Chapter 17 Chapter 1 On the Detection and Quantification of Nonlinearity via Statistics of the Gradients of a Black-Box Model Georgios Tsialiamanis and Charles R. Farrar Abstrac t Detection and identification of nonlinearity is a task of high importance for structural dynamics. On the one hand, identifying nonlinearity in a structure would allow one to build more accurate models of the structure. On the other hand, detecting nonlinearity in a structure, which has been designed to operate in its linear region, might indicate the existence of damage within the structure. Common damage cases which cause nonlinear behaviour are breathing cracks and points where some material may have reached its plastic region. Therefore, it is important, even for safety reasons, to detect when a structure exhibits nonlinear behaviour. In the current work, a method to detect nonlinearity is proposed, based on the distribution of the gradients of a data-driven model, which is fitted on data acquired from the structure of interest. The data-driven model selected for the current application is a neural network. The selection of such a type of model was done in order to not allow the user to decide how linear or nonlinear the model shall be, but to let the training algorithm of the neural network shape the level of nonlinearity according to the training data. The neural network is trained to predict the accelerations of the structure for a time-instant using as input accelerations of previous time-instants, i.e. one-step-ahead predictions. Afterwards, the gradients of the output of the neural network with respect to its inputs are calculated. Given that the structure is linear, the distribution of the aforementioned gradients should be unimodal and quite peaked, while in the case of a structure with nonlinearities, the distribution of the gradients shall be more spread and, potentially, multimodal. To test the above assumption, data from an experimental structure are considered. The structure is tested under different scenarios, some of which are linear and some of which are nonlinear. More specifically, the nonlinearity is introduced as a column-bumper nonlinearity, aimed at simulating the effects of a breathing crack and at different levels, i.e. different values of the initial gap between the bumper and the column. Following the proposed method, the statistics of the distributions of the gradients for the different scenarios can indeed be used to identify cases where nonlinearity is present. Moreover, via the proposed method one is able to quantify the nonlinearity by observing higher values of standard deviation of the distribution of the gradients for lower values of the initial column-bumper gap, i.e. for “more nonlinear” scenarios. Keyword s Structural health monitoring (SHM) · Structural dynamics · Nonlinear dynamics · Machine learning · Neural networks 1.1 Introduction In the pursuit of making everyday life safer, humans have extensively tried to model the environment around them. Structures are an important part of the environment, in which humans live. They are man-made and should be safe throughout their lifetime. Structures are exposed to numerous environmental factors, which may cause them to fail. Moreover, during operation, structures are subjected to dynamic loads, which, in time, may cause failure. Such failures will most probably result in economic damage to society and may even result in loss of human lives. Therefore, for the purpose of maintaining structures safe, the field of structural health monitoring (SHM) [1] has emerged. G. Tsialiamanis ( ) Dynamics Research Group, Department of Mechanical Engineering, University of Sheffield, Sheffield, UK e-mail: g.tsialiamanis@sheffield.ac.uk C. R. Farrar Engineering Institute, MS T-001, Los Alamos National Laboratory, Los Alamos, NM, USA e-mail: farrar@lanl.gov © The Society for Experimental Mechanics, Inc. 2024 M. R. W. Brake et al. (eds.), Nonlinear Structures & Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-36999-5_1 1 Physics-Informed Machine Learning for Advanced Structural Damage Detection and Localization Zixin Wang, Mohammad R. Jahanshahi, and Shirley J. Dyke Structural damage detection and localization are crucial aspects of structural health monitoring. Unsupervised anomaly detection can predict whether the structure is healthy or damaged, but it may face challenges in localizing the damage. In contrast, supervised damage detection can localize damage when the damage diagnosis model is trained with data from a finite element model (FEM) that accurately represents the actual structure. However, mismatches between the FEM and the actual structure are inevitable due to modeling errors, real-world uncertainties, and varying operational conditions. Consequently, a damage diagnosis model trained on the FEM cannot be directly applied to the actual structure because of discrepancies in training and testing data distributions. To bridge this gap, we propose a novel approach using hierarchical physics-informed domain adaptation. Our approach begins by detecting structural anomalies using deep autoencoders and information fusion. If the structure is identified as damaged, physics-informed domain adaptation is then employed to localize the damage. A convolutional neural network (CNN) is pre-trained to predict the frequency response functions of the structure. The weights of this pre-trained CNN model are then used to initialize a domain adaptation neural network, specifically a discriminator-free adversarial learning network (DALN). This modal-based weight initialization imposes physical constraints on the DALN. We evaluate the proposed approach using both numerical and experimental ASCE benchmark setups. The results demonstrate that our approach outperforms existing state-of-the-art methods, even when the damaged state data from the target structure is excluded from training, thereby enhancing the real-world applicability of the proposed solution. Keywords Structural health monitoring · Structural damage detection · Physics-informed neural networks · Domain adaptation· Deep learning Introduction Structural damage detection is critical for ensuring the safety and health of civil infrastructures. Data-driven models based on machine learning have been widely utilized in this field. For example, unsupervised learning uses data in the healthy state of the structure to train a model, which can then be used for structural anomaly detection (i.e., determining whether the structure is damaged). In addition to anomaly detection, damage localization is important for pinpointing the damaged area and retrofitting the structure. Supervised learning requires labeled data in both healthy and damaged states to train a model, which can then be used to predict the damage location. However, collecting massive amounts of data from an actual structure in various damage scenarios is impractical. Therefore, a digital twin, consisting of a physical structure and a corresponding FEM as its virtual representation, can be constructed. The FEM can generate the data needed to train the supervised learning model. It is worth noting that supervised learning relies on the assumption that the training and testing datasets have the ZixinWang· Mohammad R. Jahanshahi · Shirley J. Dyke Lyles School of Civil and Construction Engineering, Purdue University, West Lafayette, IN 47907 e-mail: wang4591@purdue.edu; jahansha@purdue.edu; sdyke@purdue.edu Mohammad R. Jahanshahi Elmore Family School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907 e-mail: jahansha@purdue.edu Shirley J. Dyke School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907 e-mail: sdyke@purdue.edu © The Author(s), under exclusive license to River Publishers 2025 137 Brian Damiano et al. (eds.), Structural Health Monitoring & Machine Learning, Vol. 12, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.13052/97887-438-0157-3 17
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