Wave And Finite Element Method Research Articles

Longitudinal wave propagation analysis in Entangled metallic wire materials using an improved wave and finite element method

This paper proposes a novel concept and methodology to study longitudinal wave propagation in entangled metallic wire material (EMWM), a porous structural network consisting of spatial helix wires. The numerical model of EMWM is constructed using representative volume elements (RVE) obtained through X-ray tomography and skeletonization of a standard specimen. By increasing the approximate periodic boundary, the improved wave and finite ele-ment method (WFE) is employed to model EMWM as approximately periodic structures, allowing for numerical simulations of wave propagation characteristics. Frequency bands and resonance modes of EMWM are calculated to analyze the characteristics of one-dimensional wave propagation. The numerical results show that EMWM exhibits unique frequency band characteristics compared to typical periodic porous materials. In the frequency range of 15 to n n 35 kHz, the first pass band corresponds to the tensile-compressive motion of the entire specimen, while the higher pass bands correspond to localized vibrations within the spatial wires. Additionally, the frequency bands can be adjusted across a wide range based on the relative density of EMWM specimens. The improved WFE method and obtained numerical data can facilitate the application of EMWM on high frequency vibration isolation.

A wave and Rayleigh–Ritz method to compute complex dispersion curves in periodic lossy acoustic black holes

In this work we first propose a wave and Rayleigh–Ritz method (WRRM) to analyze the vibrations of periodic systems. It is well-known that one of the main difficulties of the Rayleigh–Ritz method (RRM) is to find suitable basis functions that satisfy the problem boundary conditions. In the WRRM, this is done by expanding the system response as a superposition of the basis of the nullspace of the matrix defining the system periodic conditions. Therefore, the WRRM widens the potential of the RRM and constitutes a computationally efficient alternative to the wave and finite element method (WFEM) for structures requiring very fine meshes. Although the method is of general application, in this paper it is presented for an infinite periodic beam consisting of acoustic black holes (ABHs) cells with damping layers. Periodic ABHs exhibit broadband vibration reduction thanks to bandgap formation at low frequencies and to the ABH effect in the mid-high frequency range. The WRRM allows one to obtain the equations of motion of the periodic ABH beam and to define linear and quadratic eigenvalue problems to respectively obtain the system’s real and complex dispersion curves. While the former have been studied at extent, the latter have been barely analyzed for periodic ABH beams or plates. And they are of critical importance given that ABHs are strongly damped systems and that damping layer properties may be frequency-dependent. Therefore, the second contribution of this work is to carefully inspect the role played by the imaginary part of the complex dispersion curves on the functioning of the ABH periodic beam. Its value below and above the ABH cut-on frequency is inspected and a parametric analysis considering variations on the ABH profile and damping layer geometry is carried out. The underlying physics is explained and, finally, a transmissibility analysis on a finite ABH periodic beam with five cells is provided to confirm the mechanisms found for infinite periodic damped ABH beams.

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.

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.

Wave Electromechanical Coupling Factor for the Guided Waves in Piezoelectric Composites

A novel metrics termed the ‘wave electromechanical coupling factor’ (WEMCF) is proposed in this paper, to quantify the coupling strength between the mechanical and electric fields during the passage of a wave in piezoelectric composites. Two definitions of WEMCF are proposed, leading to a frequency formula and two energy formulas for the calculation of such a factor. The frequency formula is naturally consistent with the conventional modal electromechanical coupling factor (MEMCF) but the implementation is difficult. The energy formulas do not need the complicated wave matching required in the frequency formula, therefore are suitable for computing. We demonstrated that the WEMCF based on the energy formula is consistent with the MEMCF, provided that an appropriate indicator is chosen for the electric energy. In this way, both the theoretical closure and the computational feasibility are achieved. A numerical tool based on the wave and finite element method (WFEM) is developed to implement the energy formulas, and it allows the calculation of WEMCF for complex one-dimensional piezoelectric composites. A reduced model is proposed to accelerate the computing of the wave modes and the energies. The analytical findings and the reduced model are numerically validated against two piezoelectric composites with different complexity. Eventually an application is given, concerning the use of the shunted piezoelectric composite for vibration isolation. A strong correlation among the WEMCF, the geometric parameters and the energy transmission loss are observed. These results confirm that the proposed WEMCF captures the physics of the electromechanical coupling phenomenon associated with the guided waves, and can be used to understand, evaluate and design the piezoelectric composites for a variety of applications.

Model reduction schemes for the wave and finite element method using the free modes of a unit cell

The wave and finite element method (WFEM) is an efficient numerical tool for analysing wave propagation characteristics and forced response at intermediate and high frequencies. In this work, we introduce free-interface component modal synthesis (CMS) methods into WFEM to accelerate the calculation while maintaining the accuracy. Several free-interface CMS methods with different approximations of the residual effects are implemented and compared. A new eigenvalue scheme based on the dynamic compliance matrix is proposed. A periodic open thin-wall structure is considered as an application for which both free-wave characteristics and forced responses are computed. Aspects such as accuracy, efficiency, and convergence of the proposed method are discussed and compared with those of the Craig-Bampton fixed-interface CMS method. The methods and main findings are further verified by using another more complex periodic structure. Among the implemented models, the minimum model size is achieved by the exact CMS method. The exact CMS method only requires the modes below the maximum analysing frequency, thereby reducing the model size of the open thin-wall structure from 4416 to 16. The most numerically efficient model for WFEM is MacNeal’s CMS method, where the CPU time of free-wave analysis can be reduced by 97% for the open thin-wall structure.

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.

Enhanced wave and finite element method for wave propagation and forced response prediction in periodic piezoelectric structures

As a promising numerical tool of structural dynamics in mid- and high frequencies, the wave and finite element method (WFEM) is receiving increasingly attention and applications. In this paper, an enhanced WFEM has been developed with a reduced model and a new eigenvalue scheme. The reduced model is applicable for structures with piezoelectric shunts or local dampers; the new eigenvalue scheme can mitigate the ill-conditioning when the wave basis is calculated. The enhanced WFEM is applied to a thin-wall structure with periodically distributed piezoelectric materials (PZT). Both free wave characteristics and forced response are analyzed and the influences of the suggested enhancements are presented. It is shown that if the control factors are properly chosen, these enhancements can improve the accuracy while accelerating the calculation. Resulting from the complexity of the application, these enhancements are not optional but imperative.

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.

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.

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.

Periodically distributed piezoelectric patches optimization for waves attenuation and vibrations damping

The deformation of a structure can be understood as a superposition of waves which are induced by the excitation and reflected by the boundaries. According to this regard, attenuations of waves can lead to a strong reduction of the structural response and prevent energy to be propagated. In this work we consider periodically distributed piezoelectric patches onto the host structure so that by designing shunted electric circuits the properties of the waves can be modified. The success of the idea is also directly related to the extent of electromechanical coupling. In terms of structural modes, the coupling factor can be estimated by the open-circuit (OC) and short-circuit (SC) natural frequencies. However, in terms of waves, few criteria are available. In this work, a wave-based criterion is proposed to evaluate the coupling factor of the piezoelectric composite. To do this, enhanced Wave and Finite Element Method (WFEM) is employed to obtain the dispersion relations and the shapes of the waves. Then, the factor can be calculated in three different but equivalent formulas. An example is given thereafter, where a piezoelectric waveguide with semi-active circuits is used to control the energy flow from a source to the far-field. We show that the coupling factor is frequency dependent and it is strongly related to the geometric parameters; therefore, it significantly changes the optimal performance of the piezoelectric waveguide and it capacity to dampen vibrations.

Energy flow prediction in built-up structures through a hybrid finite element/wave and finite element approach

This paper presents a rapid and accurate numerical tool for the energy flow evaluation in a periodic substructure from the near-field to the far-field domain. Here we suppose that the near-field part contains a point source characterized by the injected power in the structure. The near-field part is then modeled by Finite Element Method (FEM) while the periodic structure and the far-field part are regarded as waveguides and modeled by an enhanced Wave and Finite Element Method (WFEM). Enhancements are made on the eigenvalue scheme, the condensation of the unit cell and the consideration of a reduced wave basis. Efforts are made to adapt substructures modeled by different strategies in a multi-scale manner such that the final matrices dimensions of the built-up structure are largely reduced. The method is then validated numerically and theoretically. An application is presented, where a structural dynamical system coupled with periodic resistive piezoelectric shunts is discussed.

Calculation of reflection and transmission coefficients of joints using a hybrid finite element/wave and finite element approach

Often structures comprise waveguides connected by joints. The knowledge of the reflection and transmission properties of these joints is important for many applications. This paper presents a hybrid approach, combining a finite element (FE) and a wave and finite element (WFE) model, for calculating the reflection and transmission coefficients of a joint. First, the joint is modelled using standard FE. Then, the waveguide is modelled using the WFE method, where the FE model of a small segment of the waveguide, whose cross-section could be arbitrarily complex, is post-processed to yield the wave properties of the whole waveguide. The two models are coupled to find the reflection and transmission matrices of the joint. This hybrid approach allows for developing a model of the structure where the basis functions are the waves travelling through the structure's various waveguides. Numerical examples are presented.

Numerical issues concerning the wave and finite element method for free and forced vibrations of waveguides

The wave and finite element (WFE) method is a numerical approach to the calculation of the wave properties of structures of arbitrary complexity. The method starts from a finite element (FE) model of only a short segment of the structure, typically by using existing element libraries and commercial FE packages. The dynamic stiffness matrix of the segment is obtained, a periodicity condition applied and an eigenvalue problem formed whose solution gives the dispersion equations and wave mode shapes. These define a wave basis from which the forced response can be found straightforwardly. Although straightforward in application, the WFE method is prone to numerical difficulties. These are discussed in this paper and methods to avoid or remove them described. Attention is focused on 1-dimensional waveguide structures, for which numerical problems are most severe. Three ways of phrasing the eigenvalue problem for free wave propagation are presented and a method based on singular value decomposition is proposed to determine eigenvectors at low frequencies. Discretisation errors are seen to occur if the segment is too large, while round-off errors occur if the segment is too small. This can be overcome by forming a super-segment from the concatenation of two or more segments. The forced response is then considered. The use of a reduced wave basis removes many problems. Direct calculation of the waves excited by a point force is very prone to poor numerical conditioning but can be circumvented by exploiting the orthogonality of the left and right eigenvectors. Numerical examples are presented.