The tension between accuracy and computational cost is a common thread throughout computational simulation. One such example arises in the modeling of mechanical joints. Joints are typically confined to a physically small domain and yet are computationally expensive to model with a high-resolution finite element representation. A common approach is to substitute reduced-order models that can capture important aspects of the joint response and enable the use of more computationally efficient techniques overall. Unfortunately, such reduced-order models are often difficult to use, error prone, and have a narrow range of application. In contrast, we propose a new type of reduced-order model, leveraging machine learning, that would be both user-friendly and extensible to a wide range of applications.
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.
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.
Proceedings of ISMA 2018 - International Conference on Noise and Vibration Engineering and USD 2018 - International Conference on Uncertainty in Structural Dynamics
Complex mechanical structures are often subjected to random vibration environments. One strategy to analyze these nonlinear structures numerically is to use finite element analysis with an explicit solver to resolve interactions in the time domain. However, this approach is impractical because the solver is conditionally stable and requires thousands of iterations to resolve the contact algorithms. As a result, only short runs can be performed practically because of the extremely long runtime needed to obtain sufficient sampling for long-time statistics. The proposed approach uses a machine learning algorithm known as the Long Short-Term Memory (LSTM) network to model the response of the nonlinear system to random input. The LSTM extends the capability of the explicit solver approach by taking short samples and extending them to arbitrarily long signals. The efficient LSTM algorithm enables the capability to perform Monte Carlo simulations to quantify model-form and aleatoric uncertainty due to the random input.
Proceedings of ISMA 2018 - International Conference on Noise and Vibration Engineering and USD 2018 - International Conference on Uncertainty in Structural Dynamics
Structural dynamic models of mechanical, aerospace, and civil structures often involve connections of multiple subcomponents with rivets, bolts, press fits, or other joining processes. Recent model order reduction advances have been made for jointed structures using appropriately defined whole joint models in combination with linear substructuring techniques. A whole joint model condenses the interface nodes onto a single node with multi-point constraints resulting in drastic increases in computational speeds to predict transient responses. One drawback to this strategy is that the whole joint models are empirical and require calibration with test or high-fidelity model data. A new framework is proposed to calibrate whole joint models by computing global responses from high-fidelity finite element models and utilizing global optimization to determine the optimal joint parameters. The method matches the amplitude dependent damping and natural frequencies predicted for each vibration mode using quasi-static modal analysis.
An assessment of two methodologies used at Sandia National Laboratories to model mechanical interfaces is performed on the Ministack finite element model. One method uses solid mechanics models to model contacting surfaces with Coulomb frictional contact to capture the physics. The other, termed the structural dynamics reduced order model, models the interface with a simplified whole joint model using four-parameter Iwan elements. The solid mechanics model resolves local kinematics at the interface while the simplified structural dynamics model is significantly faster to simulate. One of the current challenges to using the whole joint model is that it requires calibration to data. A novel approach is developed to calibrate the reduced structural dynamics model using data from the solid mechanics model to match the global dynamics of the system. This is achieved by calibrating to amplitude dependent frequency and damping of the system modes, which are estimated using three different approaches.