Structural Dynamics Problems Research Articles

Model order reduction for structural nonlinear dynamic analysis based on Isogeometric analysis

Model order reduction approach generates lower dimensional approximations to the original system while preserving model’s essential information and computational accuracy. For nonlinear structural dynamic problems, where the stiffness matrix is configuration dependent, an iterative solution procedure is inevitable and a revisit to all the elements is essential for updating the stiffness matrix. In this paper, the nonlinear dynamics of the planar curved beams and 3D cylindrical shells are studied based on the isogeometric analysis and their model order reductions are investigated based on the proper orthogonal decomposition and discrete empirical interpolation method (POD-DEIM). Numerical results show that IGA-based POD-DEIM method significantly improves the computational efficiency of the nonlinear dynamic analysis of the beam and shell structures.

A novel constrained UKF method for both updating structural parameters and identifying excitations for nonlinear structures

Finite element-based model updating of civil structures has been attracting increasing attention in structural health assessment and fragility analysis since several decades ago. However, the previously proposed finite element (FE) model updating methods for structures are either primarily limit to linear models under given external excitation or unknown excitation which should satisfy some assumptions. In this regard, this study proposes a novel constrained unscented Kalman filter (UKF) method for both updating structural parameters and identifying unknown external excitations without any assumption for strongly nonlinear structures. The unknown excitation (e.g., ground acceleration) can be preliminarily extracted from the state and measurement equations and then can be estimated by a recursive nonlinear least-square algorithm. To implement the recursive nonlinear least-square algorithm, a combined Matlab-OpenSees recursion platform herein is specially developed. It should be noted that certain constraints are necessary to be imposed on the sigma points to guarantee the propagation of the sigma points at each recursion step when the recursive nonlinear least-square algorithm is running. A three-story three-span steel frame and a three-span continuous reinforced concrete (RC) bridge, as two examples, are investigated for verifying the applicability of the novel constrained UKF method. The parameters which are most sensitive to the dynamic responses of structures are determined for identification. Subsequently, these structural parameters and the unknown excitation (e.g., ground motion) are jointly identified using the novel constrained UKF method based on partially measured noise-contaminated responses. In such a way, the feasibility of the Matlab-OpenSees recursion platform for structural dynamic inverse problems and the effectiveness of the novel constrained UKF for nonlinear model updating of structures are both verified.

RETRACTED ARTICLE: An Implicit Unconditionally Stable Integration Method for Nonlinear Structural Dynamics and Earthquake Engineering Problems

A novel time integration procedure is designed in order to solve the differential equation of motion of dynamics and earthquake engineering problems. The procedure is constituted on the principle of impulse-momentum, leading to a lesser number of assumed fields. The algorithmic properties of the procedure are determined by stability and accuracy analyses. Overshooting tendency, which is not related to stability and accuracy characteristic of a method, and order of accuracy are also examined. It is displayed that the new method is unconditionally stable and non-dissipative. Also, the numerical dispersion of the proposed algorithm appears to be much less than the commonly used integration methods. The method has no tendency to overshoot both the displacement and velocity response solutions. Its order of accuracy is around four as compared to two of the other methods considered in the study. A few numerical examples consisting of both single and multi-degree of freedom systems with linear and nonlinear characteristics are performed to see the overall behavior of the method in various practical problems. The numerical results of the proposed method obtained from these examples coincide well with the simulated exact results.

Unsolved structural dynamic mathematical models via Novel technique approach

Dynamic problems are solved using the Novel technique, considering the forcing term is free forced vibration, constantly forced vibration and harmonic function with mass and stiffness terms zeros. Two different interpolation functions have been considered for solving such problems, the first one × = a + bt and the second one × = acosωt + bsinωt. The results achieved from the novel technique using these interpolation functions are compared with the precise solutions. Not only this method other methods such as the Newmark-beta method, Runge-Kutta fourth-order and Laplace transforms have been applied to the same structural dynamic problems. The paper employs the novel technique method to various initial value problems such as emotional problems. This method has been adapted in various fields of Engineering. Limited research has been done in the past in the dynamics field using this method. So, an attempt has been made to examine the validity of such a method. The method gave exact solutions for most of the problems. It is observed that the complementary function of the solution obtained using the interpolation function × = acosωt + bsinωt is to be divided by the number of iterations to obtain the precise solution. The Novel technique gave an exact solution for cases where there is no force acting and in cases where a constant force is acting. In the cases in which a harmonic force is acting, when the mass term is zero, the novel technique method is overestimated. In the case of stiffness term zero, it is underestimated. The paper aims to obtain approximate dependent variable quantities for distinct displacement interpolation functions for various structural dynamic natural problems.

Solved structural dynamic mathematical models via novel technique approach

Dynamic problems are solved using this technique, considering the forcing term is free forced vibration, constantly forced vibration and harmonic function. Two different interpolation functions have been considered for solving such problems, the first one x = a + bt and the second one x = acosωt + bsinωt. The results achieved from this method using these interpolation functions are compared with the precise solutions. Not only this method other methods such as the Newmark-beta method, Runge-Kutta fourth-order and Laplace transforms have been applied to the same structural dynamic problems. The method gave exact solutions for most of the problems. It is observed that the complementary function of the solution obtained using the interpolation function × = acosωt + bsinωt is to be divided by the number of iterations to obtain the precise solution. This technique gave an exact solution for cases where there is no force acting and in cases where a constant force is acting. In cases where a harmonic force is acting, this method is overestimated. The paper aims to obtain approximate dependent variable quantities for distinct displacement interpolation functions for various structural dynamic natural problems.

CFD-FEM Modeling of a Floating Foundation under Extreme Hydrodynamic Forces Generated by Low Sea States

The high development of the offshore industry for supporting new marine and renewable energy projects requires a constant improvement of methods for structure designing. Because recent studies warned that maximum environmental loads occur during low sea states and not during extreme sea states as recommend by the offshore standards (e.g., RP 2AWSD-2014), this study used measured wave and current data for analyzing that warning. The Colombian Caribbean coast was selected as the study area, and in situ ADCP data combined with Reanalysis and numerical data was used for identifying proper sea states for the analysis. Then, two low and one extreme sea states were selected and their associated current profiles were extracted, for providing input data for Computational Fluid Dynamics (CFD) and Finite Element Method (FEM) simulations to evaluate the effect of the hydrodynamic forces over a floating structure. The results showed that low sea states generated maximum loads and rotations in the floating structure, and the extreme sea states caused high-frequency vibrations that could provoke structural dynamics problems such as failures due to fatigue or sudden collapse by resonance and amplification.

One-Parameter Controlled Non-Dissipative Unconditionally Stable Explicit Structure-Dependent Integration Methods with No Overshoot

Although many families of integration methods have been successfully developed with desired numerical properties, such as second order accuracy, unconditional stability and numerical dissipation, they are generally implicit methods. Thus, an iterative procedure is often involved for each time step in conducting time integration. Many computational efforts will be consumed by implicit methods when compared to explicit methods. In general, the structure-dependent integration methods (SDIMs) are very computationally efficient for solving a general structural dynamic problem. A new family of SDIM is proposed. It exhibits the desired numerical properties of second order accuracy, unconditional stability, explicit formulation and no overshoot. The numerical properties are controlled by a single free parameter. The proposed family method generally has no adverse disadvantage of unusual overshoot in high frequency transient responses that have been found in the currently available implicit integration methods, such as the WBZ-α method, HHT-α method and generalized-α method. Although this family method has unconditional stability for the linear elastic and stiffness softening systems, it becomes conditionally stable for stiffness hardening systems. This can be controlled by a stability amplification factor and its unconditional stability is successfully extended to stiffness hardening systems. The computational efficiency of the proposed method proves that engineers can do the accurate nonlinear analysis very quickly.

Isogeometric analysis of 3D solids in boundary representation for problems in nonlinear solid mechanics and structural dynamics

AbstractThis contribution presents an isogeometric approach in three dimensions for nonlinear problems in solid mechanics and structural dynamics. The proposed approach aligns with the boundary representation modeling technique in CAD. Isogeometric analysis is combined with the parameterization of the scaled boundary finite element method. In this way, the boundary description of 3D solids in CAD can be directly utilized for the analysis in an isogeometric framework. The approximation of the solution on the boundary is based on bi‐variate NURBS. We also approximate the interior of the solid with uni‐variate B‐splines to facilitate the solution of nonlinear problems. The nonlinear case is extended to three dimensions in this present work. The method is further extended to dynamic problems. The mass and damping matrices are derived using the same basis functions. The solution of the entire solid is obtained with the Galerkin method. We study several nonlinear and dynamic problems with simple geometries and arbitrary number of boundaries. Moreover, a fiber reinforced composite serves as demonstration for complex shapes.

Structural dynamic reliability analysis of super large-scale lattice domes during earthquakes using the stochastic finite element method

For large-scale dome structures subjected to earthquake ground motions, the need for accurate and efficient approaches that account for uncertainties in design, material properties, loads, damping, and manufacturing processes has grown significantly. In uncertainty analyses, the theory of probability, uncertainty quantification, and reliability analysis approaches are commonly used. However, because of computing efficiency and capability, the space scale of dome structures is severely limited. As a result, the computational issues of the complex dynamic reliability that super-large-scale dome structures encounter during earthquakes are challenging. The uncertainties of the dynamic demands of a super-large-scale lattice dome with multiple variables are quantified using a stochastic finite element method (SFEM) with nonlinear time-history analysis developed in this paper, and this method can consider the randomness of variables in more detail for a structure. The dynamic reliability is investigated in depth using the efficient Latin hypercube sampling (LHS) technique to reduce the number of repetitive time-consuming calculations. To improve the efficiency and accuracy of the analysis, an optimization approach based on the genetic algorithm (GA) is used to obtain the probability of structural failure and the reliability index. The results show that these methods are highly efficient for super-large-scale structures. Finally, a sensitivity analysis based on variable randomness is conducted to further evaluate the effects of variables on structural performance using various evaluation methods. This paper provides an efficient solution for structural dynamic reliability problems of super-large-scale structures in the SFEM framework.

A novel polynomial dimension decomposition method based on sparse Bayesian learning and Bayesian model averaging

The polynomial dimensional decomposition (PDD), an orthogonal polynomial-based metamodel, has received increasing attention in uncertainty quantification (UQ). Nevertheless, for complex high-dimensional problems, its computational burden may become unaffordable, which is usually called curse of dimensionality. To solve this problem, sparse regression methods can be considered to establish a sparse PDD model. However, when the design samples are limited, their computational accuracy may be low due to the enormous size of the polynomial bases. Aimed at this issue, we proposed a novel sparse PDD metamodel based on the Bayesian LASSO (least absolute shrinkage and selection operator) method and an adaptive candidate basis selection and model updating method (CBSMU). Firstly, to improve the CPU efficiency, an analytical Bayesian LASSO based on the sparse Bayesian learning is used for the regression analysis, which replaces the time-consuming Markov chain Monte Carlo sampling of the traditional method with the efficient iteration algorithm for calculating the posterior estimations. Then, to reduce the size of the polynomial bases, this study proposes the adaptive CBSMU for screening the significant candidate polynomial bases and updating the metamodel sequentially. The CBSMU can find out the candidate bases that contributes to improve the prediction accuracy in the view of Bayesian model averaging. Thus, during the process of the sparse PDD modeling, the size of candidate bases is relatively small, which facilitates to improve the final computational accuracy when the design samples are limited. We verify the proposed method using three high-dimensional numerical examples, and apply it to solve one complex high-dimensional engineering problem. The results show that the proposed method is more accurate for UQ than the two common methods with the same computational costs, and is well-suited for solving complex high-dimensional structural dynamic problem.

Combining set propagation with finite element methods for time integration in transient solid mechanics problems

The Finite Element Method (FEM) is the gold standard for spatial discretization in numerical simulations for a wide spectrum of real-world engineering problems. Prototypical areas of interest include linear heat transfer and linear structural dynamics problems modeled with partial differential equations (PDEs). While different algorithms for direct integration of the equations of motion exist, exploring all feasible behaviors for varying loads, initial states and fluxes in models with large numbers of degrees of freedom remains a challenging task. In this article we propose a novel approach, based in set propagation methods and motivated by recent advances in the field of Reachability Analysis. Assuming a set of initial states and inputs, the proposed method consists in the construction of a union of sets (flowpipe) that enclose the infinite number of solutions of the spatially discretized PDE. We present the numerical results obtained in five examples to illustrate the capabilities of our approach, and compare its performance against reference numerical integration methods. We conclude that, for problems with single known initial conditions, the proposed method is accurate. For problems with uncertain initial conditions included in sets, the proposed method can compute all the solutions of the system more efficiently than numerical integration methods.

A novel predictor–corrector explicit integration scheme for structural dynamics

A novel predictor–corrector explicit scheme is presented to solve structural dynamic problems. It is a three sub-steps method, in which the previous two sub-steps are set as predictors and the last sub-step is regarded as correctors. The explicit scheme is third-order accuracy and can achieve forth-order accuracy in the absent of physical damping. The stability limit of the proposed scheme is much larger than the existing methods. Also, the numerical dissipation and dispersion of the explicit scheme can be controlled through the different algorithm parameters. The explicit scheme not only possesses adequate numerical dissipation and dispersion in high-frequency responses, but also gets small numerical errors in the whole frequency domain for dynamic system. These performances of the proposed explicit scheme are further highlighted in comparison with other typical explicit schemes.

Model Bridge Span Traversed by a Heavy Mass: Analysis and Experimental Verification

In this work, we investigate the transient response of a model bridge traversed by a heavy mass moving with constant velocity. Two response regimes are identified, namely forced vibrations followed by free vibrations as the moving mass goes past the far support of the simply supported span of the bridge. Despite this being a classical problem in structural dynamics, there is an implicit assumption in the literature that moving loads possess masses that are at least an order of magnitude smaller than the mass of the bridge span that they traverse. This alludes to interaction problems involving secondary systems, whose presence does not alter the basic characteristics of the primary system. In our case, the dynamic properties of the bridge span during the passage of a heavy mass change continuously over time, leading to an eigenvalue problem that is time dependent. During the free vibration regime, however, the bridge recovers the expected dynamic properties corresponding to its original configuration. Therefore, the aim here is the development of a mathematical model whose numerical solution is validated by comparison with experimental results recovered from an experiment involving a scaled bridge span traversed by a rolling mass. Following that, the target is to identify regions in the transient response of the bridge span that can be used for recovering the bridge’s dynamic properties and subsequently trace the development of structural damage. In closing, the present work has ramifications in the development of structural health monitoring systems applicable to critical civil engineering infrastructure, such as railway and highway bridges.

Asynchronous iterations of HSS method for non-Hermitian linear systems

A general asynchronous alternating iterative model is designed, for which convergence is theoretically ensured both under classical spectral radius bound and, then, for a classical class of matrix splittings for -matrices. The computational model can be thought of as a two-stage alternating iterative method, which well suits to the well-known Hermitian and skew-Hermitian splitting (HSS) approach, with the particularity here of considering only one inner iteration. Experimental parallel performance comparison is conducted between the generalized minimal residual (GMRES) algorithm, the standard HSS and our asynchronous variant, on both real and complex non-Hermitian linear systems, respectively, arising from convection–diffusion and structural dynamics problems. A significant gain on execution time is observed in both cases.

Discrepancy-based control for positioning of large gantry crane

This paper is concerned with nonlinear control for positioning of large gantry cranes. An important structural dynamics problem of this type of cranes are flexible structural vibrations in the direction of trolley travel. They lead to faster wear of the crane construction and worsen the performance of the crane operation. Since they are mainly caused by movement of the trolley, they can be taken in consideration by redesign of the trolley motion control system. To design an appropriate control law, a generalized error measure, known as the discrepancy, is applied. Utilizing the associated stability with respect to two discrepancies, a nonlinear stabilizing control for large gantry cranes is obtained. The proposed control has been evaluated on a mathematical model and an experimental laboratory crane.

The modified weighted residual formulation in the wave based method for plate bending problems: A general formulation for different types of edge restraints

Wave based method is an efficient numerical technique that can be used to solve the structural dynamic problems, especially the vibration of plates. The approach expands the dynamic field variables to a weighted sum of wave functions, and the contribution factors are determined by minimizing the errors on the boundary conditions. While the governing equation is satisfied a priori, the boundary conditions are satisfied in the integral sense, using preferably a weighted residual formulation. However, the weighted residual formulation is not always suitable for all kinds of boundary conditions. In plate bending problems, the plate edges can be modelled as clamped, simply supported, free, symmetric or (visco) elastically restrained against translation or/and rotation, depending on how they are restrained in reality. But the currently available weighted residual formulations are suitable either only for the non-elastic restraints (clamped, simply supported, free and symmetric) or only for the general (visco) elastic restraints with both translational and rotational elasticities. In this paper, a general formulation is proposed for all the aforementioned types of edge restraints. It is equivalent to the conventional one when the plate edges are non-elastically restrained, but performs better with the introduction of corner residuals for the plate with general elastically restrained edges. Apart from the ideal non-elastic and the general full-elastic restraints, this modified formulation also works well for the partial-elastic ones with only either translational or rotational elasticity. Numerical cases are studied on a rectangular plate for all the restraint types, where the proposed formulation is validated. The restraint types can be identical or different among the plate edges, and how to apply the formulation to the non-uniformed situation is also demonstrated. Throughout the considered cases, this paper shows the accuracy and efficiency of the wave based method by comparing with the finite element method in computing the forced vibration of the thin plate. With the modified weighted residual formulation, the method further shows its great convenience in changing the restraint types or stiffness (and damping) parameters of plate edges.

A weak form temporal quadrature element formulation for linear structural dynamics

PurposeVarious time integration methods and time finite element methods have been developed to obtain the responses of structural dynamic problems, but the accuracy and computational efficiency of them are sometimes not satisfactory. The purpose of this paper is to present a more accurate and efficient formulation on the basis of the weak form quadrature element method to solve linear structural dynamic problems.Design/methodology/approachA variational principle for linear structural dynamics, which is inspired by Noble's work, is proposed to develop the weak form temporal quadrature element formulation. With Lobatto quadrature rule and the differential quadrature analog, a system of linear equations is obtained to solve the responses at sampling time points simultaneously. Computation for multi-elements can be carried out by a time-marching technique, using the end point results of the last element as the initial conditions for the next.FindingsThe weak form temporal quadrature element formulation is conditionally stable. The relation between the normalized length of element and the suggested number of integration points in one element is given by a simple formula. Results show that the present formulation is much more accurate than other time integration methods and its dissipative property is also illustrated.Originality/valueThe weak form temporal quadrature element formulation provides a choice with high accuracy and efficiency for solution of linear structural dynamic problems.

Symplectic perturbation series methodology for non-conservative linear Hamiltonian system with damping

In this paper, the symplectic perturbation series methodology of the non-conservative linear Hamiltonian system is presented for the structural dynamic response with damping. Firstly, the linear Hamiltonian system is briefly introduced and its conservation law is proved based on the properties of the exterior products. Then the symplectic perturbation series methodology is proposed to deal with the non-conservative linear Hamiltonian system and its conservation law is further proved. The structural dynamic response problem with eternal load and damping is transformed as the non-conservative linear Hamiltonian system and the symplectic difference schemes for the non-conservative linear Hamiltonian system are established. The applicability and validity of the proposed method are demonstrated by three engineering examples. The results demonstrate that the presented methodology is better than the traditional Runge–Kutta algorithm in the prediction of long-time structural dynamic response under the same time step.

Updating structured matrix pencils with no spillover effect on unmeasured spectral data and deflating pair

This paper is devoted to the study of perturbations of a matrix pencil, structured or unstructured, such that a perturbed pencil will reproduce a given deflating pair while maintaining the invariance of the complementary deflating pair. If the latter is unknown, it is referred to as no spillover updating. The specific structures considered in this paper include symmetric, Hermitian, ⋆-even, ⋆-odd and ⋆-skew-Hamiltonian/Hamiltonian pencils. This study is motivated by the well-known Finite Element Model Updating Problem in structural dynamics, where the given deflating pair represents a set of given eigenpairs and the complementary deflating pair represents the remaining larger set of eigenpairs. Analytical expressions of structure preserving no spillover updating are determined for deflating pairs of structured matrix pencils. Besides, parametric representations of all possible unstructured perturbations are obtained when the complementary deflating pair of a given unstructured pencil is known. In addition, parametric expressions are obtained for structured updating with certain desirable structures which relate to existing results on structure preservation of a symmetric positive definite or semi definite matrix pencil.

Two-Step Unconditionally Stable Noniterative Dissipative Displacement Method for Analysis of Nonlinear Structural Dynamics Problems

When solving structural dynamic problems, the displacement algorithm needs only calculating and storing structure’s displacements in the main calculation process, which makes the displacement algorithm have advantages over multivariable algorithms in calculation efficiency and storage requirements. By using a novel approach based on dimensional analysis firstly given by the first author, a one-parameter family of two-step unconditionally stable noniterative displacement algorithms, referred to as the CQ-2x method, is developed. Compared with other unconditionally stable noniterative multivariable algorithms such as the representative KR- α method, the proposed method has advantages in several aspects. The CQ-2x method is unconditionally stable regardless of stiffness hardening or stiffness weakening, while the KR- α method is only conditionally stable in case of stiffness hardening. The CQ-2x method needs only one solver within one time step, while the KR- α method needs two solvers within one time step, which makes the CQ-2x method show higher efficiency. Numerical examples are presented to demonstrate the potential of the proposed method.