Spectral Finite Element Method Research Articles

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.]

Digital-Image-Driven Stochastic Homogenization for Recycled Aggregate Concrete Based on Material Microstructure

In this paper, a digital-image-driven stochastic homogenization method is developed to analyze elastic heterogeneous media such as recycled aggregate concrete (RAC), etc. This linking can be accomplished in an efficient manner by exploiting the excellent synergy of finite pixel-element method and Monte Carlo simulation for the computation of the effective properties of the random five-phase composite. The pixel-point discretization of system geometry is used for the approximation of the mechanical response of the elastic heterogeneous microstructure. Using nanoindentation technique and scanning electron microscopy, tens of digital images of modulus map of the five-phase heterogeneous material are made. Using the moving window technique and the Monte Carlo method, the random elastic moduli of the five phases at micro-scale are obtained. Then the effective elastic modulus of the meso-scale representative volume element (RVE) is computed based on spectral stochastic finite element method. Finally, the effective modulus is used to analyze the global behavior of RVE at macro-scale. Then the finite pixel-element method is used to investigate the effect of microscopic covariance noise on the global material properties, as well as the computational efficiency. The results show that the digital image method is an accurate and efficient tool to investigate the random material properties across scales.

Vibration and noise attenuation performance of compounded periodic struts for helicopter gearbox system

To reduce the helicopter cabin noise generated from a gearbox, a novel compounded periodic strut is designed with periodic structures connected in series/parallel in the axial direction. In this paper, the emphasis is placed on exploring the vibration transfer performance of the strut in both the axial and lateral directions, along with its attenuation effects on fuselage vibration and cabin noise. First, considering all directions, a complete dynamic model of the designed strut is established based on the spectral finite element method and the transfer matrix method and is extended to the model of the gearbox/struts system. Based on this model, simulation predictions indicate that the designed strut exhibits better broadband vibration attenuation in the lateral directions compared to that in the axial direction. Under this combined action, the translational vibration transmitted to the fuselage floor can be effectively isolated. The maximum acceleration attenuation can be more than 40 dB. These conclusions are verified by further experimental studies, which confirm that the numerical model can accurately predict the strut's vibration control results in the helicopter model. Subsequently, the noise transfer performance from the gearbox to the cabin noise field is experimentally studied using the helicopter model. Similar to the vibration control results, satisfactory broadband noise attenuation is obtained when using the designed compounded periodic strut. Among the frequencies of the stop bands, the maximum noise attenuation is more than 30 dB.

A three-dimensional periodic beam for vibroacoustic isolation purposes

Abstract This paper presents results of investigations on a three-dimensional (3-D) isotropic periodic beam. The beam can represent a vibroacoustic isolator of optimised dynamic characteristics in the case of its longitudinal, flexural and torsional behaviour. The optimisation process concerned both the widths as well as the positions of particular frequency band gaps that are present in the frequency spectrum of the beam. Since the dynamic behaviour of the beam is directly related to its geometry, through an optimisation process of the beam geometry, desired dynamic characteristics of the beam were successfully obtained. For the purpose of the optimisation process a new numerical model of the beam, based on the spectral finite element method in the time domain (TD-SFEM), was developed by the authors. This model enabled the authors to investigate the beam behaviour not only in a wide frequency spectrum, but also ensured a high accuracy of the model predictions. The accuracy of this modelling approach was checked against well-known analytical formulas. However, in the case of the optimised geometry of the beam for the verification of the correctness of the modelling approach a commercial finite element method (FEM) package was used. Finally, based on the results of numerical predictions and optimised geometry of the beam a sample for experimental verification was prepared. Experimental measurements were carried out by the authors by the application of one-dimensional (1-D) laser Doppler scanning vibrometry (LDSV). The results of experimental measurements obtained by the authors confirmed the correctness of the numerical predictions, showing a high degree of correspondence.

A macro‐element strategy based upon spectral finite elements and mortar elements for transient wave propagation modeling. Application to ultrasonic testing of laminate composite materials

SummaryProposing efficient numerical modeling tools for high‐frequency wave propagation in realistic configurations, such as the one appearing in ultrasonic testing experiments, is a major challenge, especially in the perspective of inversion loops or parametric studies. We propose a numerical methodology addressing this challenge and based upon the combination of the spectral finite element method and the mortar element method. From a prior decomposition of the scene of interest into “macro‐elements,” we show how one can improve the performances of the standard finite element procedures in terms of memory footprint and computational load. Additionally, using this decomposition, we are able to efficiently reconstruct important modeling features on‐the‐fly, such as orientations of anisotropic materials or splitting directions of perfectly matched layers formulations, altogether in a robust and efficient manner. We believe that this strategy is particularly suitable for parametric studies and sensitivity analysis. We illustrate our strategy by simulating the propagation of an ultrasonic wave into an immersed and curved anisotropic laminate 3D specimen flawed with an internal circular delamination of varying size, thus showing the efficiency and the robustness of our approach.

Analysis and Modelling of Non-Fourier Heat Behavior Using the Wavelet Finite Element Method.

Non-Fourier heat behavior is an important issue for film material. The phenomenon is usually observed in some laser induced thermal responses. In this paper, the non-Fourier heat conduction problems with temperature and thermal flux relaxations are investigated based on the wavelet finite element method and solved by the central difference scheme for one- and two-dimensional media. The Cattaneo–Vernotte model and the Dual-Phase-Lagging model are used for finite element formulation, and a new wavelet finite element solving formulation is proposed to address the memory requirement problem. Compared with the current methodologies for the Cattaneo–Vernotte model and the Dual-Phase-Lagging model, the present model is a direct one which describe the thermal behavior by one equation about temperature. Compared with the wavelet method proposed by Xiang et al., the developed method can be used for arbitrary shapes. In order to address the efficient computation problems for the Dual-Phase-Lagging model, a novel iteration updating methodology is also proposed. The proposed iteration algorithms on time avoids the use the global stiffness matrix, which allows the efficient calculation for title issue. Numerical calculations have been conducted in the manner of comparisons with the classical finite element method and spectral finite element method. The comparisons from accuracy, efficiency, flexibility, and applicability validate the developed method to be an effective and alternative tool for material thermal analysis.

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.

A stochastic spectral finite element method for solution of faulting-induced wave propagation in materially random continua without explicitly modeled discontinuities

This paper proposes a new efficient stochastically adapted spectral finite element method to simulate fault dislocation and its wave propagation consequences. For this purpose, a dynamic form of the split node technique is formulated and developed to stochastic spectral finite element method in order to model fault dislocation happening within a random media without increasing computational demand caused by discontinuities. As discontinuities are not modeled explicitly herein, no additional degrees of freedom are implemented in the proposed method due to the discontinuities, while effects of these discontinuities are preserved. Therefore, the present method simultaneously includes merits of stochastic finite element method, spectral finite element method and the spilt node technique, thereby providing a new numerical tool for analysis of wave propagation under fault dislocation in random media. Several numerical simulations are solved by the proposed method, which present stochastic analysis of fault slip-induced wave propagation in layered random media. Formulations and numerical results demonstrate capability, application and efficiency of this novel method.

Use of the spectral-finite element method for infrasound propagation in a 3D heterogeneous environment

The use of the spectral element method (SEM), as implemented in the open-source software SpecFEM3D, is explored for the application of longitudinal wave propagation. Infrasound, <20 Hz, propagation using local atmospheric data and a numerical weather forecast model through a heterogeneous environment is examined. The spectral-finite element method simulates acoustic and seismic waves by particle displacement in the earth and velocity potential through the atmosphere. This method also allows for inclusion of air-to-ground coupling plus realistic topography. Modifications to the code that allow for a moving atmosphere are described. Models are designed at local distances no greater than 15 km. Furthermore, high density/velocity regions for lateral heterogeneities are implemented. Absorbing boundary conditions are applied to each of the model’s sides. Simulation results are compared to real-world data collected at separate test sites and discussed. This presentation will discuss the applicability of SpecFEM3D to realistic infrasound modeling.

Symplectic Approach on the Wave Propagation Problem for Periodic Structures with Uncertainty

An effective method is proposed in this paper for the symplectic eigenvalue problem of a periodic structure with uniform stochastic properties. Since structural parameters are uncertain, the symplectic transfer matrix of a typical substructure, which is described in the state space, also has stochastic properties. An effective spectral stochastic finite element method is adopted to ensure the symplectic orthogonality of the random symplectic matrix on the premise of certain precision. By means of the Rayleigh quotient method, the symplectic eigenvalue problem of random symplectic matrix is investigated. On the basis of this, the mean value and the standard deviation of the random eigenvalues are fully discussed. The comparison between the numerical results derived from the proposed method and the Monte-Carlo simulation indicates that the proposed method has high precision. This research provides a useful guidance for the dynamic analysis of periodic structures with stochastic properties.

Peridynamic modeling of Lamb wave propagation in bimaterial plates

In this study the nonlocal bond-based peridynamic (PD) theory was applied to simulate Lamb wave transmission in 2D bimaterial plates. Plates of unlike materials were jointed end-to-end to make a bimaterial plate. The surface of the plate was then excited tangentially by single-frequency and multi-frequency force signals. The influence of changing material properties on travelling of the symmetric and antisymmetric Lamb waves were studied in detail. The phase and group velocities of travelling wave in bimaterial plates were calculated and it was found that dissimilar material properties of two joint plates can significantly alter the amplitude and arrival time of the wave. Moreover, Fast Fourier Transform (FFT) of the time history of symmetric Lamb wave velocity was calculated and verified. Accuracy and consistency of PD wave model was confirmed entirely using Spectral Finite Element Method (SFEM). The comparison between the results of two applied methods shows a good agreement.

Telegraph equation: two types of harmonic waves, a discontinuity wave, and a spectral finite element

The telegraph equation $$\tau \partial ^2 u/\partial t^2 + \partial u/ \partial t = \tau c^2 \partial ^2 u / \partial x^2$$ arises in studies of waves in dissipative media with a damping coefficient $$1/\tau $$ , or from a Maxwell–Cattaneo type heat conduction with a relaxation time $$\tau $$ . To elucidate basic properties of this equation, two harmonic wave solutions are compared: (1) temporally attenuated and spatially periodic (TASP) and (2) spatially attenuated and temporally periodic (SATP). The phase velocities of both waves are equal to the energy velocities and less than the group velocities. The phase velocities of the two waves are different, and less than c, but both naturally lead to a speed c for the propagation of discontinuities. The two harmonic wave solutions are suitable for different initial-boundary value problems: TASP for those with space periodicity and SATP for those with time periodicity. The asymptotic behaviors of the harmonic wave solutions when the telegraph equation transitions into a nondissipative wave equation or into a parabolic diffusion equation are presented. Only the SATP waves survive when the equation turns parabolic. The spectral finite element method is formulated for 1d Maxwell–Cattaneo heat conduction based on the SATP wave solutions. The element thermal conductivity matrix is reduced to that for a conventional (nonspectral) finite element when the frequency tends to zero.

Efficient Spectral Stochastic Finite Element Methods for Helmholtz Equations with Random Inputs

The implementation of spectral stochastic Galerkin finite element approximation methods for Helmholtz equations with random inputs is considered. The corresponding linear systems have matrices represented as Kronecker products. The sparsity of such matrices is analysed and a mean-based preconditioner is constructed. Numerical examples show the efficiency of the mean-based preconditioners for stochastic Helmholtz problems, which are not too close to a resonant frequency.

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.

An iterative domain decomposition, spectral finite element method on non-conforming meshes suitable for high frequency Helmholtz problems

The purpose of this research is to describe an efficient iterative method suitable for obtaining high accuracy solutions to high frequency time-harmonic scattering problems. The method allows for both refinement of local polynomial degree and non-conforming mesh refinement, including multiple hanging nodes per edge. Rather than globally assemble the finite element system, we describe an iterative domain decomposition method which can use element-wise fast solvers for elements of large degree. Since continuity between elements is enforced through moment equations, the resulting constraint equations are hierarchical. We show that, for high frequency problems, a subset of these constraints should be directly enforced, providing the coarse space in the dual-primal domain decomposition method. The subset of constraints is chosen based on a dispersion criterion involving mesh size and wavenumber. By increasing the size of the coarse space according to this criterion, the number of iterations in the domain decomposition method depends only weakly on the wavenumber. We demonstrate this convergence behaviour on standard domain decomposition test problems and conclude the paper with application of the method to electromagnetic problems in two dimensions. These examples include beam steering by lenses and photonic crystal waveguides, as well as radar cross section computation for dielectric, perfect electric conductor, and electromagnetic cloak scatterers.

Investigation of wave propagation in piezoelectric helical waveguides with the spectral finite element method

The dispersion behaviors of wave propagation in waveguides of piezoelectric helical structures are investigated. By using the tensor analysis in the helical curve coordinate, the general strain-displacement relationship of piezoelectric helix is firstly considered. This paper's formulation is based on the spectral finite element which just requires the discretization of the cross-section with high-order spectral elements. The eigenvalue matrix of the dispersion relationship between wavenumbers and frequencies is obtained. Numerical examples on PZT5A and Ba2NaNb5O15 helical waveguides of a wide range of lay angles are presented. The effects of the piezoelectric on the dispersive properties and the variation tendency of dispersion curves on helix angles are shown. The mechanism of mode separation in piezoelectric helical waveguides is further analyzed through studying waves structures of the flexural modes.

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.

An efficient multi-time step FEM–SFEM iterative coupling procedure for elastic–acoustic interaction problems

An iterative coupling methodology between the finite element method (FEM) and the spectral finite element method (SFEM) for the modeling of coupled elastic–acoustic problems in the time domain is presented here. Since the iterative coupling procedure allows the use of a nonconforming mesh at the interface between the subdomains, the difference in the element sizes concerning the FEM and SFEM is handled in a straightforward and efficient manner, thereby retaining all the advantages of the SFEM. By means of the HHT time integration method, controllable numerical damping can be introduced in one of the subdomains, increasing the robustness of the method and improving the accuracy of the results; besides, independent time step sizes can be considered within each subdomain, resulting in a more efficient algorithm. In this work, a modification in the subcycling procedure is proposed, ensuring not only an efficient and accurate methodology but also avoiding the computation of a relaxation parameter. Numerical simulations are presented in order to illustrate the accuracy and potential of the proposed methodology.

A stochastic spectral finite element method for wave propagation analyses with medium uncertainties

• Wave propagation in a medium with stochastic properties is studied. • A stochastically enriched spectral finite element method (StSFEM) is applied. • StSFEM accelerates temporal integration schemes and decreases numerical dispersion. • It provides an efficient numerical solution for Fredholm equation of KL Expansion. • The StSFEM is developed to large-scale stochastic wave propagation phenomena. A stochastically enriched spectral finite element method (StSFEM) is developed to solve wave propagation problems in random media. This method simultaneously includes all features of spectral finite element and stochastic finite element methods, which leads to excellent accuracy and convergence by implementing Gauss–Lobatto–Legendre collocation points permitting to generate coarser meshes. In addition, the proposed StSFEM leads to diagonal mass matrices, which accelerates temporal integration schemes and provides desirable accuracy. Furthermore, it numerically solves the Fredholm integral equation arising from Karhunen–Loève Expansion with favorable accuracy and computing time. Here, the StSFEM is examined and developed to stochastic wave propagation phenomena through several numerical simulations. Results demonstrate successful performance of the StSFEM in the solved problems so that one can accomplish uncertainty quantification of time domain wave propagation within random continua by incorporating the StSFEM.

A new model order reduction method based on global kernelk-means clustering: Application in health monitoring of plate using Lamb wave propagation and impedance method

In this paper, a new method based on global kernel k-means clustering (GKKMC) has been developed as model order reduction to model electromechanical impedance and Lamb wave in a plate structure. First, a model, based on the spectral finite element method, is developed to simulate piezoelectric wafer active sensor (PWAS) induced electromechanical impedance and Lamb wave propagation. Second, the related coupled electromechanical field equations are solved in 3 dimensions, then an electric voltage signal is applied to PWAS actuator, and finally the produced voltage in PWAS sensor is calculated. In reality, high frequency impedance and Lamb wave simulation in a plate need high degrees of freedom, which leads to a very slow simulation in time and frequency domain calculations. To overcome this problem, GKKMC as a model order reduction approach is proposed and applied. For this purpose, in Lamb wave modeling, a time snapshot method based on the Rayleigh damping effect has been introduced. This proposed method not only has a high accuracy and less time to simulate the problem but also needs lower memory than a full order model. A comparison between the results obtained by GKKMC, the conventional balanced proper orthogonal decomposition method, and a full order method with experimental results have been carried out demonstrating a good agreement. The results show that using this new model order reduction based on GKKMC, simulation time speedups can be reached up to 6 times faster compared with applying a conventional balanced proper orthogonal decomposition method.