Structural Dynamics Problems Research Articles

On application of the relative entropy concept in reliability assessment of some engineering cable structures

The main research problem studied in this work is an uncertain response and reliability assessment of the spatial cable structures due to the environmental stochasticity as well as material and geometrical imperfections. Some popular cable structures are analyzed for this purpose using the Stochastic Finite Element Method (SFEM) implemented with the use of three different techniques, namely the iterative generalized perturbation method, semi-analytical approach as well as the Monte-Carlo simulation. Uncertainty quantification delivered in this study is based on the series of FEM analyses of both static and dynamic structural problems. They enable the Least Squares Method determination of the structural polynomial responses linking extreme stresses and deformations with several uncorrelated uncertainty sources. Reliability assessment, fundamental in durability and Structural Health Monitoring, is completed using a comparison of the First Order Reliability Method (FORM) with probabilistic distance formulated by Bhattacharyya. Input uncertainties are assumed to be Gaussian according to the Maximum Entropy Principle. They have specific expected values following engineering design demands or the provisions of designing codes, whereas their standard deviations do not exceed the 10% level. The methods presented and the results obtained in this study may serve for further reliability analyses of large-scale civil engineering structures completed with both steel cables and also reinforced concrete plates like suspended bridges, for instance.

A Non-Probabilistic Strategy to Investigate Imprecisely Defined Nonlinear Eigenvalue for Structural Dynamic Problems

A Non-Probabilistic Strategy to Investigate Imprecisely Defined Nonlinear Eigenvalue for Structural Dynamic Problems

Implementation of three-dimensional contact algorithm in numerical manifold method for the structural impact simulation

Structural impact often accompanies large amounts of contacts and leads to complex mechanical phenomena. In solid mechanics, the numerical manifold method (NMM) is proposed to address problems featuring continuous-discontinuous transitions by utilizing a dual coverage system encompassing both mathematical and physical covers. In the present work, a penalty contact algorithm for 3DNMM based on cover-based contact theory is programmed and applied to impact mechanics problems. The accuracy of the developed contact algorithm is firstly calibrated through free-falling blocks and collision blocks. The influence of contact parameters on contact convergence is systematically studied, and three preliminary criteria for how to set contact parameters are provided. The effectiveness of the contact algorithm is verified by conserving system momentum during block collisions. Subsequently, the contact algorithm is applied to Taylor rod and car-streetlight impact simulation, further confirming its effectiveness in modeling high-speed collisions, large displacements, and large deformations of structures. By comparing the 3DNMM results with those from Abaqus, the contact algorithm developed here performs exceptionally well in solving collision problems and produces results consistent with commercial software. The research results in the present work verify the applicability and accuracy of the proposed contact algorithm in solving structural dynamic impact problems. The present work also provides guidance for contact parameter setting in impact problems.

A Multistep Method for Integration of Perturbed and Damped Second-Order ODE Systems

Based on the Ψ-functions series method, a new numerical integration method for perturbed and damped second-order systems of differential equations is presented. This multistep method is defined for variable step and variable order (VSVO) and maintains the good properties of the Ψ-functions series method. In addition, it incorporates a recurring algebraic procedure to calculate the algorithm’s coefficients, which facilitates its implementation on the computer. The construction of Ψ-functions and the Ψ-functions series method are presented to address the construction of both explicit and implicit multistep methods and a predictor–corrector method. Three problems analogous to those solved by the Ψ-functions series method are analyzed, contrasting the results obtained with the exact solution of the problem or with its first integral. The first example is the integration of a quasi-periodic orbit. The second example is a Structural Dynamics problem associated with an earthquake, and the third example studies an equatorial satellite with perturbation J2. This allows us to compare the good behavior of the new code with other prestige codes.

A data-driven force-thermal coupling load identification method considering multi-source uncertainties of structural characteristics and measuring noises

The problem of load identification denotes identifying loads based on the measurement of structural responses, which is the inverse problem in structural dynamics. With advancements in aeronautical technology, the working environment of aircraft becomes even more complex. To accurately monitor and forecast the load environment where the structure works can provide guidance for the structural design of aircraft and predict potential structural damage. However, with the development of hypersonic aircraft, the influence of thermal field also cannot be ignored besides external forces that are applied to the structure, thus, there is an urgent need for effective methods to identify both force and thermal loads. In this paper, firstly, A data-driven load identification method and various load identification strategies for the identification of force-thermal load based on Artificial Neural Networks are proposed, and in different loading cases the relative error of identification is less than 5 %, with the training process converging efficiently within a short time; additionally, multi-source uncertainties, including Gaussian white noises and structural uncertainties, are simulated, and their influence on load identification is also evaluated; moreover, sensor placement optimization based on particle swarm optimization is carried out to enhance accuracy of load identification, and the number of sensors and the corresponding optimal placement of sensors are determined, it can be proved via numerical examples that the optimized sensor placement can reduce the error of load identification by more than 90 % of that of a random sensor placement.

Model order reduction based on low-rank approximation for parameterized eigenvalue problems in structural dynamics

This study introduces a novel reduced-order modeling approach for parameterized eigenvalue problems. The proposed framework represents eigenvectors as a low-rank approximation and is implemented using an offline-online strategy. In the offline stage, nonlinear equations are formulated by imposing a stationary condition on the weighted Rayleigh quotient over sampled points and eigenvalues of interest. A variant of the Krylov subspace approach is applied to progressively identify the dominant response characteristics at sampled points. In the online stage, a reduced-order model is obtained through Galerkin projection, and eigensolutions at unsampled points are determined. The performance of the proposed framework is validated through numerical examples, where perturbation, least-angle regression, and Gaussian process are considered for comparison. The obtained results demonstrate that the proposed framework can accurately estimate eigensolutions, even when eigenvalues are adjacent to each other. In terms of efficiency, the a priori construction of the low-dimensional basis is possible without solving the full-order model, resulting in substantial CPU time savings during the offline computation process. Lastly, the applicability of the proposed framework is investigated through an uncertainty quantification example using numerical and experimental results.

Efficient stochastic modal decomposition methods for structural stochastic static and dynamic analyses

AbstractThis article presents unified and efficient stochastic modal decomposition methods to solve stochastic structural static and dynamic problems. We extend the idea of deterministic modal decomposition method for structural dynamic analysis to stochastic cases. Standard/generalized stochastic eigenvalue equations are adopted to calculate the stochastic subspaces for stochastic static/dynamic problems and they are solved by an efficient reduced‐order method. The stochastic solutions of both static and dynamic equations are approximated by stochastic bases of the stochastic subspaces. Original stochastic static/dynamic equations are then transformed into a set of single‐degree‐of‐freedom (SDoF) stochastic static/dynamic equations, which are efficiently solved by the proposed non‐intrusive methods. Specifically, a non‐intrusive stochastic Newmark method is developed for the solution of SDoF stochastic dynamic equations, and the element‐wise division of sample vectors is used to solve the SDoF stochastic static equations. All of these methods have low computational effort and are weakly sensitive to the stochastic dimension, thus the proposed methods avoid the curse of dimensionality successfully. Two numerical examples, including two‐ and three‐dimensional spatial problems with low and high stochastic dimensions, are given to show the efficiency and accuracy of the proposed methods.

Accelerating structural dynamics simulations with localised phenomena through matrix compression and projection‐based model order reduction

AbstractIn this work, a novel approach is introduced for accelerating the solution of structural dynamics problems in the presence of localised phenomena, such as cracks. For this category of problems, conventional projection‐based Model Order Reduction (MOR) methods are either limited with respect to the range of system configurations that can be represented or require frequent solutions of the Full Order Model (FOM) to update the low‐dimensional spaces, in which solutions are represented. In the proposed approach, low‐dimensional spaces, constructed for the healthy structure, are enriched with appropriately selected columns of the flexibility matrix of the system. It can be shown that these spaces contain the solution to the original problem for the static case, while their dimension is much smaller. In order to allow their online construction for arbitrary localised features, the full flexibility matrix of the system should be available. To this end, a hierarchical representation is used for the matrices involved, allowing to compute the flexibility matrix efficiently and with reduced memory requirements. The resulting method offers significant speedups, without sacrificing the flexibility and accuracy of the full order model. The performance and limitations of the approach are studied through a series of examples in structural dynamics.

An Enhanced Circumferential Winkler Contact Force Model for Cylindrical Clearance Joints

In deployable space structures, the clearance size of joints is small, and the contact between the journal and bearing should sometimes be treated as a conformal contact, resulting in the imprecision of the contact force models in evaluating contact forces. This study proposes an enhanced hybrid theoretical and numerical contact force model to calculate the clearance joints’ contact force. Firstly, a basic formula of a dimensionless contact force is proposed, and an optimized surrogate function of the shape correction coefficient is obtained using finite element method (FEM) and response surface method (RSM). Then, the proposed model is validated by comparing it with various contact force models. Finally, this paper discussed the applicability of the proposed model in detail. These results indicate that the dimensionless expression of the proposed model is reasonable and can reasonably describe the relationship between contact force and penetration depth. The proposed model has been proven to have an excellent correlation with FEM and a minor relative error. The application scope of the proposed model is unlimited to the clearance size and the joint size. The proposed model provides a foundation for accurate impact models and is vital for the structural dynamic problems of high-precision space structures.

A new time integration method based on state formulations for dynamic analysis of nonviscously damped systems

The mode superposition method (MSM), which is used for dynamic response calculations of nonviscously damped systems, requires state–space eigensolutions, which incur a high computational cost. This study proposed an alternative symmetric formulation-based time integration method for analyzing structural dynamic systems. An anelastic displacement field (ADF) model was introduced to capture damping mechanisms in nonviscous systems. The differential equation describing the ADF model was transformed into an augmented coupled system via the introduction of dissipative variables. Two symmetric state formulations based on coupled augmented systems were derived for nonviscous systems. Consequently, alternative asymmetric state formulations were derived for nonviscous systems involving the ADF model. Thereafter, a new method was developed based on these state–space formulations by assuming linear approximations. A numerical implementation of the new method for dynamic systems was demonstrated. Three benchmark examples of discrete and continuous systems were considered to evaluate the asymmetric formulation and the computational efficiency, accuracy, stability, and time-step-size sensitivity of the new method. Further, the response of the new method was validated using MSM and precise time integration (PTI) methods. An asymmetric state formulation was also validated against MSM and direct frequency response methods. The results revealed that the new method accurately matched the MSM and PTI methods at various time steps. The results also showed that the new method was more efficient than the MSM and PTI methods, rendering it suitable for solving large-scale structural dynamics problems.

Failure Probability Prediction for Offshore Floating Structures Using Machine Learning

Summary Accurately estimating the failure probability is crucial in designing civil infrastructure systems, such as floating offshore platforms for oil and gas processing/production, to ensure their safe operation throughout their service periods. However, as a system becomes complex, the evaluation of a limit state function may involve the use of an external computer solver, resulting in a significant computational burden to perform Monte Carlo simulations (MCS). Moreover, the high-dimensionality of the limit state function may limit efficient sampling of input variables due to the “curse of dimensionality.” To address these issues, an efficient machine learning framework is proposed, combining polynomial chaos expansion (PCE) and active subspace. This will enable the accurate and efficient evaluation of the failure probability of an offshore structure, which typically involves a large number of uncertain parameters. Unlike conventional PCE schemes that use the original random variable space or the auxiliary variable space for building a surrogate model, the proposed method utilizes a reduced-dimension space to circumvent the “curse of dimensionality.” An appropriate coordinate transformation is first sought so that most of the variability of a limit state function can be accounted for. Next, a PCE surrogate limit state function is constructed on the derived low-dimensional “active subspace.” The Gram-Schmidt orthogonalization process is used for making basis polynomial functions, which is particularly effective when input random parameters do not follow the Askey scheme and/or when a dependence structure between the input parameters exists. Therefore, a nonlinear iso-probabilistic transformation, which makes the convergence of a surrogate to the true model difficult, is not required, unlike traditional PCE. Numerical examples, including limit state functions related to structural dynamics problems, are presented to illustrate the advantages of the proposed method in estimating failure probabilities for complex structural systems. Specifically, the method exhibits significantly improved efficiency in estimating the failure probability of an offshore floating structure without compromising accuracy as compared to traditional PCE and MCS.

A hierarchical Bayesian modeling framework for identification of Non-Gaussian processes

Non-Gaussian processes are frequently encountered in engineering problems, posing a challenge when it comes to identification. The main challenge in the identification arises from the fact that a non-Gaussian process can be treated as a collection of infinite dimensional non-Gaussian variables. The application of the hierarchical Bayesian modeling (HBM) framework is constrained due to the inherent complexity of dimensionality and non-Gaussian characteristics associated with these variables. To tackle the issue of dimensionality, the improved orthogonal series expansion (iOSE) representing a non-Gaussian process by time functions with non-Gaussian coefficients, which are readily obtained from discretizing the process at some specific time points, is introduced within the HBM framework. In particular, the iOSE is embedded to convert the identification of a non-Gaussian process into a finite number of non-Gaussian coefficients. Regarding their non-Gaussian characteristics, polynomial chaos expansion (PCE) is used to quantify the uncertainty of the non-Gaussian coefficients with parameters in PCE treated as hyper parameters to be estimated by the HBM framework. The proposed framework is applicable to the identification of both stationary and nonstationary non-Gaussian processes. The effectiveness of non-Gaussian process quantification by the proposed framework is demonstrated using simulated data of a non-stationary extreme value process. Theapplicability of this approach for non-Gaussian process identification is validated by accurately identifying a stochastic load in a structural dynamic problem. Furthermore, it is successfully applied to the reconstruction of random mode shapes of a building arising from different environmental conditions.

Robust scalable initialization for Bayesian variational inference with multi-modal Laplace approximations

Predictive modeling typically relies on Bayesian model calibration to provide uncertainty quantification. Variational inference utilizing fully independent (“mean-field”) Gaussian distributions are often used as approximate probability density functions. This simplification is attractive since the number of variational parameters grows only linearly with the number of unknown model parameters. However, the resulting diagonal covariance structure and unimodal behavior can be too restrictive to provide useful approximations of intractable Bayesian posteriors that exhibit highly non-Gaussian behavior, including multimodality. High-fidelity surrogate posteriors for these problems can be obtained by considering the family of Gaussian mixtures. Gaussian mixtures are capable of capturing multiple modes and approximating any distribution to an arbitrary degree of accuracy, while maintaining some analytical tractability. Unfortunately, variational inference using Gaussian mixtures with full-covariance structures suffers from a quadratic growth in variational parameters with the number of model parameters. The existence of multiple local minima due to strong nonconvex trends in the loss functions often associated with variational inference present additional complications, These challenges motivate the need for robust initialization procedures to improve the performance and computational scalability of variational inference with mixture models.In this work, we propose a method for constructing an initial Gaussian mixture model approximation that can be used to warm-start the iterative solvers for variational inference. The procedure begins with a global optimization stage in model parameter space. In this step, local gradient-based optimization, globalized through multistart, is used to determine a set of local maxima, which we take to approximate the mixture component centers. Around each mode, a local Gaussian approximation is constructed via the Laplace approximation. Finally, the mixture weights are determined through constrained least squares regression. The robustness and scalability of the proposed methodology is demonstrated through application to an ensemble of synthetic tests using high-dimensional, multimodal probability density functions. The practical aspects of the approach are demonstrated with inversion problems in structural dynamics.

Development of an explicit time integration algorithm based on optimal integration parameter updating

AbstractIntegration algorithm is an important mean to solve the discrete time equations of structural dynamics in earthquake engineering. In this paper, a new model‐based explicit integration algorithm, featured by optimization‐based integration parameter updating, is developed for a more accurate and efficient solution of structural dynamic problems. The new integration algorithm is designed to yield minimum error and controllable numerical dispersion under the premise of stability by defining objective functions and constraints. Numerical properties of the new integration algorithm are investigated in details using the amplification matrix method for a linear elastic single degree‐of‐freedom system. Meanwhile, representative numerical examples and complex structure examples are analyzed and compared with other representative algorithms. As a result, the proposed algorithm yields comparable numerical characteristics with other methods, but exhibits pronounced superiority in controllable algorithmic dissipation, higher precision, and better efficiency. Therefore, the new integration algorithm possesses extensive application prospect in solving complicated linear and nonlinear structural dynamic problems.

Regression-based identification and order reduction method for nonlinear dynamic structural models

In this paper, we propose a regression-based nonlinear reduced-order model for nonlinear structural dynamics problems, called the Nonlinear Identification and Dimension-Order Reduction (NLIDOR) algorithm. We evaluate the algorithm using a simple toy model, a chain of coupled oscillators and an actual three-dimensional flat plate. The results show that NLIDOR can accurately identify the natural frequencies and modes of the system and capture the nonlinear dynamical features, while the linear Dynamic Mode Decomposition (DMD) method can only capture linear features and is influenced by nonlinear terms. Compared with the full-order model (FOM), NLIDOR can effectively reduce computational cost, while compared with DMD, NLIDOR significantly improves computational accuracy. The results demonstrate the effectiveness and potential of NLIDOR for solving nonlinear dynamic problems in various applications.

Atomistic and continuum length scale coupling in materials using quasicontinuum method

We propose coupling atomistic and continuum length scale models in materials to develop a multiscale modeling framework. High fidelity models like Molecular Dynamics and Density Functional Theory exist at atomic scales capable of explaining various small-scale occurrences. However, these models cannot be scaled up efficiently to explain a range of observations occurring at multiple length and time scales. Since continuum properties can be considered emergent from numerous smaller-scale phenomena, models capable of effectively relating multiple length and time scales are needed. Therefore, we propose using existing models at differing yet individual length scales and suitably coupling them for tackling such multiscale problems. This coupling method should facilitate information exchange in real-time and demonstrate rapid convergence without adding significant computational overhead. Together with single-scale models, this coupling could help us relate continuum properties like stiffness, damping, and thermal conductivity to a material's electronic structure and atomic arrangement. As an illustration, we choose a simple one-dimensional problem with a limited region of interest which we wish to study at an atomic scale. We rely upon the Quasicontinuum method to model this limited atomic region, which significantly reduces the degrees of freedom involved for the atomistic subdomain. Preliminary results for this static problem indicate that the displacement error is relevant on the atomic scale. Importance is put on deriving a method capable of effectively coupling atomic and continuum subdomains for solving general static or dynamic structural problems.

An Unconditionally Stable Integration Method for Structural Nonlinear Dynamic Problems

This paper presents an unconditionally stable integration method, which introduces a linearly implicit algorithm featuring an explicit displacement expression. The technique that is being considered integrates one Newton iteration into the mean acceleration method. The stability of the proposed algorithm in solving equations of motion containing nonlinear restoring force and nonlinear damping force is analyzed using the root locus method. The objective of this investigation was to assess the accuracy and consistency of the proposed approach in contrast to the Chang method and the CR method. This is achieved by analyzing the dynamic response of three distinct structures: a three-layer shear structure model outfitted with viscous dampers, a three-layer shear structure model featuring metal dampers, and an eight-story planar frame structure. Empirical evidence indicates that the algorithm in question exhibits a notable degree of precision and robustness when applied to nonlinear dynamic problem-solving.

Development of three-dimensional numerical manifold method with cover-based contact theory

Numerical manifold method (NMM) is a typical continuous-discontinuous coupling method. It unifies the continuous and discontinuous theories in the same program framework. In the present work, based on the cover-based contact theory, the specific solution of contact covers under five different positional relationships is given in detail, and the contact judgement conditions and contact cover solution formulas under each positional relationship are proposed. By converting the contact into the relationship between reference point a0 and the outer boundary of entrance block E (A, B), the contact algorithm between 3D blocks is implemented and programmed. The developed code is initially calibrated by two simple models, and its correctness is quantitatively verified by the motion of the inclined sliding model. Finally, it is applied to more complex structural deformation and dynamic collision problems. The 3D-NMM developed in the present work can be used for structural kinematics, structural deformation and dynamic collision problems, and has broad application prospects in the field of structural mechanics.

High-order implicit time integration scheme with controllable numerical dissipation based on mixed-order Padé expansions

A single-step high-order implicit time integration scheme with controllable numerical dissipation at high frequencies is presented for the transient analysis of structural dynamic problems. The amount of numerical dissipation is controlled by a user-specified value of the spectral radius ρ∞ in the high frequency limit. Using this user-specified parameter as a weight factor, a Padé expansion of the matrix exponential solution of the equation of motion is constructed by mixing the diagonal and sub-diagonal expansions. An efficient time-stepping scheme is designed where systems of equations, similar in complexity to the standard Newmark method, are solved recursively. It is shown that the proposed high-order scheme achieves high-frequency dissipation, while minimizing low-frequency dissipation and period errors. The effectiveness of the provided dissipation control and the efficiency of the scheme are demonstrated by numerical examples. A simple guideline for the choice of the controlling parameter and time step size is provided. The source codes written in MATLAB and FORTRAN are available for download at: https://github.com/ChongminSong/HighOrderTimeIntegration.

Reduced order modeling of parametrized systems through autoencoders and SINDy approach: continuation of periodic solutions

Highly accurate simulations of complex phenomena governed by partial differential equations (PDEs) typically require intrusive methods and entail expensive computational costs, which might become prohibitive when approximating steady-state solutions of PDEs for multiple combinations of control parameters and initial conditions. Therefore, constructing efficient reduced order models (ROMs) that enable accurate but fast predictions, while retaining the dynamical characteristics of the physical phenomenon as parameters vary, is of paramount importance. In this work, a data-driven, non-intrusive framework which combines ROM construction with reduced dynamics identification, is presented. Starting from a limited amount of full order solutions, the proposed approach leverages autoencoder neural networks with parametric sparse identification of nonlinear dynamics (SINDy) to construct a low-dimensional dynamical model. This model can be queried to efficiently compute full-time solutions at new parameter instances, as well as directly fed to continuation algorithms. These aim at tracking the evolution of periodic steady-state responses as functions of system parameters, avoiding the computation of the transient phase, and allowing to detect instabilities and bifurcations. Featuring an explicit and parametrized modeling of the reduced dynamics, the proposed data-driven framework presents remarkable capabilities to generalize with respect to both time and parameters. Applications to structural mechanics and fluid dynamics problems illustrate the effectiveness and accuracy of the proposed method.