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.

Future generations of advanced spacecraft and aircraft will have a digital twin, or a model that is used to predict life and that is updated as the vehicle ages. A key component of the digital twin concept is the structural dynamics surrogate model that is used to simulate the response of structural components to the loads that the vehicle experiences. The structural surrogate will be used to predict responses, stress and ultimately the estimated life of the vehicle. The most common approach for modeling structures in the aerospace industry is the Finite Element (FE) method, which can accurately simulate the response of the structural components due to various loading conditions. Many of these advanced vehicles will also be operating in extreme environments where certain components may behave nonlinearly such as the large deformations of thin panels. As a result, to use an FE model to simulate the response would be extremely expensive. To circumvent the use of FE models to compute the nonlinear response of structures, methods have been sought to create reduced order models (ROM)s, which capture the essential characteristics of the parent FE model but at a greatly reduced computational cost. One disadvantage of ROMs is that it tends to be more difficult to ensure that they are accurate. Regardless of the numerical modeling approach used they are unlikely to exactly represent the physical structure. As a result, model correlation and updating procedures are required to ensure they are accurate representations of the real hardware. Unfortunately, linear model correlation and validation techniques that are commonly used in the aerospace industry are no longer valid in the nonlinear response regime so a new set of tools is required to validate nonlinear models. Both the nonlinear FE models and nonlinear ROMs can be highly sensitive to boundary conditions, imperfections and pre-stresses which are difficult to account for in the initial model and thus model updating is required. Furthermore, as the structure ages, changes to the structure can occur and must be properly accounted for to ensure life predictions remain accurate. The primary contribution of this work is the development of a model correlation and updating procedure applied to both FE models and ROMs based upon Nonlinear Normal Modes (NNMs) computed using the Multi-Harmonic Balance method. The NNMs serve as a strong metric to correlate the numerical models, because they represent the dynamics of the nonlinear system over a range of amplitudes and they are independent of the loading applied to the system. NNMs can be extracted from experiments so that numerical models can be correlated and validated with test data. This work presents a novel method of computing analytical gradients of the NNMs' solutions with respect to system parameters greatly accelerating the model updating procedure. The procedure is applied to both FE models and ROMs for several experimental systems demonstrating the capabilities of the model updating procedure for the two representations of geometrically nonlinear systems.

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.

Several reduced-order modeling strategies have been developed to create low-order models of geometrically nonlinear structures from detailed finite element models, allowing one to compute the dynamic response of the structure at a dramatically reduced cost. But, the parameters of these reduced-order models are estimated by applying a series of static loads to the finite element model, and the quality of the reduced-order model can be highly sensitive to the amplitudes of the static load cases used and to the type/number of modes used in the basis. Our paper proposes to combine reduced-order modeling and numerical continuation to estimate the nonlinear normal modes of geometrically nonlinear finite element models. Not only does this make it possible to compute the nonlinear normal modes far more quickly than existing approaches, but the nonlinear normal modes are also shown to be an excellent metric by which the quality of the reduced-order model can be assessed. Hence, the second contribution of this work is to demonstrate how nonlinear normal modes can be used as a metric by which nonlinear reduced-order models can be compared. Moreover, various reduced-order models with hardening nonlinearities are compared for two different structures to demonstrate these concepts: a clamped-clamped beam model, and a more complicated finite element model of an exhaust panel cover.

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

The focus of this investigation includes three aspects. First, the development of nonlinear reduced order modeling techniques for the prediction of the response of complex structures exhibiting "large" deformations, i.e. a geometrically nonlinear behavior, and modeled within a commercial finite element code. The present investigation builds on a general methodology, successfully validated in recent years on simpler panel structures, by developing a novel identification strategy of the reduced order model parameters, that enables the consideration of the large number of modes needed for complex structures, and by extending an automatic strategy for the selection of the basis functions used to represent accurately the displacement field. These novel developments are successfully validated on the nonlinear static and dynamic responses of a 9-bay panel structure modeled within Nastran. In addition, a multi-scale approach based on Component Mode Synthesis methods is explored. Second, an assessment of the predictive capabilities of nonlinear reduced order models for the prediction of the large displacement and stress fields of panels that have a geometric discontinuity; a flat panel with a notch was used for this assessment. It is demonstrated that the reduced order models of both virgin and notched panels provide a close match of the displacement field obtained from full finite element analyses of the notched panel for moderately large static and dynamic responses. In regards to stresses, it is found that the notched panel reduced order model leads to a close prediction of the stress distribution obtained on the notched panel as computed by the finite element model. Two enrichment techniques, based on superposition of the notch effects on the virgin panel stress field, are proposed to permit a close prediction of the stress distribution of the notched panel from the reduced order model of the virgin one. A very good prediction of the full finite element results is achieved with both enrichments for static and dynamic responses. Finally, computational challenges associated with the solution of the reduced order model equations are discussed. Two alternatives to reduce the computational time for the solution of these problems are explored.

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.

When vibrating with large amplitudes, thin structures experience geometric nonlinearity due to the nonlinear relationship between strains and displacements. Because full-order nonlinear analysis on geometrically nonlinear models are computationally very expensive, the derivation of efficient reduced-order models (ROMs) has always been a topic of interest.In this thesis, nonlinear reduction methods for building ROMs with geometric nonlinearity in the framework of the Finite Element (FE) procedure, are investigated. Three non-intrusive nonlinear reduction methods are specifically investigated and systematically compared. They are: implicit condensation and expansion (ICE), modal derivatives (MD), and the reduction to invariant manifold. Theoretical analysis shows that the first two methods can give reliable results only if a slow/fast assumption between slave and master coordinates holds. On the other hand, reduction to invariant manifolds allows proposing a simulation-free reduction method that can be applied without restricting assumptions on the frequencies of the slave modes.Numerical comparisons and numerous applications to continuous structures discretized with the FE procedure, are given subsequently. For application of the invariant manifold-based method, the computation is based on a direct application of the normal form to the physical space and hence to the nodes of the FE mesh, a method recently developed. The examples show the advantages and drawbacks of each reduction method when deriving ROM, and the results of the theoretical comparison are validated.Finally, the analysis of the dynamics of a system with 1:2 internal resonance and cubic nonlinearity is given in the last part of the thesis. The real normal form of the problem is first derived. Then the solution branches of the problem are investigated and compared to simpler solutions with the dynamics truncated at order two. The divergent behaviour of the hardening/softening characteristics for single-mode reduction is investigated with this more complete model.