Wave Finite Element Method Research Articles

Optimized Modular Design of Acoustic Metamaterials: Targeted Noise Attenuation

AbstractA modular design optimization method for acoustic metamaterial cells is proposed, which can ensure satisfying the targeted noise attenuation by using only the objective function of an arbitrary noise source and the constraints of the application at hand. The core algorithm is a fusion of the generalized particle swarm algorithm and wave finite element method specialized for structural and targeted optimization of metamaterials. The output is the optimized metamaterial cell and its sound barrier structure. The desired noise reduction performance is verified via acoustic testing of a 3D‐printed prototype. The simulation and experimental results of test cases show that the designed acoustic metamaterial can perfectly block the target noise band and have significant advantages in terms of design efficiency, light weight, and noise attenuation.

Study on vibro-acoustic characteristics of an adaptive piezoelectric locally resonant structure

Adaptively reducing low-frequency broadband vibration and noise is a challenge in the engineering and needs to be resolved. A piezoelectric locally resonant structure with periodic piezoelectric units and shunt damping circuits of synthetic impedance was studied to provide an option to deal with the problem. The z-scores method was applied to identify the peak frequencies used in the setting of control parameters. The band-gap performance and vibro-acoustic response were analyzed based on the plane wave expansion method and finite element method, respectively. The results show that the considered locally resonant piezoelectric structure with the z-scores peak-detection method can adaptively suppress the vibration and acoustic radiation of the plate, providing an effective method to reduce vibration and noise in time-varying excitation situations.

Kriging metamodeling approach for predicting the dispersion curves for wave propagating in complex waveguide

This study deals with the prediction and analysis of dispersion curves of waveguides, which are known to be critical steps within structural health monitoring (SHM). Despite using more sophisticated methods, such as wave finite element method (WFEM), the determination of the dispersion curves still presents severe numerical difficulties, which can be manifested by high computation time. Then, this paper proposes the metamodeling approach based on kriging theory to overcome the mentioned difficulties. Therefore, the main aim of the present study is to investigate the effectiveness of kriging metamodels when used to predict the WFEM-defined waveguide dispersion curve. Based on numerical simulations, the strategy, which consists of constructing the learning sets for the kriging metamodels using WFEM representations, is shown to be very promising regarding the convenience of the obtained compromise between the accuracy and the computing time.

Investigation of wave propagation and attenuation in periodic supported rails using wave finite element method

Investigation of wave propagation and attenuation in periodic supported rails using wave finite element method

A Homogenization Method for Replacement Stator Models in an Aero-Engine

Generally, the high-fidelity finite element models of aero-engines comprise millions of degrees of freedom (DOFs). Although they can provide precise predictions of structural dynamics, the computational cost will be often unaffordable if appropriate dimension reduction techniques are not adopted. The homogenization of the substructure, also termed as the physical replacement, reduces the model scale by simplifying the unnecessary details of the substructure, thus speeding up the dynamic analysis of the whole engine. In this study, we design the physical replacements for the stators of an aero-engine based on the long-wave assumption. These replacements have the same wave features as the stators in long-wave cases while possessing fewer DOFs. The core steps include the analytical description of the stators and the corresponding physical replacement design through two homogenizations. Specifically, we first investigate the wave characteristics of the stators using the wave finite element method and find two dominant waves: flexural and flexural–torsional coupled waves. The first homogenization introduces two analytical Timoshenko beams to describe the two wave motions of the stators. These two analytical beams are subsequently solidified into physical replacements with I, box, and open cross-sections in the second homogenization. The mechanical and geometric parameters are identified through a combination of the static analysis and the genetic algorithm (GA). The search processes are of great efficiency, because all the descriptions are analytical. Results show that the relative errors of the natural frequencies between the pristine stators and the physical replacements associated with the nodal diameters 6–15 are less than 5%.

Low-frequency vibration suppression of metastructure beam with high-static–low-dynamic stiffness resonators employing magnetic spring

The low/ultra-low-frequency structural vibrations exist widely in some engineering structures, for example, aerospace, naval, and building structures and so on. They could seriously affect the working performance and even cause the destruction of these engineering structures. In order to restrain the low/ultra-low-frequency structural vibration, a locally resonant (LR) metastructure beam equipped with high-static–low-dynamic stiffness (HSLDS) resonators employing negative stiffness magnetic spring is proposed. The analytical model of magnetic negative stiffness spring is firstly derived and the design method of HSLDS resonators is presented based on parametric optimization. Then, the dynamic model of the LR metastructure beam with HSLDS resonators is established using the wave finite element method. The effects of the number of local resonator units on low-frequency band gap are analyzed. Finally, the low-frequency vibration control performance of LR metastructure beam is validated experimentally. The experimental result is in agreement with the theoretical analysis, which demonstrates that the HSLDS resonators are conductive to suppress vibration of the metastructure beam in the low-frequency region (f < 10 Hz). The results also showed that the more the HSLDS resonators are used, the better the vibration suppression effect is.

Multi-mode propagation and diffusion analysis using the three-dimensional second strain gradient elasticity

The multi-mode propagation and diffusion properties are crucial informations when studying complex waveguides. In this paper, firstly, the three-dimensional modeling of micro-sized structures is introduced by using the second strain gradient theory. The constitutive relation is deduced while the six quintic Hermite polynomial shape functions are employed for the displacement field. The weak formulations including element stiffness and mass matrices and the force vector are calculated through the Hamilton’s principle and the global dynamic stiffness matrix of a unit cell is assembled. Then, free wave propagation characteristics are analyzed by solving eigenvalue problems within the direct wave finite element method framework. The dispersion relations of positive going waves considering the size effects are illustrated. Furthermore, the effects of higher order parameters on the dispersion curves are discussed and the forced responses with two boundary conditions are expounded. Eventually, the wave diffusion including reflection and transmission coefficients are illustrated through simple and complex coupling conditions, respectively. The dynamic analysis of coupled waveguides through the wave finite element method equipped with the second strain gradient is a novel work. The results show that the proposed approach is of significant potential for investigating the wave propagation and diffusion characteristics of micro-sized structures.

Defect localization in waveguide assemblies with curved joints via wave finite elements and time of flight analysis

A numerical approach is proposed to localize defects in elastic waveguides connected by curved elastic joints. 2D assemblies involving straight waveguides with a curved joint and a defect are specifically dealt with, where the joint is placed between the measurement point (output signals) and the defect. Such an analysis requires assessing wave conversion phenomena and times of flight for wave packets when they are transmitted through the joint and reflected by the defect. In this paper, the wave finite element (WFE) method is used to help understand these phenomena. An original strategy is proposed where the times of flight, for transmitted or reflected wave packets, are defined from the frequency derivatives of the arguments of the scattering matrices of the joint and the defect. The procedure to localize a defect follows from comparing the theoretical expressions of the times of flight with those recorded at the measurement point. Also, the proposed approach enables the identification of the types of waves which are truly transmitted through the joint and reflected by the defect. Numerical experiments are carried out which highlight the relevance, in terms of accuracy and robustness, of the proposed approach.

Elastic metamaterial composite made of molded pulp and steel for suppression of low-frequency vibration in thin-plate structures

This study proposes EMMs made of molded pulp and steel composite structures; molded pulp can be used to manufacture periodic shapes with low cost and environmental burden, which can help in the application of EMMs to steel structures that are widely used in engineering products. Toward this end, the frequency-dependent mechanical properties of the molded pulp are first identified via the vibration testing of pulp-steel laminate structures; the measured data confirm the high damping properties of molded pulp. Subsequently, wave propagation within the EMM is analyzed using the wave finite element method via dispersion curves, which suggest that the EMMs fabricated using molded pulp can attenuate wave propagation in wide frequency ranges by combining the effects of geometrical periodicity (i.e., band gap) and the material damping of the molded pulp. Further, the effects of geometrical parameters on band gaps are investigated; the results indicate that the proposed EMMs can open up band gaps in frequency ranges lower than 1000 Hz for thin plates used in engineering products. Finally, the frequency response functions of the EMMs are experimentally measured; the results confirm that EMMs can suppress vibrations in wide frequency ranges corresponding to the wave attenuation obtained by the dispersion analysis.

Study on the Propagation Process Characteristics of Anisotropic Acoustic Waves in Shale Gas Well with the Reflection Rule of Lateral Fractures

Based on the acoustic wave finite element (AWFE) method, one can establish an AWFE method and study the influence of mechanical parameters on the shale reservoir acoustic wave propagation characteristics. The different crack characteristics and different crack multi-physical coupling phenomena are studied by using the AWFE method on shale gas reservoir cracks. To calculate the shape and position along the crack near a side borehole, the model parameters are compared with the simulation results. The reflection waveform characteristics of adjacent cracks are studied by using the AWFE method. By considering the borehole axis of symmetry, for an acoustic impedance discontinuous interface on one side of the two-dimensional axisymmetric AWFE, one can establish a borehole cross-crack and an arc cross-crack reflection interface model with the AWFE method. By processing the waveform data received by different receiving points under the same source distance, the parameters, such as the reflection wave time and the distribution laws of the crack in the shale reservoir, are obtained. To verify the validity of the research method, the propagation of a reflected acoustic wave from the reservoir fracture by the filling with different media was also studied. The results show that a reflection wave arrival time changing with the source ordinate and present law, by side borehole fracture morphology, showed a suitable consistency. The well cross-crack angle range is 10~20°, according to the wave arrival time calculated by the side borehole fracture dip. For the acoustic signal propagation in the shale formation anisotropy, they found that an acoustic signal is always in the direction of the elastic modulus, with a further larger spread, a location of maximum amplitude, and a 45-degree direction to the axis of symmetry. In the lateral and longitudinal distance from an acoustic source of the same two receiver signal waves, the receiver vibration amplitude is bigger, and there is less attenuation. With the increase in the anisotropic index, the inside ovality amplitude distribution of the signal amplitude in this model is higher. When a side borehole has an arc crack and a reflected wave to time to obtain the coordinates of a reflecting interface and to compare with the results of the calculation model, the crack in the center position, and the reflection point coordinates, the relative error is less than 5%. Finally, the AWFE method could provide a new idea for the identification of the crack properties and also could be an inverse calculation of the position and morphological characteristics of fractures near the side borehole.

Parametric studies on SEA parameters for coupled plates made of composite laminate

Statistical Energy Analysis (SEA) has been used to compute velocity responses and Coupling Loss Factors (CLF) for two composite laminates joined in a 'L' junction configuration. Two case studies were carried on SEA. The effect of fibre orientation on SEA parameters was studied. The effects of internal damping were also determined. The computation of SEA parameters has been done using classical wave approach and Finite Element Method using Nastran/Patran. CLF determined using classical wave approach is independent of fibre orientation and internal damping factor. This study using FEM reveals that the SEA parameters vary in comparison with classical wave approach as fibre orientation and internal damping are also considered in the analysis. Also, by classical wave approach, the CLF varied linearly with the increase in frequency while there was some scatter in CLF determined by finite element method.

Finite element modal analysis of wave propagation in homogeneous and periodic waveguides

In this paper, we demonstrate how conventional finite element modal analysis of a slice of an infinite homogeneous or periodic waveguide provides essential information on its properties. The novelty aspects are the rigorous derivation of requested boundary conditions from the analytical bi-orthogonality relation for free waves in a waveguide and conversion of these ‘class consistent’ conditions to the finite element format. Eigenfrequencies and mode shapes obtained from the modal analysis are used to reconstruct dispersion diagram for propagating waves in a homogeneous waveguide. A novel method is proposed to assess their decay rates in the presence of material losses. For a periodic waveguide, the modal analysis of a symmetric unit periodicity cell with the ‘class consistent’ boundary conditions is innovatively used to identify partial (modal) and full stop-bands. Computational efficiency of the proposed modal analysis-based methodology as compared with standard Wave (and) Finite Element method is discussed.

Elastic wave attenuation in a metaplate with periodic hollow shapes for vibration suppression

Low noise and vibrations are requirements for engineering structures to realize comfort and ensure systems are environmental friendly. Periodic structures such as phononic crystals have been investigated to suppress noise and vibrations because they inhibit elastic wave propagation in the frequency ranges referred to as the band gap. This study led to the proposal of a metaplate (MP) with periodic hollow structures to form band gaps by changing the surface shapes of the thin plates. MPs such as this can be beneficial from the viewpoints of manufacturing cost and structural reliability when applied to engineering structures. To evaluate the potential ability of the proposed MP to suppress vibrations, we obtained the band gap frequency of out-of-plane deformation waves using the wave finite element method. To efficiently evaluate the band gaps, the internal degrees of freedom in a unit cell are reduced based on the modal analysis technique. Furthermore, the out-of-plane and in-plane deformation waves are separated using kinematic energy to extract the band gap related to the out-of-plane deformation waves. Based on this numerical framework, we demonstrate that periodic hollow shapes can form a band gap in the frequency range of interest in engineering problems. Moreover, the potential of the MP was investigated using hollow shape design and material selection, and the results indicate that the proposed MP is beneficial for tuning the band gap frequency in a unit cell of which the size is acceptable for engineering structures. Finally, vibration suppression caused by band gap is experimentally demonstrated via impact testing of an MP fabricated using additive manufacturing.

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.

Negative dispersion of a form birefringence in subwavelength gratings.

An achromatic response is required in most optical systems for wideband and straightforward configurations. The chromatic response of the optical system depends on the optical dispersion of the elements in the system. Here we study the dispersion of subwavelength grating (SWG) known to have a form birefringence. The birefringence of SWG was numerically analyzed with Bloch wave analysis (BWA) and finite element method (FEM). The sandwiched SWG with two identical substrates was studied for practical applications. We successfully demonstrated the negative dispersion form birefringence of SWG with an optimal duty cycle. This extraordinary dispersion was also shown considering the intrinsic dispersion of materials. Dispersion- and the angular response were in a tradeoff relationship while they depended on periodicity. The optical interference between the grating and the substrates can be eliminated by controlling the duty cycle. Our analysis offers optimal SWG with achromatic birefringence and high transparency, promising in the widespread applications of polarization control devices.

Logic gate and optical half-adder designed by photonic crystal based on BPSK signals

We design a logic gate and an optical half-adder based on photonic crystal using BPSK signals. The logic gate is composed of a silicon rod array and a microcavity containing a SiO2 dielectric column, which can realize XOR and XNOR logic. The optical half adder is formed by introducing a reference beam with a fixed initial phase to the proposed logic gate. The initial phase of incident light determines the logic state of the BPSK signals. The intensity of the output light determines the output logic state. We use the plane wave expansion and finite element methods to analyze logic gates and optical half adders. The proposed logic gate and optical half-adder cover 160 µm2 and 270 µm2, the response time is 0.6 ps and 0.9 ps, respectively, and the margin difference of the output signals is 103% and 81%, respectively. Their compact structure, short response time, and large margin difference of the output signals have potential application prospects in all-optical information processing.

Two-dimensional periodic structures modeling based on second strain gradient elasticity for a beam grid

Higher order gradient elasticity theories are widely applied to determine the wave propagation characteristics of micro-sized structures. The novelty of this paper, firstly, is using the Second Strain Gradient (SSG) theory to explore the mechanism of a micro-sized 2D beam grid. The strong formulas of continuum model including governing equations and boundary conditions are derived by using the Hamilton principle. Then, a valuable long-range Lattice Spring Model (LSM) is elaborated, providing a reasonable explanation for the model based on SSG theory. The dynamic continuum equations from LSM are calculated through the Fourier series transform approach. Finally, the dynamic properties of 2D beam grid are analyzed within the Wave Finite Element Method (WFEM) framework. The band structure and slowness surfaces, confined to the irreducible first Brillouin zone, are studied in frequency spectrum. The energy flow vector fields and wave beaming effects are discussed through SSG theory and Classical Theory (CT) of elasticity. The results show that the proposed approach is of significant potential for investigating the 2D wave propagation characteristics of complex micro-sized periodic structures.

A scaled wave finite element method for computing scalar wave radiation and scattering in exterior domains

The calculation of high-frequency wave radiation in exterior domains by finite element methods can lead to large computations. In this paper, it is shown that the solution in the exterior domain can be decomposed as a series expansion of functions with an analytical part made from the product of harmonic waves and polynomials in a scaled variable and a numerical part made of a finite element approximation vector. The solution of the radiation or scattering problem can be found by solving a sparse linear system which is set from the dynamic stiffness matrices of several scaled layers around the radiating body. These dynamic stiffness matrices are classical finite element matrices obtained from any finite element software. Moreover, accurate results can be obtained from a small number of terms in the series expansion. Several examples are given to estimate the efficiency of the proposed method.

A wave finite element approach for modelling wave transmission through laminated plate junctions

We present a numerical method for computing reflection and transmission coefficients at joints connecting composite laminated plates. The method is based on modelling joints with finite elements with boundary conditions given by the solutions of the wave finite element method for the plates in the infinite half-spaces connected to the joint. There are no restrictions on the number of plates, inter-plate angles, and material parameters of individual layers forming the composite. An L-shaped laminated plate junction is discussed in more detail. Comparisons of numerically predicted scattering coefficients with semi-analytical solutions for the selected structures are presented. The results obtained are essential for statistical energy analysis and dynamical energy analysis based calculations of the wave energy distribution in full built-up structure.

The plane wave finite element method for the wave scattering by a 2-D phononic crystal

The plane wave finite element method for the wave scattering by a 2-D phononic crystal