Time-domain Spectral Finite Element Research Articles

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.

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.

Time-domain Spectral Finite Element Method for Wave Propagation Analysis in Structures with Breathing Cracks

Guided waves are generally considered as a powerful approach for crack detection in structures, which are commonly investigated using the finite element method (FEM). However, the traditional FEM has many disadvantages in solving wave propagation due to the strict requirement of mesh density. To tackle this issue, this paper proposes an efficient time-domain spectral finite element method (SFEM) to analyze wave propagation in cracked structures, in which the breathing crack is modeled by defining the spectral gap element. Moreover, novel orthogonal polynomials and Gauss–Lobatto–Legendre quadrature rules are adopted to construct the spectral element. Meanwhile, a separable hard contact is utilized to simulate the breathing behavior. Finally, a comparison of the numerical results between the FEM and the SFEM is conducted to demonstrate the high efficiency and accuracy of the proposed method. Based on the developed SFEM, the nonlinear features of waves and influence of the incident mode are also studied in detail, which provides a helpful guide for a physical understanding of the wave propagation behavior in structures with breathing cracks.

Ultrasonic guided wave scattering due to delamination in curved composite structures

Wave propagation in a curved composite structure having delamination is simulated using time domain spectral finite element (TSFE) method that enables fast computation with higher-order field interpolation. Curved structures are very common in aerospace, marine and other composite structural components and understanding ways to detect delamination in these curved structures with the help of ultrasonic guided wave simulation is essential. Guided wave interaction with curved region progressively causes mode converted waves, which are present in both reflected as well as transmitted wave packets. The details are poorly understood. The additional wave packets due to interaction cause difficulty in identification of damaged induced responses. Mode conversion and reflection from the curved section reduce the useful signal strength to interrogate any delamination in the curved region. Guided wave interaction with the curved section in an L-shaped structure and a structure with T-joint are studied using TSFE simulation. Simulation results are validated using analytical solutions. Mode conversion and transmission in T-joint is studied using numerical simulation with experimental validation. Signal loss due to mode conversion and reflections at different frequencies is investigated in terms of geometric and wave parameters, which promises to identify the appropriate frequencies and choice of wave mode for monitoring.

Broadband monostatic acoustic scattering simulations of underwater unexploded ordnance using time-domain spectral-elements

Acoustic detection of unexploded ordnance (UXO) that contaminate the world’s waterways is vital. Critical to the implementation of detection methods is the numerical simulation of the scattering response of these targets using finite element methods. We introduce a parametric approach based on the time-domain spectral finite element method (SEM). This technique allows for the computation of broadband acoustic response of complex heterogeneous 3-D fluid-solid objects, in particular a 5 inch rocket UXO used for this study. The scattered field is obtained at arbitrary distances using the Kirchhoff–Helmholtz integral. The main interest of the SEM is that it is particularly adapted to high-performance computing (CPU or GPU). Contrary to most of the other finite element methods it scales perfectly; using twice as many processors leads to a halving of computing time. In addition, working in the time domain is a direct simulation of common sonars and experiments. The method is first benchmarked against the commercial finite-element code COMSOL on the monostatic response of a rigid target over a full 180 deg. Finally, results obtained for the 5 inch rocket are compared to actual measurements obtained in the NRL underwater acoustics tank facility. [Work partially supported by the Office of Naval Research.]

Time-domain spectral finite element method for analysis of torsional guided waves scattering and mode conversion by cracks in pipes

Abstract This paper presents a computationally efficient time-domain spectral finite element method (SFEM) and a crack model to take into account guided wave propagation, scattering and mode conversion in pipes. The proposed SFEM couples torsional and flexural motions of guided waves. A cracked element is proposed to predict the scattering and mode conversion effect of guided wave interaction with the crack in the pipes. The proposed SFEM and cracked element are verified by 3D finite element and experimental data. The results show that the proposed SFEM is able to predict the torsional guided wave propagation, scattering and mode conversion accurately. A series of numerical and experimental case studies are carried out to investigate the effect of the crack size on the scattering and mode converted guided waves. The findings of the study provide physical insights into the guided wave scattering and mode conversion and further advance the development of damage detection using guided waves.

Wave-based impact localization on laminated composite plates using a coarse network of sensors

This article demonstrates a methodology for the detection of foreign object impact locations on laminated composite plates using a low number of sensors. The proposed approach exploits the wave-dominated transient response of the target structure and addresses the challenges induced by the non-uniform wave propagation due to the anisotropy of composite plates on the impact localization. Captured sensor signals are processed and their frequency content is identified. Semi-analytical wave dispersion models and time-domain spectral finite element impact models encompassing the effects of laminate anisotropy on wave dispersion characteristics are employed to extract the wave velocity from their frequency content. The method for localizing the impact site is formulated as an optimization problem and involves signal correlation from different sensors, refinement of the candidate locations, and finally accurate estimation of the impact site. Numerical and experimental validations of the developed method are presented for low-velocity impacts on glass/epoxy laminated plates.

Time-domain Spectral Element Method for Impact Identification of Frame Structures using Enhanced GAs

This paper develops an Enhanced Genetic Algorithm (GA) strategy in conjunction with a time-domain Spectral Finite Element Method (SFEM) for impact identification of framed structures. For this purpose, a spatial truss spectral element is proposed for impact response simulation. In this regard, Gauss-Lobatto-Legendre quadrature rules and points configuration are adopted to construct a diagonal matrix and to gain an optimum computational demands. A decimal and a mixed GA coding system are implemented to locate and re-construct the impact features, respectively. An improved GA-based fitness assessment is designed to significantly accelerate the convergence rate of the optimization strategy. The impact identification of two frame structures is investigated taking into account the influence of externally applied loading. It is concluded that, the proposed mixed coding GA strategy effectively overcomes the main drawbacks of classical GA approach. The developed SFEM is superior to conventional FEM because of its high order interpolation and integration rules. For large structures, impact localization is successfully accomplished very fast, which provides the excellent ability in developing an on-line health monitoring system. It is included that, the robustness of the proposed SFEM lies on the considerably higher computational efficiency in achieving the most accurate results with the less computational costs.

Lamb wave interaction with composite delamination

Elastic guided wave based evaluation is useful to detect delamination in composites. This paper reports a detailed account of guided wave interaction with delamination in laminated composites modeled using time domain spectral finite element (TSFE) method. Wave scattering due to different delamination positions is studied. Wave field near the delamination in different layer-wise positions is investigated, which is useful to further characterize and correlate the far field wave packet carrying the parametric signature of delamination. Off-axis delamination in composite laminate creates an asymmetry in the elastodynamic stress transfer. It leads to wave mode conversions. Sensitivity of the delamination to the wavelengths is studied. The outcome of this study indicates potential possibilities to use frequency-wavelength information to discriminate various sizes of delamination. Energy of the scattered waves and dissipation/conversion of the wave energy due to the defects depend on the resonance characteristics of the sub-laminates. Wave scattering effect due to length-wise multiple delaminations and edge delamination is also analyzed. The resonance patterns in the signals are analyzed with reference to the defect quantification problem.

A higher order transversely deformable shell-type spectral finite element for dynamic analysis of isotropic structures

This paper deals with certain aspects related to the dynamic behaviour of isotropic shell-like structures analysed by the use of a higher order transversely deformable shell-type spectral finite element newly formulated and the approach known as the Time-domain Spectral Finite Element Method (TD-SFEM). Although recently this spectral approach is reported in the literature as a very powerful numerical tool used to solve various wave propagation problems, its properties make it also very well suited to solve static and dynamic modal problems. The robustness and effectiveness of the current spectral approach has been successfully demonstrated by the authors in the case of thin-walled spherical shell structures through a series of numerical tests comprising the analysis of natural frequencies and modes of vibration of an isotropic spherical shell as well as the wave propagation analysis in the case of the same spherical shell and a half-pipe shell-like structure.

Numerical simulation of elastic wave propagation in functionally graded cylinders using time-domain spectral finite element method

Functionally graded material is a new class of composite materials with gradually varying material contents. In this article, a two-dimensional axisymmetric time-domain spectral finite element meth...

High frequency dynamics of an isotropic Timoshenko periodic beam by the use of the Time-domain Spectral Finite Element Method

In this work results of numerical simulations and experimental measurements related to the high frequency dynamics of an aluminium Timoshenko periodic beam are presented. It was assumed by the authors that the source of beam structural periodicity comes from periodical alterations to its geometry due to the presence of appropriately arranged drill-holes. As a consequence of these alterations dynamic characteristics of the beam are changed revealing a set of frequency band gaps. The presence of the frequency band gaps can help in the design process of effective sound filters or sound barriers that can selectively attenuate propagating wave signals of certain frequency contents. In order to achieve this a combination of three numerical techniques were employed by the authors. They comprise the application of the Time-domain Spectral Finite Element Method in the case of analysis of finite and semi-infinite computational domains, damage modelling in the case of analysis of drill-hole influence, as well as the Bloch reduction in the case of analysis of periodic computational domains. As an experimental technique the Scanning Laser Doppler Vibrometry was chosen. A combined application of all these numerical and experimental techniques appears as new for this purpose and not reported in the literature available.

A cubic spline layerwise time domain spectral FE for guided wave simulation in laminated composite plate structures with physically modeled active piezoelectric sensors

A novel theoretical framework is presented enabling enhanced simulation of guided waves generated by piezoelectric actuators in laminated composite plates. A new multi-field multi-physics layerwise theory is formulated for laminated plates with piezoelectric actuators and sensors which captures symmetric and anti-symmetric stress waves. Third-order Hermite splines are employed in the approximation of displacements and electric potential through the thickness. The piezoelectric actuators and sensors are physically modeled though coupled electromechanical governing equations. An explicit time-domain spectral finite element is formulated entailing displacement and electric degrees of freedom at the nodes which are further collocated with Gauss–Lobatto–Legendre integration points. The stiffness, consistent semi-diagonal mass, piezoelectric and electric permittivity matrices are calculated. Furthermore, an explicit time integration scheme is presented for the calculation of the coupled transient electromechanical response including the free sensor response. Numerical guided wave predictions of the developed multi-node time domain spectral finite elements are correlated with well established numerical tools and with experiments conducted on laminated carbon/epoxy plates with active piezoceramic sensor networks. Important effects introduced by the physical presence of the active actuator/sensor system on guided wave propagation, such as wave reflections, mode conversions and sensor signal attenuation, are successfully captured by the developed finite element and correlated and evaluated with numerical and experimental results.

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.

A Hermite Spline Layerwise Time Domain Spectral Finite Element for Guided Wave Prediction in Laminated Composite and Sandwich Plates

A new time domain spectral plate finite element (FE) is developed to provide fast numerical calculations of guided waves and transient phenomena in laminated composite and sandwich plates. A new multifield layerwise laminate theory provides the basis for the FE, which incorporates cubic Hermite polynomial splines for the approximation of the in-plane and transverse displacement fields through the thickness of the plate, enabling the modeling of symmetric and antisymmetric wave modes. The time domain spectral FE with multi-degrees-of-freedom (DOF) per node is subsequently formulated, which uses integration points collocated with the nodes to yield consistent diagonal lumped mass matrix which expedites the explicit time integration process. Numerical simulations of wave propagation in aluminum, laminated carbon/epoxy and thick sandwich plates are presented and validated with an analytical solution and a three-dimensional (3D) solid element; moreover, the capability to accurately and rapidly predict antisymmetric and symmetric guided waves is demonstrated.

Experimental and numerical investigation of contact laws for the rapid simulation of low-energy impacts on laminated composite plates

A computationally efficient model for the impact simulation of low energy impacts on composite laminated plate structures is developed, which combines a contact law with a new time domain spectral shear plate finite element. Various contact laws are numerically and experimentally investigated based on elastoplastic and fully-plastic indentation theory. The contact laws are correlated with quasi-static indentation test data performed on carbon/epoxy cross-ply laminates and with numerical results of a 3-D finite element analysis (FEA) indentation model with Hill failure criterion. The extracted contact laws are combined with a first-order shear plate time domain spectral finite element with explicit time integration. Obtained impact simulation results are compared with a 3-D continuum explicit FEA model and experimental results. The results demonstrate excellent accuracy of the present method, and drastically improved numerical simulation speed.

Analysis of mode conversion and scattering of guided waves at cracks in isotropic beams using a time-domain spectral finite element method

Detecting damages in its early stage, and hence, to ensure the safety and reliability of structures is of vital important. Guided waves have been recognised as one of the promising damage detection techniques that are sensitive to small and different types of damages. The understanding of guided wave propagation and scattering phenomena at the damages is one of the fundamental elements to facilitate the development of this technique for damage characterisation. This paper presents a study of scattering characteristics and mode conversion effects of guided waves at cracks in isotropic beams. An efficient time-domain spectral finite element method using one-dimensional (1D) beam element is developed to solve this problem. The developed model is then used to carry out a series of case studies that consider different crack sizes in the beams. These parametric studies provide a fundamental physical insight into the mode conversion phenomena and scattering characteristics of the guided waves at the cracks.

A probabilistic approach for quantitative identification of multiple delaminations in laminated composite beams using guided waves

In this study a probabilistic approach is proposed to identify multiple delaminations in laminated composite beams using guided waves. The proposed method is a model-based approach, which provides a quantitative identification of the delaminations. This study puts forward a practical damage identification method, and hence, it can identify multiple delaminations using guided wave signal measured at a single measurement point on the laminated composite beams. The proposed method first determines the number of delaminations using Bayesian model class selection method. The Bayesian statistical framework is then employed to not only identify the delamination locations, lengths and through-thickness locations, but also quantify the associated uncertainties, which provides valuable information for engineers in making decision on necessary remedial work. In addition the proposed method employs the time-domain spectral finite element method and Bayesian updating with Subset simulation to further improve the computational efficiency. The proposed probabilistic approach is verified and demonstrated using data obtained from numerical simulations, which consider both measurement noise and modeling error, and experimental data. The results show that the proposed method can accurately determine the number of delaminations, and the identified delamination locations, lengths and through-thickness locations are closed to the true values.

A time domain spectral layerwise finite element for wave structural health monitoring in composite strips with physically modeled active piezoelectric actuators and sensors

A new explicit, two-dimensional plane strain, time domain spectral finite element is developed to enhance the simulation of guided waves generated by active piezoelectric sensors in laminated composite strips. A new multi-field layerwise theory is formulated for composite laminates with piezoelectric actuators and sensors which captures straight-crested symmetric and anti-symmetric Lamb waves. Third-order Hermite polynomial splines are employed for the approximation of displacements and electric potential through the thickness, and the piezoelectric actuators and sensors are physically modeled through coupled electromechanical governing equations. A multi-node finite element formulation is presented entailing displacement and electric degrees of freedom at nodes collocated with Gauss–Lobatto–Legendre integration points. Stiffness, diagonal mass, piezoelectric, and electric permittivity matrices are described, and the coupled transient electromechanical response is predicted by a properly formulated explicit time integration scheme. The numerical results of a nine-node time domain spectral finite element are correlated with the reported numerical results and with measured Lamb wave data generated by piezoceramic active sensor pairs in carbon/epoxy plate strips. Important effects introduced by the stiffness and mass of the active actuator/sensor system on Lamb wave propagation are captured by the developed finite element and quantified.

Analysis of mode conversion and scattering of guided waves at cracks in isotropic beams using a time-domain spectral finite element method

Detecting damages in its early stage, and hence, to ensure the safety and reliability of structures is of vital important. Guided waves have been recognised as one of the promising damage detection techniques that are sensitive to small and different types of damages. The understanding of guided wave propagation and scattering phenomena at the damages is one of the fundamental elements to facilitate the development of this technique for damage characterisation. This paper presents a study of scattering characteristics and mode conversion effects of guided waves at cracks in isotropic beams. An efficient time-domain spectral finite element method using one-dimensional (1D) beam element is developed to solve this problem. The developed model is then used to carry out a series of case studies that consider different crack sizes in the beams. These parametric studies provide a fundamental physical insight into the mode conversion phenomena and scattering characteristics of the guided waves at the cracks.