Physics-Based Reduced Order Models (ROMs) tend to rely on projection-based reduction. This family of approaches utilizes a series of responses of the full-order model to assemble a suitable basis, subsequently employed to formulate a set of equivalent, low-order equations through projection. However, in a nonlinear setting, physics-based ROMs require an additional approximation to circumvent the bottleneck of projecting and evaluating the nonlinear contributions on the reduced space. This scheme is termed hyper-reduction and enables substantial computational time reduction. The aforementioned hyper-reduction scheme implies a trade-off, relying on a necessary sacrifice on the accuracy of the nonlinear terms’ mapping to achieve rapid or even real-time evaluations of the ROM framework. Since time is essential, especially for digital twins representations in structural health monitoring applications, the hyper-reduction approximation serves as both a blessing and a curse. Our work scrutinizes the possibility of exploiting machine learning (ML) tools in place of hyper-reduction to derive more accurate surrogates of the nonlinear mapping. By retaining the POD-based reduction and introducing the machine learning-boosted surrogate(s) directly on the reduced coordinates, we aim to substitute the projection and update process of the nonlinear terms when integrating forward in time on the low-order dimension. Our approach explores a proof-of-concept case study based on a Nonlinear Auto-regressive neural network with eXogenous Inputs (NARX-NN), trying to potentially derive a superior physics-based ROM in terms of efficiency, suitable for (near) real-time evaluations. The proposed ML-boosted ROM (N3-pROM) is validated in a multi-degree of freedom shear frame under ground motion excitation featuring hysteretic nonlinearities.
The development of reduced-order models remains an active research area, despite advances in computational resources. The present work develops a novel order-reduction approach that is designed to incorporate isolated regions that contain, for example, nonlinearitites or accumulating damage. The approach is designed to use global modes of the overall system response, which are then naturally coupled to the response in the isolated region of interest. Two examples are provided to demonstrate both the accuracy and the computational efficiency of the proposed approach. The performance of this approach is compared to the exact response corresponding to a finite element simulation for the chosen problems. In addition, the accuracy and computational efficiency are shown relative to a standard Galerkin reduction based on the linear normal modes. It is found that the proposed reduction offer computational efficiency comparable to a Galerkin reduction, but more accurately represents the response of the system when both are compared to the finite element simulation.
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 wind turbine tower under a ground motion excitation. 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.
Neural networks (NNs) are known as universal function approximators and can interpolate nonlinear functions between observed data points. However, when the target domain for deployment shifts from the training domain and NNs must extrapolate, the results are notoriously poor. Prior work Martinez et al. (2019) has shown that NN uncertainty estimates can be used to correct binary predictions in shifted domains without retraining the model. We hypothesize that this approach can be extended to correct real-valued time series predictions. As an exemplar, we consider two mechanical systems with nonlinear dynamics. The first system consists of a spring-mass system where the stiffness changes abruptly, and the second is a real experimental system with a frictional joint that is an open challenge for structural dynamicists to model efficiently. Our experiments will test whether 1) NN uncertainty estimates can identify when the input domain has shifted from the training domain and 2) whether the information used to calculate uncertainty estimates can be used to correct the NN’s time series predictions. While the method as proposed did not significantly improve predictions, our results did show potential for modifications that could improve models’ predictions and play a role in structural health monitoring systems that directly impact public safety.
Neural networks (NNs) are known as universal function approximators and can interpolate nonlinear functions between observed data points. However, when the target domain for deployment shifts from the training domain and NNs must extrapolate, the results are notoriously poor. Prior work Martinez et al. (2019) has shown that NN uncertainty estimates can be used to correct binary predictions in shifted domains without retraining the model. We hypothesize that this approach can be extended to correct real-valued time series predictions. As an exemplar, we consider two mechanical systems with nonlinear dynamics. The first system consists of a spring-mass system where the stiffness changes abruptly, and the second is a real experimental system with a frictional joint that is an open challenge for structural dynamicists to model efficiently. Our experiments will test whether 1) NN uncertainty estimates can identify when the input domain has shifted from the training domain and 2) whether the information used to calculate uncertainty estimates can be used to correct the NN’s time series predictions. While the method as proposed did not significantly improve predictions, our results did show potential for modifications that could improve models’ predictions and play a role in structural health monitoring systems that directly impact public safety.
Soft ferromagnetic alloys are often utilized in electromagnetic applications due to their desirable magnetic properties. In support of these applications, the ferromagnetic alloys are also required to bear mechanical load under various loading and environmental conditions. In this study, a Fe–49Co–2V alloy was dynamically characterized in tension with a Kolsky tension bar and a Drop–Hopkinson bar at various strain rates and temperatures. Dynamic tensile stress–strain curves of the Fe–49Co–2V alloy were obtained at strain rates ranging from 40 to 230 s−1 and temperatures from − 100 to 100 °C. All dynamic tensile stress–strain curves exhibited an initial linear elastic response to an upper yield followed by Lüders band response and then a nearly linear work-hardening behavior. The yield strength of this material was found to be sensitive to both strain rate and temperature, whereas the hardening rate was independent of strain rate or temperature. The Fe–49Co–2V alloy exhibited a feature of brittle fracture in tension under dynamic loading with no necking being observed.
Bolted interfaces are a major source of uncertainty in the dynamic behavior of built-up assemblies. Contact pressure distribution from a bolt’s preload governs the stiffness of the interface. These quantities are sensitive to the true curvature, or flatness, of the surface geometries and thus limit the predictive capability of models based on nominal drawing tolerances. Fabricated components inevitably deviate from their idealized geometry; nominally flat surfaces, for example, exhibit measurable variation about the desired level plane. This study aims to develop a predictive, high-fidelity finite element model of a bolted beam assembly to determine the modal characteristics of the preloaded assembly designed with nominally flat surfaces. The surface geometries of the beam interface are measured with an optical interferometer to reveal the amount of deviation from the nominally flat surface. These measurements are used to perturb the interface nodes in the finite element mesh to account for the true interface geometry. A nonlinear quasi-static preload analysis determines the contact area when the bolts are preloaded, and the model is linearized about this equilibrium state to estimate the modal characteristics of the assembly. The linearization assumes that nodes/faces in contact do not move relative to each other and are enforced through multi-point constraints. The structure’s natural frequencies and mode shapes predicted by the model are validated by experimental measurements of the actual structure.
Fe-Co-2V is a soft ferromagnetic alloy used in electromagnetic applications due to excellent magnetic properties. However, the discontinuous yielding (Luders bands), grain-size-dependent properties (Hall-Petch behavior), and the degree of order/disorder in the Fe-Co-2V alloy makes it difficult to predict the mechanical performance, particularly in abnormal environments such as elevated strain rates and high/low temperatures. Thus, experimental characterization of the high strain rate properties of the Fe-Co-2V alloy is desired, which are used for material model development in numerical simulations. In this study, the high rate tensile response of Fe-Co-2V is investigated with a pulse-shaped Kolsky tension bar over a wide range of strain rates and temperatures. Effects of temperature and strain rate on yield stress, ultimate stress, and ductility are discussed.
Cylindrical dog-bone (or dumbbell) shaped samples have become a common design for dynamic tensile tests of ductile materials with a Kolsky tension bar. When a direct measurement of displacement between the bar ends is used to calculate the specimen strain, the actual strain in the specimen gage section is overestimated due to strain in the specimen shoulder and needs to be corrected. The currently available correction method works well for elastic-perfectly plastic materials but may not be applicable to materials that exhibit significant work-hardening behavior. In this study, we developed a new specimen strain correction method for materials possessing an elastic-plastic with linear work-hardening stress–strain response. A Kolsky tension bar test of a Fe-49Co-2V alloy (known by trade names Hiperco and Permendur) was used to demonstrate the new specimen strain correction method. This new correction method was also used to correct specimen strains in Kolsky tension bar experiments on two other materials: 4140 alloy, and 304L-VAR stainless steel, which had different work-hardening behavior.
Experiments, modeling and simulation were used to study the nonlinear dynamics of a jointed-structure in a shock tube. The structure was a full-span square cylinder with internal bolted connections excited by fluid loading. The width-based Reynolds number was ≈105. The cylinder was exposed to an impulsive force associated with the incident shock followed by transverse loading imposed by vortex shedding. In the experiment, aerodynamic loading was characterized with high-speed pressure sensitive paint (PSP). Digital image correlation (DIC) concurrently measured the structural response. The maximum displacement occurred when the vortex shedding frequency most closely matched the structural mode of the beam associated with a rocking motion at the joint. A finite element model was developed using Abaqus, where the nonlinear contact dynamics of the joint were simulated using Coulomb friction. The PSP data loaded the model and the interaction was treated as one-way coupled. The simulations well-matched the trends observed in the experiment. Overall, the root-mean-square values of the transverse displacement agreed to within 24% of the experiment. The modeling showed rocking about the joint during vortex shedding was critical to the nonlinear damping and energy dissipation in the structure. We conclude this campaign highlights the importance of jointed-connections to energy dissipation in structures under aerodynamic loading.