Spectral Finite Element Method Research Articles

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.

Addressing biases in spectral databases for increasing accuracy and computational efficiency of crystal plasticity computations

Spectral databases of crystal plasticity solutions stored as discrete Fourier transforms (DFTs) have now been used for over a decade to speed-up the crystal plasticity simulations. These approaches have demonstrated impressive computational advantages over the conventional numerical approaches that explicitly solve the governing constitutive equations. However, their implementation in finite element models (FEM) referred to as spectral crystal plasticity finite element method (SCPFEM) framework has encountered significant hurdles mainly related to instabilities in jacobian calculations leading to convergence issues in solving the governing equilibrium equations. In this work, these main hurdles encountered are tracked to certain biases introduced in the formulation of the spectral databases. These biases are related to the choices made in the selection of the independent diagonal components of the deviatoric stress tensor and the ordering of the eigenvalues of the plastic stretching tensor. Specific strategies have been designed and implemented to address these biases such as, generating separate DFT database for all three diagonal components of deviatoric stress tensor, and standardizing the assignment of eigenvalues of the plastic stretching tensor among the other improvements. These improvements to the spectral databases have been shown to improve markedly the accuracy of the database solutions for both the stress-strain responses as well as the jacobians needed in finite element computations. The benefits of the improved databases are demonstrated with selected case studies.

Spectral extended finite element method for band structure calculations in phononic crystals

In this paper, we compute the band structure of one- and two-dimensional phononic composites using the extended finite element method (X-FEM) on structured higher-order (spectral) finite element meshes. On using partition-of-unity enrichment in finite element analysis, the X-FEM permits use of structured finite element meshes that do not conform to the geometry of holes and inclusions. This eliminates the need for remeshing in phononic shape optimization and topology optimization studies. In two dimensions, we adopt a rational Bézier representation of curved (circular) geometries, and construct suitable material enrichment functions to model two-phase composites. A Bloch-formulation of the elastodynamic phononic eigenproblem is adopted. Efficient computation of weak form integrals with polynomial integrands is realized via the homogeneous numerical integration scheme—a method that uses Euler's homogeneous function theorem and Stokes's theorem to reduce integration to the boundary of the domain. Ghost penalty stabilization is used on finite elements that are cut by a hole. Band structure calculations on perforated (circular holes, elliptical holes, and holes defined as a level set) materials as well as on two-phase phononic crystals are presented that affirm the sound accuracy and optimal convergence of the method on structured, higher-order spectral finite element meshes. Several numerical examples are presented to demonstrate the advantages of p-refinement made possible by the spectral extended finite element method. In these examples, fourth-order spectral extended finite elements deliver O(10−8) accuracy in frequency calculations with more than thirty-fold fewer degrees-of-freedom when compared to quadratic finite elements.

Spectral Element Methods a Priori and a Posteriori Error Estimates for Penalized Unilateral Obstacle Problem

The purpose of this paper is the determination of the numerical solution of a classical unilateral stationary elliptic obstacle problem. The numerical technique combines Moreau-Yoshida penalty and spectral finite element approximations. The penalized method transforms the obstacle problem into a family of semilinear partial differential equations. The discretization uses a non-overlapping spectral finite element method with Legendre–Gauss–Lobatto nodal basis using a conforming mesh. The strategy is based on approximating the solution using a spectral finite element method. In addition, by coupling the penalty and the discretization parameters, we prove a priori and a posteriori error estimates where reliability and efficiency of the estimators are shown for Legendre spectral finite element method. Such estimators can be used to construct adaptive methods for obstacle problems. Moreover, numerical results are given to corroborate our error estimates.

Simplified spectral stochastic finite element formulations for uncertainty quantification of engineering structures

In general, the structure inputs including loading and physical specifications can be deterministic or stochastic and uncertainties can be modeled using a random variable, random field, or a stochastic process. A well-known method developed to analyze structures with uncertainties is the spectral stochastic finite element method (SSFEM). To improve the efficiency of SSFEM, this paper has developed three simplified formulations to separately solve problems involving stochastic loading and constant material properties, stochastic loading and random material properties and stochastic loading and stochastic material properties the main feature of which is that they need fewer finite element (FE) analyses to assess the statistical specifications of the response. To examine their accuracy and efficiency, three practical engineering problems: 1) Euler-Bernoulli beam, 2) square solid and 3) modeling the corrosion effect in a concrete structure were solved and the results were compared with those of the Monte Carlo simulation (MCS). To see if the proposed formulations can be used in structural reliability analyses, a limit state function (LSF) was defined for each problem and the results were compared with those of the corresponding MCS; their significant efficiency and enough potential was, thus, confirmed.

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.

Analysis of nonlinear frequency mixing in Timoshenko beams with a breathing crack using wavelet spectral finite element method

The nonlinear interaction of a dual-frequency Lamb wave with a breathing edge-crack leads to the generation of frequency sidebands in the response spectrum; a phenomenon referred to as the nonlinear frequency mixing. These sidebands appear at algebraic combinations of the two central interrogation frequencies and can serve as a reliable precursor to the existence of an incipient damage. In this paper, an iterative use of the wavelet spectral finite element method is presented for analyzing the phenomenon of nonlinear frequency mixing in a beam with a transverse edge-crack. The beam is modeled using the Timoshenko hypothesis while the local flexibility caused by the crack is modeled by introducing two springs corresponding to the bending and shear deformations, respectively. The intermittent contact between the two crack surfaces is simulated by switching between a defect-free beam configuration and the one containing an open-crack. The underlying steps involved in deriving the element level equations for healthy and damage spectral finite elements, together with an iterative procedure to solve the resulting set of nonlinear equations, are presented in detail. Using this numerical framework, it is exposited that relative strengths of the frequency sidebands are influenced strongly by the temporal overlap that the two constituent wave envelopes have when they propagate through the breathing crack. A modulation parameter is defined for quantifying this dependency. For a simultaneous passage (100% temporal overlap), the modulation parameter attains its maxima, while it reduces to zero when the two constituent waves propagate separately through the breathing crack with zero temporal overlap. Premised on this rationale, an operationally viable damage localization strategy, based on tuning the temporal overlap between the two constituent wave envelopes and further monitoring the modulation parameter, is proposed. The efficacy of the proposed strategy is demonstrated by considering an illustrative numerical example. The present investigation can be of potential use in the analyses concerning nonlinear wave-damage interactions and their effective use in localizing a damage.

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.

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.

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.

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.

Stochastic Modeling and Reliability Analysis of Wing Flutter

In this work, a physics-based first-order reliability method (FORM) algorithm is proposed for the flutter reliability analysis of an aircraft wing in the frequency domain. The limit state function, which is an implicit function of random variables, is defined in terms of the damping ratio of the aeroelastic system in a conditional sense on flow velocity. Two aeroelastic cases, namely, an airfoil section model and a cantilever wing model, are considered for carrying out the studies. These aeroelastic models have well separated mean bending and mean torsional modal frequencies. The geometric, structural, and aerodynamic parameters of airfoil and wing systems are modeled as independent Gaussian random variables. The effects of these on the statistics of frequency and damping ratio, and the cumulative distribution functions (CDFs) of flutter velocity are studied. In the case of the wing, the effects of modeling stiffness parameters as Gaussian random fields on the CDFs of flutter velocity are also studied. Here, spectral stochastic finite element method (SFEM) based on Karhunen–Loeve (K–L) expansion is used to discretize the random fields into random variables. From the study of an airfoil system, it is observed that parameters like torsional stiffness, elastic axis location, free stream density, and mass moment of inertia are more sensitive as compared with other parameters. However, in the case of the wing parameters such as torsional stiffness, free stream density, mass moment of inertia, and mass are observed to be more sensitive. The CDFs of flutter velocity obtained using the proposed algorithm are compared with Monte Carlo simulations (MCS) and found to be accurate. A comparative study of aeroelastic reliability for the wing is also carried out by treating stiffness parameters as random variables and random fields. It is observed that the CDFs of flutter velocity in the tail region are conservative when stiffness parameters are treated as random variables.

Computational heat transfer with spectral graph theory: Quantitative verification

The objective of this paper is to quantify the precision of a novel approach for computational heat transfer modeling based on spectral graph theory. Two benchmark heat transfer problems with planar boundaries, for which exact analytical solutions are available, are used to determine the precision of temperature predictions obtained from spectral graph, finite difference (one-dimensional), and finite element (three-dimensional) methods. These studies show that the spectral graph approach captures the temperature trends in the benchmark case studies with error in the range of 2% to 10% depending on the location in the body. These verification studies also provide an approach to calibrate the numerical parameters in the new method. The spectral graph approach is applied for predicting the thermal history of a complex three-dimensional additive manufactured (3D printed) part. The temperature trends in a 50-layer part are computed 2.3 times faster than a commercial finite-element software package, and the results differ by less than 7.5%. Further improvements in the computational speed of the spectral graph approach are expected through code optimization and parallelization. This work has far-reaching practical implications for predicting thermal-induced defects in a variety of manufacturing processes including casting and additive manufacturing.

Finite Element Approaches to Model Electromechanical, Periodic Beams

Periodic structures have some interesting properties, of which the most evident is the presence of band gaps in their frequency spectra. Nowadays, modern technology allows to design dedicated structures of specific features. From the literature arises that it is possible to construct active periodic structures of desired dynamic properties. It can be considered that this may extend the scope of application of such structures. Therefore, numerical research on a beam element built of periodically arranged elementary cells, with active piezoelectric elements, has been performed. The control of parameters of this structure enables one for active damping of vibrations in a specific band in the beam spectrum. For this analysis the authors propose numerical models based on the finite element method (FEM) and the spectral finite element methods defined in the frequency domain (FDSFEM) and the time domain (TDSFEM).

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

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

The impact of topography on seismic amplification during the 2005 Kashmir earthquake

Abstract. Ground surface topography influences the spatial distribution of earthquake-induced ground shaking. This study shows the influence of topography on seismic amplification during the 2005 Kashmir earthquake. Earth surface topography scatters and reflects seismic waves, which causes spatial variation in seismic response. We performed a 3-D simulation of the 2005 Kashmir earthquake in Muzaffarabad with the spectral finite-element method. The moment tensor solution of the 2005 Kashmir earthquake was used as the seismic source. Our results showed amplification of seismic response on ridges and de-amplification in valleys. It was found that slopes facing away from the source received an amplified seismic response, and that 98 % of the highly damaged areas were located in the topographically amplified seismic response zone.

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.

Modal pencil method for the radiation of guided wave fields in finite isotropic plates validated by a transient spectral finite element method

Elastic guided waves (GW) can be profitably used in non-destructive evaluation and in structural health monitoring of plate-like structures. Nevertheless, the multi-modal and dispersive behaviour of GW often leads to difficult interpretation of typically measured time-dependent signals. The development of efficient simulation tools appears necessary to better understand complex phenomena involved and to optimize testing configurations. Here, a semi-analytical modal method is proposed to compute GW displacement fields in finite plates radiated by an arbitrary finite-sized source of surface stresses. It takes into account GW reflections and mode conversions at plate boundaries. As far as computation efficiency is concerned, this method is independent of the length of propagation paths, allowing to efficiently address configurations involving long range propagation. Predicted results are given as sums of modal contributions to ease their interpretation. The model is validated by comparing its predictions to those computed by a transient finite-element code.

A natural frequency sensitivity-based stabilization in spectral stochasticfinite element method for frequency response analysis

In applying the spectral stochastic finite element methods to the frequency response analysis, the conventional methods are known to give unstable and inaccurate results near the natural frequencies. To address this issue, a new sensitivity based stabilized formulation for stochastic frequency response analysis is proposed in this paper. The main difference over the conventional spectral methods is that the polynomials of random variables are applied to both numerator and denominator in approximating the harmonic response solution. In order to reflect the resonance behavior of the structure, the denominator polynomials is constructed by utilizing the natural frequency sensitivity and the random mode superposition. The numerator is approximated by applying a polynomial chaos expansion, and its coefficients are obtained through the Galerkin or the spectral projection method. Through various numerical studies, it is seen that the proposed method improves accuracy, especially in the vicinities of structural natural frequencies compared to conventional spectral methods.

Analysis of High-order Approximations by Spectral Interpolation Applied to One- and Two-dimensional Finite Element Method

The implementation of high-order (spectral) approximations associated with FEM is an approach to overcome the difficulties encountered in the numerical analysis of complex problems. This paper proposes the use of the spectral finite element method, originally developed for computational fluid dynamics problems, to achieve improved solutions for these types of problems. Here, the interpolation nodes are positioned in the zeros of orthogonal polynomials (Legendre, Lobatto, or Chebychev) or equally spaced nodal bases. A comparative study between the bases in the recovery of solutions to 1D and 2D elastostatic problems are performed. Examples are evaluated, and a significant improvement is observed when the SFEM, particularly the Lobatto approach, is used in comparison to the equidistant base interpolation.