Spectral Finite Element Method Research Articles

Deep-learning-based inverse design of phononic crystals for anticipated wave attenuation

Bandgaps of phononic crystals dominating the propagation of evanescent waves have received significant attention recently, which can be determined and tuned by the topology of a unit cell. Predicting a band structure and designing topological structures with desirable characteristics have become a research hotspot. In this study, a data-driven deep learning framework is applied to arrive at the prediction of the band structure and the inverse design of topology. A convolutional neural network is trained to predict band structures of phononic crystals. After training a generative adversarial network, the generator is concatenated with the convolutional neural network for inverse design. Meanwhile, a complex band structure of phononic crystals is computed by the periodic spectral finite element method to present the spatial decay of evanescent waves. The topology with the greater spatial attenuation is screened from the ground truth topology and the inversely designed topology. Finally, an optimized topological phononic crystal with an anticipated bandgap is obtained, which has the potential for better acoustic insulation and vibration isolation.

Subsurface polycrystalline reconstruction based on full waveform inversion - A 2D numerical study

Microstructural mapping of polycrystalline metallic alloys is a key component in predicting macroscopic material behavior, and non-destructive subsurface polycrystalline imaging is a valuable yet challenging field that offers promise for metallic material characterization. This paper proposes a computational ultrasonic imaging method for high-resolution, subsurface polycrystalline reconstruction using signals measured on a sample’s boundary and a full waveform inversion technique. For this 2D numerical study, the anisotropic elastic coefficient parameterization is introduced and connected to the corresponding polycrystalline orientation. The forward wave propagation simulation is performed using a spectral finite element method. An inversion framework is developed for inverting the anisotropic elastic coefficients, and the crystalline orientations. Reconstruction performance benchmarking is systematically conducted for both rectangular regions and a representative polycrystal. Overall, the reconstructions have broad correspondence to the scanned, unknown models. We further discuss artifact mitigation strategies and the potential to extend this work.

Band gap characteristics of two-dimensional functionally graded periodic grid structures with local resonators

This paper aims to control the in-plane wave propagation characteristics of two-dimensional functionally graded periodic grid structures by adding local resonators made of the spring oscillator. The in-plane wave dispersion relations of the periodic grid structures with local resonators are calculated by spectral element method combined with Bloch theorem. Meanwhile, the effectiveness of the band gap calculation method is verified by the vibration transmission of the finite-length grid structures calculated based on spectral element method and finite element method. According to the spectral element method, the spectral stiffness matrix of the functionally graded beam element is established at first, then assembled with the additional stiffness matrix of the spring oscillator subsystem to obtain the complete spectral stiffness matrix of the functionally graded oscillator coupled beam. Finally, the whole stiffness matrix of grid structures with local resonators can be obtained by the coordinate transformation matrix, to form the in-plane wave dispersion relations equation based on the Bloch periodic boundary conditions. In addition, the effects of structural and material parameters on the in-plane wave propagation characteristics are analyzed, which can be applied to the vibration reduction design of periodic grid structures.

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.

Modeling of spatial permittivity variations using Karhunen–Loève expansion for stochastic electromagnetic problems

Fabrication tolerance often causes spatial variations in material properties in electromagnetic systems. In this paper, these spatial variations are modeled using Karhunen–Loéve expansion, which is truncated for a finite dimensional representation. Uncertainty quantification is performed by incorporating this model into an intrusive polynomial chaos expansion based spectral stochastic finite element method. Using a numerical example, it has been shown that the stochastic response obtained by the proposed approach is accurate and computationally efficient. The impact of correlation length of the covariance function used in the spectral expansion is analyzed.

Applying the spectral stochastic finite element method in multiple-random field RC structures

Applying the spectral stochastic finite element method in multiple-random field RC structures

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).

Study on Tunable Band Gap of Flexural Vibration in a Phononic Crystals Beam with PBT

Low-frequency flexural vibration plays a significant role in beam vibration control. To efficiently attenuate the propagation of flexural vibration at a low-frequency range, this paper proposes a new type of a phononic crystals beam with an adjustable band gap. The governing equations of flexural vibration in a periodic beam are established based on the Euler theory and Timoshenko theory. The band structures are calculated by the plane wave expansion method, the attenuation properties and transmission response curves with a finite periodic beam are calculated by the spectral element method and finite element method. The effects of the elastic foundation and axial stress on band gaps are discussed in detail, and the regulation of the temperature field on the band gap is emphatically studied. The theoretical and numerical results show that the elastic foundation and axial stress have significant influence on the band gap, and the location and width of the band gaps can be adjusted effectively when the Young’s modulus of PBT is changed by a varying temperature. The results are very useful for understanding and optimizing the design for composite vibration isolation beams.

A registration-free approach for statistical process control of 3D scanned objects via FEM

Recent work in on-line Statistical Process Control (SPC) of manufactured 3-dimensional (3-D) objects has been proposed based on the estimation of the spectrum of the Laplace-Beltrami (LB) operator, a differential operator that encodes the geometrical features of a manifold and is widely used in Machine Learning (i.e., Manifold Learning). The resulting spectra are an intrinsic geometrical feature of each part, and thus can be compared between parts to assess the state of the manufacturing process while avoiding part to part registration (or “part localization”) pre-processing or the need for equal size meshes, characteristics which are required in previous approaches for SPC of 3D parts. The recent spectral SPC methods, however, are limited to monitoring surface data from objects such that the scanned meshes have no boundaries, holes or missing portions. In this paper we extend spectral methods by first considering a more accurate and general estimator of the LB spectrum that is obtained by application of Finite Element Methods (FEM) to the solution of Helmholtz's equation with boundaries. It is shown how the new spectral FEM approach, while it retains the advantages of not requiring part localization/registration or equal size datasets scanned from each part, it provides more accurate spectrum estimates, which results in faster detection of out of control conditions than earlier methods, can be applied to both mesh or volumetric (solid) scans, and furthermore, it is shown how it can be applied to partial scans that result in open meshes (surface or volumetric) with boundaries, increasing the practical applicability of the methods. The present work brings SPC methods closer to contemporary research in Computer Graphics and Manifold Learning. MATLAB code that reproduces the examples of this paper is provided in the supplementary materials.

Advances in nonlinear ultrasonic detection of microcracks

<p indent="0mm">A large number of key components are widely operated under extreme conditions of high temperature and high pressure in power, nuclear energy, aerospace and other important industrial fields. Microdefects are inevitably generated in these key components during the process of manufacturing. Due to the periodic loading and thermal stress, microcracks are generally initiated from these microdefects at the location with stress concentration. It has been found that the initiation and propagation of fatigue microcracks usually occur prior to 70%–90% of fatigue life. It is difficult to observe the evidence before the formation of macrocracks, which may result in catastrophic accidents. Thus, the risk of sudden failure of key components increases greatly. Therefore, the accurate detection of microcracks is crucial for guaranteeing the safe operation of key components. Based on the diffraction, reflection and transmission, traditional ultrasonic testing method is merely validated to effectively detect macroscopic defects within half a wavelength. The nonlinear ultrasonic testing method is found to be sensitive to microcracks, as high order harmonics, sub-harmonics, DC components and frequency mixing waves can be generated due to the strong nonlinear interaction microcracks and ultrasonic waves. With the outstanding effectiveness to detect microcracks, the nonlinear ultrasonic testing method has attracted much attention in recent years. In this work, the research on detection of microcracks using nonlinear ultrasonic testing method is systematically reviewed in terms of theoretical models, numerical simulations and experimental measurements. Firstly, according to the breathing effect and sliding mechanism of microcrack surfaces, we present several theoretical models of nonlinear interaction between ultrasonic waves and microcracks, namely contact acoustic nonlinearity (CAN), bi-linear stiffness model, nonlinear spring model, Hertz contact theory, and rough contact surface model. The advantages and disadvantages of these theoretical models are discussed comparatively. Two mechanisms are mainly considered for the nonlinear interaction of microcracks and ultrasonic waves, namely the macroscopic opening and closing of crack surfaces and the microscopic contact of rough microcracks surfaces. Secondly, we introduce numerical simulation studies on the nonlinear ultrasonic generation induced by microcracks. The applicability is analyzed for several simulation methods, such as finite element method, spectral finite element method, finite difference method, boundary element method and local interaction simulation approach, and the generation and propagation of nonlinear ultrasonic waves are analyzed as well. The effect of microcrack geometries on the nonlinear ultrasonic waves is then summarized. The microcrack geometries are considered mainly clouding the microcrack length, width, depth, density and direction. Furthermore, we present the experimental measurements on the generation of nonlinear ultrasonic waves induced by microcracks and on the localization and imaging of microcracks based on the nonlinear ultrasonic waves. The generation of high order harmonics, sub-harmonics, DC components and frequency mixing waves was observed in the specimens with microcracks. The nonlinear ultrasonic detection is presented to evaluate fatigue microcracks with various microcrack length, width, depth, density and direction. Based on this phenomenon, nonlinear ultrasonic mixing method, transducer array imaging method and nonlinear ultrasonic phased array method are introduced to localize the microcracks. Finally, further studies on evaluation of microcracks using nonlinear ultrasonic waves are proposed, including the comprehensive physical model for actual microcracks in engineering components, robust detection method, and quantitative evaluation of microcracks and service life.

A Novel Method for Intrusive Stochastic Estimation of Geometric Tolerance Effects in Finite Element Electromagnetic Analysis

The robust design of microwave, millimeter-wave, or Terahertz components and structures incorporating manufacturing process tolerances would enhance the fabrication yield and thereby reduce the overall production cost. Based on the spectral stochastic finite element method (SSFEM) for material variations, a novel geometrical SSFEM (GSSFEM) is proposed to analyze dimensional uncertainty in 3-D microwave models. The Jacobian for Piola mapping is represented as a function of stochastic variables to capture mesh level uncertainties. Polynomial chaos expansion is used to approximate the electric field as a random process. The technique is validated by applying it to waveguide problems with uncertainty in geometric dimensions. GSSFEM results are compared with analytical formulations of the transmission coefficient for a physical insight and further with the Monte Carlo simulations for quantitative evaluation. The results using the proposed approach are in excellent agreement with conventional methods and are found to be faster than sparse grid stochastic collocation, while the scaling of the computational requirements with the number of degrees of freedom and stochastic dimension is as good as this efficient scheme. Furthermore, this approach has been employed to analyze uncertainty in multiple geometric parameters to compare their sensitivity. The proposed GSSFEM can be employed for analyzing the impact of fabrication tolerance of passive waveguide components at microwave frequencies and beyond.

Complete vibration band gap characteristics of two-dimensional periodic grid structures

In this article, the spectral element method combined with Bloch theorem is applied to calculate complete vibration band gap characteristics of two-dimensional periodic grid structures, which is verified by structural vibration transmission results based on spectral element method and finite element method. According to the differential equation of motion, complete spectral stiffness matrix of beam element is established at first, then stiffness matrix of grid structure can be obtained by coordinate transformation and assembly method, so as to form governing equation of motion based on Bloch boundary conditions. The band gap characteristics of periodic structures can be calculated by solving motion equation. Through band gap distribution analysis of two-dimensional periodic grid structures, the dispersion relations of different unit arrangements and effects of material and structural parameters on band gap characteristics are obtained which can be applied to acoustic design or vibration and noise reduction areas.

Transient wave propagations with the Noh-Bathe scheme and the spectral element method

The spectral elements of the Lobatto family provide desirable characteristics of convergence and accuracy using a diagonal mass matrix in dynamic analysis. These characteristics might be expected to render the spectral element method with the use of an explicit time integration scheme effective for the transient analysis of wave propagations. In this paper we study the use of the central difference method and the recently proposed Noh-Bathe method for explicit time integration in the use of the spectral finite element method. The Noh-Bathe scheme is a second-order accurate procedure with small solution errors in the required frequency range while suppressing spurious high frequencies. We calculate appropriate CFL numbers for the time integrations and give an analysis of the dispersion errors for different orders of spectral elements. Finally, we demonstrate the capabilities of the Noh-Bathe scheme compared to the central difference method through the solution of several numerical examples of wave propagations.

Vertical to Horizontal Spectral Ratio (VHSR) Response of Seismic Wave Propagation in a Homogeneous Elastic – Poroelastic Medium Using The Spectral Finite Element Method

&lt;p&gt;Numerical modeling of 2D seismic wave propagation using spectral finite element method to estimate the response of seismic waves passing through the poroelastic medium from a hydrocarbon reservoir has been carried out. A hybrid simple model of the elastic - poroelastic - elastic with a mesoscopic scale element size of about 50cm was created. Seismic waves which was in the form of the ricker function are generated on the first elastic medium, propagated into the poroelastic medium and then transmitted to the second elastic medium. Pororoelastic medium is bearing hydrocarbon fluid in the form of gas, oil or water. Vertical and horizontal component of velocity seismograms are recorded on all mediums. Seismograms which are recorded in the poroelastic and second elastic medium show the existence of slow P compressional waves following fast P compressional waves that do not appear on the seismogram of the first elastic medium. The slow P wave is generated when the fast P wave enters the interface of the elastic - poroelastic boundary, propagated in the poroelastic medium and is transmited to the second elastic medium. The curves of Vertical to horizontal spectrum ratio (VHSR) which are observed from seismograms recorded in the poroelastic and the second elastic medium show that the peak of VHSR values at low frequency correlated with the fluid of poroelastic reservoir. The highest VHSR value at the low frequency which is recorded on the seismogram is above the 2.5 Hz frequency for reservoirs containing gas and oil in the second elastic medium, while for the medium containing water is the highest VHSR value is below the 2.5 Hz frequency.&lt;/p&gt;

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.