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This paper presents a study of wave propagation through an infinite periodic structure that consists of elastic Timoshenko beams interconnected with rigid bodies. This is a generalized approach in which the beams are not coaxial and the centre of mass of each rigid body is placed away from the intersection of their neutral axes. An analytical approach is used by applying the transfer matrix method (TMM), along with the Floquet–Bloch theorem for elastic wave propagation. Subsequent parametric analysis is performed with visualization of resulting band diagrams of a representative structure. These results are verified through comparison with solutions obtained using the finite-element method (FEM). In this manner, a comprehensive dynamical analysis of tailored metastructures is provided. This article is part of the theme issue 'Current developments in elastic and acoustic metamaterials science (Part 2)'. [ABSTRACT FROM AUTHOR]

This paper communicates an analytical study on computing the natural frequencies and in-plane deflections caused by static forces in the panel walls using Euler-Bernoulli, Timoshenko, Timoshenko and Goodier, Couple-stress, and Micropolar-Cosserat theory. The study highlights the formulation of the transfer matrix via the state-space method in the spatial domain; from coupled governing equations of motion that arises from the Micropolar-Cosserat theory. This theory captures the novel curvature of edges and moments of the panels at energy density level due to its unique feature of asymmetric shear stresses; that emphasizes the loss of ellipticity of governing equations. The analytical solution of the Micropolar-Cosserat theory yield appropriate results compared to plane-stress simulation of the panels using finite element analysis. [ABSTRACT FROM AUTHOR]

A machine learning assisted efficient, yet comprehensive characterization of the dynamics of coronaviruses, in conjunction with finite element (FE) approach, is presented. Without affecting the accuracy of prediction in low‐frequency vibration analysis, an equivalent model for the FE analysis is proposed, based on which the natural frequencies corresponding to first three non‐rigid modes are analyzed. To quantify the inherent system‐uncertainty efficiently, Monte Carlo simulation is proposed in conjunction with the machine learning based FE computational framework for obtaining complete probabilistic descriptions considering both individual and compound effect of stochasticity. A variance based sensitivity analysis is carried out to enumerate the relative significance of different material parameters corresponding to various constituting parts of the coronavirus structure. Using the modal characteristics like natural frequencies and mode shapes of the virus structure including their stochastic bounds, it is possible to readily identify coronaviruses by comparing the experimentally measured dynamic responses in terms of the peaks of frequency response function. Results from this first of its kind study on coronaviruses along with the proposed generic machine learning based approach will accelerate the detection of viruses and create efficient pathways toward future inventions leading to cure and containment in the field of virology. [ABSTRACT FROM AUTHOR]

The finite element method has been employed to study the effects of different boundary conditions on the axial buckling of multiwall carbon nanotubes (MWCNTs). Unlike previous works, both homogeneous and heterogeneous end constraints are considered for the constituent tubes of various MWCNTs comprising shell-type (i.e., the length-to-diameter ratio L/D<10), beam-type (i.e., L/D>10), and the two different types of constituent tubes. The results show that clamping the individual tubes of simply supported or free MWCNTs exerts a variety of influences on their buckling behaviors depending on the type of the MWCNTs, the position, and the number of the clamped tubes. Clamping the outermost tube can enhance the critical buckling strain up to four times of its original value and can shift the buckling modes of those MWCNTs consisting both shell- and beam-type tubes. In contrast, little difference can be observed when simply supported ends of MWCNTs are replaced by free ends or vice versa. Explicit buckling mode shapes obtained using the finite element method for various physically realistic cases have been shown in the paper. [ABSTRACT FROM AUTHOR]

Most piezoelectric energy harvesting research has focused on developing on-resonance harvesters that work at low frequencies, even though higher frequencies can generate more power. In addition, conventional resonant harvesters have low efficiency when the excitation frequency is away from resonance. Using mechanical impacts has the potential to improve the overall harvested energy since high frequencies are excited during impacts. Also, the presence of impacts reduces the influence of the base excitation frequency and the requirement to exactly match the resonance frequency. To take advantage of the higher frequency response, an impact energy harvester is designed and validated experimentally. The harvester consists of a cantilever beam with a piezoelectric patch attached to its base which impacts with a stiff object. The harvester is modelled using finite element analysis and a Hertzian contact law. The model is tested and validated in the laboratory using an in-house manufactured demonstrator. Good agreement with the experimental data is obtained, setting the basis for future optimisation of the harvester geometry and piezoelectric properties. [ABSTRACT FROM AUTHOR]

A mathematical form for the response of the stochastic finite element analysis of elliptical partial differential equations has been established through summing products of random scalars and random vectors. The method is based upon the eigendecomposition of a system’s stiffness matrix. The computational reduction is achieved by only summing the dominant terms and by approximating the random eigenvalues and the random eigenvectors. An error analysis has been conducted to investigate the effect of the truncation and the approximations. Consequently, a novel error minimisation technique has been applied through the Galerkin error minimisation approach. This has been implemented by utilising the orthogonal nature of the random eigenvectors. The proposed method is used to solve three numerical examples: the bending of a stochastic beam, the flow through a porous media with stochastic permeability and the bending of a stochastic plate. The results obtained through the proposed random eigenfunction expansion approach are compared with those obtained by using direct Monte Carlo Simulations and by using polynomial chaos. [ABSTRACT FROM AUTHOR]

Bayesian techniques have been widely used in finite element model (FEM) updating. The attraction of these techniques is their ability to quantify and characterize the uncertainties associated with dynamic systems. In order to update an FEM, the Bayesian formulation requires the evaluation of the posterior distribution function. For large systems, this function is difficult to solve analytically. In such cases, the use of sampling techniques often provides a good approximation of this posterior distribution function. The hybrid Monte Carlo (HMC) method is a classic sampling method used to approximate high-dimensional complex problems. However, the acceptance rate of HMC is sensitive to the system size, as well as to the time step used to evaluate the molecular dynamics trajectory. The shadow HMC technique (SHMC), which is a modified version of the HMC method, was developed to improve sampling for large system sizes by drawing from a modified shadow Hamiltonian function. However, the SHMC algorithm performance is limited by the use of a non-separable modified Hamiltonian function. Moreover, two additional parameters are required for the sampling procedure, which could be computationally expensive. To overcome these weaknesses, the separable shadow HMC (S2HMC) method has been introduced. This method uses a transformation to a different parameter space to generate samples. In this paper, we analyse the application and performance of these algorithms, including the parameters used in each algorithm, their limitations and the effects on model updating. The accuracy and the efficiency of the algorithms are demonstrated by updating the finite element models of two real mechanical structures. It is observed that the S2HMC algorithm has a number of advantages over the other algorithms; for example, the S2HMC algorithm is able to efficiently sample at larger time steps while using fewer parameters than the other algorithms. [ABSTRACT FROM PUBLISHER]

An analytical formulation has been developed in this article for predicting the equivalent elastic properties of irregular honeycombs with spatially random variations in cell angles. Employing unit-cell based approaches, closed-form expressions of equivalent elastic properties of regular honeycombs are available. Closed-form expressions for equivalent elastic properties of irregular honeycombs are very scarce in available literature. In general, direct numerical simulation based methods are prevalent for this case. This paper proposes a novel analytical framework for predicting equivalent in-plane elastic moduli of irregular honeycombs using a representative unit cell element (RUCE) approach. Using this approach, closed-form expressions of equivalent in-plane elastic moduli (longitudinal and transverse Young’s modulus, shear modulus, Poisson’s ratios) have been derived. The expressions of longitudinal Young’s modulus, transverse Young’s modulus, and shear modulus are functions of both structural geometry and material properties of irregular honeycombs, while the Poisson’s ratios depend only on structural geometry of irregular honeycombs. The elastic moduli obtained for different degree of randomness following the proposed analytical approach are found to have close proximity to direct finite element simulation results. [ABSTRACT FROM AUTHOR]

An analytical framework has been developed for predicting the equivalent in-plane elastic moduli (longitudinal and transverse Young’s modulus, shear modulus, Poisson’s ratios) of irregular auxetic honeycombs with spatially random variations in cell angles. Employing a bottom up multi-scale based approach, computationally efficient closed-form expressions have been derived in this article. This study also includes development of a highly generalized finite element code capable of accepting number of cells in two perpendicular directions, random structural geometry and material properties of irregular auxetic honeycomb and thereby obtaining five in-plane elastic moduli of the structure. The elastic moduli obtained for different degree of randomness following the analytical formulae have been compared with the results of direct finite element simulations and they are found to be in good agreement corroborating the validity and accuracy of the proposed approach. The transverse Young’s modulus, shear modulus and Poisson’s ratio for loading in transverse direction (effecting the auxetic property) have been found to be highly influenced by the structural irregularity in auxetic honeycombs. [ABSTRACT FROM AUTHOR]

This paper introduces the idea of nonlocal normal modes arising in the dynamic analysis of nanoscale structures. A nonlocal finite element approach is developed for the axial vibration of nanorods, bending vibration of nanobeams and transverse vibration of nanoplates. Explicit expressions of the element mass and stiffness matrices are derived in closed-form as functions of a length-scale parameter. In general the mass matrix can be expressed as a sum of the classical local mass matrix and a nonlocal part. The nonlocal part of the mass matrix is scale-dependent and vanishes for systems with larger lengths. Classical modal analysis and perturbation method are used to understand the dynamic behaviour of discrete nonlocal systems in the light of classical local systems. The conditions for the existence of classical normal modes for undamped and damped nonlocal systems are established. Closed-form approximate expressions of nonlocal natural frequencies, modes and frequency response functions are derived. Results derived in the paper are illustrated using examples of axial and bending vibration of nanotubes and transverse vibration of graphene sheets. [ABSTRACT FROM AUTHOR]

A model order reduction scheme of the transient response of large-scale randomly parametrized linear finite element system in state space form has been proposed. The reduced order model realization is aimed at preserving the invariant properties of the dynamic system model based on the dominant coupling characteristics of the specified system inputs and outputs. An a - priori model reduction strategy based on the balanced truncation method has been proposed in conjunction with the stochastic spectral Galerkin finite element method. Approximation of the dominant modes of the controllability Gramian has been performed with iterative Arnoldi scheme applied to Lyapunov equations. The reduced order representation of the randomly parametrized dynamical system has been obtained with Arnoldi–Lyapunov vector basis using an implicit time stepping algorithm. The performance and the computational efficacy of the proposed scheme has been illustrated with examples of randomly parametrized advection–diffusion–reaction problem under the action of transient external forcing functions. The convergence of the proposed reduced order scheme has been shown with a-posteriori error estimates. [ABSTRACT FROM AUTHOR]

Stiffness matrices of beams with stochastic distributed parameters modelled by random fields are considered. In stochastic finite element analysis, deterministic shape functions are traditionally employed to derive stiffness matrices using the variational principle. Such matrices are not exact because the deterministic shape functions are not derived from the exact solution of the governing stochastic partial differential equation with the relevant boundary conditions. This paper proposes an analytical method based on Castigliano's approach for a beam element with general spatially varying parameters. This gives the exact and a simple closed-form expression of the stiffness matrix in terms of certain integrals of the spatially varying function. The expressions are valid for any integrable random fields. It is shown that the exact element stiffness matrix of a stochastically parametered beam can be expressed by three basic random variables. Analytical expressions of the random variables and their associated coefficient matrices are derived for two cases: when the bending rigidity is a random field and when the bending flexibility is a random field. It is theoretically proved that the conventional stochastic element stiffness matrix is a first-order perturbation approximation to the exact expression. A sampling method to obtain the basic random variables using the Karhunen–Loève expansion is proposed. Results from the exact stiffness matrices are compared with the approximate conventional stiffness matrix. Gaussian and uniform random fields with different correlation lengths are used to illustrate the numerical results. The exact closed-form analytical expression of the element stiffness matrix derived here can be used for benchmarking future numerical methods. [ABSTRACT FROM AUTHOR]

SUMMARY The problem of representing random fields describing the material and boundary properties of the physical system at discrete points of the spatial domain is studied in the context of linear stochastic finite element method. A randomly parameterized diffusion system with a set of independent identically distributed stochastic variables is considered. The discretized parametric fields are interpolated within each element with multidimensional Lagrange polynomials and integrated into the weak formulation. The proposed discretized random-field representation has been utilized to express the random fluctuations of the domain boundary with nodal position coordinates and a set of random variables. The description of the boundary perturbation has been incorporated into the weak stochastic finite element formulation using a stochastic isoparametric mapping of the random domain to a deterministic master domain. A method for obtaining the linear system of equations under the proposed mapping using generic finite element weak formulation and the stochastic spectral Galerkin framework is studied in detail. The treatment presents a unified way of handling the parametric uncertainty and random boundary fluctuations for dynamic systems. The convergence behavior of the proposed methodologies has been demonstrated with numerical examples to establish the validity of the numerical scheme. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: Uncertainty propagation in complex engineering systems with fuzzy variables constitutes a significant challenge. This paper proposes a Polynomial Chaos type spectral approach based on orthogonal function expansion. A fuzzy variable is represented as a set of interval variables via the membership function. The interval variables are further transformed into the standard interval . Smooth nonlinear functions of standard interval variables are projected in the basis of Legendre polynomials by exploiting its orthogonal properties over the interval . The coefficients associated with the basis functions are obtained by a Galerkin type of error minimisation. The method is first illustrated using scalar functions of multiple fuzzy variables. Later the method is proposed for elliptic type finite element problems where the technique is extended to vector valued functions with multiple fuzzy variables. The response of such systems can be expressed in the complete basis of multivariate Legendre polynomials. The coefficients, obtained by Galerkin type of error minimisation, can be calculated from the solution of an extended set of linear algebraic equations. An eigenfunction based model reduction technique is proposed to obtain the coefficient vectors in an efficient way. A numerical example of axial deformation of a rod with fuzzy axial stiffness is considered to illustrate the proposed methods. Linear and nonlinear membership functions are used and the results are compared with direct numerical simulation results. [Copyright &y& Elsevier]

Abstract: A novel dynamic finite element method is carried out for a small-scale nonlocal rod which is embedded in an elastic medium and undergoing axial vibration. Eringen's nonlocal elasticity theory is employed. Natural frequencies are derived for general boundary conditions. An asymptotic analysis is carried out. The stiffness and mass matrices of the embedded nonlocal rod are obtained using the proposed finite element method. Nonlocal rods embedded in an elastic medium have an upper cut-off natural frequency which is independent of the boundary conditions and the length of the rod. Dynamic response for the damped case has been obtained using the conventional finite element and dynamic finite element approaches. The present study would be helpful for developing nonlocal finite element models and study of embedded carbon nanotubes for future nanocomposite materials. [Copyright &y& Elsevier]

Abstract: The probabilistic characterisation of a frequency response function can be very valuable for structural design and control. Unfortunately, obtaining a sufficiently large sample for wide ranges of vibration can easily become unaffordable. A variety of modelling assumptions can dramatically add to the computational cost: nonproportional damping, multiscale material properties, and high-resolution finite element analysis are some examples. This paper explores Bayesian emulators as surrogates for expensive finite element models in structural dynamics. We demonstrate the effectiveness of the method by performing uncertainty analysis of the frequency response of a nonproportionally damped plate made of a carbon fibre/epoxy composite material. [Copyright &y& Elsevier]

Abstract: Free and forced axial vibrations of damped nonlocal rods are investigated. Two types of nonlocal damping models, namely, strain-rate-dependent viscous damping and velocity-dependent viscous damping, are considered. A frequency-dependent dynamic finite element method is developed to obtain the forced vibration response. Frequency-adaptive complex-valued shape functions are proposed to obtain the dynamic stiffness matrix in closed form. The stiffness and mass matrices of the nonlocal rod are also obtained using the conventional finite element method. Results from the dynamic finite element method and conventional finite element method are compared. Using an asymptotic analysis it is shown that, unlike its local counterpart, a nonlocal rod has a maximum cut-off frequency. A closed-form exact expression for this maximum frequency as a function of the nonlocal parameter has been obtained for undamped and damped systems. The frequency response function obtained using the proposed dynamic finite element method shows extremely high modal density near the maximum frequency. This leads to clustering of resonance peaks which is not easily obtainable using classical finite element analysis. [Copyright &y& Elsevier]

Abstract: Uncertainty propagation engineering systems possess significant computational challenges. This paper explores the possibility of using correlated function expansion based metamodelling approach when uncertain system parameters are modeled using Fuzzy variables. In particular, the application of High-Dimensional Model Representation (HDMR) is proposed for fuzzy finite element analysis of dynamical systems. The HDMR expansion is a set of quantitative model assessment and analysis tools for capturing high-dimensional input–output system behavior based on a hierarchy of functions of increasing dimensions. The input variables may be either finite-dimensional (i.e., a vector of parameters chosen from the Euclidean space ) or may be infinite-dimensional as in the function space . The computational effort to determine the expansion functions using the alpha cut method scales polynomially with the number of variables rather than exponentially. This logic is based on the fundamental assumption underlying the HDMR representation that only low-order correlations among the input variables are likely to have significant impacts upon the outputs for most high-dimensional complex systems. The proposed method is integrated with a commercial Finite Element software. Modal analysis of a simplified aircraft wing with Fuzzy parameters has been used to illustrate the generality of the proposed approach. In the numerical examples, triangular membership functions have been used and the results have been validated against direct Monte Carlo simulations. [Copyright &y& Elsevier]

The stochastic finite element analysis of elliptic type partial differential equations is considered. A reduced method of the spectral stochastic finite element method using polynomial chaos is proposed. The method is based on the spectral decomposition of the deterministic system matrix. The reduction is achieved by retaining only the dominant eigenvalues and eigenvectors. The response of the reduced system is expanded as a series of Hermite polynomials, and a Galerkin error minimization approach is applied to obtain the deterministic coefficients of the expansion. The moments and probability density function of the solution are obtained by a process similar to the classical spectral stochastic finite element method. The method is illustrated using three carefully selected numerical examples, namely, bending of a stochastic beam, flow through porous media with stochastic permeability and transverse bending of a plate with stochastic properties. The results obtained from the proposed method are compared with classical polynomial chaos and direct Monte Carlo simulation results. [ABSTRACT FROM AUTHOR]

Abstract: A complex system can be modeled using various fidelities with the finite element method. A high-fidelity model is expected to be more computationally expensive compared to a low-fidelity model and in general may contain more degrees of freedom and more elements. This paper proposes a novel multi-fidelity approach to solve boundary value problems using the finite element method. A Bayesian approach based on Gaussian process emulators in conjunction with the domain decomposition method is developed. Using this approach one can seamlessly assimilate a low-fidelity model with a more expensive high-fidelity model. The idea is illustrated using elliptic boundary value problems. [Copyright &y& Elsevier]

Abstract: This paper proposes a hyperelastic finite element-based lattice approach for the description of buckling behaviour in single wall carbon nanotubes (SWCNTs). A one-term incompressible Ogden-type hyperelastic model is adopted to describe the equivalent mechanical response of C–C bonds in SWCNTs under axial compression. The material constants of the model are chosen by matching the linearised response with the elastic constants adopted in the AMBER force field and by establishing equivalence between the Ogden strain energy and the variation of the interatomic strain energy obtained from molecular mechanics simulations. Numerical experiments are carried out and the results are compared to atomistic simulations, demonstrating the predictive capabilities of the present model in capturing initial buckling strain, deformation mechanisms and post-buckling behaviour under very large compressive deformations. [Copyright &y& Elsevier]

Abstract: Estimating the natural frequencies of a wind turbine system consisting rotor, nacelle, tower, foundation and surrounding soil is one of the important design considerations. This paper experimentally investigates the behaviour of a model wind turbine supported on a particular type of foundation called a monopile. Monopile is a single large diameter (2.5–4m) long slender column inserted deep into the ground. This can be thought of as an extension of the wind turbine tower. In particular, the role of soil/foundation in the dynamics of wind turbines has been investigated. Analytical methods are developed incorporating the rotational and translation flexibility of the foundation. Novel experimental techniques have been developed to obtain the parameters necessary for the analytical model. The analytical model is validated using a finite element approach and experimental measurements. In total, results from 17 test cases is reported in the paper. Experimental results show that the natural frequencies and the damping factors of the wind turbine tower change significantly with the type of soil/foundation. Analytical results for the natural frequencies agree reasonably well to the experimental results and finite element results. [Copyright &y& Elsevier]

Abstract: The stochastic finite element analysis of elliptic type partial differential equations with non-Gaussian random fields are considered. A novel approach by projecting the solution of the discretized equation into a reduced finite dimensional orthonormal vector basis is investigated. It is shown that the solution can be obtained using a finite series comprising functions of random variables and orthonormal vectors. These functions, called as the spectral functions, can be expressed in terms of the spectral properties of the deterministic coefficient matrices arising due to the discretization of the governing partial differential equation. Based on the projection in a reduced orthonormal vector basis, a Galerkin error minimization approach is proposed. The constants appearing in the Galerkin method are solved from a system of linear equations which has much smaller dimension compared to the original discretized equation. A hybrid analytical and simulation based computational approach is proposed to obtain the moments and probability density function of the solution. The method is illustrated using the stochastic nanomechanics of a zinc oxide (ZnO) nanowire deflected under the atomic force microscope (AFM) tip. The results are compared with the results obtained using direct Monte Carlo simulation, classical Neumann expansion and polynomial chaos approach for different correlation lengths and strengths of randomness. [Copyright &y& Elsevier]

Abstract: Reliability analysis of uncertain dynamical systems is considered. The excitations are modeled as non-stationary Gaussian processes, whereas parametric uncertainties due to structural randomness are modeled as non-Gaussian random variables. The structural responses are, therefore, non-Gaussian processes. The limit state is formulated in terms of the extreme value distribution of the response process. Developing these extreme value statistics analytically is not straightforward, which makes failure probability estimations difficult. An alternative procedure is investigated for computing exceedance probabilities. Proposed approach involves generating a full functional operational model, which approximates the original limit surface. Once the approximate form of the original limit state is defined, the failure probability can be obtained by statistical simulation. Thus, the method can be integrated with commercial finite element software, which permits uncertainty analysis of large structures with complexities that include material and geometric nonlinear behavior. [Copyright &y& Elsevier]

Abstract: Uncertainty propagation in multi-parameter complex structures possess significant computational challenges. This paper investigates the possibility of using the High Dimensional Model Representation (HDMR) approach when uncertain system parameters are modeled using fuzzy variables. In particular, the application of HDMR is proposed for fuzzy finite element analysis of linear dynamical systems. The HDMR expansion is an efficient formulation for high-dimensional mapping in complex systems if the higher order variable correlations are weak, thereby permitting the input–output relationship behavior to be captured by the terms of low-order. The computational effort to determine the expansion functions using the method scales polynomically with the number of variables rather than exponentially. This logic is based on the fundamental assumption underlying the HDMR representation that only low-order correlations among the input variables are likely to have significant impacts upon the outputs for most high-dimensional complex systems. The proposed method is first illustrated for multi-parameter nonlinear mathematical test functions with fuzzy variables. The method is then integrated with a commercial finite element software (ADINA). Modal analysis of a simplified aircraft wing with fuzzy parameters has been used to illustrate the generality of the proposed approach. In the numerical examples, triangular membership functions have been used and the results have been validated against direct Monte Carlo simulations. It is shown that using the proposed HDMR approach, the number of finite element function calls can be reduced without significantly compromising the accuracy. [ABSTRACT FROM AUTHOR]

Abstract: This paper investigates the hyperelastic behaviour of single wall carbon nanotubes (SWCNTs) by means of a finite element-based lattice approach. A one-term incompressible Ogden-type hyperelastic model is chosen to describe the mechanical response of SWCNTs under tensile loading. In order to determine the material constants of the model, numerical tests are conducted on a representative arrangement of carbon atoms, establishing equality between the Ogden strain–energy and the variation of the Tersoff–Brenner interatomic potential. The material constants determined here are then used in numerical simulations carried out on SWCNTs models. A good predictive capability of the present model is found when the obtained results are compared to published data. A first conclusion obtained from the present work suggests that a value of 0.147nm for the Cuivalent diameter is suitable for the hyperelastic description of SWCNTs. A second conclusion reveals a prediction of 0.51 for the breaking strain of SWCNTs under tension, which is in excellent agreement with results obtained from molecular dynamics simulations and continuum theory. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents a generic high dimensional model representation (HDMR) method for approximating the system response in terms of functions of lower dimensions. The proposed approach, which has been previously applied for problems dealing only with random variables, is extended in this paper for problems in which physical properties exhibit spatial random variation and may be modelled as random fields. The formulation of the extended HDMR is similar to the spectral stochastic finite element method in the sense that both of them utilize Karhunen–Loève expansion to represent the input, and lower-order expansion to represent the output. The method involves lower dimensional HDMR approximation of the system response, response surface generation of HDMR component functions, and Monte Carlo simulation. Each of the low order terms in HDMR is sub-dimensional, but they are not necessarily translating to low degree polynomials. It is an efficient formulation of the system response, if higher-order variable correlations are weak, allowing the physical model to be captured by the first few lower-order terms. Once the approximate form of the system response is defined, the failure probability can be obtained by statistical simulation. The proposed approach decouples the finite element computations and stochastic computations, and consecutively the finite element code can be treated as a black box, as in the case of a commercial software. Numerical examples are used to illustrate the features of the extended HDMR and to compare its performance with full scale simulation. [Copyright &y& Elsevier]

Abstract: In stochastic finite element problems the solution of a system of coupled linear random algebraic equations is needed. This problem in turn requires the calculation of the inverse of a random matrix. Over the past four decades several approximate analytical methods and simulation methods have been proposed for the solution of this problem in the context of probabilistic structural mechanics. In this paper we present a new solution method for stochastic linear equations. The proposed method is based on Neumann expansion and the recently developed joint diagonalisation solution strategy. Unlike the classical Neumann expansion, here only the inversion of a diagonal matrix is needed. Numerical examples are given to illustrate the use of the expressions derived in the paper. [ABSTRACT FROM AUTHOR]

Abstract: We demonstrate that the mechanical behaviour of bilayer graphene can be predicted using a first order deformation approach typical of sandwich structures. The mechanical transverse deformation of bilayer graphene under central loading is simulated using a mixed atomistic continuum – Finite Element technique, and the results post-processed using a First Order Sandwich Structures (FOSS) deformation approach. The proposed technique provides good agreement with experimental and theoretical data available in open literature. [Copyright &y& Elsevier]

Abstract: The potential of single-walled carbon nanotubes (SWCNTs) as a mass sensor is examined using continuum mechanics based approach. The carbon nanotube resonators are assumed to be either in cantilevered or in bridged configurations. Simple analytical formulas are developed for CNT-based nanoresonators with attached mass. A closed-form expression has been derived to detect the mass of biological objects from the frequency-shift. A simple linear approximation of the nonlinear sensor equation has been investigated. The validity and the accuracy of these formulas are examined for a wide range of cases. The results indicate that the new sensor equations can be used for CNT-based biosensors with reasonable accuracy. [Copyright &y& Elsevier]

In the modeling of complex dynamical systems, high-resolution finite element models are routinely adopted to reduce the discretization error. Such an approach may fail to enhance the confidence in simulation-based predictions when the dynamical systems exhibit significant variability in the parameterization and description of the mathematical model due to parametric (data) uncertainty and model uncertainty. In contrast to parametric uncertainty, model uncertainty poses significant challenges as no explicit parameter is available a priori to characterize its behavior. In the current investigation, a set of sprung-mass oscillators randomly attached to a baseline system (namely, a homogeneous cantilever plate) gives rise to model uncertainty. Contrary to model parametric uncertainty traditionally tackled using stochastic finite element method (SFEM), the model uncertainty arising from randomly attached sprung-masses gives rise to a new variant of dynamical system for every sample. The feasibility of adopting a Wishart random matrix to represent such ensemble of dynamical systems derived from model perturbation is investigated in this paper. The experiments conducted on a vibrating plate with randomly attached sprung-mass oscillators demonstrate that the Wishart random matrix model may provide a reasonable representation of model uncertainty in linear dynamical systems in the medium and high-frequency region. [Copyright &y& Elsevier]

A simple and easy-to-implement algorithm to identify a generalized proportional viscous damping matrix is developed in this work The chief advantage of the proposed technique is that only a single drive-point frequency response function (FRF) measurement is needed. Such FRFs are routinely measured using the standard techniques of an experimental modal analysis, such as impulse test. The practical utility of the proposed identification scheme is illustrated on three representative structures: (1) a free-free beam in flexural vibration, (2) a quasiperiodic three-cantilever structure made by inserting slots in a plate in out-of-plane flexural vibration, and (3) a point-coupled-beam system. The finite element method is used to obtain the mass and stiffness matrices for each system, and the damping matrix is fitted to a measured variation of the damping (modal damping factors) with the natural frequency of vibration. The fitted viscous damping matrix does accommodate for any smooth variation of damping with frequency, as opposed to the conventional proportional damping matrix. It is concluded that a more generalized viscous damping matrix, allowing for a smooth variation of damping as a function of frequency, can be accommodated within the framework of standard finite element modeling and vibration analysis of linear systems. [ABSTRACT FROM AUTHOR]

Abstract: Significant uncertainties have been reported in the material properties of carbon nanotubes. In this paper, a probabilistic approach to take account of these uncertainties is proposed. The probability density function of the natural frequencies is derived in closed-form. The new results are compared with high-fidelity stochastic finite element simulations. [Copyright &y& Elsevier]

Analyzes multiple-degree-of-freedom linear systems whose damping characteristics may be successfully modified. Use of classical modal analysis to treat lumped-parameter nonviscously damped linear systems; Application of time-domain finite-element formulation; Dynamic response analysis of stationary random vibration.

An enrichment scheme based upon the Neumann expansion method is proposed to augment the deterministic coefficient vectors associated with the polynomial chaos expansion method. The proposed approach relies upon a split of the random variables into two statistically independent sets. The principal variability of the system is captured by propagating a limited number of random variables through a low-ordered polynomial chaos expansion method. The remaining random variables are propagated by a Neumann expansion method. In turn, the random variables associated with the Neumann expansion method are utilised to enrich the polynomial chaos approach. The effect of this enrichment is explicitly captured in a new augmented definition of the coefficients of the polynomial chaos expansion. This approach allows one to consider a larger number of random variables within the scope of spectral stochastic finite element analysis in a computationally efficient manner. Closed-form expressions for the first two response moments are provided. The proposed enrichment method is used to analyse two numerical examples: the bending of a cantilever beam and the flow through porous media. Both systems contain distributed stochastic properties. The results are compared with those obtained using direct Monte Carlo simulations and using the classical polynomial chaos expansion approach. [ABSTRACT FROM AUTHOR]

Free and forced bending vibration of damped nonlocal nano-beams resting on an elastic foundation is investigated. Two types of nonlocal damping models, namely, strain-rate-dependent viscous damping and velocity-dependent viscous damping are considered. A frequency-dependent dynamic finite element method is developed to obtain the forced vibration response. Frequency-adaptive complex-valued shape functions are used for the derivation of the dynamic stiffness matrix. It is shown that there are six unique coefficients which define the general dynamic stiffness matrix. These complex-valued coefficients are obtained exactly in closed-form and illustrated numerically as functions of the frequency. It is proved that the general dynamic stiffness matrix reduces to the well known special cases under different conditions. The stiffness and mass matrices of the nonlocal beam are also obtained using the conventional finite element method. A numerical algorithm to extract the eigenvalues from the dynamic stiffness matrix with a transcendental element for the special case when the system is undamped is suggested. Results from the dynamic finite element method and the conventional finite element method are compared. The application of the dynamic stiffness approach is shown through forced response analysis of a double-walled carbon nanotube in pinned-pinned and cantilever configurations. Explicit closed-form expressions of the dynamic response for both the cases have been obtained and the role of crucial system parameters such as, the damping factors, the nonlocal parameter and the foundation stiffness have been investigated. • A novel dynamic finite element method is developed for damped, nonlocal nano-beams resting on an elastic foundation. • Frequency-adaptive complex-valued shape functions are used to obtain coefficients of the dynamic stiffness matrix. • The general dynamic stiffness matrix reduces to classical beam finite element matrices under appropriate limiting cases. • Closed-form expressions have been derived to obtain the exact dynamic response of the nonlocal beam. • Analytical results are validated using classical full-scale finite element simulations. [ABSTRACT FROM AUTHOR]

Graphene and its associated nanostructures (GANS) have been widely investigated by means of experimental and numerical approaches over the last decade. GANS and GANS reinforced composite materials show exceptional promise towards superior mechanical and thermal properties along with limitless opportunity to tailor, control, design, modify and manipulate such properties. These attributes make graphene and its associated nanostructures as one of the most important future material technologies in aerospace, automotive, medical, civil and military sectors of the 21st century. Among the various numerical methods used to analyse GANS and GANS reinforced composite materials, the finite element method (FEM) plays a prominent role. The FEM has been the standard analysis and simulation method for conventional structural and mechanical problems over the past half a century. However, its growing role and impact in atomistic-scale numerical simulation in general, and GANS, in particular, is not well known within the wider scientific and engineering modelling and simulation research community. There is a compelling need to document the expansive use of the finite element method, its advantages, shortcomings, relevance and purpose in a way which is pertinent to both material science and numerical simulation researchers. This paper serves this need by discussing the current state of the art of finite element methodologies available to study GANS and GANS reinforced composites in the most comprehensive manner. A detailed description of the popular space frame based numerical simulation strategy widely used to represent GANS is given. An extensive survey is conducted on more than 600 research papers in order to examine the finite element predictions of the mechanical and thermal properties of graphene and its associated composite materials. These properties are selected in view of their direct relevance to crucial future technologies, such as high-performance automotive components, aerospace and bioengineering systems, energy technologies, and advanced therapeutic and surgical devices. Omissions of some fundamental mechanical and thermal modelling issues for GANS have been identified and insightful guidance towards future research directions to comprehensively address them is given. By reviewing a significant breadth of publications across several academic disciples, a large scatter in the numerical predictions of essential material constants arising from the differences in fundamental assumptions and approximations has been reported. The origin of such discrepancies has been identified, analysed and established. The paper further focuses on the idealization of nanostructures and nanocomposites by means of representative volume elements (RVEs). The need for this multiscale modelling strategy to mature in order to include the simultaneous description of different material length scales within multiphysics simulation problems has been discussed. This paper will serve as standalone reference material for future research works and will pave the way for novel investigations in the context of atomistic simulations and their potential applications to the development of next-generation engineering devices and cutting-edge technological applications. [ABSTRACT FROM AUTHOR]

In this study an analytical model of offshore wind turbines (OWTs) supported on flexible foundation is presented to provide a fast and reasonably accurate natural frequency estimation suitable for preliminary design or verification of Finite Element calculations. Previous research modelled the problem using Euler–Bernoulli beam model where the foundation is represented by two springs (lateral and rotational). In contrast, this study improves on previous efforts by incorporating a cross-coupling stiffness thereby modelling the foundation using three springs. Furthermore, this study also derives the natural frequency using Timoshenko beam model by including rotary inertia and shear deformation. The results of the proposed model are also compared with measured values of the natural frequency of four OWTs obtained from the literature. The results show that the Timoshenko beam model does not improve the results significantly and the slender beam assumption may be sufficient. The cross-coupling spring term has a significant effect on the natural frequency therefore needs to be included in the analysis. The model predicts the natural frequency of existing turbines with reasonable accuracy. [ABSTRACT FROM AUTHOR]

Abstract: A novel approach for uncertainty propagation and response statistics estimation of randomly parametrized structural dynamic systems is developed in this paper. The frequency domain response of a stochastic finite element system is resolved at randomly sampled design points in the input stochastic space with an infinite series expansion using preconditioned stochastic Krylov bases. The system response is expressed in the eigenvector space of the structural system weighted with finite order rational functions of the input random variables, termed spectral functions. The higher the order of the spectral functions, the more accurate is the order of approximation of the stochastic system response. However, this increased accuracy comes at a computational cost. This cost is mitigated by using a Bayesian metamodel. The proposed approach is used to the analyze the stochastic vibration response of a corrugated panel with random elastic parameters. The results obtained with the proposed hybrid approach are compared with direct Monte Carlo simulations, which have been considered as the benchmark solution. [Copyright &y& Elsevier]

Abstract: This paper investigates the effects of spatially uncertain material properties on the aeroelastic response predictions (e.g., rotating frequencies, vibratory loads, etc.) of composite helicopter rotor. Initially, the spatial uncertainty is modeled as discrete random variables along the blade span and uncertainty analysis is performed with direct Monte Carlo simulations (MCS). Uncertainty effects on the rotating frequencies vary with the higher-order modes in a non-linear way. Each modal frequency is found to be more sensitive to the uncertainty at certain sections of the rotor blade than uncertainty at other sections. Uncertainty effects on the vibratory hub load predictions are studied in the next stage. To reduce the computational expense of stochastic aeroelastic analysis, a high-dimensional model representation (HDMR) method is developed to approximate the aeroelastic response as functions of blade stiffness properties which are modeled as random fields. Karhunen–Loève expansion and a lower-order expansion are used to represent the input and outputs, respectively, in the HDMR formulation which is similar to the spectral stochastic finite element method. The proposed method involves the approximation of the system response with lower-dimensional HDMR, the response surface generation of HDMR component functions, and Monte Carlo simulation. The proposed approach decouples the computationally expensive aeroelastic simulations and uncertainty analysis. MCS, performed with computationally less expensive HDMR models, shows that spatial uncertainty has considerable influence on the vibratory hub load predictions. [Copyright &y& Elsevier]

Abstract: In-plane elastic instability of bilayer graphene sheets is investigated using atomistic finite element approaches. The equivalent homogenised properties of graphene sheet are expressed in terms of the thickness, equilibrium lengths and force-field models used to represent the C–C bonds of the graphene lattice. The covalent bonds are represented as structural beams with stretching, bending, torsional and shear deformation, and the strain energies associated to affine deformation mechanisms. The overall mechanical properties and geometric configurations of the nano-structures represented as truss assemblies are then calculated minimising the total potential energy associated to the loading, thickness and average equilibrium lengths of the bonds. Different boundary conditions and aspect ratios are considered for both bilayer and single-layer graphene sheets. The bilayer graphene sheets are found to be offering remarkably higher buckling strengths as compared to single-layer sheets. [Copyright &y& Elsevier]

Abstract: The sliding oscillations of multiwall carbon nanotubes (MWCNTs) with open ends are characterized. The interlayer van der Waals interaction provides a restoring force for the oscillatory system whereas the ultralow interlayer friction was ignored due to its negligible influence on the oscillation. Based on this model, various oscillation modes were obtained with the associated frequency of the order 1–100GHz by using the finite element method. The radius and length dependence and the effect of end constraints have been further studied and quantified for the sliding oscillation of MWCNTs. The results are interpreted in terms of the mass change and the restriction on the motion of the individual tubes of MWCNTs. In view of these analyses, MWCNTs with open ends may expect to serve as nanoscale oscillatory devices where the oscillation modes and frequency can be controlled by choosing the tube geometry and imposing desirable end constraints. [Copyright &y& Elsevier]

The stochastic dynamic stability analysis of laminated composite curved panels under non-uniform partial edge loading is studied using finite element analysis. The system input parameters are randomized to ascertain the stochastic first buckling load and zone of resonance. Considering the effects of transverse shear deformation and rotary inertia, first order shear deformation theory is used to model the composite doubly curved shells. The stochasticity is introduced in Love's and Donnell's theory considering dynamic and shear deformable theory according to the Sander's first approximation by tracers for doubly curved laminated shells. The moving least square method is employed as a surrogate of the actual finite element model to reduce the computational cost. The results are compared with those available in the literature. Statistical results are presented to show the effects of radius of curvatures, material properties, fibre parameters, and non-uniform load parameters on the stability boundaries. [ABSTRACT FROM AUTHOR]

This article presents the application of finite-element fuzzy model updating to the DLR AIRMOD structure. The proposed approach is initially demonstrated on a simulated mass-spring system with three degrees of freedom. Considering the effect of the assembly process on variability measurements, modal tests were carried out for the repeatedly disassembled and reassembled DLR AIRMOD structure. The histograms of the measured data attributed to the uncertainty of the structural components in terms of mass and stiffness are utilised to obtain the membership functions of the chosen fuzzy outputs and to determine the updated membership functions of the uncertain input parameters represented by fuzzy variables. In this regard, a fuzzy parameter is introduced to represent a set of interval parameters through the membership function, and a meta model (kriging, in this work) is used to speed up the updating. The use of non-probabilistic models, i.e. interval and fuzzy models, for updating models with uncertainties is often more practical when the large quantities of test data that are necessary for probabilistic model updating are unavailable. [ABSTRACT FROM AUTHOR]

This paper presents a stochastic approach to study the natural frequencies of thin-walled laminated composite beams with spatially varying matrix cracking damage in a multi-scale framework. A novel concept of stochastic representative volume element (SRVE) is introduced for this purpose. An efficient radial basis function (RBF) based uncertainty quantification algorithm is developed to quantify the probabilistic variability in free vibration responses of the structure due to spatially random stochasticity in the micro-mechanical and geometric properties. The convergence of the proposed algorithm for stochastic natural frequency analysis of damaged thin-walled composite beam is verified and validated with original finite element method (FEM) along with traditional Monte Carlo simulation (MCS). Sensitivity analysis is carried out to ascertain the relative influence of different stochastic input parameters on the natural frequencies. Subsequently the influence of noise is investigated on radial basis function based uncertainty quantification algorithm to account for the inevitable variability for practical field applications. The study reveals that stochasticity/ system irregularity in structural and material attributes affects the system performance significantly. To ensure robustness, safety and sustainability of the structure, it is very crucial to consider such forms of uncertainties during the analysis. [ABSTRACT FROM AUTHOR]

The present computational study investigates on stochastic natural frequency analyses of laminated composite curved panels with cutout based on support vector regression (SVR) model. The SVR based uncertainty quantification (UQ) algorithm in conjunction with Latin hypercube sampling is developed to achieve computational efficiency. The convergence of the present algorithm for laminated composite curved panels with cutout is validated with original finite element (FE) analysis along with traditional Monte Carlo simulation (MCS). The variations of input parameters (both individual and combined cases) are studied to portray their relative effect on the output quantity of interest. The performance of the SVR based uncertainty quantification is found to be satisfactory in the domain of input variables in dealing low and high dimensional spaces. The layer-wise variability of geometric and material properties are included considering the effect of twist angle, cutout sizes and geometries (such as cylindrical, spherical, hyperbolic paraboloid and plate). The sensitivities of input parameters in terms of coefficient of variation are enumerated to project the relative importance of different random inputs on natural frequencies. Subsequently, the noise induced effects on SVR based computational algorithm are presented to map the inevitable variability in practical field of applications. [ABSTRACT FROM AUTHOR]

This paper presents the quantification of uncertain natural frequency for laminated composite plates by using a novel surrogate model. A group method of data handling in conjunction to polynomial neural network (PNN) is employed as surrogate for numerical model and is trained by using Latin hypercube sampling. Subsequently the effect of noise on a PNN based uncertainty quantification algorithm is explored in this study. The convergence of the proposed algorithm for stochastic natural frequency analysis of composite plates is verified and validated with original finite element method (FEM). Both individual and combined variation of stochastic input parameters are considered to address the influence on the output of interest. The sample size and computational cost are reduced by employing the present approach compared to direct Monte Carlo simulation (MCS). [ABSTRACT FROM AUTHOR]

This paper presents an efficient uncertainty quantification (UQ) scheme for frequency responses of laminated composite plates. A bottom up surrogate based approach is employed to quantify the variability in free vibration responses of composite cantilever plates due to uncertainty in ply orientation angle, elastic modulus and mass density. The finite element method is employed incorporating effects of transverse shear deformation based on Mindlin’s theory in conjunction with a random variable approach. Parametric studies are carried out to determine the stochastic frequency response functions (SFRF) along with stochastic natural frequencies and modeshapes. In this study, a surrogate based approach using General High Dimensional Model Representations (GHDMR) is employed for achieving computational efficiency in quantifying uncertainty. Subsequently the effect of noise is investigated in GHDMR based UQ algorithm. This paper also presents an uncertainty quantification scheme using commercial finite element software (ANSYS) and thereby comparative results of stochastic natural frequencies are furnished for UQ using GHDMR approach and ANSYS. [ABSTRACT FROM AUTHOR]

Recent research in the field of finite element model updating (FEM) advocates the adoption of Bayesian analysis techniques to dealing with the uncertainties associated with these models. However, Bayesian formulations require the evaluation of the Posterior Distribution Function which may not be available in analytical form. This is the case in FEM updating. In such cases sampling methods can provide good approximations of the Posterior distribution when implemented in the Bayesian context. Markov Chain Monte Carlo (MCMC) algorithms are the most popular sampling tools used to sample probability distributions. However, the efficiency of these algorithms is affected by the complexity of the systems (the size of the parameter space). The Hybrid Monte Carlo (HMC) offers a very important MCMC approach to dealing with higher-dimensional complex problems. The HMC uses the molecular dynamics (MD) steps as the global Monte Carlo (MC) moves to reach areas of high probability where the gradient of the log-density of the Posterior acts as a guide during the search process. However, the acceptance rate of HMC is sensitive to the system size as well as the time step used to evaluate the MD trajectory. To overcome this limitation we propose the use of the Shadow Hybrid Monte Carlo (SHMC) algorithm. The SHMC algorithm is a modified version of the Hybrid Monte Carlo (HMC) and designed to improve sampling for large-system sizes and time steps. This is done by sampling from a modified Hamiltonian function instead of the normal Hamiltonian function. In this paper, the efficiency and accuracy of the SHMC method is tested on the updating of two real structures; an unsymmetrical H-shaped beam structure and a GARTEUR SM-AG19 structure and is compared to the application of the HMC algorithm on the same structures. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents a multiscale approach for vibration frequency analysis of graphene/polymer composites. The graphene is modelled at the atomistic scale, and the matrix deformation is analysed by the continuum finite element method. Inter-connectivity between graphene and polymer matrix are assumed to be bonded by van der Waals interactions at the interface. The impact of geometrical configuration (armchair and zigzag), boundary conditions and length on the overall stiffness of the graphene reinforced plastics (GRP) is studied. The natural frequency and vibrational mode shapes of GRP studied have displayed dependence on the length and also the boundary conditions. The exceptional vibrational behaviour and large stiffness displayed by GRP makes them a potential replacement for conventional composite fibres such as carbon and glass fibres. [Copyright &y& Elsevier]