Time-domain Spectral Element Method Research Articles

On the emergence of the second harmonic shear horizontal wave in presence of tangential prestress

Nonlinear ultrasonic techniques have significant advantages in characterizing incipient defects of materials. The classical nonlinear ultrasonic theory indicates that a primary longitudinal wave can generate a second harmonic longitudinal wave due to the material nonlinearity (MN) or contact acoustic nonlinearity (CAN). However, in both MN and CAN, a second harmonic shear horizontal (SH) component generated from a normally-incident SH wave has not been reported. In this work, a time domain spectral finite element method combined with the bi-potential contact theory is extended to simulate the CAN problems with prestresses. Numerical verifications are performed to show the robustness and accuracy of the developed method. After that, the nonlinear interactions between the SH wave and frictional interfaces under complex prestresses are investigated. It is found for the first time that when there are tangential prestresses at the frictional interfaces or crack surfaces, a primary SH wave of normal incidence can generate the second harmonic SH wave. A theoretical model is then proposed to interpret this new phenomenon. It is found that the even SH wave harmonics result from the asymmetric sliding of the frictional interface induced by the tangential prestresses. This asymmetric sliding mechanism can be compared with the breathing behavior of a crack caused by an incident longitudinal wave. The findings of this work could enrich the understanding of contact acoustic nonlinearity behavior of the SH wave and promote the development of related nondestructive evaluation techniques.

Seismic analysis of a I-sectioned steel frame based on an improved time-domain spectral element method

An improved time-domain spectral element method (ITSEM) is developed for seismic analysis of I-sectioned steel structures in this paper. The developed method considers the effect of section type on spectral stiffness using shear correction coefficient. In the calculation process of the time-domain spectral stiffness and mass matrices, the invariant integral operations of the element matrices are extracted and saved in advance. Thus, the global mass and stiffness matrices of the structures can be obtained only by simple addition, subtraction, multiplication and division operations, which greatly improves computational efficiency. Next, the natural frequencies of a I-sectioned beam are computed and compared with the finite element method (FEM) and analytical results to validate the ITSEM. Then, the seismic responses of a ten-story steel frame using ITSEM are obtained and compared with those by FEM, which proves the effectiveness, accuracy and high efficiency of ITSEM in elastic seismic analysis.

Spectral element modeling of ultrasonic guided wave propagation in optical fibers

Recent advancements in fiber optic methods have enabled their use for guided wave sensing. It opens up new possibilities for Structural Health Monitoring. The aim of this paper is to provide insight for the physics related to guided wave propagation and coupling between the optical fiber and solid structure. For this purpose, a new approach for non-matching interface based on Lagrange multipliers and the time domain spectral element method was developed. A parallelized code has been implemented in order to simulate the guided wave propagation in the structure, its coupling into the optical fiber and the propagation in the fiber in a computationally efficient way. The paper presents four studies showing the efficacy of the modeling approach. The paper first shows the improvement in the computation speed through the use of parallelization and a more efficient implementation. Then the results of the simulation of wave propagation in the fiber are compared with results from previous simulation studies using commercially available software. The third study shows that the spectral element method is able to capture the directional sensitivity of optical fiber based sensors. Lastly, the simulation is used for detection of simulated damage using the spectral element method based simulation. The results indicate that indeed the spectral element implementation is able to recreate the wave coupling phenomena, capture the physics of the system including directional sensitivity and reflections from damage.

Model-Assisted Guided-Wave-Based Approach for Disbond Detection and Size Estimation in Honeycomb Sandwich Composites.

One of the axioms of structural health monitoring states that the severity of damage assessment can only be done in a learning mode under the supervision of an expert. Therefore, a numerical analysis was conducted to gain knowledge regarding the influence of the damage size on the propagation of elastic waves in a honeycomb sandwich composite panel. Core-skin debonding was considered as damage. For this purpose, a panel was modelled taking into account the real geometry of the honeycomb core using the time-domain spectral element method and two-dimensional elements. The presented model was compared with the homogenized model of the honeycomb core and validated in the experimental investigation. The result of the parametric study is a function of the influence of damage on the amplitude and energy of propagating waves.

3-D multiparameter full-waveform inversion for ocean-bottom seismic data using an efficient fluid–solid coupled spectral-element solver

SUMMARYOcean-bottom seismic acquisition systems deployed on the seabed give access to three-component geophone data and hydrophone data. Compared with conventional streamer acquisitions, the separation of sources and receivers makes it possible to increase the maximum offset and azimuth coverage for improving the illumination at depth. Furthermore, the three-component geophones naturally capture elastic wave propagation effects. While this information is mostly overlooked up to now, reconstructing jointly P- and S-wave velocities would significantly improve the subsurface characterization. To achieve a 3-D high-resolution multiparameter reconstruction, we design an efficient 3-D fluid–solid coupled full waveform modelling and inversion engine. In this engine, fluid and solid domains are divided explicitly and handled with the acoustic and elastic wave equations, respectively. The numerical implementation is based on a time-domain spectral-element method (SEM) with a flexible 3-D Cartesian-based hexahedral mesh, which contributes to an accurate coupling of the acoustic and elastic wave equations and high computational efficiency through domain-decomposition based parallelization. We select the best acoustic–elastic coupled formulations among 4 possibilities with criteria based on numerical accuracy and implementation efficiency. Moreover, we propose a specific hybrid approach for the misfit gradient building so as to use a similar modelling solver for both forward and adjoint simulations. Synthetic case studies on a 3-D extended Marmousi-II model and a 3-D deep-water crustal-scale model illustrate how our modelling and inversion engine can efficiently extract information from ocean-bottom seismic data to simultaneously reconstruct both P- and S-wave velocities within a full waveform inversion framework.

Removing the Courant-Friedrichs-Lewy stability criterion of the explicit time-domain very high degree spectral-element method with eigenvalue perturbation

The explicit time-domain spectral-element method (SEM) for synthesizing seismograms has gained tremendous credibility within the seismological community at all scales. Although the recent introduction of nonperiodic homogenization has addressed the spatial meshing difficulty of the mechanical discontinuities, the Courant-Friedrichs-Lewy (CFL) stability criterion strictly constrains the maximum time step, which still puts a great burden on the numerical simulation. In the explicit time-domain SEM, the source of instability of using a time step beyond the stability criterion is that some unstable eigenvalues of the updated matrix are larger than what can be accurately simulated. We have succeeded in removing the CFL stability condition in explicit time-domain SEM by combining the forward time dispersion-transform method, the eigenvalue perturbation, and the inverse time dispersion-transform method. Our theoretical analyses and numerical experiments in the homogeneous, moderate, and strong heterogeneous models show that this combination can accurately simulate waveforms with time steps several times the size of the CFL limit even toward the Nyquist limit especially for the efficient very high degree SEM, which abundantly saves iteration times without suffering from time-dispersion error. It demonstrates a potential application prospect in some situations such as full-waveform inversion that require multiple numerical simulations for the same model.

Elastic constants identification of fibre-reinforced composites by using guided wave dispersion curves and genetic algorithm for improved simulations

There is a great potential for model-based approaches in which guided waves are utilised for damage detection, localisation and size estimation. However, to be effective, the model must accurately represent guided wave propagation behaviour, especially wave velocity dependence on the angle of propagation. Unfortunately, improperly assumed or derived through homogenisation elastic material properties can lead to larger wave velocity changes than caused by damage. In order to account for this problem, the elastic constants are estimated based on experimental measurements of Lamb waves and optimisation techniques.In the proposed approach, experimental measurements of full wavefield data of propagating Lamb waves are conducted by scanning laser Doppler vibrometer. The obtained wavefield data is processed by using the 3D Fourier transform. Slicing the data at selected frequencies yields wavenumber profiles. These profiles can be directly translated to wave propagation velocity dependence on the angle of propagation. Data can also be sliced at selected angles giving dispersion curve images. At the same time, wavenumber profiles or dispersion curves can be calculated by using a semi-analytical model for the range of material properties. A genetic algorithm is used in which the best fit between experimental data and the semi-analytical model is optimised.Experiments were conducted on a unidirectional CFRP panel and the above-mentioned procedure was applied. Identified material properties were used in the in–house code of the time domain spectral element method for simulations of the propagating waves. Finally, numerical results were compared to the experimental full wavefield data showing much better accuracy than in case of application of homogenisation techniques.Furthermore, the correctness of elastic constants determined by the proposed method was validated against the elastic constants obtained through static test standards.

Time domain spectral element-based wave finite element method for periodic structures

In this work, a time domain spectral element-based wave finite element method is proposed to analyze periodic structures. Time domain spectral element-based formulation reduces the total degrees of freedom and also renders a diagonal mass matrix resulting in substantial reduction in computation time for the wave finite element method. The formulation is then considered to obtain the stop band characteristics for a periodic bar and a Timoshenko beam considering geometric as well as material periodicity. The impact of geometric parameters on the stop bands of 1-D structures is then investigated in detail. It is shown that the stop bands can be obtained in the frequency range of interest, and its width can be varied by tuning those geometric parameters. Also, the effect of material uncertainty is studied in detail on the stop band characteristics of periodic 1-D structures, and the same formulation is utilized for Monte Carlo simulations. Results show that randomness in density influences more the bandwidth of the stop bands than that of elastic parameters.

3D Airborne EM Forward Modeling Based on Time-Domain Spectral Element Method

Airborne electromagnetic (AEM) method uses aircraft as a carrier to tow EM instruments for geophysical survey. Because of its huge amount of data, the traditional AEM data inversions take one-dimensional (1D) models. However, the underground earth is very complicated, the inversions based on 1D models can frequently deliver wrong results, so that the modeling and inversion for three-dimensional (3D) models are more practical. In order to obtain precise underground electrical structures, it is important to have a highly effective and efficient 3D forward modeling algorithm as it is the basis for EM inversions. In this paper, we use time-domain spectral element (SETD) method based on Gauss-Lobatto-Chebyshev (GLC) polynomials to develop a 3D forward algorithm for modeling the time-domain AEM responses. The spectral element method combines the flexibility of finite-element method in model discretization and the high accuracy of spectral method. Starting from the Maxwell's equations in time-domain, we derive the vector Helmholtz equation for the secondary electric field. We use the high-order GLC vector interpolation functions to perform spectral expansion of the EM field and use the Galerkin weighted residual method and the backward Euler scheme to do the space- and time-discretization to the controlling equations. After integrating the equations for all elements into a large linear equations system, we solve it by the multifrontal massively parallel solver (MUMPS) direct solver and calculate the magnetic field responses by the Faraday's law. By comparing with 1D semi-analytical solutions for a layered earth model, we validate our SETD method and analyze the influence of the mesh size and the order of interpolation functions on the accuracy of our 3D forward modeling. The numerical experiments for typical models show that applying SETD method to 3D time-domain AEM forward modeling we can achieve high accuracy by either refining the mesh or increasing the order of interpolation functions.

Ultrasonic Guided Wave Signal Based Nondestructive Testing of a Bonded Composite Structure Using Piezoelectric Transducers

This paper presents ultrasonic guided wave (UGW) propagation-based nondestructive testing (NDT) of an adhesively bonded composite structure (ACS). In the process, a series of scanning laser Doppler vibrometry (SLDV)-based laboratory experiments and time-domain spectral element method (SEM)-based numerical simulations were carried out on an ACS with barely visible impact damage (BVID) and a hole. A good agreement was observed between the numerical and experimental UGW signals in the cases studied. Finally, a full-field and elliptical signal processing method-based NDT strategy was proposed that uses differential damage features of the registered UGW signals to identify different types of BVIDs in the ACS.

Crack orientation detection in plate-like structures via Lamb wave specular reflection analysis

The determination of crack orientation plays an important role in the non-destructive evaluation (NDE) of plate-like structures. In this paper, the interaction between Lamb waves and cracks is investigated using the time domain spectral element method (SEM) as the simulation tool. On that basis, we analyse the geometric principle of specular reflections and propose an efficient crack angle detection algorithm. During the process, four sensors take turns to excite the Lamb wave which would result in twelve crack reflection signals. Using the corresponding actuator as the centre and the reflection length as the radius, we can obtain twelve reflection circles in total. When these reflections are specular, the mirror points of the actuators with respect to the crack line are exactly the intersection points of the circles. Thus, we can obtain the crack angle by this geometric relation. Validations on different cracks are made and the result of the algorithm shows a good agreement with the real value.

On the Influence of Slopes, Source, Seabed and Water Column Properties on T Waves: Generation at Shore

The term T waves is generally associated with acoustic waves generated by seismic events that subsequently travel horizontally in the ocean at the speed of sound. In this paper, we use a time-domain spectral-element method to perform a parametric study of the influence of seafloor slope, source position and media properties for a typical (downslope) T-wave generation scenario. We find that the energy and duration of these waves are particularly sensitive to the environment. In particular, the slopes and physical characteristics of the seabed play a crucial role for both the generation and the conversion of these waves. Likewise, the depth and position of the earthquake relative to the slope is of great importance, with the presence of privileged areas for the generation of T waves, which we map.

A time-domain spectral element method with C1 continuity for static and dynamic analysis of frame structures

This paper presents a new C1-type beam spectral element method, which aims to ensure the continuity of the deflection and its first derivative at the junction of the adjacent spectral elements. The proposed method is based on the Euler-Bernoulli beam theory. The main novelty of the method is that two different sets of interpolation functions are adopted for the axial displacement and deflection in order to meet the C1 continuous requirement. By removing the rotational degrees of freedom of the internal nodes, a lumped mass matrix can be formed in this spectral element method. Static and dynamic analysis of various beams and frame structures are performed to examine the validity of the proposed spectral beam element model. The numerical results show that the proposed method is accurate and effective. A single element for each component of the structure is able to provide good results.

Uncertainty analysis of higher-order sandwich beam using a hybrid stochastic time-domain spectral element method

A stochastic time-domain spectral element method (STSEM) is applied to sandwich beam structures. STSEM couples the computational efficiencies of the spectral stochastic finite element method and the time-domain spectral element method. A higher-order sandwich panel theory incorporating the core flexibility is considered for the analysis. The material properties are modeled as random fields and represented by the Karhunen-Loéve expansion. The stochastic nodal displacements are expressed using the polynomial chaos expansion, and the expansion coefficients are computed by solving the spectral stochastic finite element equations. The computational efficiency of the proposed method is compared with Monte-Carlo simulation.

Local wave propagation analysis in concrete-filled steel tubes with spectral element method using absorbing layers – Part II: Application in coupling system

In this paper, a two-dimensional (2D) time domain spectral element method (SEM) is developed for concrete-filled steel tube (CFST) and piezoelectric lead zirconate titanate (PZT) patches coupling system, where the piezoelectric and inverse piezoelectric effect of PZT actuator and sensor and their coupling effect with CFST are considered. The local wave field and the response of the embedded PZT sensor was numerically studied in a substructure of the CFST coupled with PZT patches and surrounded by absorbing layers with increasing damping (ALID) and the performance of the ALID in attenuating wave reflection at boundaries of the coupling substructure model is further investigated. Then, numerical simulation on the effect of interface debonding defect on local wave propagation within the coupling substructure model surrounded by the designed ALID was carried out. By comparing the displacement-based wave fields and the response of the embedded PZT sensor within the substructure, the effect of interface debonding defect on the wave propagation along the steel tube and in the concrete core, and the time history of the response voltage of the embedded PZT sensor is investigated in detail. Numerical results show that interface debonding defects in the substructure leads to changes in the wave propagation path, the local wave field, the time duration of wave travelling to the embedded PZT sensor from the PZT actuator mounted on the surface of the steel tube of CFST, as well as response voltage of the embedded PZT sensor. In addition, PZT sensor measurement is sensitive to the length of interface debonding defects but insensitive to their depth. SEM for the coupling substructure model with ALID instead of the model of whole cross section of CFST provides an efficient way to simulate the local elastic wave propagation for understanding the mechanism of the interface debonding defects detection approach based on wave measurement.

Stochastic strain and stress computation of a higher-order sandwich beam using hybrid stochastic time domain spectral element method

In this work, a combination of the spectral stochastic finite element method (SSFEM) and the time-domain spectral element method (TSEM), referred to as the stochastic time domain spectral element method (STSEM), is presented to compute the stochasticity of strain and stress of a higher-order sandwich composite beam with spatial variability in the material properties. The method proposed in this work employs the efficiencies of both SSFEM and TSEM for the uncertainty analysis of a sandwich beam. The material properties of face sheets and core are considered as Gaussian random fields, which are discretized using the Karhunen-Loéve expansion, and polynomial chaos expansion is used to represent the response quantity. A numerical example is considered for which, first, a sensitivity analysis is performed to identify the most sensitive material properties. Then, the proposed STSEM is used to demonstrate the computational efficiency and numerical accuracy in comparison with Monte-Carlo simulation. Moreover, the effect of core depth on strain and stress variability is also examined.

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.

Local wave propagation analysis in concrete-filled steel tube with spectral element method using absorbing layers – Part I: Approach and validation

Concrete-Filled Steel Tube (CFST) members have been widely employed in high-rise buildings or long-span bridges as vertical or axial load-carrying components and the development of efficient interface debonding detection approach for CFST members has been a common concern. In recent years, wave propagation measurement based interface debonding defect approaches have shown their efficiency and efficient numerical simulation techniques on elastic wave propagation within CFST members are critical to understand the mechanism of the approaches. In this study, in order to efficiently investigate the effect of interface debonding on local elastic wave propagation within CFST members, a two-dimensional (2D) time domain spectral element method (SEM), where absorbing layers with increasing damping (ALID) is introduced to attenuate the wave reflection effect on boundaries, is developed at first. Then, the developed 2D time domain SEM with ALID is employed to simulate the wave propagation in a steel plate and an isotropic panel mimicking concrete core in CFST members. The effect of ALID on the attenuation of reflected Lamb waves in the steel plate and the bulk waves in the isotropic panel is validated. Finally, the local wave propagation in a substructure of the CFST model is numerically studied with the proposed 2D SEM with ALID. The dimensions and parameters of the ALID are determined and their effect on eliminating the reflected waves within the substructure excited by a single point load and a surface-mounted PZT patch actuator is demonstrated, respectively. The developed 2D time domain SEM with ALID provides efficient tool to simulate the local wave propagation in CFST specimens with interface debonding defects for understanding the mechanism of local stress wave measurement based interface debonding detection method to be discussed in the accompanying paper.

On the possible role of gravity waves in the ocean in T-phase excitation by earthquakes

Recent applications of a time-domain spectral-element method to obtain full-wave solutions of seismo-acoustic problems by Bottero et al. [J. Acoust. Soc. Am., 144, EL222–EL228 (2018)] and especially their investigation of T-phases in the Mediterranean [Bottero et al., J. Acoust. Soc. Am., 141, 4045 (2017)] challenge the existing understanding of the mechanisms of seismic energy conversion into T-waves, which propagate at the speed of sound in water. Motivated by these results, this paper aims to evaluate the contribution of scattering by hydrodynamic waves into T-phase generation. Ocean is modeled as a range-independent waveguide with superimposed volume inhomogeneities due to internal gravity waves and surface roughness due to wind waves and sea swell. Guided acoustic waves are excited by volume and surface scattering of ballistic body waves. The size of the effective source of T-waves and their amplitudes are calculated in the single scattering—multiple reflections approximation. Efficiency of the gravity wave-mediated conversion of seismic energy will be compared to that of the conversion that occurs on a seafloor slope.

2D–3D interface coupling in the time domain spectral element method for the adhesive layer effects on guided wave propagation in composite plates

A new approach for numerical simulation of the wave propagation in a composite plate is presented in this paper. Problem is realized by the 2D–3D coupled time domain spectral element method. All components used in the simulation are decomposed from each other. A connection between them is guaranteed by the interface elements realized by Lagrange multipliers. With this type of scheme, one can avoid a modelling of all the components by the three-dimensional (3D) spectral elements. In particular, the computation time can be reduced by using two-dimensional (2D) elements for adhesive layer modelling. Otherwise, 3D elements are very inefficient because a convergence of Central Difference Time Integration Method applied for solving an equation of motion is strongly correlated with the size of the smallest element. Numerical calculations were performed for the composite plate modelled by the 3D solid elements with a piezoelectric transducer bonded to the plate by the adhesive layer modelled by 2D shell elements. The simulations were carried out for an adhesive layer of various thickness for the excitation frequency in the range of 50–200 kHz. The presented model was verified by experimental data obtained by the laser vibrometer.