On the Coupling of Reduced Order Modeling with Substructuring of Structural Systems with Component Nonlinearities

Abstract

The emergence of digital virtualization has brought Reduced Order Models (ROM) into the spotlight. A successful reducedorder representation should allow for modeling of complex effects, such as nonlinearities, and ensure validity over a domain of inputs. Parametric Reduced Order Models (pROMs) for nonlinear systems attempt to accommodate both previous requirements [1]. Our work addresses a physics-based reduced representation of structural systems with localized nonlinear features. Via implementation of the approach described in [2], we achieve a substructuring formulation similar to Component Mode Synthesis. However, instead of individual component modes, reduction modes of a global nature are obtained from the corresponding linear monolithic system. In turn, this technique allows for a divide and conquer strategy that naturally couples the response between linear and nonlinear subdomains. A pROM able to exploit this modular formulation is developed as a next step. The framework treats each component independently, and individual projection bases are assembled by applying a Proper Orthogonal Decomposition (POD) on snapshots of each subdomain's response. Parametric dependencies on the boundary conditions and the structural and excitation traits are injected into the pROM utilizing clustering, following the methodology in [3]. To this end, a modified strategy is employed, relying on the Modal Assurance Criterion as a measure to indicate optimal training samples while defining regions with similar underlying dynamics on the domain of parametric inputs. A numerical case study of a 3D wind turbine tower featuring material nonlinearities exemplifies our approach. The derived pROM offers an accelerated approximation of the underlying high fidelity response and can be utilized for numerous tasks, including vibration control, residual life estimation, and condition assessment of hotspot locations.

A Physics-based Reduced Order Model with Machine Learning Boosted Hyper-Reduction.

“…In Figure 1, the MAC guided clustering converged based on the maximum number of clusters allowed. As demonstrated in [23], the random sampling sequence does not influence the overall results. Figure 1b highlights that although our sampling assumes no prior knowledge of how dynamics evolve in the input domain, the MAC measure is able to indicate regions of relatively low accuracy and add samples on the training set.…”

“…The geometric configuration, damping and material properties correspond to the published standardized template. Previous versions of the example can be found in [19,12,23].…”

“…In our previous work we have already successfully utilized the Modal Assurance Criterion (MAC) [22] ( also known as vector cosine or cosine similarity measure ) as a clustering metric to perform greedy sampling during the training phase of the reduction framework [23]. MAC is defined as a scalar measure of consistency between two modal vectors, namely φ r and φ s , and provides a form of confidence factor to evaluate information arising from different sources as follows:…”

“…With these assumptions and definitions in place, the algorithmic framework of the parametric ROM is now in place. For a schematic representation the reader is referred to [23].…”

Neural modal ordinary differential equations: Integrating physics-based modeling with neural ordinary differential equations for modeling high-dimensional monitored structures

“…To achieve this, we propose to blend a dynamical version (Girin et al, 2020) of a variational autoencoder (VAE) (Kingma and Welling, 2013), with a projection basis containing the eigenmodes that are derived from the linearization of a physics-based model, termed as neural modal ODEs. We justify these components in the proposed architecture as follows: (a) the majority of the aforementioned projection-based methods, which commonly rely on proper orthogonal decomposition (POD (Liang et al, 2002), have been applied for the reduction of nonlinear models/simulators (Abgrall and Amsallem, 2016; Amsallem et al, 2015; Balajewicz et al, 2016; Peherstorfer and Willcox, 2016; Marconia et al, 2021; Vlachas et al, 2022). In this case, we rely on the availability of actual measured data but not simulations of full-order models, which may bear with model bias.…”