Spectral Finite Element Formulation Research Articles

Performance of 2D-spectral finite element method in dynamic analysis of concrete gravity dams

This article presents a two-dimensional spectral finite element formulation for dynamic analysis of concrete gravity dams. Finite element method (FEM) is the one of the most extensively used analysis tools for solving problems of dynamic analysis of structures. However, it needs extensive computational resources and time for large structures, leading to the development of alternate computationally efficient modeling techniques over the past few decades. Some of these techniques are broadly termed as “Spectral Finite Element Methods” (SFEMs). Frequency domain-based SFEM (FDSFEM) and time domain-based SFEM (TDSFEM) are the most commonly used SFEMs found in the literature. This article provides a brief review of these methods identifying their key advantages and disadvantages; and explores the feasibility of applying such methods in the dynamic analysis of concrete gravity dams. Here, the TDSFEM has been used to perform the dynamic time history analysis of a concrete gravity dam due to its relative advantages over FDSFEM, which is challenging for irregular geometry and finite domain. Sensitivity analysis and convergence studies have been performed using 4-noded and 9-noded elements. The foundation of the dam has been modeled using two-dimensional infinite elements in both FEM and TDSFEM analysis, which is a novel application in the case of TDSFEM. This article quantifies the computational efficiency of TDSFEM over the conventional FEM by comparing the computation time in both methods and thus emphasizes the applicability of TDSFEM in dynamic analysis, especially in case of complex geometries and large two-dimensional structures (like concrete gravity dam). The results demonstrate the computational efficiency of TDSFEM over FEM when higher order elements are considered. Modal analysis and time history analysis results point that the use of higher order elements of TDSFEM can be a viable alternative to the use of the conventional FEM leading to significant saving in computational time with reasonable accuracy for dynamic analysis of large structures like concrete gravity dams.

An efficient computational framework for hailstone impacts on composite plates utilizing a semi-empirical viscoplastic contact law

The dynamic response of composite structures under impact loading conditions is a complex problem which is attracting substantial attention. Specifically, the case of hailstone impact introduces additional challenges that are urging to be encountered. Hence, the main goal of this paper is to present the development and validation of a computationally efficient impact model that encompasses a novel semi-empirical viscoplastic contact law for crushable ice impactors together with a new time domain spectral shear plate finite element formulation including nonlinear effects due to large displacements and rotations. The new viscoplastic contact law, which provides the coupling between the hailstone impactor and the targeted composite plate, is derived from analytical expressions and observations based on experiments and high-fidelity finite element simulations which utilize a fully integrated ice failure material model. High-velocity spherical hailstone impact experiments on woven glass/epoxy composite plate target structures are conducted using a high pressure pre-charged gas gun Obtained results are validated against high fidelity finite element models and experimental measurements. The results demonstrate the adequacy of the proposed contact law to capture the impact loading, as well as the accuracy, computational efficiency and additional benefits of the presented computational framework compared to other well-established numerical tools.

Control of wave propagation response using quasi crystals: A formulation based on spectral finite element

ABSTRACTThis paper presents wave propagation studies in quasi crystal structures and quasi crystal reinforced aluminium structures. The analysis is performed using frequency domain spectral finite element formulation. The analysis considers different 2-D decagonal and 3-D icosahedral quasi crystals. First, wave propagation analysis of quasi crystal structure alone is performed and the propagation of phonon and phason modes for different quasi crystals are studied. The study includes the propagation of axial and transverse wave responses in these quasi crystals. The study has found that the amplitude of the phason modes is very small compared to the phonon modes and the increase of the phason mode content (through increase in R) increases the phason mode amplitude, without affecting the phonon mode amplitudes. It is shown that the dominant axial phonon mode is non-dispersive and the dominant flexural phonon mode is dispersive. In the next study, the aluminium beam structure is reinforced with different quasi crystals in different configurations and the wave propagation of axial and transverse responses are studied. For all the combinations of quasi crystal aluminium beam combination, there is substantial suppression of responses both for the axial and the bending responses. Unsymmetrical configuration produces substantial non-dominant phonon modes which propagate dispersively. It is found that for a symmetric bi-morph configuration, the response is reduced significantly, about 68% and 75% for axial loading and 80% and 78% for flexural loading, respectively, for the 2-D decagonal quasi crystal and the 3-D icosahedral quasi crystal.

Stochastic finite element method for random harmonic analysis of composite plates with uncertain modal damping parameters

Damping parameters of fiber-reinforced composite possess significant uncertainty due to the structural complexity of such materials. Considering the parameters as random variables, this paper uses the generalized polynomial chaos (gPC) expansion to capture the uncertainty in the damping and frequency response function of composite plate structures. A spectral stochastic finite element formulation for damped vibration analysis of laminate plates is employed. Experimental modal data for samples of plates is used to identify and realize the range and probability distributions of uncertain damping parameters. The constructed gPC expansions for the uncertain parameters are used as inputs to a deterministic finite element model to realize random frequency responses on a few numbers of collocation points generated in random space. The realizations then are employed to estimate the unknown deterministic functions of the gPC expansion approximating the responses. Employing modal superposition method to solve harmonic analysis problem yields an efficient sparse gPC expansion representing the responses. The results show while the responses are influenced by the damping uncertainties at the mid and high frequency ranges, the impact in low frequency modes can be safely ignored. Utilizing a few random collocation points, the method indicates also a very good agreement compared to the sampling-based Monte Carlo simulations with large number of realizations. As the deterministic finite element model serves as black-box solver, the procedure can be efficiently adopted to complex structural systems with uncertain parameters in terms of computational time.

Vibration analysis of cracked Timoshenko beam under moving load with constant velocity and acceleration by spectral finite element method

This study describes spectral finite element (SFE) formulation and solution for vibration of cracked Timoshenko beam subjected to moving load. The crack is modeled by two massless springs at the crack position. The dynamic stiffness matrix of cracked beam together with both constant velocity and constant acceleration moving load vectors is evaluated for spectral elements in frequency-domain. The accuracy of results obtained from SFE formulation is compared with that of finite element (FE) method. The SFE results display remarkable superiority compared to FE ones in reducing the number of elements together with increasing numerical accuracy.

Spectrally formulated one-dimensional element for analysis of wave propagation in pretwisted anisotropic strips

A spectrally formulated one-dimensional element is developed to study the wave propagation in pretwisted anisotropic strips. The element is based on linear sectional analysis of pretwisted anisotropic strips with small pretwist. Firstly, the governing equations of the strip are obtained through dimensional reduction of laminated shell theory to a linear one dimensional (1-D) theory using the variational asymptotic method (VAM). Next, an exact dynamic stiffness matrix is derived in the frequency-wavenumber domain using spectral finite element (SFE) method. In SFE formulation the mass distribution is modeled exactly and as a result a single element is sufficient to capture the exact frequency response of a regular structure. For numerical validation of the proposed model, the natural frequencies of the strip are compared to the modal behavior of pretwisted composite strips available in literature. The model is used to predict wave responses due to modulated sinusoidal input. The different wave modes present, namely axial, flexural, and torsional modes are seen and their velocities are compared to that obtained from the dispersion plot. This proposed modeling strategy is a first step towards the development of a comprehensive model to do structural health monitoring of structures idealized as pretwisted strips.

Spectral finite element for wave propagation analysis of laminated composite curved beams using classical and first order shear deformation theories

A frequency domain spectral finite element formulation is presented for the wave propagation analysis of laminated composite curved beams using the first order shear deformation theory (FSDT) and the classical laminate theory (CLT). The elements are derived from the exact solution of the governing equation of motion in frequency domain, obtained through Fourier transformation of the time domain equation. The formulation is validated by comparing the results for natural frequencies with the published results. The new elements are then employed to perform dispersion and wave propagation analyses of curved composite beams. The numerical results reveal that the wavenumbers predicted by the CLT show large deviation from those of the FSDT even for thin beams, and the deviation increases and occurs at lower frequencies with the increase in the thickness to radius ratio. The orthotropy ratio of the composite has a significant effect on the wavenumbers for tangential and mid-surface rotation modes. The wave propagation response predicted by the CLT differs widely from the FSDT prediction, for thin and thick, and shallow and deeply curved beams at both low and high frequencies. Thus, the CLT should not be used for wave propagation analysis of even thin curved laminated beams.

Formulation of 36-noded piezoelectric spectral finite element scheme with active/passive layers coupled by Lagrange multipliers

A novel spectral finite element formulation scheme is presented for modeling a plate structure with surface-mounted piezoelectric transducers. Surface-mounted piezoelectric transducers may asymmetrically distribute the mass in the thickness direction of the plate/panel structure, resulting in a coupled mass matrix in spectral element formulation. A new procedure is developed by equating the layer-wise kinematics of the element using undetermined Lagrange multipliers to achieve the diagonal mass matrix. To demonstrate the effectiveness of the element formulation scheme, a two-dimensional piezoelectric spectral element is constructed with 36 nodes and five active/passive layers (layers: transducer/bond/plate/bond/transducer). The performance of the developed element is illustrated by (a) simulation of Lamb wave propagation and estimation of its velocity, and (b) simulation of the effect of transducer size, its dynamics and shear lag on sensor’s response. The results presented highlight the importance of modeling the dynamics of transducers and understanding the effects on sensor response. The presented technique has relevance in the field of structural health monitoring, wherein it can be used to model and simulate aircraft panels with surface-mounted piezoelectric transducers.

Stochastic response of an axially loaded composite Timoshenko beam exhibiting bending–torsion coupling

A spectral finite element method is proposed to investigate the stochastic response of an axially loaded composite Timoshenko beam with solid or thin-walled closed section exhibiting bending–torsion materially coupling under the stochastic excitations with stationary and ergodic properties. The effects of axial force, shear deformation (SD) and rotary inertia (RI) as well as bending–torsion coupling are considered in the present study. First, the damped general governing differential equations of motion of an axially loaded composite Timoshenko beam are derived. Then, the spectral finite element formulation is developed in the frequency domain using the dynamic shape functions based on the exact solutions of the governing equations in undamped free vibration, which is used to compute the mean square displacement response of axially loaded composite Timoshenko beams. Finally, the proposed method is illustrated by its application to a specific example to investigate the effects of bending–torsion coupling, axial force, SD and RI on the stochastic response of the composite beam.

Frequency-dependent vibration analysis of symmetric cross-ply laminated plate of Levy-type by spectral element and finite strip procedures

This research describes spectral finite element formulation for vibration analysis of rectangular symmetric cross-ply laminated composite plates of Levy-type based on classical lamination plate theory (CLPT). Formulation based on SFEM includes partial differential equations of motion, spectral displacement field, dynamic shape functions, and spectral element stiffness matrix (SESM). In this paper, vibration analysis of composite plate is investigated in two sections: free vibrations and forced vibrations. In free vibrations, natural frequencies are calculated for different Young’s moduli ratios and boundary conditions. In forced vibrations, plate vibrations are investigated under high-frequency concentrated impulsive loads. The resulting responses due to spectral element formulation are compared with those of (time-domain) finite element and analytical formulations, whenever available. The results demonstrate the superiority of SFEM with respect to FEM, in reducing computational burden, simultaneously increasing numerical accuracy, specifically for excitations of high-frequency content. By reducing the time duration of impulsive loads, and consequently increasing the modal contribution of higher modes in the transient response of plate, the accuracy of FEM responses decreases substantially accompanied with a high volume of computations, while the accuracy of the SFEM response results is very high and simultaneously, with a low computational burden. Practically, SFEM follows very closely exact analytical solutions.

High-Order Spectral Stochastic Finite Element Analysis of Stochastic Elliptical Partial Differential Equations

This study presents an experiment of improving the performance of spectral stochastic finite element method using high-order elements. This experiment is implemented through a two-dimensional spectral stochastic finite element formulation of an elliptic partial differential equation having stochastic coefficients. Deriving this spectral stochastic finite element formulation couples a two-dimensional deterministic finite element formulation of an elliptic partial differential equation with generalized polynomial chaos expansions of stochastic coefficients. Further inspection of the performance of resulting spectral stochastic finite element formulation with adopting linear and quadratic (9-node or 8-node) quadrilateral elements finds that more accurate standard deviations of unknowns are surprisingly predicted using quadratic quadrilateral elements, especially under high autocorrelation function values of stochastic coefficients. In addition, creating spectral stochastic finite element results using quadratic quadrilateral elements is not unacceptably time-consuming. Therefore, this study concludes that adopting high-order elements can be a lower-cost method to improve the performance of spectral stochastic finite element method.

Spectral finite element and nonlocal continuum mechanics based formulation for torsional wave propagation in nanorods

Because of future promising exploration of nanotechnology, focus is being put in the miniaturization of mechanical and electromechanical devices. Attention is sought toward the development of nanodevices and nanomachines. The length scales associated with nanostructures like are such that to apply any classical continuum techniques, we need to consider the small length scales such as lattice spacing between individual atoms, surface properties, grain size, etc. This makes a physically consistent classical continuum model formulation very challenging. So this work presents Eringen's nonlocal elasticity theory, that has been incorporated into classical torsional rod model to capture unique properties of the nanorods under the umbrella of continuum mechanics theory. The strong effect of the nonlocal scale has been obtained which leads to substantially different torsional wave behaviors of nanorods from those of macroscopic rods. Nonlocal torsional rod model is developed for nanorods. Explicit expressions are derived for torsional wavenumbers and wave speeds of nanorods. The analysis shows that the wave characteristics are highly over estimated by the classical rod model, which ignores the effect of small-length scale. The studies also shows that the nonlocal scale parameter introduces certain band gap region in torsional wave mode where no wave propagation occurs. This is manifested in the spectrum cures as the region where the wavenumber tends to infinite or wave speed tends to zero. Next, the Spectral Finite Element formulation of nanorods is performed. The exact frequency dependent shape functions and the dynamic stiffness matrix for the nanorod are obtained as a function of nonlocal scale parameter. It has been found that the nonlocal small scale has significant effect on the exact shape functions and the elements of the dynamic stiffness matrix. These effects are also captured in the present work. The results presented in this paper can provide useful guidance for the study and design of the next generation of nanodevices that make use of the wave dispersion properties of carbon nanotubes.

Free and transient vibration analysis of an un-symmetric cross-ply laminated plate by spectral finite elements

In this study, a spectral finite element formulation for free and transient vibration analysis of a rectangular un-symmetric cross-ply laminated composite plate of Levy-type is developed. Plate kinematics obeys classical lamination plate theory. The formulation is based on dynamic shape functions in frequency-domain derived from the exact solution of the governing wave equations. Based on Levy-type solution and finite strip method, 2/D space-time partial differential equations (PDEs) of motion are transformed to 1/D space-time PDEs. Spectral elements are used to transform 1/D space-time PDEs to temporal ordinary differential equations. For investigating validity and accuracy of the model presented, numerical solutions are established for both free and transient vibrations. Comparison of spectral finite element method (SFEM) time responses with an exact analytical method demonstrates the superiority of SFEM in both reducing computational cost and increasing numerical accuracy. It is correct especially for excitations of high-frequency contents. A comparison of SFEM and FEM shows the same result.

Dynamics of rotating composite beams: A comparative study between CNT reinforced polymer composite beams and laminated composite beams using spectral finite elements

This paper presents a spectral finite element formulation for uniform and tapered rotating CNT embedded polymer composite beams. The exact solution to the governing differential equation of a rotating Euler–Bernoulli beam with maximum centrifugal force is used as an interpolating function for the spectral element formulation. Free vibration and wave propagation analysis is carried out using the formulated spectral element. The present study shows the substantial effect of volume fraction and L/D ratio of CNTs in a beam on the natural frequency, impulse response and wave propagation characteristics of the rotating beam. It is found that the CNTs embedded in the matrix can make the rotating beam non-dispersive in nature at higher rotation speeds. Embedded CNTs can significantly alter the dynamics of polymer-nanocomposite beams. The results are also compared with those obtained for carbon fiber reinforced laminated composite rotating beams. It is observed that CNT reinforced rotating beams are superior in performance compared to laminated composite rotating beams.

Spectral Finite Element Formulation for Nanorods via Nonlocal Continuum Mechanics

In this article, the Eringen’s nonlocal elasticity theory has been incorporated into classical/local Bernoulli-Euler rod model to capture unique properties of the nanorods under the umbrella of continuum mechanics theory. The spectral finite element (SFE) formulation of nanorods is performed. SFE formulation is carried out and the exact shape functions (frequency dependent) and dynamic stiffness matrix are obtained as function of nonlocal scale parameter. It has been found that the small scale affects the exact shape functions and the elements of the dynamic stiffness matrix. The results presented in this paper can provide useful guidance for the study and design of the next generation of nanodevices that make use of the wave dispersion properties of carbon nanotubes.

A Legendre spectral element model for sloshing and acoustic analysis in nearly incompressible fluids

A new spectral finite element formulation is presented for modeling the sloshing and the acoustic waves in nearly incompressible fluids. The formulation makes use of the Legendre polynomials in deriving the finite element interpolation shape functions in the Lagrangian frame of reference. The formulated element uses Gauss-Lobatto-Legendre quadrature scheme for integrating the volumetric stiffness and the mass matrices while the conventional Gauss-Legendre quadrature scheme is used on the rotational stiffness matrix to completely eliminate the zero energy modes, which are normally associated with the Lagrangian FE formulation. The numerical performance of the spectral element formulated here is examined by doing the inf-sup test on a standard rectangular rigid tank partially filled with liquid. The eigenvalues obtained from the formulated spectral element are compared with the conventional equally spaced node locations of the h-type Lagrangian finite element and the predicted results show that these spectral elements are more accurate and give superior convergence. The efficiency and robustness of the formulated elements are demonstrated by solving few standard problems involving free vibration and dynamic response analysis with undistorted and distorted spectral elements, and the obtained results are compared with available results in the published literature.

Time–transient response for ultrasonic guided waves propagating in damped cylinders

Herein, an enhanced spectral finite element (SFE) formulation to calculate the time–transient response in cylindrical waveguides is proposed. The original aspect over SFE-based formulations consists in the possibility to account for the effect of material absorption, i.e. guided waves attenuation, on the calculation of the time–transient response. First, the damped steady-state response is constructed by a weighted superposition of the waveguide modal properties obtained from the spectral decomposition of the governing wave equation. To this purpose an enhanced spectrally formulated finite element is developed, in which material damping is included allowing for complex stress–strain viscoelastic constitutive relations in force of the correspondence principle. Dispersive modal properties for the damped waveguide (phase velocity, energy velocity, attenuation and wavestructures) follow straightforwardly by simple formulae. Next, the frequency response of the problem is calculated by weighting the modal data and the spectrum of the applied time-dependent force via Cauchy residue theorem. Finally, the inverse Fourier transform of the frequency response leads to the time–transient response for propagative damped guided waves. The approach is not restricted to any anisotropy degree, holds for any linear viscoelastic constitutive relation that can be characterized and formulated in the frequency domain and it can be applied to SFE formulations for arbitrary cross-section waveguides. A study on guided waves propagating in a scheduled 4.in-40 ANSI steel pipe is presented, where the steel is considered first as perfectly elastic and then as an hysteretic viscoelastic medium, in order to show the effect of material absorption on the time–transient response.

Spectral stochastic finite element analysis for laminated composite plates

A spectral stochastic finite element formulation with consideration of multi-layer effect and spatial variability of material properties is developed for probabilistic analysis of laminated composite plates. The material properties of each lamina are modeled as a set of random fields and represented by the Karhunen–Loève expansion. The expansion is incorporated into a spatial discretization in accordance with a standard finite element procedure based on the first-order shear deformation theory. A spectral expansion with use of polynomial chaos is employed to represent the stochastic nodal displacements in terms of standard normal random variables. A preconditioning matrix is then proposed for the solution of spectral stochastic finite element equations with use of a preconditioning conjugate gradient technique. The various statistics of interest of the system responses are finally obtained by means of the coefficients of the spectral expansion. The numerical accuracy and the computational efficiency of the method are demonstrated by comparison with Monte-Carlo simulation.

Perturbation technique for wave propagation analysis in a notched beam using wavelet spectral element modeling

In this paper, spectral finite element is formulated for an Euler–Bernoulli beam with through-width notch type defect. In spectral finite element modeling, exact shape functions are derived and finite element procedure is followed in the transformed frequency domain. Here spectral finite element formulation is done using Daubechies scaling function bases for temporal approximation. In comparison to the conventional Fourier transform based spectral finite element method, the use of localized bases functions in the Daubechies scaling function based spectral finite element method allows accurate wave propagation analysis of finite length structures. The wave propagation response of the damaged beam is considered as a perturbation of the undamaged beam response within the restriction of small damage. First, numerical experiments are performed with narrow banded modulated pulse loading to obtain the location of damage from wave arrival time. Next, a broad banded impulse load is considered and effects of parameters like damage width, depth, and location on the responses are studied in time and frequency domains.

Numerical modelling of ground-borne noise and vibration in buildings due to surface rail traffic

This paper deals with the numerical computation of the structural and acoustic response of a building to an incoming wave field generated by high-speed surface railway traffic. The source model consists of a moving vehicle on a longitudinally invariant track, coupled to a layered ground modelled with a boundary element formulation. The receiver model is based on a substructuring formulation and consists of a boundary element model of the soil and a finite element model of the structure. The acoustic response of the building's rooms is computed by means of a spectral finite element formulation. The paper investigates the structural and acoustic response of a multi-story portal frame office building up to a frequency of 150 Hz to the passage of a Thalys high-speed train at constant velocity. The isolation performance of three different vibration countermeasures: a floating-floor, a room-in-room, and base-isolation, are examined.