Wave Finite Element Method Research Articles

Active vibration control by a tunable piezoelectric shunt metamaterial

Acoustic metamaterials are composite materials exhibiting effective properties and acoustic behavior not found in traditional materials. Primarily through periodic subwavelength resonant inclusions, acoustic metamaterials can enable steering, cloaking, lensing, and frequency band control of acoustic waves. However, a common drawback of acoustic metamaterials is that effectiveness is limited to narrow frequency bands. Thus, investigation of practical active and adaptable acoustic metamaterials is valuable in achieving wider operation frequency bands. Here, a metamaterial consisting of active tunable piezoelectric shunts is investigated numerically and experimentally from the unit cell level. A physical model of the unit cell is developed using the finite element method. From the finite element model, the wave finite element method is applied to compute the dispersion and forced response of the periodic structure. It is demonstrated that the shunts introduce an additional degree of freedom by which adaptable bending wave attenuation can be accomplished. Since the periodic shunts are only effective at certain frequency bands, a known optimization method is implemented to tune the shunts. Additionally, a new optimization scheme is compared to the existing scheme found in literature.

Shielding Effectiveness Simulation of SMT EMI Gaskets Utilizing ANSYS HFSS

Abstract As more components populate smaller board areas, the risk of unwanted coupling between components increases. Shielding and radiated emissions tests outlined by IEEE standards such as IEC 61000-4-21 and ASTM4935D identify measurement techniques to quantify shielding effectiveness. In a PCB topology, shields enclose components to create a Faraday cage, but necessitate a method to ground and fasten the shield to the PCB. Typical methods include the use of gaskets, sealants and clips. This paper reviews a gasket methodology and validates a parameterized ANSYS HFSS simulation with the measured relationship between gasket configuration and shielding effectiveness. Surface mount, rectangular, compressible gaskets manufactured by W.L. Gore & Associates, Inc. are used in several configurations to shield a PCB radiator. Radiated emissions measurements are performed in an ETS-Lindgren reverberation chamber to evaluate shielding effectiveness for each configuration. ANSYS HFSS, a full wave Finite Element Method (FEM) electromagnetic simulation solver, is used to simulate each geometric configuration and is verified with measured results.

Wave-based SHM of sandwich structures using cross-sectional waves

Summary The identification of structural damage in composite waveguides is a critical issue in aerospace and transportation industries. Frequently, these structures involve periodic patterns or dissipative components that considerably reduce the range, robustness, and available bandwidth of ultrasonic structural health monitoring techniques. On the other hand, wave-based methods provide more accurate information on a defect's type, size, and location than modal analysis techniques. This paper focuses on a low-frequency wave-based method for structural integrity assessment of complex waveguides. The wave finite element method is employed to compute the dispersion curves of non-standard cross-sectional waves exhibiting increased strain energy. The spectral results are used to analyse the diffusion of guided elastic waves through representative localized defects in a laminated sandwich panel. To validate the diffusion model, reflection and transmission coefficients are determined for several wave pulses on typical defects using time-domain virtual experiments and cross-sectional energy acquisition. Results demonstrate that using cross-sectional waves provides a sensitivity to damage up to 2.8× higher than flexural waves in the low-frequency range. These results are explained by the presence of local resonances within the cross section, producing wavelengths in the transverse direction of propagation. These waves may prove suitable for cost-effective structural health monitoring applications because they can travel long distances through heterogeneous and periodic structures.

Simplified elastic wave modeling in seven-wire prestressed parallel strands

This paper considers elastic wave propagation in seven-wire parallel strands. Under a transverse distributed force, the stress state of seven-wire strands is derived from the Hertzian theory, which is regarded as the prestress state for the dynamic analysis. According to the wave finite element method, the elastodynamic equation with prestress is discretized corresponding to harmonic wave motion. Firstly, the results are verified by the semi-analysis finite element method. Then, dispersion properties in seven-wire parallel strands are computed and compared to those of double cylindrical rods. Besides, the results from parallel strands and helical strands indicate that wave propagation in the former is significantly different from that in the latter. For the deep understanding of wave characteristics, the displacement vectors are used to identify the propagating modes. It is found that there is only one torsional-like mode which rotates with a twist center in the whole region. Meanwhile, it is found that the variation of prestress has large influence on the torsional-like mode for low frequencies, but little on propagating modes for high frequencies. In addition, there appears the notch frequency and the effect of prestress on it is discussed in detail. The notch frequency decreases with the rising of the prestress in a certain range.

Multi-Scale Homogenization of Transversal Waves in Periodic Composite Beams

This paper deals with the deduction of new homogenized models for the flexural wave in bi-periodic beams. According to the homogenization theory, the long-wave assumption is used and the valid frequency range of homogenized models is limited to the first Bragg band gap. However, the classical homogenization method, whose idea is taking the component’s mean values as effective material properties, has limitations in mimicking the dispersive behavior and the real valid frequency range is far less than the limit. Thus, enriched homogenized models, derived by the multi-scale asymptotic homogenization method, are proposed to provide more accurate homogenization models with larger real valid frequency range. The new homogenized models are validated by investigating the dispersion relation in the infinite case and the frequency response function in the finite case. Wave finite element method (WFEM) are used to provide associated references. A parametric study is carried out in the infinite case while two different boundary conditions are considered in the finite case.

Using PWE/FE method to calculate the band structures of the semi-infinite beam-like PCs: Periodic in z-direction and finite in x–y plane

This paper couples the plane wave expansion (PWE) and finite element (FE) methods to calculate the band structures of the semi-infinite beam-like phononic crystals (PCs) with the infinite periodicity in z-direction and finiteness in x–y plane. Explicit matrix formulations are developed for the calculation of band structures. In order to illustrate the applicability and accuracy of the proposed coupled plane wave expansion and finite element (PWE/FE) method to beam-like PCs, several examples are displayed. At first, PWE/FE method is applied to calculate the band structures of the Pb/rubber beam-like PCs with circular and rectangular cross sections, respectively. Then, it is used to calculate the band structures of steel/epoxy and steel/aluminum beam-like PCs with the same geometric parameters. Last, the band structure of the three-component beam-like PC is also calculated by the proposed method. Moreover, all the results calculated by PWE/FE method are compared with those calculated by finite element (FE) method, and the corresponding results are in good agreement.

Stochastic wave finite element method in uncertain elastic media through the second order perturbation

In this work, the stochastic wave finite element (SWFE) method for uncertain media through the second-order perturbation is formulated. A parametric approach for uncertainties is considered and combined to the finite element technique. The stochastic state space formulation is detailed in this work. The originality of this paper is the study of the second-order perturbation. The sensitivity and the precision of the SWFE approach are treated through the second-order perturbation introduced in the structural parameters. The question of the statistics of the propagation constants and the wave modes is considered. Comparisons with analytical results and Monte Carlo simulations are performed.

Stochastic Wave Finite Element Method in Uncertain Elastic Media Through the Second Order Perturbation

Stochastic Wave Finite Element Method in Uncertain Elastic Media Through the Second Order Perturbation

Assessments of shear core effects on sound transmission loss through sandwich panels using a two-scale approach

The present work proposes a modeling strategy based on a two-scale approach to deal with the flexural vibroacoustic behavior of sandwich panels notably in the low-mid frequency range. The vibroacoustic indicators of interest here are primarily the sound transmission loss and the bending-shear transition frequency of the sandwich panels. The stated indicators depend mainly on the geometrical and mechanical parameters of the studied cases. The parametric analyses, therefore, bring out the effects of the different meso-scale parameters on the macro-scale responses of the sandwich panels by performing three-dimensional model of a representative elementary cell. The two-scale approach is mainly based on a numerical method known as the wave finite element method. In the developed approach, an expression of the sound transmission loss was derived wherein the shear core effect of sandwich panels has been included. The obtained results were compared with two models and experimental data to check the accuracy of the prediction and showed a good agreement.

Guided wave propagation in rotating functionally graded annular plates

Elastic guided wave propagation in a rotating functionally graded material (FGM) annular plate is presented in this paper. The material properties are assumed to vary continuously along the radial direction. The elastodynamic equation of annular plates which take into account initial hoop, centrifugal and Coriolis effects is derived, and the wave finite element method is extended to model wave motion related to rotation with 3D-Chebyshev spectral elements. Firstly, wave properties in a straight bar are computed and compared to that of the Rayleigh–Ritz method. Then, wave propagation in the FGM annular plate with various material gradient indexes is considered and the results indicate that the index has large influence on wave characteristics. With contour profiles of transverse sections, propagating wave modes in the plate can be identified distinctly. Besides, the effects of rotation on wave propagation are discussed, which show that the extensional-like and shearing-like wave modes are very sensitive to the rotation at low frequencies but the flexural are not. In addition, the curve veering phenomenon existing in FGM annular plates is also found, which analyzes the influences of material gradient index and rotating speed and points out the variations in the relative critical frequencies.

Wave Finite Element Formulation of the Acoustic Transmission Through Complex Infinite Plates

A finite element-based derivation of the transmission loss (TL) of anisotropic layered infinite plates is presented in this paper. The wave-finite element method (WFE) is used to represent the plate with a finite element model of a single unit cell. The incident acoustic field is a known plane wave, and the reflected and transmitted pressures are supposed to be plane waves with unknown amplitudes and phases. The periodicity conditions on the unit cell allow to find a simple matrix equation linking the amplitudes of the transmitted and reflected fields as a function of the incident one. This approach is validated for several cases against classical analytical models for thin plates and sandwich constructions, where the results agree perfectly for a reasonable mesh size. The method is then used to study the effect of stacking order in a laminated composite plate. The main interest of the method is the use of finite elements, which enables a relative easy modelling since most packages readily include different formulations, compared to analytical models, where different formulations have to be implemented for every kind of material.

Modeling ellipsometric measurement of three-dimensional structures with rigorous coupled wave analysis and finite element method simulations

Using rigorous coupled wave analysis (RCWA) and finite element method (FEM) simulations together, many interesting ellipsometric measurements can be investigated. This work specifically focuses on simulating copper grating structures that are plasmonically active. Looking at near-field images and Mueller matrix spectra, understanding of physical phenomena is possible. A general strategy for combatting convergence difficulties in RCWA simulations is proposed and applied. The example used is a copper cross-grating structure with known slow convergence. Baseline simulations on simple samples are provided for comparison and determination of FEM accuracy.

Application of the wave finite element method to reinforced concrete structures with damage

Vibration based methods are commonly deployed to detect structural damage using sensors placed remotely from potential damage sites. Whilst many such techniques are modal based there are advantages to adopting a wave approach, in which case it is essential to characterise wave propagation in the structure. The Wave Finite Element method (WFE) is an efficient approach to predicting the response of a composite waveguide using a conventional FE model of a just a short segment. The method has previously been applied to different structures such as laminated plates, thinwalled structures and fluid-filled pipes. In this paper, the WFE method is applied to a steel reinforced concrete beam. Dispersion curves and wave mode shapes are first presented from free wave solutions, and these are found to be insensitive to loss of thickness in a single reinforcing bar. A reinforced beam with localised damage is then considered by coupling an FE model of a short damaged segment into the WFE model of the undamaged beam. The fundamental bending, torsion and axial waves are unaffected by the damage but some higher order waves of the cross section are significantly reflected close to their cut-on frequencies. The potential of this approach for detecting corrosion and delamination in reinforced concrete beams will be investigated in future work.

A wave finite element-based approach for the modeling of periodic structures with local perturbations

The wave finite element (WFE) method is investigated to describe the dynamic behavior of finite-length periodic structures with local perturbations. The structures under concern are made up of identical substructures along a certain straight direction, but also contain several perturbed substructures whose material and geometric characteristics undergo arbitrary slight variations. Those substructures are described through finite element (FE) models in time-harmonic elasticity. Emphasis is on the development of a numerical tool which is fast and accurate for computing the related forced responses. To achieve this task, a model reduction technique is proposed which involves partitioning a whole periodic structure into one central structure surrounded by two unperturbed substructures, and considering perturbed parts which are composed of perturbed substructures surrounded by two unperturbed ones. In doing so, a few wave modes are only required for modeling the central periodic structure, outside the perturbed parts. For forced response computation purpose, a reduced wave-based matrix formulation is established which follows from the consideration of transfer matrices between the right and left sides of the perturbed parts. Numerical experiments are carried out on a periodic 2D structure with one or two perturbed substructures to validate the proposed approach in comparison with the FE method. Also, Monte Carlo (MC) simulations are performed with a view to assessing the sensitivity of a purely periodic structure to the occurrence of arbitrarily located perturbations. A strategy is finally proposed for improving the robustness of periodic structures. It involves artificially adding several “controlled” perturbations for lowering the sensitivity of the dynamic response to the occurrence of other uncontrolled perturbations.

On the use of the wave finite element method for passive vibration control of periodic structures

In this work, a strategy for passive vibration control of periodic structures is proposed which involves adding a periodic array of simple resonant devices for creating band gaps. It is shown that such band gaps can be generated at low frequencies as opposed to the well known Bragg scattering effects when the wavelengths have to meet the length of the elementary cell of a periodic structure. For computational purposes, the wave finite element (WFE) method is investigated, which provides a straightforward and fast numerical means for identifying band gaps through the analysis of dispersion curves. Also, the WFE method constitutes an efficient and fast numerical means for analyzing the impact of band gaps in the attenuation of the frequency response functions of periodic structures. In order to highlight the relevance of the proposed approach, numerical experiments are carried out on a 1D academic rod and a 3D aircraft fuselage-like structure.

Acoustic transmission properties of pressurised and pre-stressed composite structures

This work was focused on the examination of the effect of the pre-stress, namely tension and pressure, on the wave propagation and acoustic behaviour of composite laminates. The dispersion characteristics of two dimensional layered and sandwich structures were predicted using Wave Finite Element Method (WFEM). The structures were examined in non-stressed and pre-stressed scenarios. After extracting the mass and stiffness matrix of a small periodic segment of the structure using commercially available Finite Elements software, a polynomial eigenvalue problem was formed, the solutions of which consisted of the propagation constants of the waves of the structure. This way the wavenumbers and eigenvectors of the out of plane structural displacements were extracted. These wave propagation magnitudes were then used to calculate important Statistical Energy Analysis (SEA) quantities, such as modal density and radiation efficiency. The effect of pre-stress on these quantities, along with its effect on loss factor of the structure were examined.

Vibration band gaps for elastic metamaterial rods using wave finite element method

Band gaps in elastic metamaterial rods with spatial periodic distribution and periodically attached local resonators are investigated. New techniques to analyze metamaterial systems are using a combination of analytical or numerical method with wave propagation. One of them, called here wave spectral element method (WSEM), consists of combining the spectral element method (SEM) with Floquet–Bloch׳s theorem. A modern methodology called wave finite element method (WFEM), developed to calculate dynamic behavior in periodic acoustic and structural systems, utilizes a similar approach where SEM is substituted by the conventional finite element method (FEM). In this paper, it is proposed to use WFEM to calculate band gaps in elastic metamaterial rods with spatial periodic distribution and periodically attached local resonators of multi-degree-of-freedom (M-DOF). Simulated examples with band gaps generated by Bragg scattering and local resonators are calculated by WFEM and verified with WSEM, which is used as a reference method. Results are presented in the form of attenuation constant, vibration transmittance and frequency response function (FRF). For all cases, WFEM and WSEM results are in agreement, provided that the number of elements used in WFEM is sufficient to convergence. An experimental test was conducted with a real elastic metamaterial rod, manufactured with plastic in a 3D printer, without local resonance-type effect. The experimental results for the metamaterial rod with band gaps generated by Bragg scattering are compared with the simulated ones. Both numerical methods (WSEM and WFEM) can localize the band gap position and width very close to the experimental results. A hybrid approach combining WFEM with the commercial finite element software ANSYS is proposed to model complex metamaterial systems. Two examples illustrating its efficiency and accuracy to model an elastic metamaterial rod unit-cell using 1D simple rod element and 3D solid element are demonstrated and the results present good approximation to the experimental data.

Thermoelastic wave characteristics in a hollow cylinder using the modified wave finite element method

This paper represents a modified formulation of the wave finite element (WFE) method for propagating analysis of thermoelastic waves in a hollow cylinder without energy dissipation. The 2D-high-order spectral element with the Gauss–Legendre–Lobatto integration is applied into the WFE method, which produces the diagonal mass matrix. Based on the assumption of harmonic displacement fields by Fourier series expansion, the general discretization wave equation is simplified from the 3D problem to 2D. Dispersion properties of elastic wave propagation in the hollow cylinder are computed considering the choice of the spectral element orders, and the results indicate the high efficiency and high accuracy of the modified formulation compared with that of the software Disperse. Then, using the modified formulation, the thermoelastic dynamic equation of the cylinder is derived from the generalized thermoelasticity theory. The propagation of the thermoelastic wave (including two kinds of wave modes) in the cylinder without energy dissipation is discussed in different cases. Finally, wave structures along the radial direction of thermoelastic wave modes are shown at the nondimensional frequency 1.25, which can be used for the recognition of different modes.

On the Variability of the Sound Transmission Loss of Composite Panels Through a Parametric Probabilistic Approach

A robust model for the prediction of the variability of the vibro-acoustic response is presented in this paper. The dynamic response of composite panels is treated using a Statistical Energy Analysis (SEA) approach. One of the basic input parameters is the propagating flexural wavenumber of the modeled panel. The Wave Finite Element Method (WFEM) is used to investigate the dispersion characteristics of the layered panel. It is based on the evaluation of the mass and the stiffness matrices of a periodic segment of the structure. A polynomial eigenvalue problem is then formed for calculating the wavenumbers and the wave mode shapes. The main novelty in this paper consists in evaluating the influence of the variability of the mechanical parameters of the composite panel on its vibro-acoustic response, that is on its sound transmission loss (STL). This influence is quantified using the generalized polynomial chaos expansion. The efficiency of the approach is exhibited for isotropic and orthotropic panels.

Low-loss Y-junction two-dimensional magneto-photonic crystals circulator using a ferrite cylinder

A new defect structure of two-dimensional magnetic-photonic crystal is given and a new three-port Y-junction circulator using a ferrite cylinder is proposed in this short paper. Based on the agreements between the resonance frequency of micro-cavity constructed by a point defect in the 2D triangular lattice photonic crystal with those numerical results of corresponding literatures, external characteristics of the circulator were calculated by the plane wave expansion method and finite element method. According to the scaling theory of Maxwell's equations, a 3cm Y-junction 2D MPCs circulator is developed by scaling the radius of the center ferrite cylinder. The results show that there is a significant improvement for insertion loss 0.062dB and isolation 26.2dB compared with those in literatures.