Free Wave Propagation Research Articles

Physics-informed neural networks for fully non-linear free surface wave propagation

This study proposed fully nonlinear free surface physics-informed neural networks (FNFS-PINNs), an advancement within the framework of PINNs, to tackle wave propagation in fully nonlinear potential flows with the free surface. Utilizing the nonlinear fitting capabilities of neural networks, FNFS-PINNs offer an approach to addressing the complexities of modeling nonlinear free surface flows, broadening the scope for applying PINNs to various wave propagation scenarios. The improved quasi-σ coordinate transformation and dimensionless formulation of the basic equations are adopted to transform the time-dependent computational domain into the stationary one and align variable scale changes across different dimensions, respectively. These innovations, alongside a specialized network structure and a two-stage optimization process, enhance the mathematical formulation of nonlinear water waves and solvability of the model. FNFS-PINNs are evaluated through three scenarios: solitary wave propagation featuring nonlinearity, regular wave propagation under high dispersion, and an inverse problem of nonlinear free surface flow focusing on the back-calculation of an initial state from its later state. These tests demonstrate the capability of FNFS-PINNs to compute the propagation of solitary and regular waves in the vertical two-dimensional scenarios. While focusing on two-dimensional wave propagation, this study lays the groundwork for extending FNFS-PINNs to other free surface flows and highlights their potential in solving inverse problems.

Elastoacoustic wave propagation in a biphasic mechanical metamateriala).

Humans are sensitive to air-borne sound as well as to mechanical vibrations propagating in solids in the frequency range below 20 kHz. Therefore, the development of multifunctional filters for both vibration reduction and sound insulation within the frequency range of human sensitivity is a research topic of primary interest. In this paper, a high-contrast biphasic mechanical metamaterial, composed of periodic elastic solid cells with air-filled voids, is presented. By opening intercellular air-communicating channels and introducing channel-bridging solid-solid couplings, the frequency dispersion spectrum of the metamaterial can be modified to achieve complete and large bandgaps for acoustic and elastic waves. From a methodological viewpoint, the eigenproblem governing the free wave propagation is solved using a hybrid analytical-computational technique, while the waveform classification is based on polarization factors expressing the fraction of kinetic and elastic energies stored in the solid and fluid phases. Based on these theoretical results, a mechanical metafilter consisting of an array of a finite number of metamaterial cells is conceived to provide a technical solution for engineering applications. The forced response of the metafilter is virtually tested in a computational framework to assess its performance in passively controlling the propagation of broadband sound and vibration signals within solid and fluid environments. Quantitative results synthesized by transmission coefficients demonstrate that the metafilter can remarkably reduce the transmitted response in the frequency band of human sensitivity.

A study of the non-linear properties and wave generation of the multi-layer non-hydrostatic wave model SWASH

Despite the increasing use of non-hydrostatic models in the study of wave processes in coastal regions, there is still limited understanding of the non-linear properties of the governing equations and how they improve with increased vertical resolution. In this study, the governing equations of the non-hydrostatic wave model SWASH are analysed and the linear and non-linear solutions up to third-order of all dependent variables are derived, considering one to four vertical layers. The analysis concludes that the model can achieve excellent non-linear properties with respect to the Stokes theory over a large range of water depths using only a few vertical layers. Furthermore, deriving solutions for all variables enables the formulation of improved wave generation and absorption boundary conditions for non-hydrostatic models. A well-known issue of non-linear wave models is related to the generation and propagation of spurious free waves, resulting to non-homogeneous wave fields. In this study, it is proven that by imposing the derived exact mathematical solutions of the governing equations at the model’s boundaries, the target first- and second-order wave profiles can be generated with high accuracy, while the spurious waves can be entirely eliminated.

Rotation impact on the radial vibrations of frequency equation of waves in a magnetized poroelastic medium

This research delves into the propagation of radial free harmonic waves in a poroelastic cylinder, conceptualized as a magnetically and rotationaly influenced hollow structure. The primary aim is to elucidate the magnetic field's and rotation impact on the vibrational behavior of such systems. The investigative method encompasses the resolution of motion equations, formulated as partial differential equations, through the application of Lame's potential theory. This analytical process is augmented by the implementation of fitting boundary conditions, culminating in the derivation of a comprehensive expression for the complex dispersion equation, predicated on the premise that the wavenumber embodies a complex entity. The precision of the model is corroborated through a comparative analysis with established literature, underpinned by an exploration of diverse scenarios. The research employed MATLAB for both numerical and graphical assessments, focusing on the dispersion and displacement attributes. Dispersion relations within the poroelastic medium were computed, considering varied magnitudes of magnetic field intensity, rotation and angular velocities. The outcomes are articulated through complex-valued dispersion relations, transcendental formulations, and numerical resolutions employing MATLAB's bisection technique. These insights hold substantial significance for the theoretical advancement in orthopedic research, particularly concerning cylindrical poroelastic media. This study deduces that the radial vibrational patterns and the corresponding frequency equation within a poroelastic medium are profoundly modified by the magnetic field's interference and rotation. This study formulate a novel governing equation for a poroelastic medium, highlighting the significance of radial vibrations and investigating the impact of magnetic field and rotation.

Simulation of elastic guided waves in concrete structures: A parametric study for construction materials

Concrete buildings constructed globally are susceptible to diverse circumstances of use and exposure to environmental elements. Diverse varieties of construction materials are used in the building sector to fabricate edifices and constructions. Architects and construction project managers use these classifications of materials and products to precisely determine the materials and techniques employed in building projects. This study presents a simulation of elastic guided waves in concrete structures, which is motivated by the vast variety of uses of such structures. The elastic-guided wave technique is a technology used for nondestructive assessment. This approach utilizes acoustic waves that travel along an extended structure, directed by its limits. This enables the propagation of waves over a significant distance while minimizing energy dissipation. Currently, the Elastic guided wave method is extensively used for the examination and evaluation of various engineering structures, namely for inspecting metallic pipelines globally. From a single place, it is possible to investigate distances of several hundred meters in some instances. Additionally, there are apps available for the examination of rail tracks, rods, and metal plate constructions. In this research, a novel mathematical model is introduced for accurately replicating elastic guided waves in concrete structures in practical applications. To reinforce the current concrete structure using mathematics, graphene oxide powders with improved mechanical properties are obtained using the role of the mixture, and the Halpin-Tsai micromechanical model is used. After obtaining the mathematical modeling of the current concrete structure using the theory of shear deformation with four unknown high-order terms and Hamilton’s principle, the equations are solved using free and forced wave propagation approaches. Finally, some applicable suggestions for improving the efficiency and dynamic stability of the presented concrete structure reinforced by graphene oxide powders are presented in detail in the results section.

MULTI-LINEAR-ELEMENT DEPTH-INTEGRATED MODELS FOR FLOWS WITH A FREE SURFACE

In this study, we derive a new set of depth-integrated models for solving unsteady flows that satisfy the Euler equations and nonlinear free surface boundary conditions in the sigma -coordinates. One of the obvious differences between the present approach and the ones directly solving the 3D Euler equations using Finite Element Method (FEM) is that in the present model the vertical velocity and the pressure field are eliminated by integrating the continuity equation and vertical momentum equation, respectively. Therefore, the present models only solve the horizontal velocity components and the free surface displacement in the two-dimensional horizontal (2DH) space. The new models are also more advantageous in using fewer elements in the vertical direction while achieving better performance. At this stage, the capability and application of the new models in dealing with free surface wave propagation problems are addressed.

Wave propagation in periodic nano structures through second strain gradient elasticity

The study of wave propagation in the periodic nano-waveguides is of importance to advance the development of energy and wave controlling systems. In this paper, firstly, the element discretization of nano structures is introduced based on the second strain gradient theory. The displacement field is determined through the utilization of six quintic Hermite polynomial interpolating functions. By applying Hamilton’s principle, the weak form, which incorporates the element matrices, is determined and the global dynamic stiffness matrix of a unit cell is assembled. Then, the wave finite element method, based on the inverse form, direct form and contour integral solution, is used to analyze the two-dimensional free wave propagation properties of complex nano structures by solving the eigenvalue problems. The interpretation of the frequency spectrum is confined to the irreducible first Brillouin zone, encompassing both the dispersion relation and band structure. Furthermore, the slowness surfaces and energy flow vector fields are discussed via the second strain gradient theory. The novel work, which utilizes the second strain gradient theory equipped with the wave finite element method to investigate two-dimensional complex nano structures, has revealed significant potential in elucidating the two-dimensional wave propagation characteristics of the complex nano-waveguides.

A Power-Based Framework for Quantifying Parameter Uncertainties in Finite Vibroacoustic Metamaterial Plates.

Vibroacoustic metamaterials (VAMMs) are artificial materials that are specifically designed to control, direct, and manipulate sound waves by creating a frequency gap, known as the stop band, which blocks free wave propagation. In this paper, a new power-based approach that relies on the active structural intensity (STI) for predicting the stop band behavior of finite VAMM structures is presented. The proposed method quantifies the power loss in a locally resonant finite VAMM plate in terms of percentage, such as STI99% and STI90%, for stop band prediction. This allows for the quantitative analysis of the vibration attenuation capabilities of a VAMM structure. This study is presented in the context of a two-dimensional VAMM plate with 25 resonators mounted in the middle section of the plate. It has been demonstrated that this method can predict the stop band limits of a finite VAMM plate more accurately than using negative effective mass, unit cell dispersion analysis, or the frequency response function methods. The proposed approach is then implemented to establish a framework for investigating the influence of parameter uncertainties on the stop band behavior of the VAMM plate. Based on the STI99% method, which aims for significant vibration reduction, stricter tolerances in the mass fabrication process are required to ensure the robustness of VAMM. Conversely, the STI90% method suggests that larger fabrication tolerances can be leveraged to achieve a broader stop band range while still meeting the desired performance level, leading to cost savings in manufacturing.

Development and Application of an Infragravity Wave (InWave) Driver to Simulate Nearshore Processes

AbstractInfragravity waves are key components of the hydro‐sedimentary processes in coastal areas, especially during extreme storms. Accurate modeling of coastal erosion and breaching requires consideration of the effects of infragravity waves. Here, we present InWave, a new infragravity wave driver of the Coupled Ocean‐Atmopshere‐Waves‐Sediment Transport (COAWST) modeling system. InWave computes the spatial and temporal variation of wave energy at the wave group scale and the associated incoming bound infragravity wave. Wave group‐varying forces drive free infragravity wave growth and propagation within the hydrodynamic model of the coupled modeling system, which is the Regional Ocean Modeling System (ROMS) in this work. Since ROMS is a three‐dimensional model, this coupling allows for the combined formation of undertow currents and infragravity waves. We verified the coupled InWave‐ROMS with one idealized test case, one laboratory experiment, and one field experiment. The coupled modeling system correctly reproduced the propagation of gravity wave energy with acceptable numerical dissipation. It also captured the transfer of energy from the gravity band to the infragravity band, and within the different infragravity bands in the surf zone, the measured three‐dimensional flow structure, and dune morphological evolution satisfactorily. The idealized case demonstrated that the infragravity wave variance depends on the directional resolution and horizontal grid resolution, which are known challenges with the approach taken here. The addition of InWave to COAWST enables novel investigation of nearshore hydro‐sedimentary dynamics driven by infragravity waves using the strengths of the other modeling components, namely the three‐dimensional nature of ROMS and the sediment transport routines.

Semi-Analytical Finite-Element Analysis for Free and Forced Wave Propagation Using COMSOL and LiveLink for Matlab

The Semi-Analytical Finite-Element (SAFE) method represents one of the most established numerical approaches for predicting the propagation of elastic waves in one-dimensional structures of arbitrary cross-sections. Its implementation in the commercial finite-element software COMSOL Multiphysics has been proposed in recent years; however, it is limited to only the free wave propagation for computing dispersion curves. To overcome this limitation, this paper proposes an extension of this approach that combines COMSOL and its Livelink for Matlab tool. This enables the extraction from COMSOL of the assembled mass and stiffness SAFE matrices to solve problems of both free and forced wave propagation in the Matlab environment. The resulting customised software takes advantage of both the potential of commercial FE software and the power of Matlab without worrying about compatibility issues. A model of a simply supported plate strip and that of a more complex geometry are implemented to validate, respectively, the SAFE matrix extraction procedure and the implemented forced response formulation. The results agree well with corresponding analytical and numerical results validating the proposed implementation of the SAFE method.

Strategic distribution of resonators' parameters for broadband vibration mitigation

Locally resonant metamaterials have recently come to the fore as novel lightweight and compact Noise, Vibrations and Harshness (NVH) solutions. They can be designed/obtained by assembling nominally equal resonant elements, which are tuned to the desired frequency range, onto a host structure in a sub-wavelength manner. This combination leads to tunable frequency ranges, known as stop bands i.e. frequency regions where free wave propagation is not allowed. Nonetheless, these stop bands are typically only effective in a limited frequency region. In order to broaden the frequency range of noise and vibration reduction, an optimization procedure could be applied in order to obtain the ideal distribution of resonance frequencies and masses to achieve a minimal sound radiation over a certain frequency region. However, this approach typically requires a considerable computational time. For this reason, this paper proposes a set of guidelines that define a strategy for broadband NVH reduction without the use of optimization. The proposed strategy is applied to a finite plate with a predefined added grid for the resonator positions. The performance of the strategy is numerically analyzed by assessing the vibration response of the plate for multiple excitation locations and within different frequency ranges. The obtained responses are then compared to the response of the plate with an optimized grid of resonators. This study shows that the proposed strategy leads to a reasonable vibration reduction with similar levels with respect to the response obtained from optimization.

Natural waves in a spatial viscoelastic cylinder with a radial crack

The investigation of the dispersion of waves in elastic bodies with different cutouts is of great interest in various scientific and technological fields. The objective of this research is to examine the propagation of free damped waves in elastic dissipative bodies with longitudinal notches. The dynamic behavior of cylindrical bodies with different cutouts is described by the equations of viscoelasticity. The solution of a system of differential equations is expressed by cylindrical Bessel and Hankel functions. The frequency equation is solved using the Muller, Gauss, and orthogonal run methods. It is found that with a decrease in the oscillation frequency in a cylinder of a given radius, the phase and group velocities of the longitudinal wave tend to the common limit - the phase velocity of the rod waves. It is found that as the oscillation frequency reduces in a cylinder of a given radius, the phase and group velocities of the longitudinal wave tend to converge towards the common limit. Additionally, it was found that for a specific value of the Poisson's ratio (0.2833), the frequencies of the shear and radial longitudinal resonances coincide in the cylinder.

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.

Multifield nested metafilters for wave propagation control

The present work proposes a novel class of multifield nested tunable metadevices that serve as high performance acoustic metafilters. The designed metafilter is characterized by a multiscale, hierarchical structure. At the mesoscale, the metamaterial consists of a sequence of two different periodically alternating layers: a polymeric homogeneous layer, which exhibits a viscoelastic constitutive response, and a microstructured one. The latter is based on the periodic repetition of a multiphase microscale cell that is composed by a stiff elastic external coating, a viscoelastic phase and an internal disk of piezoelectric material shunted by an external electrical circuit having a tunable impedance/admittance This tuning parameter affects the constitutive elastic properties of the piezoelectric phase and, in turn, the overall response of the microscale cell, thereby ultimately enabling to achieve an optimal filtering performance for the metadevice. Due to the periodicity of the multiphase cell at the microscale, a two-scale variational-asymptotic homogenization technique is exploited in the frequency domain in order to obtain the frequency-dependent overall constitutive properties of the microstructured layer. Subsequently, in-plane free wave propagation inside the periodic multilayered metamaterial at the mesoscale is investigated by means of Floquet–Bloch theory, together with the transfer matrix method. By triggering the shunting effect, a stiffening of the piezoelectric phase can be achieved, which is demonstrated to open low frequency band gaps in the metamaterial frequency spectrum. The filtering capability of the metadevice has been assessed as a function of its geometrical features and the tuning parameter, thus paving the way towards the design of sophisticated and topologically optimized acoustic filters.

Radial and Axial Displacement of the Initially-Tensioned Orthotropic Arterial Wall Under the Influence of Harmonics and Wave Reflection

Abstract This study examines radial and axial displacement of the arterial wall under the influence of harmonics and wave reflection for the role of axial wall displacement in pulsatile wave propagation. The arterial wall is modeled as an initially-tensioned thin-walled orthotropic tube. In conjunction with three pulsatile parameters in blood flow, a free wave propagation analysis is conducted on the governing equations of the arterial wall and no-slip conditions at the blood-wall interface to obtain the frequency equation and pulsatile parameter expressions under different harmonics. The influence of wave reflection is then added to pulsatile parameter expressions. With the harmonic values of measured pulsatile pressure and blood flow rate at the ascending aorta in the literature, the waveforms of radial wall displacement, axial wall displacement, and wall shear stress are calculated under different orthotropicity and axial initial tension. The developed theory and calculated results indicate that (1) difference in waveform between blood flow rate, wall shear stress, and axial wall displacement is caused by harmonics, rather than wave reflection; (2) Axial wall displacement does not affect blood flow rate, radial wall displacement, and wall shear stress; (3) Besides wall shear stress, radial wall displacement gradient also contributes to axial wall displacement and its contribution is adjusted by axial initial tension; (4) different wave reflections only noticeably affect the maximum and minimum values of wall shear stress; and (5) The amplitude and waveform of axial wall displacement are predominantly dictated by axial elasticity and axial initial tension, respectively.

An Insight on the Estimation of Wave Propagation Constants in an Orthogonal Grid of a Simple Line-Supported Periodic Plate Using a Finite Element Mathematical Model

This article describes the propagation of free waves in a two-dimensional periodic plate using the finite element (FE) method. The advantage of periodic structure analysis is that all the dynamic properties of a finite structure are derived from a single phase-frequency curve or surface. Infinite plates are considered as a combination of periodic plates on an orthogonal array of simple, evenly spaced line supports. A single periodic unit of the system is represented by a more accurate high-precision arbitrary triangular shallow shell FE model to find the plane wave frequency in terms of the propagation constants of the 2D periodic plate. Only the purely propagating waves with no attenuation are considered here. The natural frequency of the infinite plate was obtained for different propagation constants in the two directions of the plate. The results are compared with the literature data. The bounding frequency of the propagation surface is compared to the data published from single square and rectangular plates with different edge boundary conditions. In addition, the natural frequency of the plate supported by finite line support with spans Nx (x-direction) and Ny (y-direction) is compared with the frequency obtained from the propagation curve by the discretization principle. The comparison is seen to be very close. It is found that the current PS-FEM approach can be used to generate dispersion relations with reasonable accuracy.

Wave propagation in viscoelastic metamaterials via added-state formulation

Mechanical metamaterials composed of a non dissipative periodic microstructure of flexible ligaments and stiff rings, viscoelastically coupled with local resonators, are considered. By following a variational approach, the linear damped dynamics are described according to a beam lattice formulation, valid for a generic coordination number. A consistent theoretical description of the viscoelastic ring-resonator coupling is introduced, leading to an integral–differential form of the governing equations. The enlargement of the model dimension by addition of viscoelastic states allows to conveniently describe the metamaterial dynamics through a system of ordinary differential equations. Therefore, the application of the Z-transform and a suitable mapping in the wavevector space allows to obtain the system of ordinary differential equations governing the free and forced propagation of damped waves. Subsequently, the polynomial eigenproblem ruling the harmonic free wave propagation in the complex frequency domain is stated by applying the bilateral Laplace transform, whereas an analytical procedure is outlined to solve the forced wave propagation problem in the time domain. The triangular beam lattice metamaterial is considered as significant benchmark for application of the theoretical and methodological framework. Considering first the free wave propagation, the properties of the dynamic stiffness matrix are discussed and the dispersion frequency spectrum is parametrically investigated, with focus on the spectral effects of the viscoelastic coupling. The dependence of the stop bandwidth amplitude between the low and high-frequency spectral branches on the relaxation time characterizing the viscous function is also highlighted and discussed. Moreover, the forced response to a harmonic external point source is numerically investigated in the time domain. The results are qualitatively compared and quantitatively discussed by distinguishing the fundamental cases of non-resonant, resonant and quasi-resonant excitation frequency.

Rotating radial vibrations in human bones (femoral, mandibular and tibia) and crystals (Mg, Co, Cd, Zn and beryl) made cylindrical shell under magnetic field and hydrostatic stress

Radial free harmonic wave propagation is considered for the investigation in cylindrical shells made of orthotropic human bones, transversely isotropic hexagonal crystals and simply isotropic materials under the effect of magnetic field and hydrostatic state of stress. Numerical investigations are performed for the frequency equation through 2D and 3D graphical visuals for the impact of rotation, hydrostatic stress, axial magnetic field, inner and outer radii of the cylindrical shell on the radial phase velocity of the wave. A comparative study has been developed for femoral, tibia, mandibular buccal and lingua bones, Mg, Co, Cd, Zn, beryl and ice hexagonal crystals.

Interval uncertainty propagation in metamaterial beams using machine learning based optimization

Metamaterials have recently emerged in the search for lightweight noise and vibration solutions. One of their appealing properties for noise control engineering is the ability to create stop bands, which are frequency ranges without free wave propagation. These stop bands arise from the sub-wavelength addition of identically tuned resonators in or on a host structure. However, when manufacturing metamaterials, variability in material properties and geometry is inevitably introduced. On the one hand, the metamaterial attenuation performance can deteriorate due to variability, while on the other hand, variability can even broaden their typically narrowband performance. During the early phases in the design, when often little information concerning the inherent variability is available, it would be desirable to be able to assess the input uncertainty effects on the performance of metamaterials. This work focuses on the numerical assessment of the impact of uncertainties in resonator properties on the vibration attenuation performance of metamaterial beams. To this end, a machine learning based optimization strategy, the so-called Bayesian optimization, is employed in a recently developed non-intrusive uncertainty propagation approach. This enables a very efficient evaluation of the upper and lower bounds on the metamaterial performance, with the uncertain resonator parameters defined as interval variables.

Prediction of diffuse sound transmission through finite-sized periodic structures

The diffuse sound transmission through a wall or floor can be efficiently computed with a hybrid approach. The wall is then modelled deterministically as finite size effects and modal behaviour can be important in the considered frequency range, while the rooms carry a diffuse field and are modelled as stochastic subsystems. The finite element method is usually employed to compute the modal behaviour of the wall. At higher frequencies, the computational cost then increases significantly as a fine mesh is required due to the short wavelength of the structural deformation. For this reason, an alternative approach was recently developed for finite-sized thick and layered walls, which allows to replace the finite element model by an analytical model. However, the application of this approach is limited to layered structures such as sandwich panels or double glazing. In the present work, the extension towards more complicated, finite-sized building elements which exhibit spatial periodicity, is considered by invoking periodic structure theory. The propagating waves resulting from the free wave propagation analysis of the periodic unit cell, are combined into standing waves, which satisfy the simply supported boundary conditions. A Fourier transform of the system of equations allows for a fast conversion of the mode shapes of the unit cell to the entire finite-sized structure. The methodology is illustrated by comparing the predicted sound insulation for two cross laminated timber panels and two periodic rib-stiffened panels with alternative predictions involving a full finite element model of the entire structure, and experimental data.