Chapter 20 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 Topology Optimization of Isolated Response Curves in 3D Geometrically-nonlinear Beam Enora Denimal Goy, Yichang Shen, Samuel Fruchard, Adrien Me´lot, and Ludovic Renson Abstract Topology optimisation is a powerful tool for designing efficient and light structures. However, classical topology optimisation methods (SIMP, LSF), which are gradient-based, are not adapted to deal with nonlinear vibrations in the context of geometrical nonlinearities as the simulation of such systems is computationally expensive, and the strong nonlinear behaviour makes the objective function non-convex with many local minima. The present work investigates the potential of using global optimisation methods to topology optimise those structures. To provide more robust nonlinear features in the optimisation, the bifurcations are directly tracked and optimised. The strategy is applied to a 3D finite element model of a beam. Keywords Bifurcation· Topology optimisation· Global optimisation· Geometric nonlinearities · Reduced-order Model Introduction A major transformation in design methods for mechanical structures (transport, aeronautics, space, etc) is taking place in order to continue to make them lighter and increase their performances while benefiting from the arrival of new technologies such as 3D printing. Optimising the dynamics of these structures is becoming extremely complex due to the intrinsic presence of non-linearities leading to dynamic phenomena rarely observed in industrial mechanical systems. For example, in the presence of non-linearities, the dynamics of structures can exhibit bifurcations. Previous studies have focused on the structural and topology optimisation of mechanical structures to limit the level of amplitudes or avoid bifurcation behaviour in the context of localised nonlinearities [1–3]. The present work has for objective to expand these works to the context of geometric nonlinearities. Such nonlinearities imply a significant increase in the computational cost, making the use of reduced-order models (ROM) as well as efficient optimisation techniques crucial. In [4], the authors employed gradient-based optimisation for nonlinear resonances mitigation in the context of geometrical nonlinearities, yet the approach is limited to structures composed of beams and can only do shape optimisation. In this work, a framework is proposed to perform topology optimisation of geometrically nonlinear structures to control the appearance of bifurcations for 3D structures. For each new geometries, a ROM is constructed to solve the nonlinear problem and bifurcations are tracked. To reduce the computational cost of the optimisation, a global optimisation algorithm coupling the CMA-ES to the PSO is proposed. The topology is parametrised with the Moving Morphable Component [5] framework and unconnected geometries are removed in the optimisation loop to reach higher efficiency. Enora Denimal Goy Inria, CMAP, Ecole Polytechnique, Palaiseau, France e-mail: enora.denimal-goy@inria.fr Yichang Shen· Ludovic Renson Dynamics Group, Department of Mechanical Engineering, Imperial College London, London, UK e-mail: yichang.shen@imperial.ac.uk; l.renson@imperial.ac.uk Samuel Fruchard· Adrien Me´lot Universite´ Gustave Eiffel, Inria, COSYS-SII, I4S, Campus Beaulieu, Rennes, France e-mail: samuel.fruchard@inria.fr; adrien.melot@inria.fr © The Author(s), under exclusive license to River Publishers 2025 157 Ludovic Renson et al. (eds.), Nonlinear Structures & Systems, Vol. 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.13052/97887-438-0146-7 20
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