Transient Wave Propagation Research Articles

Local absorbing boundary conditions to simulate wave propagation in unbounded viscoelastic domains

In this paper, Local Absorbing Boundary Conditions are presented for wave propagation through the viscoelastic media. This method is an extension of Local Absorbing Boundary Condition (ABC) proposed by Lysmer and Kuhlemeyer (1969). The proposed method does not converge for Kelvin type of viscoelastic materials but converges for Maxwell type of viscoelastic material model. This method is verified with 1D and 2D finite element models using explicit solver and the accuracy is compared with standard solutions. This study concluded that better responses can be obtained for the viscoelastic wave propagation using present approach when compared with traditional Local Absorbing Boundary Conditions. This study also concluded that the proposed method is suitable for low to medium viscous damping materials. For high viscous materials, the proposed method can be applied to transient wave propagation problems.

Composite dynamic models for periodically heterogeneous media

Propagation of elastic waves through discrete and continuous periodically heterogeneous media is studied. A two-scale asymptotic procedure allows us to derive macroscopic dynamic equations applicable at frequencies close to the resonant frequencies of the unit cells. Matching the asymptotic solutions by two-point Padé approximants, we obtain new higher-order equations that describe the dynamic behaviour of the medium both in the low and in the high frequency limits. An advantage of the proposed approach is that all the macroscopic parameters can be determined explicitly in terms of the microscopic properties of the medium. Dispersion diagrams are evaluated and the propagation of transient waves induced by pulse and harmonic loads is considered. The developed analytical models are verified by comparison with data of numerical simulations. For high-contrast media, we can observe an analogy between the propagation of waves in heterogeneous solids and in thin-walled waveguides. It is also shown that different combinations of cell resonances may result in some additional types of waves that do not appear in the classical continuous theory.

Analytical D’Alembert Series Solution for Multi-Layered One-Dimensional Elastic Wave Propagation with the Use of General Dirichlet Series

A general initial-boundary value problem of one-dimensional transient wave propagation in a multi-layered elastic medium due to arbitrary boundary or interface excitations (either prescribed tractions or displacements) is considered. Laplace transformation technique is utilised and the Laplace transform inversion is facilitated via an unconventional method, where the expansion of complex-valued functions in the Laplace domain in the form of general Dirichlet series is used. The final solutions are presented in the form of finite series involving forward and backward travelling wave functions of the d’Alembert type for a finite time interval. This elegant method of Laplace transform inversion used for the special class of problems at hand eliminates the need for finding singularities of the complex-valued functions in the Laplace domain and it does not need utilising the tedious calculations of the more conventional methods which use complex integration on the Bromwich contour and the techniques of residue calculus. Justification for the solutions is then considered. Some illustrations of the exact solutions as time-histories of stress or displacement of different points in the medium due to excitations of arbitrary form or of impulsive nature are presented to further investigate and interpret the mathematical solutions. It is shown via illustrations that the one-dimensional wave motions in multi-layered elastic media are generally of complicated forms and are affected significantly by the changes in the geometrical and mechanical properties of the layers as well as the nature of the excitation functions. The method presented here can readily be extended for three-dimensional problems. It is also particularly useful in seismology and earthquake engineering since the exact time-histories of response in a multi-layered medium due to arbitrary excitations can be obtained as finite sums.

The Tropospheric Pathway of the ENSO–North Atlantic Teleconnection

Abstract El Niño–Southern Oscillation (ENSO) exerts an influence on the North Atlantic–European (NAE) region. However, this teleconnection is nonlinear and nonstationary owing to the superposition and interaction of a multitude of influences on this region. The stratosphere is one of the major players in terms of the influence of the ENSO signal on this sector. Nevertheless, there are tropospheric dynamical links between the North Pacific and the North Atlantic that are clearly influenced by ENSO. This tropospheric pathway of ENSO to the NAE has received less attention. In view of this, the present study revisits the tropospheric pathway of ENSO to the North Atlantic using ECMWF reanalysis products. Anomalous propagation of transient and quasi-stationary waves across North America is analyzed with respect to their sensitivity to ENSO. Transient (quasi-stationary zonal waves 1–3) wave activity flux (WAF) from the Pacific to the Atlantic increases during El Niño (La Niña) conditions leading to a negative (positive) phase of the North Atlantic Oscillation (NAO). This response is observed from January to March for El Niño and only visible during February for La Niña events. However, the stratosphere strongly modulates this response. For El Niño (La Niña) conditions a weaker (stronger) stratospheric vortex tends to reinforce the negative (positive) NAO with the stratosphere and troposphere working in tandem, contributing to a stronger and more persistent tropospheric circulation response. These findings may have consequences for the prediction of the NAO during times with an inactive stratosphere.

Research on Transient Wave Propagation Across Nonlinear Joints Filled with Granular Materials

A nonlinear thin layer interface model overcoming the long-wavelength assumption of displacement discontinuity model is developed to analyze the full-wavelength wave propagation across nonlinear parallel joints. In this study, the filling material is treated as a thin layer with reduced mechanical properties to reveal multiple reflections and time shifting within the filled joint. The nonlinearity of the filling material is considered, and a recursive matrix is derived in time domain. An experimental study on wave propagation across a filled joint was carried out by SHPB test. The quartz sand layer of different filling thicknesses sandwiched between Hopkinson bars was pressured by compressional waves to investigate the wave attenuation. Comparisons of the joint thickness, the wave frequency and the incident angle are carried out between the present model and the existing displacement discontinuity model. The results indicate that the thin layer interface model considering the thickness of the joint is capable of extending the long-wavelength assumption to full-wavelength research and it is more appropriate for filled joints with thick thickness that is comparable to wavelength. Then, this model is extended to parallel joints, and the properties of the filling material (i.e., initial elastic modulus and maximum closure), impact velocity, incident angle and wave frequency on wave attenuation are discussed for a joint set. The spacing dependency of the transmission coefficient for parallel joints is compared with displacement discontinuity model.

Nonlocal theory of thermoelastic materials with voids and fractional derivative heat transfer

ABSTRACTA new nonlocal theory of generalized thermoelastic materials with voids based on Eringen’s nonlocal elasticity and Caputo fractional derivative is established. The one-dimensional form of the above theory is then applied to study transient wave propagation in an infinite thermoelastic materials with voids due to a time-dependent continuous heat sources distributed in a plane area. Laplace transform and eigenvalue approach techniques are used to obtain the closed form solution in the transform domain. Numerical inversions of the studied physical variables are carried out by Zakian algorithm in the space-time domain. Numerical results are plotted graphically in some cases and the results obtained are analyzed. Some comparisons for different cases are also noted.

Transient implicit wave propagation dynamics with overlapping finite elements

We present novel overlapping finite elements used with the Bathe time integration method to solve transient wave propagation problems. The solution scheme shows two important properties that have been difficult to achieve in the numerical solution of general wave propagations: monotonic convergence of calculated solutions with decreasing time step size and a solution accuracy almost independent of the direction of wave propagation through the mesh. The proposed scheme can be efficiently used with irregular meshes. These properties make the scheme (the combined spatial and temporal discretizations) promising to solve general wave propagation problems in complex geometries involving multiple waves. A dispersion analysis is given and various example problems are solved to illustrate the performance of the solution scheme.

A quadratic b-spline based isogeometric analysis of transient wave propagation problems with implicit time integration method

• Wave dispersion error with both the spatial isogeometric elements and direct time integration method is first studied. • New generalized numerical algorithm is proposed to calculate the dispersion errors. • New isogeometric analysis for wave problems are proposed with desirable accuracy and computation efficiency. A uniform quadratic b-spline isogeometric element is exclusively considered for wave propagation problem with the use of desirable implicit time integration scheme. A generalized numerical algorithm is proposed for dispersion analysis of one-dimensional (1-D) and two-dimensional (2-D) wave propagation problems where the quantified influence of the defined CFL number on wave velocity error is analyzed and obtained. Meanwhile, the optimal CFL (Courant–Friedrichs–Lewy) number for the proposed 1-D and 2-D problems is suggested. Four representative numerical simulations confirm the effectiveness of the proposed method and the correctness of dispersion analysis when appropriate spatial element size and time increment are adopted. The desirable computation efficiency of the proposed isogeometric method was confirmed by conducting time cost and calculation accuracy analysis of a 2-D numerical example where the referred FEM was also tested for comparison.

Improved quadratic isogeometric element simulation of one-dimensional elastic wave propagation with central difference method

Two improved isogeometric quadratic elements and the central difference scheme are used to formulate the solution procedures of transient wave propagation problems. In the proposed procedures, the lumped matrices corresponding to the isogeometric elements are obtained. The stability conditions of the solution procedures are also acquired. The dispersion analysis is conducted to obtain the optimal Courant-Friedrichs-Lewy (CFL) number or time-step sizes corresponding to the spatial isogeometric elements. The dispersion analysis shows that the isogeometric quadratic element of the fourth-order dispersion error (called the isogeometric analysis (IGA)-f quadratic element) provides far more desirable numerical dissipation/dispersion than the element of the second-order dispersion error (called the IGA-s quadratic element) when appropriate time-step sizes are selected. The numerical simulations of one-dimensional (1D) transient wave propagation problems demonstrate the effectiveness of the proposed solution procedures.

Elastodynamic and wave propagation analysis in a FG graphene platelets-reinforced nanocomposite cylinder using a modified nonlinear micromechanical model

This paper deals with the transient dynamic analysis and elastic wave propagation in a functionally graded graphene platelets (FGGPLs)-reinforced composite thick hollow cylinder, which is subjected to shock loading. A micromechanical model based on the Halpin-Tsai model and rule of mixture is modified for nonlinear functionally graded distributions of graphene platelets (GPLs) in polymer matrix of composites. The governing equations are derived for an axisymmetric FGGPLs-reinforced composite cylinder with a finite length and then solved using a hybrid meshless method based on the generalized finite difference (GFD) and Newmark finite difference methods. A numerical time discretization is performed for the dynamic problem using the Newmark method. The dynamic behaviors of the displacements and stresses are obtained and discussed in detail using the modified micromechanical model and meshless GFD method. The effects of the reinforcement of the composite cylinder by GPLs on the elastic wave propagations in both displacement and stress fields are obtained for various parameters. It is concluded that the proposed micromechanical model and also the meshless GFD method have a high capability to simulate the composite structures under shock loadings, which are reinforced by FGGPLs. It is shown that the modified micromechanical model and solution technique based on the meshless GFD method are accurate. Also, the time histories of the field variables are shown for various parameters.

Analytical solution for the transient wave propagation of a buried cylindrical P-wave line source in a semi-infinite elastic medium with a fluid surface layer

This article develops an analytical solution for the transient wave propagation of a cylindrical P-wave line source in a semi-infinite elastic solid with a fluid layer. The analytical solution is presented in a simple closed form in which each term represents a transient physical wave. The Scholte equation is derived, through which the Scholte wave velocity can be determined. The Scholte wave is the wave that propagates along the interface between the fluid and solid. To develop the analytical solution, the wave fields in the fluid and solid are defined, their analytical solutions in the Laplace domain are derived using the boundary and interface conditions, and the solutions are then decomposed into series form according to the power series expansion method. Each item of the series solution has a clear physical meaning and represents a transient wave path. Finally, by applying Cagniard's method and the convolution theorem, the analytical solutions are transformed into the time domain. Numerical examples are provided to illustrate some interesting features in the fluid layer, the interface and the semi-infinite solid. When the P-wave velocity in the fluid is higher than that in the solid, two head waves in the solid, one head wave in the fluid and a Scholte wave at the interface are observed for the cylindrical P-wave line source.

More Frequent Sudden Stratospheric Warming Events due to Enhanced MJO Forcing Expected in a Warmer Climate

Sudden stratospheric warming (SSW) events influence the Arctic Oscillation and midlatitude extreme weather. Observations show SSW events to be correlated with certain phases of the Madden–Julian oscillation (MJO), but the effect of the MJO on SSW frequency is unknown, and the teleconnection mechanism, its planetary wave propagation path, and time scale are still not completely understood. The Arctic stratosphere response to increased MJO forcing expected in a warmer climate using two models is studied: the comprehensive Whole Atmosphere Community Climate Model and an idealized dry dynamical core with and without MJO-like forcing. It is shown that the frequency of SSW events increases significantly in response to stronger MJO forcing, also affecting the averaged polar cap temperature. Two teleconnection mechanisms are identified: a direct propagation of MJO-forced transient waves to the Arctic stratosphere and a nonlinear enhancement of stationary waves by the MJO-forced transient waves. The MJO-forced waves propagate poleward in the lower stratosphere and upper troposphere and then upward. The cleaner results of the idealized model allow identifying the propagating signal and suggest a horizontal propagation time scale of 10–20 days, followed by additional time for upward propagation within the Arctic stratosphere, although there are significant uncertainties involved. Given that the MJO is predicted to be stronger in a warmer climate, these results suggest that SSW events may become more frequent, with possible implications on tropospheric high-latitude weather. However, the effect of an actual warming scenario on SSW frequency involves additional effects besides a strengthening of the MJO, requiring further investigation.

Analysis of the transient surface wave propagation in soft-solid elastic plates.

In dynamic elastography, the goal is to estimate the Young's modulus from audio-frequency wave propagation in soft-tissues. Within this frequency range, the shear wavelength is centimeter-sized while the compressional wavelength is meter-sized. Thus, the experimental data are usually collected in the near-field of the source. Near-field effects have been widely studied for bulk wave propagation. However, the near- and transient-fields of surface and guided waves have received less attention. In this work, the transient surface displacement field in soft-solid elastic plates in vacuum is analyzed. Due to the high Poisson's ratio, mode conversion has special characteristics in soft-solids. They are analyzed through this work where it is shown that the transient-field over the surface can be interpreted by tracing a few reflections. The authors show the existence of a critical distance needed for the formation of Rayleigh-Lamb modes. Below this distance, only direct surface waves propagate without contribution from reflected waves. Thus, the dispersion curve differs from that predicted by Rayleigh-Lamb modes. Instead, the authors propose a model based on the interference of surface waves, which agree with the experimental data. In addition, the conditions needed in order to retrieve the shear wave phase velocity from the surface field are given.

Transient Wave Scattering and Its Influence on Transient Analysis and Leak Detection in Urban Water Supply Systems: Theoretical Analysis and Numerical Validation

This paper investigates the impacts of non-uniformities of pipe diameter (i.e., an inhomogeneous cross-sectional area along pipelines) on transient wave behavior and propagation in water supply pipelines. The multi-scale wave perturbation method is firstly used to derive analytical solutions for the amplitude evolution of transient pressure wave propagation in pipelines, considering regular and random variations of cross-sectional area, respectively. The analytical analysis is based on the one-dimensional (1D) transient wave equation for pipe flow. Both derived results show that transient waves can be attenuated and scattered significantly along the longitudinal direction of the pipeline due to the regular and random non-uniformities of pipe diameter. The obtained analytical results are then validated by extensive 1D numerical simulations under different incident wave and non-uniform pipe conditions. The comparative results indicate that the derived analytical solutions are applicable and useful to describe the wave scattering effect in complex pipeline systems. Finally, the practical implications and influence of wave scattering effects on transient flow analysis and transient-based leak detection in urban water supply systems are discussed in the paper.

On the propagation of transient waves in a viscoelastic Bessel medium

In this paper we discuss the uniaxial propagation of transient waves within a semi-infinite viscoelastic Bessel medium. First, we provide the analytic expression for the response function of the material as we approach the wave-front. To do so, we take profit of a revisited version of the so called Buchen-Mainardi algorithm. Secondly, we provide an analytic expression for the long time behavior of the response function of the material. This result is obtained by means of the Tauberian theorems for the Laplace transform. Finally, we relate the obtained results to a peculiar model for fluid-filled elastic tubes.

Nonlinear ultrasound simulations using the discontinuous Galerkin method

Histotripsy with nonlinear ultrasound is an emerging noninvasive therapeutic modality that generates cavitation with high-intensity shock waves to precisely destroy diseased soft tissue. Numerical simulation of these shock waves in nonlinear, absorbing media is needed to characterize histotripsy systems and optimize treatments. We are developing a discontinuous Galerkin code based on the Westervelt equation to simulate transient wave propagation in the brain and skull. The discontinuous Galerkin method is a good choice for this simulation problem since this approach has high-order accuracy, geometric flexibility, low dispersion error, and excellent scalability on massively parallel machines. The Westervelt equation is formulated in a first-order flux form and discretized using a strong form of the discontinuous Galerkin method. Numerical results, both linear and nonlinear, from 1D and 2D discontinuous Galerkin codes, are presented and compared to both analytical and numerical benchmark solutions. In particular, the discontinuous Galerkin method captures nonlinear steepening of a high-intensity pulse with minimal numerical artifacts. The development of a 3D massively parallel code is also briefly discussed. [This work was supported in part by a grant from the Focused Ultrasound Foundation.]

Transient Influence Zone Based Decomposition of Water Distribution Networks for Efficient Transient Analysis

Computational efficiency and accuracy of transient analysis for urban water distribution networks (WDN) become progressively important to the design and management of the system. In addition to the improvement of numerical model and computational capacity, which has been widely studied in the literature, efficient and accurate treatment of practical and complex WDN is another potential way to enhance the transient analysis. This paper aims to develop a zonal method for effective decomposition of WDN, which is mainly based on the transient sources and their influence regions in the system, in order to achieve efficient transient analysis. A concept of transient influence zone (TIZ) is firstly proposed and implemented to demonstrate the critical influence region of transient wave propagation in the system under specific design criteria. The obtained TIZ for each transient source is then mapped by introducing appropriate and equivalent boundaries so as to separate the TIZ from the entire WDN. To this end, the efficient Lagrangian model for prior-estimating pressure fluctuation extremes, the pressure fluctuation limitation for mapping TIZ borders and the quasi-reservoir condition for representing border boundaries are applied for characterizing the TIZs. A realistic network is adopted to demonstrate the applicability and accuracy of the proposed method. The application results and analysis indicate that the developed TIZ-based decomposition method provides a considerable efficiency improvement for transient analysis with sufficient modeling accuracy.

Fluid-structure interaction in straight pipelines with different anchoring conditions

This investigation aims at assessing the fluid-structure interaction (FSI) occurring during hydraulic transients in straight pipeline systems fixed to anchor blocks. A two mode 4-equation model is implemented incorporating the main interacting mechanisms: Poisson, friction and junction coupling. The resistance to movement due to inertia and dry friction of the anchor blocks is treated as junction coupling. Unsteady skin friction is taken into account in friction coupling. Experimental waterhammer tests collected from a straight copper pipe-rig are used for model validation in terms of wave shape, timing and damping. Numerical results successfully reproduce laboratory measurements for realistic values of calibration parameters. The novelty of this paper is the presentation of a 1D FSI solver capable of describing the resistance to movement of anchor blocks and its effect on the transient pressure wave propagation in straight pipelines.

Perfectly matched layers for convex truncated domains with discontinuous Galerkin time domain simulations

This paper deals with the design of perfectly matched layers (PMLs) for transient acoustic wave propagation in generally-shaped convex truncated domains. After reviewing key elements to derive PML equations for such domains, we present two time-dependent formulations for the pressure–velocity system. These formulations are obtained by using a complex coordinate stretching of the time-harmonic version of the equations in a specific curvilinear coordinate system. The final PML equations are written in a general tensor form, which can easily be projected in Cartesian coordinates to facilitate implementation with classical discretization methods. Discontinuous Galerkin finite element schemes are proposed for both formulations. They are tested and compared using a three-dimensional benchmark with an ellipsoidal truncated domain. Our approach can be generalized to domains with corners.

An enriched finite element model for wave propagation in fractured media

In this paper a new numerical method has been developed in the context of enriched finite element methods (FEMs) to analyze wave propagation in fractured media. The method combines the advantages of global enrichment with harmonic functions via the Generalized FEM (GFEM) with the efficacy of the Phantom Node Method (PNM), an eXtended FEM (XFEM) variant, to model cracks independently of the mesh. The GFEM enrichment suppresses the spurious oscillation that appear in regular FEM analysis of transient wave propagations due to numerical dispersion and Gibb's phenomenon. For use in explicit simulations, a mass lumping methodology has been introduced with a critical time step size that is both similar to that of the underlining FEM and independent of the location of the fracture. Through three examples, the developed PNM-GFEM is demonstrated to more accurately model wave propagation in fractured media than either the FEM or the PNM/XFEM.