Time-domain Spectral Finite Element Research Articles

Effect of moisture on the sensing capabilities of piezoelectric actuator/sensor pairs on flax fibre reinforced composite laminates

The sensitivity of the piezoelectric actuators’ response to the moisture uptake has been investigated on flax fibre reinforced composite laminates. Non-woven flax fabric was impregnated with epoxy and composite laminates were fabricated using hand lay-up and cured in a compression moulding machine. The laminates were initially impacted with a drop-weight impact machine at a low energy level, with the aim to generate an artificial surface crack that will allow the moisture to enter the fibre-matrix interface more effectively. A set of piezoelectric actuators and sensors were adhered on the surfaces of the composite panels in a specific rectangular arrangement to allow for wave capturing at various directions. The wave characteristics of the samples were extracted in pristine stage and then the samples were impacted and immersed in distilled water until saturation. At regular intervals after the water immersion, the panels’ weight was measured using a laboratory scale and the wave propagation characteristics of the materials ware measured in order to investigate the gradual effect of moisture on the capabilities of the actuators and sensors. The propagation of stress waves in the dry and wet samples was investigated and the effect of moisture absorption was examined experimentally using piezoelectric wafers until the saturation was reached. Finally, a continuum damage mechanics layerwise time domain spectral finite element model, both for the composite laminate and the rich resin layers, was also developed to simulate the materials’ performance under impact and evaluate the wave propagation within the material at the dry state.

A time-domain spectral finite element method for acoustoelasticity: Modeling the effect of mechanical loading on guided wave propagation

Ultrasonic testing techniques such as guided wave-based structural health monitoring aim to evaluate the integrity of a material with sensors and actuators that operate in situ, i.e. while the material is in use. Since ultrasonic wave propagation is sensitive to environmental conditions such as pre-deformation of the structure, the design and performance evaluation of monitoring systems in this context is a complicated task that requires quantitative data and the associated modeling effort. In our work, we propose a set of numerical tools to solve the problem of mechanical wave propagation in materials subjected to pre-deformation. This type of configuration is usually treated in the domain of acoustoelasticity. A relevant modeling approach is to consider two different problems: a quasi-static nonlinear problem for the large displacement field of the structure and a linearized time-domain wave propagation problem. After carefully reviewing the modeling ingredients to represent the configurations of interest, we propose an original combination of numerical tools that leads to a computationally efficient algorithm. More specifically, we use 3D shell elements for the quasi-static nonlinear problem and the time-domain spectral finite element method to numerically solve the wave propagation problem. Our approach can represent any type of material constitutive law, geometry or mechanical solicitation. We present realistic numerical results on 3D cases related to the monitoring of both isotropic and anisotropic materials, illustrating the genericity and efficiency of our method. We also validate our approach by comparing it to experimental data from the literature.

Arbitrary-Order Sensitivity Analysis in Wave Propagation Problems Using Hypercomplex Spectral Finite Element Method

Many modern structural health monitoring (SHM) systems use piezoelectric transducers to induce and measure guided waves propagating in structures for structural damage detection. To increase the detection capabilities of SHM systems, gradient-based optimization of sensor placement is frequently necessary. However, available numerical differentiation methods for mechanical wave propagation problems suffer from truncation and subtraction errors and are difficult to extend to high-order sensitivities. This paper addresses these issues by introducing an approach to obtain highly accurate numerical sensitivities of arbitrary order in mechanical wave propagation problems. The hypercomplex time-domain spectral finite element method (ZSFEM) couples the hypercomplex Taylor series expansion method with the time-domain spectral finite element method. We show how ZSFEM can be implemented within the commercial finite element package ABAQUS/Explicit. For verification, we compared the numerical and analytical results of the displacement and its sensitivities with respect to mechanical parameters, geometry, and boundary conditions for a rod subjected to a sudden, distributed axial load. First- and second-order sensitivities were obtained with normalized root mean square deviations below 4×10−3. Mesh convergence analyses revealed that p-refinement offered better convergence rates than h-refinement for the outputs and their sensitivities. Also, the sensitivities obtained with ZSFEM were compared with finite differences showing higher accuracy and step-size independence (e.g., no iteration is needed to determine the step size that minimizes the error). For simplicity, ZSFEM was presented only for one-dimensional truss elements, but the method is general and can be applied to other elements.

Guided wave-based characterisation of cracks in pipes utilising approximate Bayesian computation

This paper proposed a probabilistic framework of damage characterisation to detect and identify early-state cracks in pipe-like structures using ultrasonic guided waves. The crack location, crack sizes (e.g., depth and width of the crack), and Young's modulus are considered as unknown parameters in the model updating using a Bayesian approach, by which their values and the associated uncertainties are quantified. The proposed framework is developed based on approximate Bayesian computation (ABC) by subset simulation, which is a likelihood-free Bayesian approach. This algorithm estimates the posterior distributions of unknown parameters by directly accessing the similarity between the measured signals from experiments and the simulated guided wave (GW) signals from the numerical model. In this case, the evaluation of likelihood functions can be smartly circumvented during Bayesian inference. A time-domain spectral finite element (SFE) method with a cracked finite element model is employed to model the pipes to enhance the computational efficiency of the simulation and model updating. Numerical and experimental case studies are carried out to evaluate the performance of the proposed likelihood-free approach. Numerical results show the accuracy and robustness of the proposed approach in identifying unknown parameters under different scenarios. The associated uncertainties of each parameter are also quantified by analysing the statistical properties of the sample set, such as mean and coefficient of variation (COV) values. Experimental results show that the proposed method can accurately identify the unknown parameters, which further verifies the accuracy and practicability of the probabilistic damage characterisation framework.

Ultrasonic guided waves through Coalesence of matrix cracks and delaminations in laminated composites

Composite fabric-matrix interface mismatch during manufacturing and applied loads cause matrix micro-cracks and their evolution toward interface delamination and stiffness reduction. Fatigue and impact loads can grow these matrix cracks further as early damage precursors, which are not easily detectable. We simulate ultrasonic guided wave propagation in laminated composite with matrix crack mixed delamination site to understand better detection possibilities for these damage precursors using the ultrasonic inspection method. Since extremely small wavelength ultrasonic waves attenuate rapidly in composites, whereas long wavelength guided waves may not produce high sensitivity to small-scale defects, therefore we attempt to analyze super-resolution quantitative scheme with suitable frequencies to correlate defect parameters. We employ a time-domain spectral finite element method for fast computation of high-frequency wavefield while accurately resolving the matrix crack-delamination geometric features. Its high-order field interpolation enables us to model the nearfield interactions of the waves with the transverse micro-cracks and further with an evolved macro-scale delamination. The computational scheme is aimed to scale to complex defect local field simulations and cover larger size composite structures. This is achieved due to a mathematically consistent diagonalized mass matrix property which makes the implicit time marching-based computation highly efficient for highly resolved and large-size finite element systems. The transient dynamic stress field around the cracks modulates the guided wave mode packets. We analyze the characteristics of the signal sensitivities due to the defect features, most notably, an interesting correlation between matrix crack density and frequency modulation or sub-band contribution. Wave interaction with the defect region creates weak but distinct reflection patterns while transmitting through the region. The study provides useful and new understanding towards defect size estimation and indicative measure of thickness wise layers of matrix cracks and its coalescence into delamination in terms of energy spectral density of the sub-bands.

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.

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.

Impact localization and force reconstruction for composite plates based on virtual time reversal processing with time-domain spectral finite element method

Composite structures are widely used in different industries as important load-carrying components due to their excellent performance. These structures are prone to be damaged under low-velocity impacts with invisible cracks. Hence, impact identification is a critical part of composite structural health monitoring. In this paper, an approach is developed, which is based on time-reversal (T-R) processing and time-domain spectral finite element method. The spatial and temporal focusing properties of T-R are utilized to locate the impact source and restore the impact force, respectively. Moreover, the impact energy is evaluated by establishing the function of impact force and kinetic energy at the impact moment. The proposed method is validated experimentally for different scenarios. The influences of the signal length and accuracy of the numerical model on the proposed method are discussed in detail. The results demonstrate that the proposed method can correctly locate the impact source in composite structures, and the identified impact force can reproduce the impact event efficiently. The impact localization is essentially independent of the effect of signal length while it is vulnerable to the mesh grid. Besides, the developed method has consistent results in the case of data perturbation of material properties in the numerical model, which shows its good robustness.

Locating of acoustic emission source for stiffened plates based on stepwise time-reversal processing with time-domain spectral finite element simulation

Stiffened thin-walled structures are widely utilized in aerospace engineering as critical load-bearing components. These structures are prone to be damaged from external impact or corrosion, and fatigue cracks. Acoustic emission (AE) is a key phenomenon accompanying damage and can be used as an efficient indicator to locate the damage in stiffened structures. In this paper, a novel AE source locating approach is proposed, which is based on combining the concepts of the time-reversal (T-R) guided wave and time-domain spectral finite element method (T-D SFEM), in which an improved T-R strategy named stepwise T-R is developed to overcome the defocus issue, and the T-D SFEM is utilized to simulate the re-emitted wavefield. Benefit from the improvement and virtual simulation, the focused wavefield can be rapidly obtained without high-cost and large bulk wavefield imagine devices. The approach is validated experimentally. In addition, the effects of the signal length, mesh size, and noise levels on locating are studied in different scenarios. The results show that the proposed approach can locate the AE source in the stiffened plate efficiently and accurately. The optimal signal length and suggested mesh size are also decided. Besides, the robustness of the proposed method is demonstrated and the results are effective in the case of the measured signals with 30% white noise.

Material property identification in composite structures using time domain spectral elements

Lightweight, highly optimized, aerospace structures like spacecraft, aircraft, and launch vehicles, are made up of composite structures. These composite structures as well as the additive manufacturing techniques used in recent times require material property identification. Also, one of the main concerns in composite structures is the degradation of the material properties due to aging of structure and moisture absorption. Identification of material properties of the degraded structure poses an important problem to determine the margins available for the safe performance of the structure. Here we propose a new method to identify the material property (like elastic properties and density estimation) nondestructively, in any general elastic structure with particular emphasis on composite structures. The proposed method requires few measured responses and a good numerical model to determine the material properties. The new procedure requires a simple input force on the structure at some known locations. This generates and propagates high-frequency stress waves in the structure. Very few measured responses are used, to compute the material properties. Time Domain Spectral Finite Element (TSFEM) is used as the numerical model since it has high convergence rates for high-frequency content transient loads. A new anisotropic Timoshenko composite time-domain spectral beam element is formulated for this purpose, which will be used to perform the material property identification for both metallic as well as composite structures. Measured responses are used in this numerical model which is coupled with, the nonlinear least square technique to get the material properties. The proposed approach is demonstrated using both numerically and experimentally obtained responses. These examples show the efficiency of the developed algorithm based on TSFEM in estimating the material property of a structure.

C1-continuous time-domain spectral finite element for modeling guided wave propagation in laminated composite strips based on third-order theory

The model-based structural health monitoring techniques for composite structures based on guided waves demand fast and accurate computation of the wave propagation response. This article presents a computationally efficient and accurate time-domain spectral finite element to simulate ultrasonic guided wave propagation in laminated composite beam and infinite panel-type structures based on Levinson–Bickford–Reddy’s refined third-order theory. The element has four degrees of freedom (DOFs) per node, irrespective of the number of layers in the laminate. The C1-continuity of the deflection as required by the theory is achieved by employing the generalized Hermite-type spectral Lobatto basis functions, developed recently by the authors, while the C0-continuous Lobatto basis functions interpolate the axial displacement and shear rotation. The governing equations are derived using Hamilton’s principle. A detailed numerical study is performed to assess the new element’s accuracy, efficiency, and convergence for free vibration, Lamb wave propagation of fundamental symmetric and anti-symmetric modes, and impact wave propagation of composite beams and infinite panels. The study reveals that the new spectral element predicts more accurate, faster converged, and efficient solutions than its conventional counterpart and other available C0-continuous spectral elements with a layer-independent number of DOFs for composite structures.

Time-domain spectral finite element based on third-order theory for efficient modelling of guided wave propagation in beams and panels

Time-domain spectral finite element based on third-order theory for efficient modelling of guided wave propagation in beams and panels

Spectral finite element method for efficient simulation of nonlinear interactions between Lamb waves and breathing cracks within the bi-potential framework

Nonlinear ultrasonic guided waves are of great importance in structural health monitoring, since they have both the large-area inspection capability of guided waves and the desirable sensitivity to incipient defects from nonlinear ultrasonic features. Efficient numerical methods play a key role in investigating the nonlinear features of guided waves. In this work, a time domain spectral finite element method (TDSFEM) combined with the bi-potential contact theory is developed to capture the nonlinear interactions between guided waves and breathing cracks. The bi-potential approach has better accuracy than the conventional penalty method, since there is no any user-defined parameter for computation in the former method while the latter one requires a user-experience-based penalty factor. In addition, a semi-explicit algorithm is proposed to integrate the wave equation, which can take full advantage of the diagonal mass matrix of TDSFEM while not sacrificing numerical stability. Numerical verifications against ANSYS show that the computational efficiency of the bi-potential method-based TDSFEM is much higher than that of the FEM via the penalty method. Numerical case studies are then performed to investigate the nonlinear interactions between Lamb waves and breathing cracks. Results show that the proposed method can successfully capture the classical nonlinear features, such as higher harmonic generation and zero frequency (DC) response. Moreover, the influence of the length and gap of a breathing crack on the contact acoustic nonlinearity (CAN) is investigated. As an efficient numerical approach, the bi-potential method-based TDSFEM may be a good tool for investigating the CAN.

Guided wave propagation analysis in stiffened panel using time-domain spectral finite element method

Stiffened panels have been widely utilized in fuselages and wings as critical load-bearing components. These structures are prone to be damaged under long-term and extreme loads, and their health monitoring has been a common concern. The guided wave-based monitoring method is regarded as an efficient approach to detect the damage in stiffened plates because of its wide monitoring range and high sensitivity to micro-damage. Efficient simulation of wave propagation can theoretically demonstrate the detection mechanism of the method. In this study, a Time-Domain Spectral Finite Element Method (TD-SFEM) is adopted to study the wavefield in stiffened plates, where continuous Absorbing Layers with Increasing Damping (ALID) strategy is proposed to circumvent the disturbance of reflected waves on boundaries. After the convergence analysis, the developed TD-SFEM with ALID is validated by the finite element method first. Then, wave scattering and the influence of the stiffener are investigated in detail by comparing the results with the non-stiffened structure. Finally, the effects of the parameters of the stiffener, such as the height and width, on wave propagation are studied, respectively. The results illustrate that the proposed TD-SFEM with ALID is an efficient approach to study the wave propagation in the stiffened plate and can reveal the mechanism of influence of the stiffener. It is found that the height of the stiffener changes the interference of wavefield in the plate, while the effects of the width are mainly in wave scattering and mode conversion.

The Influence of the Grid Density of Measurement Points on Damage Detection in an Isotropic Plate by the Use of Elastic Waves and Laser Scanning Doppler Vibrometry.

Damage detection in structural components, especially in mechanical engineering, is an important element of engineering practice. There are many methods of damage detection, in which changes in various parameters caused by the presence of damage are analysed. Recently, methods based on the analysis of changes in dynamic parameters of structures, that is, frequencies or mode shapes of natural vibrations, as well as changes in propagating elastic waves, have been developed at the highest rate. Diagnostic methods based on the elastic wave propagation phenomenon are becoming more and more popular, therefore it is worth focusing on the improvement of the efficiency of these methods. Hence, a question arises about whether it is possible to shorten the required measurement time without affecting the sensitivity of the diagnostic method used. This paper discusses the results of research carried out by the authors in this regard both numerically and experimentally. The numerical analysis has been carried out by the use of the Time-domain Spectral Finite Element Method (TD-SFEM), whereas the experimental part has been based on the measurement performed by 1-D Laser Doppler Scanning Vibrometery (LDSV).

Investigation of critical delamination characteristics in composite plates combining cubic spline piezo-layerwise mechanics and time domain spectral finite elements

The scope of the present work is to elucidate the reverberations of propagating guided waves through delaminated regions confined in composite plate structures. As it is well known, delamination between consequent plies is the most common type of damage in composite structures. A mean of investigation of delamination cracks is through guided wave propagation. It has been observed that either type of guided waves (antisymmetric or symmetric) has different repercussions inside and outside the delaminated area. Inside this work is used a well-established numerical tool developed by the authors. By making good use of its reported numerical capabilities, the authors used the Hermitian Cubic Splined Layerwise mechanics to incorporate the delamination and the physical presence of piezoelectric actuators. The two mentioned aspects were then integrated into a time domain spectral finite element. The outcome was the development of a multinode explicit layerwise finite element, able of modelling the physical presence of active piezoelectric sensors, the existence of delamination cracks through the thickness and the excitation of each fundamental wave mode separately (A0 &S0). The calculated semi-diagonal mass matrix, due to the collocation of the nodes with the grid points of Gauss–Lobatto–Legendre quadrature, boosts the speed of numerical time integration, while the use of high order polynomials as shape functions, conduce to a vast reduction of the mesh size, enabling the thorough investigation and simulation of high frequency wave propagation.

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.

A C1‐continuous time domain spectral finite element for wave propagation analysis of Euler–Bernoulli beams

AbstractA C1‐continuous time‐domain spectral finite element (SFE) is developed for efficient and accurate analysis of flexural‐guided wave propagation in Euler–Bernoulli beam‐type structures. A new C1‐continuous spectral interpolation using the Lobatto basis is proposed, which is shown to eliminate the Runge phenomenon observed in the conventional higher order Hermite interpolation. It is also able to diagonalize the mass matrix, an attractive feature of existing C0‐continuous SFEs, which enhances computational efficiency. The developed element is validated by comparing the results for natural frequencies of first 20 modes with analytical solutions, and its performance for wave propagation problems is assessed in comparison with converged ABAQUS solutions obtained with a very fine mesh using the classical beam element. It is shown that the present element yields excellent accuracy with much faster convergence, higher computational efficiency, and many‐fold reduction in computational time than the conventional FE for narrowband high‐frequency flexural guided wave propagation problems in both undamaged and damaged beams. It also shows excellent performance for wave propagation under broadband impact excitations and initial displacements. The C1‐continuous interpolation proposed here will pave the way for developing several new SFEs for elastic‐ and piezoelectric‐laminated beams using advanced higher order laminated theories, which require C1‐continuity of displacements.

Study on the Propagation of Stress Waves in Natural Fiber Composite Strips

The propagation of Lamb waves within the structure of natural fiber reinforced composite strips is investigated using a semi-analytical solution and a time domain spectral finite element numerical method. The need to monitor the structural health of natural fiber reinforced composites is becoming greater, as these sustainable composites are being increasingly used in various industrial applications in automotive and marine structures. Three different types of flax fiber composites were studied and the fundamental wave modes were excited on the structure. Both methods under consideration were able to capture the symmetric and antisymmetric wave modes for all the material configurations. Especially the complex nature of a hybrid flax/glass fiber composite was studied and results were very promising for future damage investigation. Further to this, an attempt was made to excite the hybrid strip at higher frequency and the study revealed the potential to capture all the existing wave modes.

A non-linear cubic spline layerwise time domain spectral finite element for the analysis of impacts on sandwich structures

The present work addresses the non-linear impact dynamic response of sandwich composite plates with Hertzian contact interaction between the impactor and the plate. Initially, a well-established Cubic Spline Layerwise Theory is employed for the precise description of complex through the thickness kinematics of sandwich composite plates. Furthermore, the expressions of Green Lagrange non-linear strain terms, along with the Deshpande-Fleck flow rule for polymeric crushable foams are presented for non-linear explicit impact dynamics. The material and geometric non-linearities as well as the layerwise mechanics, are integrated into a Time Domain Spectral Element which possesses the attributes of high-order Lagrangian polynomial shape functions and node collocation with the integration points due to the Gauss-Legendre-Lobbato integration scheme. The aforementioned numerical package provides a fast and accurate non-linear explicit integration dynamic module for the prediction of the impact response of sandwich plates, demonstrating the universality and validity of using Hertzian contact in sandwich and laminated composite structures. The proposed novel non-linear numerical model is correlated with experimental results and also with a high-fidelity three-dimensional solid finite element model in terms of accuracy and computationally efficiency.