Chapter 12 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 On the use of Statistical Learning Theory for model selection in Structural Health Monitoring C. A. Lindley, N. Dervilis, and K. Worden Whenever data-based systems are employed in engineering applications, defining an optimal statistical representation is subject to the problem of model selection. This paper focusses on how well models can generalise in Structural Health Monitoring (SHM). Although statistical model validation in this field is often performed heuristically, it is possible to estimate generalisation more rigorously using the bounds provided by Statistical Learning Theory (SLT). Therefore, this paper explores the selection process of a kernel smoother for modelling the impulse response of a linear oscillator from the perspective of SLT. It is demonstrated that incorporating domain knowledge into the regression problem yields a lower guaranteed risk, thereby enhancing generalisation. Keywords Structural health monitoring · statistical learning theory · structural risk minimisation · kernel smoothers · impulse response Introduction The incorporation of intelligent systems in engineering is a subject of increasing popularity in the literature. Of particular interest in the present study is the development of data-based methods tailored to addressing various challenges encountered in Structural Health Monitoring (SHM) [1]. Whether these methods are designed to detect, localise or classify damage [2], one aspect that is shared across them is the problem of model selection; namely, of deciding over a set of statistical models, which one can adequately represent the system at hand. The means to determine such a model is by a process of validation during the training phase, which is a crucial step to ensure that the model can genenralise well. Failure to do so can lead to an erroneous representation of the system, and thus inaccurate predictions upon the introduction of new observations. This consideration may arguably be of greater importance in SHM, since poor predictions could be perilous for human safety, and detrimental towards financial projections, in the event of unforeseen failures. To ensure good generalisation, standard practice is followed, often involving the division of the entire dataset into a training set and a test set. In short, the training set is used to search for optimal parameters, while the test set simulates unseen data, allowing the evaluation of the prediction error for both sets during the learning process. Some optimal model is chosen that can maintain a low error for the training set while preventing the error to grow large for the test set. This criterion is commonly met heuristically with the use of cross-validation methods [3]. The concept of generalisation, however, can be approached in a more rigorous fashion and thus assert the confidence one has about the employed statistical model. The search for an optimal model is thereby explored here by eliciting the methods found in Statistical Learning Theory (SLT) [4]; concretely, in the selection of an optimal kernel smoother for modelling the impulse response of a simple structure. The premise of the following study is to establish a rigorous mathematical framework for the model-selection problem in SHM, ensuring the selection of an optimal model that generalises well when trained with a limited amount of data. The Model-Selection Problem To illustrate the model-selection problem, one may first consider the case in which the aim is to fit a function to a set of data. If dealing with a regression problem, then a sensible approach would be to find some weighted combination of inputs that C. A. Lindley· N. Dervilis · K.Worden Dynamics Research Group, Department of Mechanical Engineering, University of Sheffield, Mappin Street, Sheffield S1 3JD, UK e-mail: c.a.lindley@sheffield.ac.uk; n.dervilis@sheffield.ac.uk; k.worden@sheffield.ac.uk © The Author(s), under exclusive license to River Publishers 2025 95 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 12
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