One-dimensional Wave Propagation Research Articles

Steady-state periodic solutions of the nonlinear wave propagation problem of a one-dimensional lattice using a new methodology with an incremental harmonic balance method that handles time delays

A new methodology is developed in this work to solve a one-dimensional (1D) nonlinear wave propagation problem. In its response, the space variable is converted to time functions at different locations, and the resultant time functions are merged with the time variable. Since the merged variables are essentially time-delay variables, the main point of the methodology is that the governing equation of the nonlinear wave propagation problem can be converted to a corresponding nonlinear delay differential equation (DDE) with multiple delays. The new methodology is shown how to formulate the converted DDE, and a modified incremental harmonic balance method is used to solve the 1D nonlinear DDE by introducing a delay matrix operator, where a formula of the Jacobian matrix is derived and can be efficiently and automatically used in the Newton method. This new methodology is demonstrated by solving examples of 1D monatomic chains under the nonlinear Hertzian contact law and with cubic springs. Results well match those in previous works, but derivation effort of solutions and calculation time can be significantly reduced here. When the algorithm of the methodology is compiled as a computer program, there is no additional derivation required to solve wave propagation problems associated with different governing equations.

Seismic harmonic response of inhomogeneous soil: scaling analysis

The study of the seismic response of inhomogeneous soil deposits, underlying rigid bedrock, has traditionally been framed as a one-dimensional wave propagation problem in a linear viscoelastic medium. When it comes to modelling soil inhomogeneity, the current trend to model the continuous variation of soil stiffness with depth employs a ‘generalised parabolic function’. Exact solutions have already been obtained, but these come in terms of Bessel functions, which muddle the interpretation of results and obstruct assessment of the parameters’ influence. This also hinders the definition of an ‘equivalent homogeneous’ soil deposit (a relevant concept utilised in many seismic codes). It is shown herein that the so-called inhomogeneity factor plays a secondary role in determining the response in many cases; thus straightforward guidelines are suggested for simplifying the problem, leading to elementary scaling relations. The scalings provide simple yet meaningful relations that reveal and explain the fundamental traits of the dynamic behaviour of these systems.

One-dimensional scalar wave propagation in multi-region domains by the boundary element method

Two boundary element method formulations are presented for the analysis of the one-dimensional scalar wave propagation problem in multi-region domains. One of the formulations employs the time-domain fundamental solution; the other, the fundamental solution related to the static problem. The problem domain is constituted of sub-domains with different material properties and, consequently, with different wave propagation velocities. Thus, the sub-region technique, akin to the boundary element method, is used to deal with the non-homogeneous problem. At the end of the article, some examples are presented, illustrating the characteristics of each formulation and their accuracy.

Vibration transfer characteristic of gradient-like structure based on magnetorheological fluid

In this paper, a gradient-like structure composed of magnetorheological (MR) fluids is proposed, and its vibration transfer characteristic is studied through the modeling, numerical calculation and experimental test. Under the action of an externally applied magnetic field, the MR fluid exhibits the liquid-solid transformation property: the process of transformation between solid and liquid in fact is the change of vibration-transfer impedance. Therefore, based on this property, the gradient-like structure is constructed by controlling the external magnetic field. Based on the wave equation of one-dimensional elastic wave propagation, the wave equation of elastic wave transfer in the gradient-like structure is established. In order to describe the relationship between the complex shear modulus and Lame constant of MR fluid and magnetic field intensity in the wave equation, the equivalent parameter model of MR fluid is established based on the theory of elasticity and viscoelastic materials. Then, the experimental set-up is built to modify this model through experiments. Afterward, the discretization method of continuous medium and transfer matrix method are adopted to solve the wave equation, and the expression of vibration level drop is obtained. Through the numerical calculation, the trend of vibration level drop varying with the frequency of incident elastic wave and the intensity of magnetic field for the gradient-like structure is obtained. Finally, the vibration transfer characteristic of the gradient-like structure is studied experimentally, and the influence of magnetic field intensity on the vibration transfer characteristic of the gradient-like structure is analyzed. The results show that the numerical results are in good accordance with the experimental results, thereby verifying that the numerical model is accurate. And the gradient-like structure has a better attenuation effect on the elastic wave than the MR fluid under the action of a uniform magnetic field, and has an excellent tunable property as well.

Time–Frequency Signal Processing for Integrity Assessment and Damage Localization of Concrete Piles

This paper presents a new integrity assessment and damage localization method for piles based on one-dimensional wave propagation theory by integrating the analytical mode decomposition (AMD), recursive Hilbert transform (RHT) and complex continuous wavelet transform (CCWT) into a single assessment tool. The AMD is first used as a band pass filter to extract the mono-component over a frequency band of interest from the response of a pile head, aimed at attenuating the interference from various noisy signals. Then, the mono-component signal is demodulated into a purely frequency-modulated signal by means of RHT, which greatly reduces the interferences from the amplitude-modulated function. Finally, the CCWT is utilized to process the frequency-modulated signal and to calculate phase angles; the latter are subsequently mapped into the time–frequency domain to localize pile damage. The methodology is verified by a numerical example, in which a concrete pile is modeled by the finite element method considering the soil-pile interaction, and by an experimental case study on an actual pile. The results from the numerical and experimental examples demonstrate that the proposed method improves the efficiency of damage identification when compared with other three methods ([Formula: see text], [Formula: see text] and CCWT). In addition, the proposed method enables the localization of damage in full-scale piles situated in soil with an acceptable engineering accuracy by mutual validation with other pile integrity assessment methods, e.g. the ultrasonic emission method.

INVERSE COMPUTATIONAL DETERMINATION OF JOHNSON-COOK PARAMETERS USING THE SHPB TEST APPARATUS

The paper describes determination of the material parameters of the Johnson-Cook constitutive model of steel S235 JR sample material by applying the inverse computational methodology using the digital twin model of the SHPB. A quasi-static tensile testing of bulk material was conducted first to determine the base material parameters. This was followed by dynamic impact testing at two different strain rates using the SHPB. A digital twin computational model was built next in the LS-Dyna explicit finite element system to carry out the necessary computer simulations of the SHPB test. The inverse determination of strain hardening material parameter of Johnson-Cook model was done by using the Nelder-Mead simplex optimisation by comparing the measured and computed stress to time signals on incident and transmission bars. The obtained Johnson-Cook material parameters much better describe the sample material behaviour at very high strain-rates in computational simulations, if compared to the parameters derived by the classic, one-dimensional wave propagation Hopkinson procedure.

A stable high-order FC-based methodology for hemodynamic wave propagation

A high-order numerical algorithm is proposed for the solution of one-dimensional arterial pulse wave propagation problems based on use of an accelerated “Fourier continuation” (FC) methodology for accurate Fourier expansion of non-periodic functions. The solver provides high-order accuracy, mild Courant-Friedrichs-Lewy (CFL) constraints on the time discretization and, importantly, results that are essentially free of spatial dispersion errors—enabling fast and accurate resolution of clinically-relevant problems requiring simulation of many cardiac cycles or vascular segments. The left ventricle-arterial model that is employed presents a particularly challenging case of ordinary differential equation (ODE)-governed boundary conditions that include a hybrid ODE-Dirichlet model for the left ventricle and an ODE-based Windkessel model for truncated vasculature. Results from FC-based simulations are shown to capture the important physiological features of pressure and flow waveforms in the systemic circulation. The robustness of the proposed solver is demonstrated through a number of numerical examples that include performance studies and a physiologically-accurate case study of the coupled left ventricle-arterial system. The results of this paper imply that the FC-based methodology is straightforwardly applicable to other biological and physical phenomena that are governed by similar hyperbolic partial differential equations (PDEs) and ODE-based time-dependent boundaries.

Simulation of fully non-stationary spatially varying ground motions considering nonlinear soil behavior

Taking the nonlinear soil behavior into consideration, a novel method for simulating fully non-stationary spatially varying earthquake ground motions (SVEGMs) was proposed. Based on the one-dimensional (1D) wave propagation model, time-varying shear modulus and damping ratio were introduced into the classical transfer function model, where the evolution characteristics of soil properties (i.e., shear modulus and damping ratio) during earthquake were obtained by using a two-dimensional (2D) site response analysis. By using an example of application, the out-of-plane and in-plane fully non-stationary SVEGMs on ground surface were simulated. The acceleration and displacement time-histories were observed, and the auto power spectral density functions (PSDFs) and coherency loss functions of simulated motions were compared with corresponding targets, respectively. The normalized time-frequency spectra of simulated motions were presented to exhibit the spectral non-stationarity. It was shown that the evolution tendency of predominant frequency of simulated motions reflected similar characteristics with that of shear modulus of soil, which demonstrated the validness of the proposed method.

One-Dimensional Transient Wave Propagation in a Dry Overlying Saturated Ground

While the propagation of stress wave generated by dynamic compaction in dry or saturated granular soil has received much coverage in the research literature, attention to situations with dry soil overlying saturated soil, or mixed-phase ground, is limited. In such cases, the compressional waves have to propagate from a dry layer above the groundwater table to saturated soil below the groundwater table. In this paper, the transient wave propagation characteristics in a mixed-phase ground with an idealized interface between the dry and saturated layer is studied. As the time domain solutions for such problems are often unavailable using analytical methods, a numerical approach based on a dual-phase coupled finite element method implemented on a commercial software platform is adopted. The wave behaviour across the interface is studied and the energy transmission and reflection mechanism from dry to saturated layer is examined. The amplitude, speed and attenuation of the compressional waves and their dependencies on the soil permeability, skeleton stiffness and load duration are quantitatively evaluated via a comprehensive parametric study. As a precursor to the numerical investigation of wave propagation in a mixed-phase ground due to dynamic compaction, the results presented in this study are likely to help in providing a better understanding of the ground improvement effect of dynamic compaction in soil involving a groundwater table.

A dispersive homogenization model for composites and its RVE existence

An asymptotic homogenization model considering wave dispersion in composites is investigated. In this approach, the effect of the microstructure through heterogeneity-induced wave dispersion is characterised by an acceleration gradient term scaled by a “dispersion tensor”. This dispersion tensor is computed within a statistically equivalent representative volume element (RVE). One-dimensional and two-dimensional elastic wave propagation problems are studied. It is found that the dispersive multiscale model shows a considerable improvement over the non-dispersive model in capturing the dynamic response of heterogeneous materials. To test the existence of an RVE for a realistic microstructure for unidirectional fiber-reinforced composites, a statistics study is performed to calculate the homogenized properties with increasing microstructure size. It is found that the convergence of the dispersion tensor is sensitive to the spatial distribution pattern. A calibration study on a composite microstructure with realistic spatial distribution shows that convergence is found although only with a relatively large micromodel.

Investigation of stress wave induced cracking behavior of underground rock mass by the numerical manifold method

The present study investigated the coupling effects of cracking and wave propagation on the underground rock mass by the numerical manifold method (NMM). The traditional NMM was developed for the simulation of dynamic cracking behavior of rock mass. One-dimensional wave propagation was utilized to validate the present code. Subsequently, the transient stress field in the rock mass with an inclined crack was investigated. Finally, two cases of underground caverns with symmetrical or unsymmetrical discontinuities under stress wave were simulated to validate the application potentials. The results show that the transient stress field is not only influenced by the stress concentration around the pre-existing crack, but also influenced by the characteristics of stress wave, e.g. reflection, transmission and diffraction, which are different from the static situation. The results also show that the cracking significantly affects the stress wave duration and amplitude. On the other hand, the stress wave induced cracking depends on the wave propagation distance significantly. Moreover, the NMM can be used to simulate the cracking behavior of underground rock mass under stress wave efficiently.

Adaptive simulation of wave propagation problems including dislocation sources and random media

An adaptive Tikhonov regularization is integrated with an h-adaptive grid-based scheme for simulation of elastodynamic problems, involving seismic sources with discontinuous solutions and random media. The Tikhonov method is adapted by a newly-proposed detector based on the MINMOD limiters and the grids are adapted by the multiresolution analysis (MRA) via interpolation wavelets. Hence, both small and large magnitude physical waves are preserved by the adaptive estimations on non-uniform grids. Due to developing of non-dissipative spurious oscillations, numerical stability is guaranteed by the Tikhonov regularization acting as a post-processor on irregular grids. To preserve waves of small magnitudes, an adaptive regularization is utilized: using of smaller amount of smoothing for small magnitude waves. This adaptive smoothing guarantees also solution stability without over smoothing phenomenon in stochastic media. Proper distinguishing between noise and small physical waves are challenging due to existence of spurious oscillations in numerical simulations. This identification is performed in this study by the MINMOD limiter based algorithm. Finally, efficiency of the proposed concept is verified by: 1) three benchmarks of one-dimensional (1-D) wave propagation problems; 2) P-SV point sources and rupturing line-source including a bounded fault zone with stochastic material properties.

On the modelling of the extraction process of rare metals from metal-containing systems by the method of filtration combustion

Numerical modelling of heterogeneous combustion in porous media with phase transitions is considered. To describe the propagation of time-dependent one-dimensional waves of heterogeneous combustion of metal-containing mixtures, a mathematical model and a numerical method are proposed. The model is based on the assumption of interacting interpenetrating continua using classical approaches of the theory of filtration combustion and includes equations of state, continuity, momentum conservation and energy for each phase. The numerical method is based on a combination of explicit and implicit finite difference schemes. The carried out numerical calculations showed the efficiency of the proposed model and allowed one to detect the concentration of the condensed metal near the front of the combustion wave.

Modeling and Analysis of Helmholtz Cavity Basing on One-dimensional Plane Wave Propagation

Propellant volumetric measurement based on the tank cavity resonant dynamics may be a promising alternate for satellite to improve its performance. The tank could be modeled as ‘Helmholtz resonator’, and volume of the cavity can be calculated from the resonant frequency according to the theoretical formula. Unlike the traditional Helmholtz resonation modelling, where cavity is taken as an ideal acoustic component, this paper gives mathematical modelling based on one-dimensional plane wave approximation, with compressibility of liquid being concerned. Comparisons with simulated and experimental results validate the present model. Good consistency could be found between theoretical and numerical results. Meanwhile, they are slightly larger than the experimental results. Results prove that present model can accurately predict the resonance resonant frequency of the tank, which may be used in the propellant volumetric measurement.

Solutions of Non-stationary Dynamic Problems of Linear Viscoelasticity

The problems of non-stationary waves propagation in linear viscoelastic bodies are considered. The issues related to the construction of the solutions of such problems by method of integral Laplace transform in time are discussed. The statements about the properties of the solution in the space of Laplace transforms, which simplify the process of finding the originals, are formulated. As an example, the solution of the problem of propagation of one-dimensional shear waves in a viscoelastic cylinder is obtained.

A wave propagation model of distributed energy absorption system for trains

The dynamic response of distributed energy absorption system used as a passive safety device in high-speed trains is investigated with a one-dimensional wave propagation model, in which carriages are modelled with elastic rods and energy absorber layers are described by a rigid-perfectly plastic-locking material model. Three collision scenarios are analyzed by taking the stresses and velocities of the interfaces as control variables and a set of ordinary deferential equations is obtained for each stage of collision, the duration of which is the time for the elastic wave to pass from one end of the rod to the other. A platform phenomenon of response in all stages is observed. A simplified approximate analysis procedure is proposed to analyze the energy absorption capability. It is found that ignoring the elastic wave effect of carriages will lead to a wrong estimate of the energy absorption capability and the contribution of individual energy absorber.

Modified Auxiliary Equation Method versus Three Nonlinear Fractional Biological Models in Present Explicit Wave Solutions

In this article, we present a modified auxiliary equation method. We harness this modification in three fundamental models in the biological branch of science. These models are the biological population model, equal width model and modified equal width equation. The three models represent the population density occurring as a result of population supply, a lengthy wave propagating in the positive x-direction, and the simulation of one-dimensional wave propagation in nonlinear media with dispersion processes, respectively. We discuss these models in nonlinear fractional partial differential equation formulas. We used the conformable derivative properties to convert them into nonlinear ordinary differential equations with integer order. After adapting, we applied our new modification to these models to obtain solitary solutions of them. We obtained many novel solutions of these models, which serve to understand more about their properties. All obtained solutions were verified by putting them back into the original equations via computer software such as Maple, Mathematica, and Matlab.

Active wave suppression in the interior of a one-dimensional domain

We consider the problem of creating an absorber in the interior of a system that supports one-dimensional wave propagation, such as a waveguide, using active control. The control goal is suppression of wave propagation beyond a prescribed region of the waveguide, but without perturbing the propagation within that region. Unlike boundary control, achieving full absorption in the interior introduces a challenge of creating both a non-transmitting and a non-reflecting sink. We overcome this challenge by introducing a near uni-directional control wave which is accomplished by using two concentrated actuators. The residual back-action wave is minimized based on the available control energy. We employ this control wave in two algorithms, a feed-forward and a feedback one, which we denote by Interior Wave Suppression. We illustrate the theoretical results through numerical simulation of an acoustic waveguide example.

Self-excited vibrations of spring-loaded valves operating at small pressure drops

Flow-induced vibration is a major issue for spring-loaded valves. However, their complex geometry, combined with their diversity of operational conditions, creates a significant challenge for modelling their dynamic response to potential excitation mechanisms. This paper proposes an analytical framework to describe flow–sound–structurecoupling in spring-loaded valves operating at subcritical conditions, therefore without the occurrence of choking at the vena contracta. The model consists of a single-degree of freedom model of the valve structure, a pseudo-force implicit algorithm to calculate impact forces, a one-dimensional unsteady Bernoulli model to describe the flow, and a one-dimensional wave propagation finite difference scheme to account for the acoustic feedback. The model has shown excellent agreement with experimental results as well as promising predictive capabilities for valve life and wear assessment.

Interface-preserving level set method for simulating dam-break flows

An interface-preserving level set method that solves advection and re-initialization equations for simulating three-dimensional dam-break flows is developed. This method solves mass transfer problems on a uniform staggered Cartesian grid. The advection equation that is used to advect the level set function for capturing the interface is discretized by a proposed fourth-order spatial discretization scheme. This scheme is dispersion-relation-preserving and is compact-reconstruction weighted essentially non-oscillatory (DRP-CRWENO4). This scheme is compared with a previous fifth-order, weighted, essentially non-oscillatory (WENO5) scheme and can represent an interface more accurately, while exactly preserving mass conservation. This level set approach introduces a mass correction term into the re-initialization equation based on local interface-preserving conditions. An explicit Adams–Bashforth algorithm on a staggered Eulerian grid is used for the Navier–Stokes solver. The point successive over-relaxation method is then employed to solve the resulting linear system. Two one-dimensional wave propagation problems are simulated to verify the proposed DRP-CRWENO4 scheme, which is shown to be capable of effectively capturing large gradients with fourth-order accuracy. To demonstrate their resolution, the two advection schemes (WENO5 and DRP-CRWENO4) are applied in two two-dimensional benchmark cases, i.e., a vortex deforming problem and Zalesak's disk problem, where simulation results of both schemes are compared against each other. Demonstration study is then further extended to three-dimensional cases of the vortex deforming problem and Zalesak's sphere problem, and simulation results agree well with those using hybrid particle level set method. Finally, several dam-break problems with and without obstacles are investigated to validate the coupled two-phase incompressible flow and level set method solver. The results for the predicted flow structure and mass conservation properties are compared with the reported experimental data or numerical simulations.