Ricker Pulse Research Articles

Time domain modeling of elastic waves using a stress-based unsplit-field perfectly matched layer with enhanced numerical stability

A stress-based unsplit-field perfectly matched layer (PML) method with enhanced numerical stability is proposed to simulate elastic waves in heterogeneous unbounded domains. The proposed PML method attenuates the waves by forcing the satisfaction of mixed-field equations constructed using a complex coordinate stretching in the frequency domain. Unlike other unsplit-field PMLs, which introduce coordinate stretching before combining the constitutive and compatibility equations, the proposed method first combines the two equations and then stretches the resulting equation. The stretched equation is converted back to the time domain using the inverse Fourier transform to yield transient elastic wave equations in the PML-truncated domain. Semi-discrete equations of motion for displacement and stresses were obtained using standard Lagrange-family finite elements. In a series of numerical examples involving semi-infinite elastic media, the accuracy of the wave solutions calculated using the proposed PML method was evaluated using a displacement L2-norm, a normalized L2-error, and an energy norm. The examples further validated the wave absorption performance of the PML and evaluated the long-term stability of solutions utilizing the Ricker pulse excitation with various central frequencies. The numerical experiments indicate that the solutions obtained using the proposed PML are significantly more stable than those obtained using the existing strain-based unsplit-field PMLs. The proposed method is particularly useful for applications requiring explicit time integration of semi-discrete equations of motion because the mass matrix can be diagonalized. The PML method and its associated solution strategies can be applied to various engineering problems, such as soil-structure interaction analysis, monitoring of structures, and geophysical probing.

Application of Robust Phase Algorithms for Seismic Emission Detection in the Area of Blasting Operations in Mines

In this paper, we developed algorithms for multichannel processing of seismic array records that allow detecting signals of weak seismic events observed against the background of natural seismic noise. The algorithm for detecting signals of weak seismic events is based on the coherence assessment of longitudinal waves (P-waves) recorded by a small aperture array. The advantage of this algorithm over the known single-channel short/long term averaging (STA/LTA) detector is the ability to detect time intervals of P-waves at a signal-to-noise ratio (SNR) less than unity. Based on the time intervals of the array records with the registered coherence measure values that ​​exceed a certain threshold, the apparent slowness vector is estimated by using the phase bimforming algorithm that is robust to noise (Zhang et al., 2008; Kushnir and Varypaev, 2017). The developed algorithms of multichannel processing were tested using real seismic records of a small aperture array installed in the Kursk Magnetic Anomaly area near the mine workings for iron ore extraction. Signals of both strong (underground industrial explosions) and weak seismic events were detected for 20 h of seismic recording. In some cases, seismic events were shown to have a complex source mechanism. Using the Capon filter (Capon, 1969), for weak seismic events, P-waves with a duration not exceeding 0.2 s and a temporal shape similar to the known function of a Ricker pulse (Bording, 1996) are found to have resemblance.

Shear building stiffness estimation by wave traveling time analysis

A novel damage identification technique to estimate stiffness in a multistory building supported on solid ground is presented. Based on a shear building model, a 1-dimensional wave equation for a vertically propagating shear wave is derived. A Ricker pulse is used as excitation signal and propagated through the building. Wave propagation in the building is based on the Thomson–Haskell method, where each story is represented as a single layer in a multiple stratum model. The wave arrival times of the pulse at each story are used to calculate the stiffness of the columns. The involved calculations in this method grow only linearly with the number of stories, as opposed to other identification methods, as modal analysis, that grow geometrically; this makes this approach an interesting alternative method to asses building integrity. Simulation result for a building with heterogeneous characteristics across the stories confirms the feasibility of the proposal.

A finite element model for seismic response analysis of deformable rocking frames

SummaryA new finite element model to analyze the seismic response of deformable rocking bodies and rocking structures is presented. The model comprises a set of beam elements to represent the rocking body and zero‐length fiber cross‐section elements at the ends of the rocking body to represent the rocking surfaces. The energy dissipation during rocking motion is modeled using a Hilber–Hughes–Taylor numerically dissipative time step integration scheme. The model is verified through correct prediction of the horizontal and vertical displacements of a rigid rocking block and validated against the analytical Housner model solution for the rocking response of rigid bodies subjected to ground motion excitation. The proposed model is augmented by a dissipative model of the ground under the rocking surface to facilitate modeling of the rocking response of deformable bodies and structures. The augmented model is used to compute the overturning and uplift rocking response spectra for a deformable rocking frame structure to symmetric and anti‐symmetric Ricker pulse ground motion excitation. It is found that the deformability of the columns of a rocking frame does not jeopardize its stability under Ricker pulse ground motion excitation. In fact, there are cases where a deformable rocking frame is more stable than its rigid counterpart. Copyright © 2016 John Wiley & Sons, Ltd.

Use of Ricker motions as an alternative to pushover testing

When undertaking centrifuge studies on seismic soil–structure interaction, it is useful to be able to define the pseudo-static ‘pushover’ response of the structure. Normally, this requires separate centrifuge experiments with horizontal actuators. This paper describes an alternative procedure, using Ricker ground motions to obtain the pushover response, thereby allowing both this and the response to seismic shaking to be determined using a centrifuge-mounted shaker. The paper presents an application of this technique to a 1:50 scale model bridge pier with two different shallow foundations, as part of a study on seismic protection using rocking isolation. The moment–rotation (‘backbone’) behaviour of the footings was accurately determined in the centrifuge to large rotations, as verified through independent three-dimensional dynamic non-linear finite-element modelling. Ricker wavelet ground motions are therefore shown to be a useful tool for the identification of pushover response without requiring additional actuators. Furthermore, a simplified analytical methodology is developed, which allows one to predict the maximum foundation rotation induced by a specific Ricker pulse. This methodology may be useful in predicting the characteristics (frequency and acceleration magnitude) of the Ricker pulse required to describe the pushover response of any (practically) rigid oscillator supported on shallow foundations.

Three dimensional non-axisymmetric transient acoustic radiation from an eccentric hollow cylinder

Theoretical analysis and detailed numerical simulations are presented for three-dimensional non-stationary vibroacoustic response of a doubly fluid-loaded infinitely long eccentric hollow elastic circular cylinder of arbitrary wall thickness, under arbitrary non-axisymmetric time-dependent on-surface mechanical drives. The formulation is based on the Navier equations of linear elasticity for the solid material, the wave equation for the internal and external fluid domains, the classical method of separation of variables, and the translational addition theorem for cylindrical wave functions. Laplace transform with respect to the time coordinate, Fourier transform with respect to the axial coordinate, and Fourier series expansion in the circumferential direction, are employed to obtain the transformed solutions in the form of partial-wave expansions of transcendental functions. A set of equally spaced virtual sources are assumed along the axial direction, and semi-analytical solutions are obtained in the Laplace domain as discrete summations of simple two dimensional solutions with different axial wave numbers. Time domain solutions are subsequently calculated by using Durbin’s numerical inverse Laplace transform scheme. The analytical results are illustrated with numerical examples in which air- or water-filled concentric and eccentric steel cylinders, submerged in water, are driven by a pair of equal and opposite external radial point loads of finite duration described by a Ricker pulse. Some interesting features of the transient fluid–solid interaction problem such as the generation of shell-induced circumnavigating waves, appearance of multiple high/low amplitude pressure spots in the internal fluid, and formation of non-propagating evanescent waves in the external fluid are noted using appropriate 2D images of the three dimensional sound fields. Limiting cases are considered and the validity of results is established with the aid of a commercial finite element package as well as by comparison with the data in the existing literature.

Analysis of Ringing and Noise in FE and FDTD Calculated Acoustic Pulse Profiles

Ringing, i.e. the emergence of an oscillatory tail behind a wave pulse as it propagates through a medium, is a pervasive artefact in FE and FDTD calculated waveforms. It is known to be a consequence of numerical dispersion arising from the discretization of the equations of motion. The use of an irregular mesh in a FE code has the further consequence of rendering the displacement field increasingly noisy with distance behind the wave front. In this paper these effects are illustrated using the commercial FE package ABAQUS with square and irregular triangular meshes to calculate the progress of a longitudinally polarized Ricker pulse along the axis of a cylindrically shaped aluminium specimen. We are able to give a precise analytical account of the evolution of ringing on the basis of a low order approximation for the dispersion relation of the discretized equations of motion. A qualitative account is provided of the generation of noise in the use of an irregular triangular mesh.

Solution of time-domain acoustic wave propagation problems using a RBF interpolation model with “a priori” estimation of the free parameter

In the last decade, Meshless Methods have found widespread application in different fields of engineering and science. Beyond novelty, their mathematical simplicity and numerical accuracy have been the key of their rapid dissemination. Among meshless techniques, RBF (Radial Basis Functions) based methods can be simple and general to solve the problems related to multiple areas of applied physics and engineering. In the specific field of acoustics, there are usually two possible approaches for solving a problem: time- and frequency-domain. In this paper, the authors propose a local time-domain approach to establish an efficient methodology for the solution of large-scale acoustic wave propagation problems. For this purpose, a local interpolation scheme, based on the reproduction of the local wave field using RBFs (MultiQuadric and Gaussian), is implemented and its accuracy is verified against known closed-form solutions. An explicit time-domain marching procedure is adopted, and the quality of the numerical results is also compared with that obtained using standard space-time Finite–Difference schemes. Additionally, the RBF interpolation model is used to simulate the propagation of a Ricker pulse in two simple test cases, and applied to simulate a more complex configuration, corresponding to an underwater sound propagation problem. In this frame, the results are also compared with those computed using a fourth-order in space and second-order in time Finite–Difference scheme.

2.5D scattering of waves by rigid inclusions buried under a fluid channel via BEM

Abstract A 2.5D Boundary Element Method (BEM) formulation, applied in the frequency domain, is developed to compute the scattering of waves by rigid inclusions buried in a semi-infinite solid under a fluid layer, when this system is excited by a spatially-sinusoidal harmonic load. The BEM algorithm includes Green's functions for a horizontal fluid layer over a semi-infinite solid, which avoids the discretrization of the horizontal surfaces, and thus only the rigid inclusion needs to be discretized by boundary elements. The model uses complex frequencies with a small imaginary part to avoid aliasing phenomena. Time domain responses are obtained by applying an inverse Fourier Transform to the frequency results. The source is modeled as a Ricker pulse. The simulations are performed for three different properties of the solid medium: a fast formation, a slow formation and a sediment formation.

FINITE ELEMENT MODELLING OF INFINITE EULER BEAMS ON KELVIN FOUNDATIONS EXPOSED TO MOVING LOADS IN CONVECTED CO-ORDINATES

The paper deals with the finite element method (FEM) solution of the problem with loads moving uniformly along an infinite Euler beam supported by a linear elastic Kelvin foundation with linear viscous damping. Initially, the problem is formulated in a moving co-ordinate system following the load using a Galilean co-ordinate transformation and subsequently the analytical solution to the homogeneous beam problem is shown. To be used in more complicated cases where no analytical solutions can be found, a numerical approach of the same problem is then suggested based on the FEM. Absorbing boundary conditions to be applied at the ends of the modelled part of the infinite beam are derived. The quality of the numerical results for single-frequency, harmonic excitation is tested by comparison with the indicated analytical solution. Finally, the robustness of the boundary condition is tested for a Ricker pulse excitation in the time domain.

Coda spectral power and apparent attenuation factor of acoustic waves in 3-D random media

The propagation of acoustic waves in three-dimensional random media is studied by means of synthetic seismograms produced by extensive finite-difference simulations. Gaussian or exponential stationary media, probed by a Ricker pulse, are considered. Coda spectral power, as well as frequency dependence of the apparent attenuation factor, are studied. When the relative velocity fluctuations are smaller than 5%, the numerical results are in good agreement with theoretical predictions, at least as far as the coda spectral power is concerned. These results are discussed and recommendations are made for future investigation.

Propagation and attenuation of a Ricker pulse in one-dimensional randomly heterogeneous media: a numerical study

Numerical simulations are extensively used to study some major features of acoustic wave propagation in one-dimensional random media. Gaussian and exponential stationary media, in which a Ricker pulse is applied, are considered. The properties numerically shown in this paper deal with the irregular progression of the wave front and the changes in its energy distribution with respect to time. Particular emphasis is given to the study of the apparent attenuation factor. Not only are the results shown to be in good agreement with existing analytical studies, but they also provide a more thorough insight into propagation phenomena, especially when the corresponding analytical predictions no longer hold true.

Measurements of acoustic wave scattering due to sea bottom roughness and sub-bottom sound-speed fluctuations.

An eight-element hydrophone array and an omnidirectional source were used for the bottom scattering measurements within the frequency range of 2–20 kHz. The reverberant tail of a Ricker pulse or a gated cw pulse were used for the backscattering measurements. Volume fluctuations of sound speed were measured as deep as 5 m by a newly developed source and hydrophone array penetrators. The penetrators were deployed 5–10 m apart to obtain a tomographic image of vertical cross section of sub-bottom. A nonlinear ray tracing and inversion scheme was used to calculate the sub-bottom sound-speed fluctuations and two-dimensional wave-number spectra. Penetration depths of the acoustic signals were also measured by using the same system. Preliminary results of the measurements of bottom backscattering and sub-bottom sound-speed fluctuations indicate that sub-bottom volume scattering may dominate the bottom roughness scattering in case of large fluctuations of sub-bottom sound speed and large penetration depths.