A local basis approximation approach for nonlinear parametric model order reduction

TL;DR

This paper develops a physics-based parametric MOR (pMOR) framework for nonlinear structural dynamics, focusing on material nonlinearity and parametric excitation. It introduces a local-basis POD approach with interpolation on the Grassmann manifold via a coefficient matrix Xi to capture unseen parameter configurations, coupled with ECSW hyper-reduction to dramatically reduce online cost. The method is validated on a Bouc-Wen two-story frame and a 3D wind-turbine tower under earthquake excitation, showing that local-basis and coefficient- interpolation pROMs can closely match high-fidelity responses while achieving substantial speedups. The work enables efficient, accurate condition assessment, monitoring, control, and residual life estimation for large-scale nonlinear structures under varying load spectra, advancing practical digital-twin and online diagnostics capabilities.

Abstract

The efficient condition assessment of engineered systems requires the coupling of high fidelity models with data extracted from the state of the system `as-is'. In enabling this task, this paper implements a parametric Model Order Reduction (pMOR) scheme for nonlinear structural dynamics, and the particular case of material nonlinearity. A physics-based parametric representation is developed, incorporating dependencies on system properties and/or excitation characteristics. The pMOR formulation relies on use of a Proper Orthogonal Decomposition applied to a series of snapshots of the nonlinear dynamic response. A new approach to manifold interpolation is proposed, with interpolation taking place on the reduced coefficient matrix mapping local bases to a global one. We demonstrate the performance of this approach firstly on the simple example of a shear-frame structure, and secondly on the more complex 3D numerical case study of an earthquake-excited wind turbine tower. Parametric dependence pertains to structural properties, as well as the temporal and spectral characteristics of the applied excitation. The developed parametric Reduced Order Model (pROM) can be exploited for a number of tasks including monitoring and diagnostics, control of vibrating structures, and residual life estimation of critical components.

Figures (11)

• Figure 1: Graphical description of the pROM local bases interpolation approach implemented on this paper. The green surface represents the Grassmann Manifold ($M$) and the circular points denoted by $\mathbf{V}_{\mathrm{i}}, i=1,2,3$ the training samples. The validation parametric sample is denoted by $\mathbf{V}_4$ and the Exp and Log notations refer to the mapping processes to and from the manifold Zimmer. The orange surface is the tangent space drawn to point $\mathbf{V_{\mathrm{ref}}}$. The representation is inspired from Amsal.
• Figure 2: Two story building with nonlinear links. Geometrical configuration depicted in grey and example deformed state in black. Domain sampling and partitioning approaches examined are also visualized.
• Figure 3: Domain error plot for Bouc Wen model for Partition A and Partition B for the Coefficients Interpolation pROM (Table \ref{['tableref']}). The $\mathbb{R}\mathbb{E}_{\mathbf{rf}}$ error of Equation \ref{['errornorm']} is evaluated with respect to the approximation of the restoring forces $rf$. A 3D and a 2D projection error plot are provided.
• Figure 4: Accuracy of the proposed Coefficients Interpolation pROM (Table \ref{['tableref']}) for Partition B on capturing the variation of the hysteretic component. The approximation accuracy on a response time history and on three different shape of the hysteresis curve are depicted to demonstrate the potential of the method.
• Figure 5: Finite Element Representation of the HFM