Non-Intrusive Data-Free Parametric Reduced Order Model for Geometrically Nonlinear Structures

TL;DR

The paper tackles the challenge of efficiently analyzing geometrically nonlinear structures under geometric variations by introducing a fully non-intrusive Parametric Reduced Order Model (PROM). It builds a database of equation-driven ROMs using linear VMs plus nonlinear companions (SMDs/DMs), then interpolates all reduced operators with Radial Basis Functions to rapidly generate PROMs for new parameters while preserving the ROM's polynomial structure and symmetry. The method provides analytical parameter sensitivities and demonstrates substantial online speed-ups (up to several thousand-fold) with good accuracy on curved-panel and wing-box examples, making it well-suited for uncertainty quantification, optimization, and model updating. The work also discusses practical considerations such as continuity of the RB across parameters, MAC-based reordering to track modal branches, and limitations tied to sampling density and topology-preserving morphing.

Abstract

We present a fully non-intrusive parametric reduced-order modeling (PROM) framework for geometrically nonlinear structures subject to geometric variations. The method builds upon equation-driven Galerkin ROMs constructed from vibration modes and modal-derivative companion vectors, while nonlinear reduced tensors are identified from standard finite element outputs. A database of such ROMs is generated over a set of training samples, and all reduced operators-including the linear stiffness matrix, the quadratic and cubic nonlinear tensors, the Rayleigh damping parameters, and the reduction basis-are interpolated using Radial Basis Functions (RBFs). A global reduced basis is obtained through a two-level POD compression, combined with a MAC-guided reordering strategy to ensure parametric smoothness. The resulting PROM preserves the symmetry and polynomial structure of the reduced equations, enabling robust and efficient adaptation to new parameter values. Analytical parameter sensitivities follow directly from the interpolation model. The approach is demonstrated on a parametrically curved panel and a wing-box with geometric variations, showing excellent agreement with high-fidelity simulations and enabling substantial reductions in computational cost for parametric analyses.

Figures (12)

• Figure 2: In (a) geometry of the flat panel. In (b) the parameters-dependent shape functions used to define the height of the plate. In (c), realization of the mesh for $p_1 = 1.3 \text{t}, \ p_2 = 0.89 \text{t}, \text{and}\ p_3 = 1.6$ (In the illustration, the out of plane $z$ coordinates are scaled by a factor of 10).
• Figure 3: Training, validation and test samples of the curved panel.
• Figure 4: Energy lost in SVD data compression using different number of left singular vectors for $\mathbf{\Phi_{G}}$ (a) and $\mathbf{\Theta_{G}}$ (b).
• Figure 5: In (a) uniform pressure load time history, in (b) out-of-plane displacement time history of the midspan (normalized to the panel thickness) for the individual parameter realizations in the test set.
• Figure 6: Out-of-plane displacement time history at panel midspan for different geometrical configurations in the test set.