Spectral Finite Element Model Research Articles

Exploitation of the Spectral Stochastic Finite Element Model for the Evaluation of Surface Defects of the CFRP Composite

Exploitation of the Spectral Stochastic Finite Element Model for the Evaluation of Surface Defects of the CFRP Composite

Analysis of the blade aeroelastic effect on the floating offshore wind turbine wake in a focusing wave with a hybrid model

With the recent trend toward larger floating offshore wind turbines (FOWT), the influence of blade deformation becomes more obvious. Modelling this fully coupled fluid-structure interaction (FSI) system, especially at extreme sea level with large-scale motion of the entire system, is an essential challenge. A hybrid numerical model integrating the potential-viscous flow model (qaleFOAM), the unstable actuator line method (UALM), the Legendre spectral finite element model (BeamDyn), and the Lumped Mass Mooring Model (MoorDyn) is employed in the current research. This model is capable of effortlessly and accurately reflecting the blade aeroelastic effect on the dynamics of a FOWT system under the circumstances of a focused wave and uniform wind. The result reveals that the aeroelastic impacts have been identified to not only increase the fatigue damage on the turbine blades but also affect the wake field distribution. Based on the temporal and spatial distribution of the velocity fields, it is found that the coupled effects of the large wave elevation and the FOWT high-frequency motions will disturb the velocity in the wake region. The slight variation of the wind speed in the wake region will be exacerbated by about 0.21% to 18.6%, and it transforms the shape of the wake region when the blade aeroelastic effect participates in the coupled effect. This phenomenon may further affect the downstream FOWT system features through the upstream FOWT wake field.

A hybrid numerical model for simulating aero-elastic-hydro-mooring-wake dynamic responses of floating offshore wind turbine

It demands many computational resources to model the coupled responses of a floating offshore wind turbine (FOWT), especially for its aero-elastic-hydro-mooring-wake dynamics and their interaction. In this paper, a new hybrid numerical model for FOWT systems is developed, which is based on the hybrid potential-viscous flow model called qaleFOAM. In this model, the aerodynamics of wind turbine are solved by the unsteady actuator line method (UALM); the elastic responses of the turbine blade are calculated by the Legendre spectral finite element model (BeamDyn); the hydrodynamics of the floating platform are dealt with by the combination of the fully nonlinear potential solver and a two-phase Navier–Stokes solver; the mooring dynamics are considered with the Lumped Mass Mooring Model (MoorDyn), and the turbine wake is solved with the large eddy simulation (LES) model. This newly formulated model can deal with wind, wave, mooring dynamics, platform motions, and turbine structural dynamics involved in the FOWT system. To demonstrate the capability of the present model, various cases with different complexities are investigated and compared with the experimental data and other numerical results. Then, the model is applied to simulation of a semi-submersible FOWT system, subjected to a regular wave and a uniform wind. The prediction of the aerodynamic performance, blade tip deflection, platform motion responses, and mooring line tension loads show good agreements with the results from other methods. In addition, the phenomenon of the coupled effects between the dynamic responses of platform, blade deformation and wake flow are captured reasonably well.

Broadband Sound Transmission Loss in a Corrugated Sound Panel Using Periodic Structure Theory

Abstract The vibration and sound radiation characteristics of a double leaf panel wall with corrugated core are analyzed. The core comprises of repeating unit cells along the length of the panel. Each cell is a combination of a straight beam with uniform cross section and curved beams with varying cross section. The vibroacoustic property of the corrugated panel is analyzed assuming plain strain condition. The analysis presented in the work shows that the proposed sandwich panel design provides broadband sound transmission loss (STL) characteristics. The dispersion analysis, forced response characteristics, and sound radiation characteristics of the structure are presented. It is found that bending frequency band gap obtained from the dispersion analysis is a prominent criterion to achieve a higher STL. The structural and acoustic behaviors of the sandwich panel are analyzed through the spectral finite element model. The STL for the core element for various taper ratios is presented. It is found that a higher taper ratio shifts the STL characteristics to the low frequency range.

Progressive debonding analysis of sandwich composite strips using a cohesive-layerwise spectral finite element model

This paper presents a novel numerical framework for the simulation of debonding growth in sandwich structures via a layerwise spectral finite element model. A variable kinematic layerwise theory is developed, which employs cubic Hermite splines for the composite facesheets and the core and linear interpolation functions for the thin resin-rich layers in connection with continuum damage mechanics for tracking of debonding propagation. The layerwise mechanics and continuum damage mechanics are integrated into a spectral finite element, termed thereafter as cohesive-layerwise spectral finite element, with integration points collocated to nodes and high-order spatial approximation in the plane of the structure. The developed computational model is applied to global mode I, mode II, and mixed-mode fracture problems and correlated with experimental results and reference high-fidelity finite element models. The evaluation cases of the present method reveal significant enhancements in computational speed and meshing process due to elimination of cohesive layer parameter calibration, alleviation of FE aspect ratio constraints and high-convergence rates.

A superconvergent elastically coupled double beam element for analysis of adhesively bonded lap joints

A novel elastically coupled double beam (ECDB) finite element with superconvergent properties is developed to primarily study wave propagation in the bonded region of an adhesively bonded single lap joint. An ECDB element consists of two axial-flexural-shear coupled beams coupled with continuously distributed linear elastic springs. A general formulation applicable to both metals and composites is proposed. An exact solution to the static part of six-coupled governing equations of motion is obtained, which is then used to formulate exact stiffness and mass matrices. Consequently, the ECDB element thus formulated has superconvergent properties and is inherently free of shear locking. It is shown that merely a single ECDB element is sufficient to accurately capture the tip deflections of a cantilever double beam type geometry subjected to static loading, following which eigenvalue analysis is performed, and comparisons are made with the commercial finite element software—Abaqus. Further, wave propagation studies are carried out to demonstrate the efficiency of the element in evaluating ultrasonic wave responses across various metallic and composite ECDBs, and comparisons are made with Abaqus and frequency-domain spectral finite element (SFEM) model described in Paunikar and Goapalakrishnan (“Wave propagation in adhesively bonded metallic and composite lap joints modelled through spectrally formulated elastically coupled double beam element,” Compos. Struct., under review). Following this, wave propagation in symmetric metallic, geometrically asymmetric metallic, and symmetric laminated composite lap joints with strong and weak bonding is studied by employing two-noded Lagrangian frame elements to mesh the adherends and superconvergent elastically coupled double beam (SECDB) elements to mesh the bonded region. The results obtained from the superconvergent finite element (SCFE) simulations are compared with those given by Abaqus and SFEM. Lastly, the SCFE models for the cases of perfectly bonded aluminium and symmetric ply laminated composite are experimentally validated. We have shown that the SECDB element developed in this work may be used to efficiently carry out static and dynamic analysis in any symmetric or asymmetric bonded joint comprising of only metals or composites or a combination of the two. It is also demonstrated that various levels of adhesion in a bonded joint can be simulated by varying the coupling spring stiffness value. Additionally, the SECDB element may also be used to study other double beam like coupled structures, for instance, space platforms or dynamic vibration absorbers.

A cubic spline layerwise spectral finite element for robust stress predictions in laminated composite and sandwich strips

A new theoretical and numerical tool, comprising of a novel cubic Hermite spline layerwise theory and a spectral strip finite element with integration points collocated to its nodes and high-order polynomial shape functions, is evaluated regarding its capability to predict intralaminar and interlaminar stresses in laminated and sandwich composite strips. Stress predictions in various laminated and thick sandwich beams are presented and compared with exact solutions, high-fidelity plane-strain FE models and reported zig-zag layerwise models. Emphasis is placed on the capability of the present model to predict interlaminar stresses on the free-edges, at ply interfaces, in resin-rich layers and in laminates with artificial delaminations. All numerical results illustrate the capabilities of the present computational tool to provide accurate and robust interlaminar stress predictions implementing coarse discretization, thus ensuring the validity and rapid convergence of the combined higher-order layerwise laminate theory and spectral finite element model.

Estimation of the coupling loss factors of structural junctions with in-plane waves by means of the inverse statistical energy analysis problem

The Coupling Loss Factors (CLF) to describe the vibration transmission at junctions in the context of Statistical Energy Analysis (SEA) are determined. Out-of-plane energy (bending waves) and in-plane energy (treating transverse shear waves and quasi longitudinal waves as a single subsystem) as well as their interactions are considered. The CLF values are obtained from the inverse SEA problem where the energies and input powers are computed numerically by means of a Spectral Finite Element model. A set of L, T and X-junctions with variable dimensions and material properties typical from heavy building structures is considered. The Transmission Loss (TL) of the junctions is computed from the CLF values. Afterwards the TL results are approximated by means least squares leading to simplified expressions for each transmission path. The TL SEA results are compared with the TL predicted by means of a semi-infinite wave-based model.

Ultrasonic guided wave based structural damage detection and localization using model assisted convolutional and recurrent neural networks

The inverse problem of damage identification involves real-time, continuous observation of structures to detect any undesired, abnormal behavior and ultrasonic guided waves are considered as one of the preferred candidates for this. A parallel implementation of a reduced-order spectral finite element model is utilized to formulate the forward problem in an isotropic and a composite waveguide. In this work, along with a time-series dataset, a 2D representation of continuous wavelet transformation based time–frequency dataset is also developed. The datasets are corrupted with several levels of Gaussian random noise to incorporate different kinds of uncertainties and noise present in the real scenario. Deep learning networks like convolutional and recurrent neural networks are utilized to numerically approximate the solution of the inverse problem. A hybrid strategy of classification and regression in a supervised setting is proposed for combined damage detection and localization. The performance of the networks is compared based on metrics like accuracy, loss value, mean absolute error, mean absolute percentage error, and coefficient of determination. The predictions from conventional machine learning algorithms, trained on feature engineered dataset are compared with the deep learning algorithms. The generalization of the trained deep networks on different excitation frequencies and a higher level of uncertainties is also highlighted in this work.

Time-Domain Spectral Finite Element Method for Modeling Second Harmonic Generation of Guided Waves Induced by Material, Geometric and Contact Nonlinearities in Beams

This study proposes a time-domain spectral finite element (SFE) method for simulating the second harmonic generation (SHG) of nonlinear guided wave due to material, geometric and contact nonlinearities in beams. The time-domain SFE method is developed based on the Mindlin–Hermann rod and Timoshenko beam theory. The material and geometric nonlinearities are modeled by adapting the constitutive relation between stress and strain using a second-order approximation. The contact nonlinearity induced by breathing crack is simulated by bilinear crack mechanism. The material and geometric nonlinearities of the SFE model are validated analytically and the contact nonlinearity is verified numerically using three-dimensional (3D) finite element (FE) simulation. There is good agreement between the analytical, numerical and SFE results, demonstrating the accuracy of the proposed method. Numerical case studies are conducted to investigate the influence of number of cycles and amplitude of the excitation signal on the SHG and its performance in damage detection. The results show that the amplitude of the SHG increases with the numbers of cycles and amplitude of the excitation signal. The amplitudes of the SHG due to material and geometric nonlinearities are also compared with the contact nonlinearity when a breathing crack exists in the beam. It shows that the material and geometric nonlinearities have much less contribution to the SHG than the contact nonlinearity. In addition, the SHG can accurately determine the crack location without using the reference data. Overall, the findings of this study help further advance the use of SHG for damage detection.

Wave propagation analysis of laminated composite shell panels using a frequency domain spectral finite element model

This paper presents a frequency domain spectral finite element model (SFEM) for studying wave propagation in laminated composite shell panels. The SFEM formulation is derived using two different theories: (i) the first order shear deformation theory (FSDT) that takes shear deformation into consideration, and (ii) the classical shell theory (CST) that neglects shear deformation. The spectral approach here is based on the use of fast Fourier transform (FFT) and inverse fast Fourier transform (IFFT). The spectral element uses exact shape functions and a frequency dependent exact dynamic stiffness matrix is obtained relating the nodal displacements to the nodal forces of an element. The formulation is validated by comparing the present results for wavenumber dispersion and natural frequencies with the published results. The present SFEM is then used to perform wavenumber dispersion, phase and group velocities, and wave propagation analyses of laminated composite shell panels. The numerical studies show that the wavenumbers obtained from the SFEM based on the CST are significantly different from those of the FSDT at high frequencies. The CST results for the group velocity and wave propagation response to tone burst excitation show significant deviation from the FSDT results for out-of-plane A0 flexural mode even at relatively low frequencies. The orthotropy ratio and lamination arrangements in a composite shell panel have significant effect on the group velocity as well as the flexural mode wave propagation response.

Impact force identification in structures using time-domain spectral finite elements

During the launch and subsequent life of a spacecraft, there are various transient loads due to stage separation and micro-meteorite impact which a spacecraft structure must be designed to withstand. However, the force histories for these transients at these locations are not available to design the structure due to practical considerations like mounting of instrumentation. To address this problem, a new algorithm for performing impact force identification based on time-domain responses measured at various locations is proposed. The time-domain spectral finite element model is adopted as it requires a fewer number of measured responses which clearly has an advantage with respect to conventional finite element method. Hence, we use time-domain spectral finite element method, to generate the mass, stiffness, and damping matrices, which are required to perform force identification. Experiments are performed to obtain the time-domain responses on the beam and portal frame structures on which force identification is performed. The efficiency of the new algorithm is demonstrated using a variety of responses and different one-dimensional structures. However, the proposed algorithm is general in nature and can be used for one-, two-, and three-dimensional structures and with conventional generalized finite element model. The results from the proposed algorithm show an excellent match between the reconstructed force histories and the experimentally measured force. © 2020, Springer-Verlag GmbH Austria, part of Springer Nature.

Spectral finite element method for wave propagation analysis in smart composite beams containing delamination

PurposeThe purpose of the study is to present a frequency domain spectral finite element model (SFEM) based on fast Fourier transform (FFT) for wave propagation analysis of smart laminated composite beams with embedded delamination. For generating and sensing high-frequency elastic waves in composite beams, piezoelectric materials such as lead zirconate titanate (PZT) are used because they can act as both actuators and sensors. The present model is used to investigate the effects of parametric variation of delamination configuration on the propagation of fundamental anti-symmetric wave mode in piezoelectric composite beams.Design/methodology/approachThe spectral element is derived from the exact solution of the governing equation of motion in frequency domain, obtained through fast Fourier transformation of the time domain equation. The beam is divided into two sublaminates (delamination region) and two base laminates (integral regions). The delamination region is modeled by assuming constant and continuous cross-sectional rotation at the interfaces between the base laminate and sublaminates. The governing differential equation of motion for delaminated composite beam with piezoelectric lamina is obtained using Hamilton’s principle by introducing an electrical potential function.FindingsA detailed study of the wave response at the sensor shows that the A0 mode can be used for delamination detection in a wide region and is more suitable for detecting small delamination. It is observed that the amplitude and time of arrival of the reflected A0 wave from a delamination are strongly dependent on the size, position of the delamination and the stacking sequence. The degraded material properties because of the loss of stiffness and density in damaged area differently alter the S0 and A0 wave response and the group speed. The present method provides a potential technique for researchers to accurately model delaminations in piezoelectric composite beam structures. The delamination position can be identified if the time of flight of a reflected wave from delamination and the wave propagation speed of A0 (or S0) mode is known.Originality/valueSpectral finite element modeling of delaminated composite beams with piezoelectric layers has not been reported in the literature yet. The spectral element developed is validated by comparing the present results with those available in the literature. The spectral element developed is then used to investigate the wave propagation characteristics and interaction with delamination in the piezoelectric composite beam.

Dynamic Analysis of Tapered Thin-Walled Beams Using Spectral Finite Element Method

Tapered thin-walled structures have been widely used in wind turbine and rotor blade. In this paper, a spectral finite element model is developed to investigate tapered thin-walled beam structures, in which torsion related warping effect is included. First, a set of fully coupled governing equations are derived using Hamilton’s principle to account for axial, bending, and torsion motion. Then, the differential transform method (DTM) is applied to obtain the semianalytical solutions in order to formulate the spectral finite element. Finally, numerical simulations are conducted for tapered thin-walled wind turbine rotor blades and validated by the ANSYS. Modal frequency results agree well with the ANSYS predictions, in which approximate 30,000 shell elements were used. In the SFEM, one single spectral finite element is needed to perform such calculations because the interpolation functions are deduced from the exact semianalytical solutions. Coupled axial-bending-torsion mode shapes are obtained as well. In summary, the proposed spectral finite element model is able to accurately and efficiently to perform the modal analysis for tapered thin-walled rotor blades. These modal frequency and mode shape results are important to carry out design and performance evaluation of the tapered thin-walled structures.

Investigation of rotary inertial dynamic effects on chatter boundary in milling process using three-dimensional Timoshenko tool model

Chatter prediction is essential to conduct stable machining. Many of the previous studies made to simulate milling process have not considered gyroscopic and rotary inertial dynamics effects. The aim of this research is to investigate the effects of rotary inertial dynamics on prediction of chatter in the milling operation. For this purpose, a three-dimensional rotating cantilever Timoshenko beam is considered for modeling the cutting tool. In the proposed model, cutting forces are applied at the end of the beam. Imposing spectral finite element model, the governing delay partial differential equations of the system reduce to nondelay ordinary differential equations. The main contributions of the current research are modeling the cutting tool as a continuous model by Timoshenko beam theory, finding and solving nondelay ordinary differential equations of the cutter via spectral finite element model, and studying the effects of gyroscopic and rotary inertial dynamics, completely. For validating, the system stability predictions obtained from the presented model are compared with experimental outcomes from the literature. Besides the common stability lobe diagram which is in terms of depth of cut and the spindle speed, other stability diagrams based on cutter length and diameter are illustrated. Using this model, influences of rotary inertial dynamics on these stability diagrams are investigated. The results show that ignoring the rotary inertial dynamics causes significant errors in prediction of chatter boundaries especially in high angular velocity of the tool. In addition, the effect of number of the cutter teeth on the stability of milling process is studied. The presented stability diagram may help a machinist to choose a better set of parameters, such as tool length and diameter, number of cutter flutes, depth of cut, and spindle velocity, for doing a stable milling process.

Dynamic analysis of a thick hollow cylinder made of two-dimensional functionally graded material using time-domain spectral element method

ABSTRACTIn this paper, the time-domain dynamic response of a thick hollow cylinder made of two-dimensional functionally graded material is studied. An axisymmetric solid spectral finite element model is proposed for the analysis of elastic wave propagation in the cylinder. The effectiveness of the proposed model is verified by comparing with the conventional finite element method. The effect of material grading pattern is analyzed. The results demonstrate the effectiveness of the present method for the analysis of elastic wave propagation in functionally graded solids with axial symmetry and the material composition variation has an important effect on structural wave propagation behavior.

Impact of the Test Device on the Behavior of the Acoustic Emission Signals: Contribution of the Numerical Modeling to Signal Processing

In a context of nuclear safety experiment monitoring with the non destructive testing method of acoustic emission, we study the impact of the test device on the interpretation of the recorded physical signals by using spectral finite element modeling. The numerical results are validated by comparison with real acoustic emission data obtained from previous experiments. The results show that several parameters can have significant impacts on acoustic wave propagation and then on the interpretation of the physical signals. The potential position of the source mechanism, the positions of the receivers and the nature of the coolant fluid have to be taken into account in the definition a pre-processing strategy of the real acoustic emission signals. In order to show the relevance of such an approach, we use the results to propose an optimization of the positions of the acoustic emission sensors in order to reduce the estimation bias of the time-delay and then improve the localization of the source mechanisms.

Modelling and analysis of nonlinear guided waves interaction at a breathing crack using time-domain spectral finite element method

This study proposes a time-domain spectral finite element (SFE) model and investigates nonlinear guided wave interaction at a breathing crack. An extended time-domain SFE method based on the Mindlin–Hermann rod and Timoshenko beam theory is proposed to predict the nonlinear guided wave generation at the breathing crack. An SFE crack element is proposed to simulate the mode-conversion effect, in which a bilinear crack mechanism is implemented to take into account the contact nonlinearity at the breathing crack. There is good agreement between the results calculated using the proposed time-domain SFE method and three-dimensional finite element simulation. This demonstrates the accuracy of the proposed SFE method in simulating contact nonlinearity at the breathing crack. Parametric studies using the fundamental symmetric (S0) and anti-symmetric (A0) modes of guided waves are also carried out to provide physical insights into the higher harmonics generated due to the contact nonlinearity at the breathing crack. The magnitude of the higher harmonics generated as a function of the crack depth is investigated in detail. The results show that the mode-converted higher harmonic guided waves provide valuable information for damage detection.

Beam damage uncertainty quantification using guided Lamb wave responses

In this article, a Bayesian inference approach is applied to conduct uncertainty quantification on notch damage in a beam structure using guided Lamb wave responses. The proposed methodology not only determines the notch damage characteristics but also quantifies associated uncertainties of these inferred values. The correlation between crack location and extent is investigated as well, because such information is essential for decision-making in the structural health monitoring applications. First, a spectral finite element model is used to characterize Lamb wave propagation responses in a beam under lead-zirconate-titanate actuation and sensing. Very few elements are required to accurately capture the wave propagation. The lead-zirconate-titanate sensor can pick up the reflected wave responses from both boundaries and damages. Total 18 simulation cases were generated by varying notch damage extent, damage location, and noise level. Second, the Markov Chain Monte Carlo techniques are employed to estimate...

Wavelet-based spectral finite element dynamic analysis for an axially moving Timoshenko beam

In this article, wavelet-based spectral finite element (WSFE) model is formulated for time domain and wave domain dynamic analysis of an axially moving Timoshenko beam subjected to axial pretension. The formulation is similar to conventional FFT-based spectral finite element (SFE) model except that Daubechies wavelet basis functions are used for temporal discretization of the governing partial differential equations into a set of ordinary differential equations. The localized nature of Daubechies wavelet basis functions helps to rule out problems of SFE model due to periodicity assumption, especially during inverse Fourier transformation and back to time domain. The high accuracy of WSFE model is then evaluated by comparing its results with those of conventional finite element and SFE results. The effects of moving beam speed and axial tensile force on vibration and wave characteristics, and static and dynamic stabilities of moving beam are investigated.