Wave And Finite Element Research Articles

Semi-Analytical Solutions for Wave Propagation of Periodically Repetitive Schwarz Primitive Triply Periodic Minimal Surface Based Structure Embedded With Acoustic Black Hole Resonators

The Schwarz Primitive triply periodic minimal surface (P-TPMS) lattice structure is attracting more attention due to its superior mechanical properties and unique topological configuration. In this study, a new periodic configuration is proposed in which the unit cell is formed by embedding the acoustic black hole (ABH) component in the primary P-TPMS component. A semi-analytical method is developed to calculate the wave propagation properties of the proposed structure. The dynamical equilibriums of the primary P-TPMS and ABH components are described by the finite element method and the WKB method respectively. By considering the compatibility at the connections between the components and using the equivalent dynamic stiffness technique, dispersion analysis can be implemented in the framework of the wave and finite element (WFE) method. Various numerical examples with different number, type and connection position of the resonators are investigated. By comparing the calculation results from the present semi-analytical method with that from the finite element method, the effectiveness of the proposed method is validated. The influences of the number, type, and connection position of the resonators on the bandgap properties are investigated.

Numerical modal analysis of bulk-reactive lined duct with shear flow

In this paper, the eigenmodes of bulk-reactive lined duct with shear flow are calculated and analyzed by a wave and finite element (WFE) method. Using this method, the eigenmodes can be calculated based on commercially available finite element (FE) matrices, which is of considerable practical interest when dealing with complicated cases. The proposed method is validated by comparing results with Chebyshev collocation method. It is shown that the axial wavenumbers obey the convergence rate of Se4 for all kinds of modes, where Se is the mesh size. The calculated eigenmodes can be divided into air-borne modes, lining-borne modes and surface modes, where air-borne modes mainly propagate in air domain; lining-borne modes propagate in both air and lining domains; surface modes propagate near the interface between air and lining domains. It is found that, for air-borne and lining-borne modes, boundary layer thickness has small effect on lower order modes, but has relatively large effect on mode shapes of higher order modes. For the surface modes, boundary layer thickness can greatly change their mode properties. In the calculated example, with perforated panel considered, two coupled modes emerge. With perforation ratio evolving from large to small, one coupled mode evolves from a lower-order air-borne mode to a higher-order air-borne mode, and the other evolves from a lining-borne mode to an air-borne mode. Generally, the perforation ratio has small effect on air-borne and lining-borne modes; while it has large effect on surface modes and the coupled mode, especially in the range of [3%, 13%].

Investigating acoustics and wave behaviour in cross-laminated timber panels

Cross-laminated timber (CLT) is a timber product that is becoming increasingly popular in construction in NZ because of the ability to prefabricate panels off-site, as well as being lightweight and sustainable compared to other building materials. There is currently a lack of information on its acoustical properties, as the complex geometry through the thickness means it is difficult to model and predict sound transmission. The WFE (wave and finite element) method has been employed as it allows for a small segment of a material to be modelled using standard FE methods and can incorporate several material layers. It then requires finding the mass and stiffness matrices of the segment and post-processing them to determine the wave behaviour of the structure as a whole. The WFE method was used to model the sound transmission of several different CLT panels and these results were compared against measurements taken by the National Research Council Canada. In-house testing was also performed to obtain experimental wavenumbers, and these were also compared to wavenumbers produced by the WFE method.

Analysis of the vibroacoustic characteristics of cross laminated timber panels using a wave and finite element method

This paper presents an analysis of the vibroacoustic characteristics of Cross-Laminated Timber (CLT) panels using a Wave and Finite Element (WFE) method. Two WFE modelling approaches are investigated: the first models each layer of the panel as a homogeneous orthotropic material, whereas the second approximates the entire panel as a single equivalent homogeneous orthotropic material. Results are also compared to those of a thin, orthotropic plate model which is a model commonly used in the literature. The three methods are used to calculate the plane-wave and diffuse-field sound transmission loss of a CLT panel. It is seen that all models are in reasonable agreement at low frequencies, but that the models predict different sound transmission loss at higher frequencies. The dispersion curves and mode shapes of the free waves propagating within the in-vacuo panel are predicted for both WFE models. These results are used to interpret the vibroacoustic behaviour of the panel. It is observed that, at high frequencies, the vibroacoustic behaviour of the panel is complicated, with many wave modes cutting on. This behaviour is not modelled by the equivalent orthotropic plate model, which is incapable of capturing those higher-order waves. The contributions to the sound transmitted by the various wave modes are estimated and the most significant modes identified. Finally, predictions of the sound transmission loss of a number of CLT panels are calculated using the layer-wise WFE modelling approach. These predictions are compared with experimental measurements, showing reasonable agreement. The predictions clearly capture the important coincidence dips present in the measured spectra, although there are some minor discrepancies between the measured and predicted levels and coincidence frequencies. These discrepancies may be partially due to the uncertainty and variability of the material properties of wood. A series of simulations are performed to investigate the effect of the uncertainty of different structural material properties on the sound transmission loss.

A fast Bayesian inference scheme for identification of local structural properties of layered composites based on wave and finite element-assisted metamodeling strategy and ultrasound measurements

Reliable verification and evaluation of the mechanical properties of a layered composite ensemble are critical for industrially relevant applications, however it still remains an open engineering challenge. In this study, a fast Bayesian inference scheme based on multi-frequency single shot measurements of wave propagation characteristics is developed to overcome the limitations of ill-conditioning and non-uniqueness associated with the conventional approaches. A Transitional Markov chain Monte Carlo (TMCMC) algorithm is employed for the sampling process. A Wave and Finite Element (WFE)-assisted metamodeling scheme in lieu of expensive-to-evaluate explicit FE analysis is proposed to cope with the high computational cost involved in TMCMC sampling. For this, the Kriging predictor providing a surrogate mapping between the probability spaces of the model predictions for the wave characteristics and the mechanical properties in the likelihood evaluations is established based on the training outputs computed using a WFE forward solver, coupling periodic structure theory to conventional FE. The valuable uncertainty information of the prediction variance introduced by the use of a surrogate model is also properly taken into account when estimating the parameters’ posterior probability distribution by TMCMC. A numerical study as well as an experimental study are conducted to verify the computational efficiency and accuracy of the proposed methodology. Results show that the TMCMC algorithm in conjunction with the WFE forward solver-aided metamodeling can sample the posterior Probability Density Function (PDF) of the updated parameters at a very reasonable cost. This approach is capable of quantifying the uncertainties of recovered independent characteristics for each layer of the composite structure under investigation through fast and inexpensive experimental measurements on localized portions of the structure.

Sensitivity analysis of generalised eigenproblems and application to wave and finite element models

The first and second order sensitivity analysis of the eigenvalue problem of generalised, non-symmetric matrices using perturbation theory is developed. These results are then applied to sensitivity analysis of wave propagation in structures modelled using the wave and finite element (WFE) method. Three formulations of the WFE eigenvalue problem are considered: the transfer matrix method, the projection method and Zhong's method. The sensitivities with respect to system parameters of wavenumbers and wave mode shapes are derived. Expressions for the group velocity are presented. Numerical results for a thin beam, a foam core panel and a cross-laminated timber panel are used to demonstrate the proposed approach. It is shown that sensitivities can be calculated at negligible computational cost.

Bayesian inference for damage identification based on analytical probabilistic model of scattering coefficient estimators and ultrafast wave scattering simulation scheme

Ultrasonic Guided Waves (GW) actuated by piezoelectric transducers installed on structures have proven to be sensitive to small structural defects, with acquired scattering signatures being dependent on the damage type. This study presents a generic framework for probabilistic damage characterization within complex structures, based on physics-rich information on ultrasound wave interaction with existent damage. To this end, the probabilistic model of wave scattering properties estimated from measured GWs is inferred based on absolute complex-valued ratio statistics. Based on the probabilistic model, the likelihood function connecting the scattering properties predicted by a computational model containing the damage parametric description and the scattering estimates is formulated within a Bayesian system identification framework to account for measurement noise and modelling errors. The Transitional Monte Carlo Markov Chain (TMCMC) is finally employed to sample the posterior probability density function of the updated parameters. However, the solution of a Bayesian inference problem often requires repeated runs of “expensive-to-evaluate” Finite Element (FE) simulations, making the inversion procedure firmly demanding in terms of runtime and computational resources. To overcome the computational challenges of repeated likelihood evaluations, a cheap and fast Kriging surrogate model built and based on a set of training points generated with an experiment design strategy in tandem with a hybrid Wave and Finite Element (WFE) computational scheme is proposed in this study. In each “numerical experiment”, the training outputs (i.e. ultrasound scattering properties) are efficiently computed using the hybrid WFE scheme which combines conventional FE analysis with periodic structure theory. By establishing the relationship between the training outputs and damage characterization parameters statistically, the surrogate model further enhances the computational efficiency of the exhibited scheme. Two case studies including one numerical example and an experimental one are presented to verify the accuracy and efficiency of the proposed algorithm.

On the numerical eigenmode analysis of acoustically lined ducts with uniform mean flow

The present work concerns the numerical eigenmode analysis of acoustically lined ducts of any constant cross-sectional shape in the presence of mean flow. The duct is lined with bulk-reactive porous material, which is separated from the flow channel by a perforated panel. A combined wave and finite element (WFE) method is introduced to investigate the effects of uniform mean flow on the eigenmodes and eigenvalues of the lined ducts. The proposed method is validated against analytical solutions for a simple case. A convergence rate of Se4 for eigenvalues can be reached for Mach number up to 0.8, with Se being the mesh size. In this study, the eigenmodes of a lined duct are divided into air-borne modes and lining-borne modes. The air-borne modes propagate mostly in the air domain, while the lining-borne modes can propagate in both air domain and porous domain. The effect of Mach number on the axial wavenumbers and mode shapes are analysed. It is found that, for air-borne modes, Mach number will strongly affect their wavenumbers, while its effect on the mode shape is marginal. On the contrary, for the lining-borne modes, Mach number has small effect on the wavenumber, but its effect on the mode shape is significant. Specifically, even the mode properties can be changed from a lining-borne mode to an air-borne mode for some left-travelling lining-borne mode. The transmission loss (TL) of a lined duct with three typical Mach numbers is calculated by the mode matching method. It is found that up-flow will generally increase TL of lined duct, compared with the case of no flow; while down-flow will generally decrease TL. In both situations, the peak frequency does not show significant change.

A wave and finite element based homogenised model for predicting sound transmission through honeycomb panels

For vibroacoustic analysis, honeycomb-cored panels are often modelled as homogenised solid plates, which involves various assumptions and approximations. In this paper, a wave and finite element (WFE) based modelling strategy is proposed to predict sound transmission through honeycomb-cored panels. In this method, a three-dimensional periodic cell of the structure is modelled using a conventional finite element (FE) method. Due to the complexity of the core geometry, the FE model can contain a large number of internal nodes. Guyan reduction is used to reduce the model size. The in-vacuo wavenumbers are found, and from them the group velocities and modal densities determined. Wave propagation in the fluids surrounding the structure is modelled analytically. The acoustic loading is modelled using equivalent external nodal forces. The relatively small-sized mass and stiffness matrices are then post-processed using periodicity theory and equilibrium conditions. An accurate homogenised model is developed for calculating the structural response to acoustic excitation. Excitation of the structure by oblique plane waves and a diffuse sound field are both considered. Various numerical examples are presented to illustrate this model. The homogenised model developed in this paper is general and accurate, and can model sound transmission through honeycomb-cored panels of any configurations.

Wave Scattering and Power Flow in Straight-Helical-Straight Waveguide Structure

The knowledge of wave scattering and power flow in waveguide structures is important for many engineering applications. In this paper, power flow and scattering in a straight-helical-straight waveguide structure are investigated using the wave and finite element (WFE) method. For simple (straight or helical) waveguides, wave scattering (and subsequently the power flow and scattering) can be resolved analytically. This is not the case for complex waveguides such as laminated or sandwiched waveguides or waveguides with noncanonical cross-sections. In such cases, the WFE method is used to model the wave behavior in each waveguide in the structure. The power flow is then studied by considering how waves reflect and transmit at the boundaries that join the straight waveguides with the helical waveguide. We present three numerical examples but analytical solutions can be obtained for the first example only; for the second and third examples, the WFE is used in earnest since the wave behavior and subsequently the power flow would at best be extremely difficult to formulate analytically.

Wave propagation in slowly varying waveguides using a finite element approach

This work investigates structural wave propagation in one dimensional waveguides with randomly varying properties along the axis of propagation, specifically when the properties vary slowly enough such that there is negligible backscattering, even if the net change is large. Wave-based methods are typically applied to homogeneous waveguides but the WKB (after Wentzel, Kramers and Brillouin) approximation can be used to find a suitable generalisation of the wave solution in terms of the change of phase and amplitude but is restricted to analytical solutions. A wave and finite element (WFE) approach is proposed to extend the applicability of the WKB method to cases where no analytical solution of the equations of motion is available. The wavenumber is expressed as a function of the position along the waveguide. A Gauss-Legendre quadrature scheme is subsequently used to obtain the phase change, while the wave amplitude is calculated using conservation of power. The WFE method is used to evaluate the wavenumbers at each integration point. Moreover, spatially correlated randomness can be included in the formulation by random field properties and in this paper is expressed by a Karhunen-Loève expansion. Numerical examples are compared to a standard FE approach and to available analytical solutions. They show good agreement when compared to either a full FE or analytical solution and require only a few WFE evaluations, providing a suitable framework for efficient stochastic analysis in waveguides.

Sound transmission through cylindrical structures using a wave and finite element method

This paper concerns the use of the Wave and Finite Element (WFE) method for predicting sound transmission through, and radiation from, an infinitely long cylindrical structure. The formulation is general and can be used to analyse acoustic radiation or transmission due to arbitrary internal or external excitation. The WFE method involves modelling a small segment of a structure using traditional Finite Element (FE) methods. The mass and stiffness matrices of the segment are then used to calculate the response of the structure to excitation. The WFE approach for calculating sound transmission is validated by comparison with analytic solutions for three example problems: (1) the internal sound field produced by an acoustic source exterior to a thin-walled cylinder, (2) the radiation from a thin-walled cylinder due to excitation from a point force acting on the inner surface and (3) sound transmission through an orthotropic cylinder. In order to demonstrate the usefulness of the approach, the WFE method is then used to predict sound transmission through, and radiation from, more complicated cylindrical structures, for which an analytical model would be difficult to develop.

Calculation of guided wave interaction with nonlinearities and generation of harmonics in composite structures through a wave finite element method

The extensive usage of composite materials in modern industrial applications implies a great range of possible structural failure modes for which the structure has to be frequently and thoroughly inspected. Nonlinear guided wave inspection techniques have been continuously gaining attention during the last decade. This is primarily due to their sensitivity to very small sizes of localised damage. A number of complex transformation phenomena take place when an elastic wave impinges on a nonlinear segment, including the generation of higher and sub-harmonics. Moreover, the transmission and reflection coefficients of each wave type become amplitude dependent. In this work, a generic Finite Element (FE) based computational scheme is presented for quantifying guided wave interaction effects with Localised Structural Nonlinearities (LSN) within complex composite structures. Amplitude dependent guided wave reflection, transmission and conversion is computed through a Wave and Finite Element (WFE) method. The scheme couples wave propagation properties within linear structural waveguides to a LSN and is able to compute the generation of higher and sub-harmonics through a harmonic balance projection. A Newton-like iteration scheme is employed for solving the system of nonlinear differential equations. Numerical case studies are presented for waveguides coupled through a joint exhibiting nonlinear mechanical behaviour.

Prediction of sound transmission through, and radiation from, panels using a wave and finite element method.

This paper describes the extension of a wave and finite element (WFE) method to the prediction of noise transmission through, and radiation from, infinite panels. The WFE method starts with a conventional finite element model of a small segment of the panel. For a given frequency, the mass and stiffness matrices of the segment are used to form the structural dynamic stiffness matrix. The acoustic responses of the fluids surrounding the structure are modelled analytically. The dynamic stiffness matrix of the segment is post-processed using periodic structure theory, and coupled with those of the fluids. The total dynamic stiffness matrix is used to obtain the response of the medium to an incident acoustic pressure. Excitation of the structure by oblique plane waves and a diffuse sound field are considered. The response to structural excitation and the consequent radiation are determined. Since the size of the WFE model is small, computational times are small. Various example applications are presented to illustrate the approach, including a thin isotropic panel, an antisymmetric, cross-ply sandwich panel and a symmetric panel with an orthotropic core.

Prediction of noise transmission through infinite panels using a wave and finite element method

The transmission of sound through simple infinite isotropic panels can be predicted in a straightforward manner using well-established analytical models. However, such models become difficult to implement for more complicated structures such as laminates etc. In these cases, numerical approaches such as the finite element method are viable alternatives. However, these methods can be computationally intensive as the entire structure must usually be meshed with the surrounding fluid being modelled using finite, infinite or boundary element methods. This paper describes the extension of a Wave and Finite Element (WFE) method for the prediction of sound transmission through infinite, two-dimensional panels. Excitation of the structure by oblique plane waves, a diffuse sound field and a point force are all considered. The WFE method involves a finite element model of a small segment of the panel from which the wave properties and the response to external excitation can be found. Because the WFE method only involves meshing a small segment of the panel, computational times are significantly less than a full finite element simulation. Two example applications of the method are described, namely a thin isotropic panel and a laminated panel.

Application of the Wave and Finite Element Method to Calculate Sound Transmission Through Cylindrical Structures

This paper concerns the prediction of sound transmission through a cylindrical structure. The problem considered is that of sound generated by a line source located exterior to a two-dimensional circular cylinder which produces sound waves which transmit through the cylinder to an internal medium. An analytical solution is presented for the case of sound transmission through a thin cylindrical shell, by modelling the shell response using the Flugge- Byrne-Lur'ye equations. This solution is then compared to calculations where the response of the cylinder is calculated using the Wave and Finite Element (WFE) method. The WFE method involves modelling a small segment of a structure using traditional finite element (FE) methods. The mass and stiffness matrices of the segment are then used to calculate the response of the structure to excitation by an acoustic field. The WFE approach for calculating sound transmission is validated by comparison with the analytic solution. Formulating analytic solutions for more complicated structures can be cumbersome whereas using a numerical technique, such as the WFE method, is relatively straightforward.

Flexural Wave Propagation in Slowly Varying Random Waveguides Using a Finite Element Approach

This work investigates structural wave propagation in waveguides with randomly varying properties along the axis of propagation, specifically when the properties vary slowly enough such that there is negligible backscattering. Wave-based methods are typically applied to homogeneous waveguides but the WKB (after Wentzel, Kramers and Brillouin) approximation can be used to find a suitable generalisation of the wave solution in terms of the change of phase and amplitude, but is restricted to analytical solutions. A wave and finite element (WFE) approach is proposed to extend the applicability of the WKB method to cases where no analytical solution is available. The wavenumber is expressed as a function of the position along the waveguide and a Gauss-Legendre quadrature scheme is used to obtain the phase change while the wave amplitude is calculated using conservation of power. The WFE method is used to evaluate the wavenumbers at each integration point. The flexural vibration example is considered with random field proprieties being expressed by a Karhunen-Loeve expansion. Results are compared to a standard FE approach and to the WKB analytical solution. They show good agreement and require only a few WFE evaluations, providing a suitable framework for spatially correlated randomness in waveguides.

Calculating the forced response of cylinders and cylindrical shells using the wave and finite element method

The dynamic response of circular cylinders can be obtained analytically in very few (and simple) cases. For complicated (thick or anisotropic) circular cylinders, researchers often resort to the finite element (FE) method. This can lead to large models, especially at higher frequencies, which translates into high computational costs and memory requirements. In this paper, the response of axially homogenous circular cylinders (that can be arbitrarily complex through the thickness) is obtained using the wave and finite element (WFE) method. Here, the homogeneity of the cylinder around the circumference and along the axis are exploited to post-process the FE model of a small rectangular segment of the cylinder using periodic structure theory and obtain the wave characteristics of the cylinder. The full power of FE methods can be utilised to obtain the FE model of the small segment. Then, the forced response of the cylinder is posed as an inverse Fourier transform. However, since there are an integer number of wavelengths around the circumference of a closed circular cylinder, one of the integrals in the inverse Fourier transform becomes a simple summation, whereas the other can be resolved analytically using contour integration and the residue theorem. The result is a computationally efficient technique for obtaining the response to time harmonic, arbitrarily distributed loads of axially homogenous, circular cylinders with arbitrary complexity across the thickness.

A Finite Element Method for Modelling Waves in Laminated Structures

The generality and complexity of laminated structures raises issues regarding modelling their dynamics and optimising their design. In this paper, a wave and finite element (WFE) method for modelling the dynamic behaviour of plane and axisymmetric laminated structures is described. A small segment of the structure is modelled using conventional finite element (FE) methods, typically using a commercial package. The mass and stiffness matrices are found, periodicity conditions are applied, and an eigenvalue problem is formulated and solved to find the dispersion relations. The frequency dependence of viscoelastic material properties and pre-stress can be taken into account straightforwardly. A hybrid FE/WFE approach for determining transmission characteristics of joints is described. Numerical examples are presented, including anisotropic, plane and cylindrical foam-cored laminated sandwich constructions with pre-stress. The method is simple in application, provides accurate results at low computational cost and is a valuable tool for evaluating the vibro-acoustic behaviour of multi-layer panels and optimising their design.

The loss-factor of pre-stressed laminated curved panels and cylinders using a wave and finite element method

In this paper a wave and finite element (WFE) post-processing technique is applied to predict the effects of pre-stress on the damping of curved panels. It is seen that pre-stress substantially reduces the global loss factor, especially of those modes of vibration which involve considerable radial displacement. This is particularly significant for aerospace structures since ground-test results typically do not include pre-loads . The WFE approach and its extension to include pre-stress effects are briefly described. Numerical examples concerning a pressurised viscoelastic cylinder and a pre-loaded curved laminated panel are presented.