the SSMs and the associated ROMs up to arbitrary polynomial orders. This non-intrusive algorithm only requires system nonlinearity as a black box and hence, enables SSM-based model reduction via generic finite-element software. Our expressions and algorithms are valid for systems with up to cubic-order nonlinearities, including velocity-dependent nonlinear terms, asymmetric damping, and stiffness matrices, and hence work for a large class of mechanics problems. We demonstrate the effectiveness of the proposed non-intrusive approach over a variety of FE examples of increasing complexity, including a micro-resonator FE model containing more than a million degrees of freedom.

Figures (12)

• Figure 1: The schematic of a shallow shell structure Jain2021HowModelspart-i.
• Figure 2: The forced response curve for the shallow shell structure with 1:2 internal resonance. Here, the solid and dashed lines denote stable and unstable periodic orbits predicted using non-intrusive SSM reduction, the red circles and magenta squares represent stable and unstable periodic orbits obtained from intrusive SSM reduction, the cyan circles denote saddle-node (SN) bifurcation points, and black squares denote Hopf bifurcation (HB) points. All results here are obtained from $\mathcal{O}(5)$ approximations of the SSM.
• Figure 3: (Left panel) Model showing the interior structure of the wing. These ribs are then covered with plates to create the model considered here Jain2021HowModels. (Right panel) The full model of the wing is covered with plates. Boundary conditions are chosen, such as fixing the end of the wing as if attached to an aircraft.
• Figure 4: Forced response curve regarding the tip deflection of the aircraft wing. In the upper-left panel, we present the results obtained from $\mathcal{O}(3)$ approximation with and without the contribution of the leading-order non-autonomous part of SSM. Specifically, TV and TI stand for time-varying and time-independent solutions with and without the contribution. The results predicted from SSM-based reductions truncated at various expansion orders are shown in the upper-right panel. The lower panel presents the results at $\mathcal{O}(7)$ expansion, obtained using both the intrusive computation in li2023nonlinear and our proposed non-intrusive algorithm.
• Figure 5: A schematic plot of the mesh for FE model of the perforated cover plate.

Theorems & Definitions (1)

• Remark 1