Reduced order models (ROMs) are a highly efficient alternative to full-order finite element models (FEM) of geometrically nonlinear structures. Many non-intrusive reduced order modeling methods have been developed over the decades to serve as a digital twin of geometrically nonlinear structures, providing accurate dynamic simulations with dramatically reduced computational cost. However, the ROM methods pose some critical issues. The existing methods are sometimes not reliable, and so expensive simulations must be run to check the accuracy and optimality of the ROMs before they can be used confidently. Also, a ROM is typically only valid for a single FEM and does not account for variations in the FEM. Thus, if the design of the structure changes so that the FEM changes in some way, one must recompute the corresponding ROM with a new set of static load-displacement solutions. This also greatly increases the cost of analysis using ROMs, making them less attractive. This dissertation proposes a new data-driven reduced order modeling method for geometrically nonlinear structures, which can resolve these issues while keeping the computational cost reasonable. The first chapter presents an application of quasi-static modal analysis (QSMA) to reduced order modeling of geometrically nonlinear structures, to investigate the effect of static versus dynamic modal coupling on the nonlinear dynamic behaviors and how that can be used to create an efficient single-mode ROM. The second study proposes a new data-driven reduced order model based on Gaussian process regression (GPR), which accurately captures how the ROM coefficients change as the FEM is changed, so that one GPR ROM can predict the behavior of a wide range of systems and also quantify its predictive confidence. Through the advances presented in these two chapters, this dissertation opens up new possibilities for efficient and reliable model updating of geometrically nonlinear structures. In this respect, the third chapter explores an application of a data-driven ROM to FE model updating. The proposed updating method incorporates a GPR ROM into the model updating procedure, allowing one to use a single GPR ROM to efficiently update the FEM parameters to match actual field data. Lastly, a case study of model updating of a curved panel subjected to non-uniform thermal fields is demonstrated, which highlights how sensitive the structural response can be with respect to the localized thermal loads. This study suggests a need for a data-driven approach for updating the thermal model to account for highly uncertain and complicated thermal effects. This could be a fruitful avenue for future research.

This unique book explores both theoretical and experimental aspects of nonlinear vibrations and stability of shells and plates. It is ideal for researchers, professionals, students, and instructors. Expert researchers will find the most recent progresses in nonlinear vibrations and stability of shells and plates, including advanced problems of shells with fluid-structure interaction. Professionals will find many practical concepts, diagrams, and numerical results, useful for the design of shells and plates made of traditional and advanced materials. They will be able to understand complex phenomena such as dynamic instability, bifurcations, and chaos, without needing an extensive mathematical background. Graduate students will find (i) a complete text on nonlinear mechanics of shells and plates, collecting almost all the available theories in a simple form, (ii) an introduction to nonlinear dynamics, and (iii) the state of art on the nonlinear vibrations and stability of shells and plates, including fluid-structure interaction problems.

Nonlinear Structures & Systems, Volume 1: Proceedings of the 40th IMAC, A Conference and Exposition on Structural Dynamics, 2022, the first volume of nine from the Conference brings together contributions to this important area of research and engineering. The collection presents early findings and case studies on fundamental and applied aspects of Nonlinear Dynamics, including papers on: Experimental Nonlinear Dynamics Jointed Structures: Identification, Mechanics, Dynamics Nonlinear Damping Nonlinear Modeling and Simulation Nonlinear Reduced-Order Modeling Nonlinearity and System Identification

Mathematical models are at the core of modern science and technology. An accurate description of behaviors, systems and processes often requires the use of complex models which are difficult to analyze and control. To facilitate analysis of and design for complex systems, model reduction theory and tools allow determining "simpler" models which preserve some of the features of the underlying complex description. A large variety of techniques, which can be distinguished depending on the features which are preserved in the reduction process, has been proposed to achieve this goal. One such a method is the moment matching approach. This monograph focuses on the problem of model reduction by moment matching for nonlinear systems. The central idea of the method is the preservation, for a prescribed class of inputs and under some technical assumptions, of the steady-state output response of the system to be reduced. We present the moment matching approach from this vantage point, covering the problems of model reduction for nonlinear systems, nonlinear time-delay systems, data-driven model reduction for nonlinear systems and model reduction for "discontinuous" input signals. Throughout the monograph linear systems, with their simple structure and strong properties, are used as a paradigm to facilitate understanding of the theory and provide foundation of the terminology and notation. The text is enriched by several numerical examples, physically motivated examples and with connections to well-established notions and tools, such as the phasor transform.

This monograph addresses the state of the art of reduced order methods for modeling and computational reduction of complex parametrized systems, governed by ordinary and/or partial differential equations, with a special emphasis on real time computing techniques and applications in computational mechanics, bioengineering and computer graphics. Several topics are covered, including: design, optimization, and control theory in real-time with applications in engineering; data assimilation, geometry registration, and parameter estimation with special attention to real-time computing in biomedical engineering and computational physics; real-time visualization of physics-based simulations in computer science; the treatment of high-dimensional problems in state space, physical space, or parameter space; the interactions between different model reduction and dimensionality reduction approaches; the development of general error estimation frameworks which take into account both model and discretization effects. This book is primarily addressed to computational scientists interested in computational reduction techniques for large scale differential problems.

FEM updating allows FEMs to be tuned better to reflect measured data. It can be conducted using two different statistical frameworks: the maximum likelihood approach and Bayesian approaches. This book applies both strategies to the field of structural mechanics, using vibration data. Computational intelligence techniques including: multi-layer perceptron neural networks; particle swarm and GA-based optimization methods; simulated annealing; response surface methods; and expectation maximization algorithms, are proposed to facilitate the updating process. Based on these methods, the most appropriate updated FEM is selected, a problem that traditional FEM updating has not addressed. This is found to incorporate engineering judgment into finite elements through the formulations of prior distributions. Case studies, demonstrating the principles test the viability of the approaches, and. by critically analysing the state of the art in FEM updating, this book identifies new research directions.

Non-linear stochastic systems are at the center of many engineering disciplines and progress in theoretical research had led to a better understanding of non-linear phenomena. This book provides information on new fundamental results and their applications which are beginning to appear across the entire spectrum of mechanics. The outstanding points of these proceedings are Coherent compendium of the current state of modelling and analysis of non-linear stochastic systems from engineering, applied mathematics and physics point of view. Subject areas include: Multiscale phenomena, stability and bifurcations, control and estimation, computational methods and modelling. For the Engineering and Physics communities, this book will provide first-hand information on recent mathematical developments. The applied mathematics community will benefit from the modelling and information on various possible applications.