Numerical Simulation Of Wave Propagation Research Articles

Memory‐Efficient Simulation of Frequency‐DependentQ

Memory‐variable methods have been widely applied to approximate frequency‐independent quality factor Q in numerical simulation of wave propagation. The frequency‐independent model is often appropriate for frequencies up to about 1 Hz but at higher frequencies is inconsistent with some regional studies of seismic attenuation. We apply the memory‐variable approach to frequency‐dependent Q models that are constant below, and follow a power‐law above, a chosen transition frequency. We present numerical results for the corresponding memory‐variable relaxation times and weights, obtained by nonnegative least‐squares fitting of the Q ( f ) function, for a range of exponent values; these times and weights can be scaled to an arbitrary transition frequency and a power‐law prefactor, respectively. The resulting memory‐variable formulation can be used with numerical wave‐propagation solvers based on methods such as finite differences (FDs) or spectral elements and may be implemented in either conventional or coarse‐grained form. In the coarse‐grained approach, we fit effective Q for low‐ Q values (<200) using a nonlinear inversion technique and use an interpolation formula to find the corresponding weighting coefficients for arbitrary Q . A 3D staggered‐grid FD implementation closely approximates the frequency–wavenumber solution to both a half‐space and a layered model with a shallow dislocation source for Q as low as 20 over a bandwidth of two decades. We compare the effects of different power‐law exponents using a finite‐fault source model of the 2008 M w 5.4 Chino Hills, California, earthquake and find that Q ( f ) models generally better fit the strong‐motion data than the constant Q models for frequencies above 1 Hz. Online Material: Figures comparing finite difference and frequency–wavenumber seismograms for an elastic layered‐model point source simulation. Median spectral acceleration centered at 1 s and Fourier amplitude centered at 0.25 and 2.25 Hz for strong ground motion recordings and synthetics from the Chino Hills earthquake.

A micro-inertia gradient visco-elastic motivation for proportional damping

In this paper, a micro-inertia gradient visco-elasticity theory is proposed and implemented for the description of wave dispersion in periodic visco-elastic composites characterised by (stiffness-)proportional damping. An expression for the internal length parameter has been derived in terms of geometry and material properties. The theory has been validated through a numerical simulation of wave propagation in a one-dimensional periodic composite bar for two different heterogeneity levels, where the proposed theory has shown good agreement with the solution obtained by explicitly modelling the material heterogeneity. The effects of both gradient enrichment and viscosity on wave propagation as well as their interaction have also been analysed.

An active–passive acoustics approach for bond-line condition monitoring in aerospace skin stiffener panels

This article demonstrates the effectiveness of acoustic nondestructive inspection methods to monitor the condition of the bond-line in carbon fiber reinforced polymer stiffened panels. In particular, acoustic emission and guided ultrasonic waves were utilized experimentally to identify the onset of debonding in the stiffened panels and track its evolution under constant amplitude fatigue loading. Before testing, numerical simulations of wave propagation were carried out to evaluate the wave characteristics and optimize the placement of sensors along the spar. A damage index based on numerically produced guided waves was developed to estimate the extent of debonding which was tested and proved effective using actual experimental data. The panels were subjected to both quasi-static and fatigue loading conditions, while continuously recording acoustic emission and triggering guided ultrasonic waves at predetermined load levels. The acoustic emission activity associated with debonding was found to have a dual dominant frequency content and a low frequency centroid, which were also confirmed with wavelet analysis. In view of potential aerospace applications of the investigation presented herein, pattern recognition algorithms were also implemented and showed great potential for real-time detection of such damage.

Numerical simulation of wave propagation in a realistic model of the human external ear

In this study, a numerical investigation is performed to evaluate the effects of high-pressure sinusoidal and blast wave's propagation around and inside of a human external ear. A series of computed tomography images are used to reconstruct a realistic three-dimensional (3D) model of a human ear canal and the auricle. The airflow field is then computed by solving the governing differential equations in the time domain using a computational fluid dynamics software. An unsteady algorithm is used to obtain the high-pressure wave propagation throughout the ear canal which is validated against the available analytical and numerical data in literature. The effects of frequency, wave shape, and the auricle on pressure distribution are then evaluated and discussed. The results clearly indicate that the frequency plays a key role on pressure distribution within the ear canal. At 4 kHz frequency, the pressure magnitude is much more amplified within the ear canal than the frequencies of 2 and 6 kHz, for the incident wave angle of 90° investigated in this study, attributable to the ‘4-kHz notch’ in patients with noise-induced hearing loss. According to the results, the pressure distribution patterns at the ear canal are very similar for both sinusoidal pressure waveform with the frequency of 2 kHz and blast wave. The ratio of the peak pressure value at the eardrum to that at the canal entrance increases from about 8% to 30% as the peak pressure value of the blast wave increases from 5 to 100 kPa for the incident wave angle of 90° investigated in this study. Furthermore, incorporation of the auricle to the ear canal model is associated with centerline pressure magnitudes of about 50% and 7% more than those of the ear canal model without the auricle throughout the ear canal for sinusoidal and blast waves, respectively, without any significant effect on pressure distribution pattern along the ear canal for the incident wave angle of 90° investigated in this study.

A comparison of finite-difference and spectral-element methods for elastic wave propagation in media with a fluid-solid interface

The numerical simulation of wave propagation in media with solid and fluid layers is essential for marine seismic exploration data analysis. The numerical methods for wave propagation that are applicable to this physical settings can be broadly classified as partitioned or monolithic: The partitioned methods use separate simulations in the fluid and solid regions and explicitly satisfy the interface conditions, whereas the monolithic methods use the same method in all the domain without any special treatment of the fluid-solid interface. Despite the accuracy of the partitioned methods, the monolithic methods are more common in practice because of their convenience. In this paper, we analyse the accuracy of several monolithic methods for wave propagation in the presence of a fluid-solid interface. The analysis is based on grid-dispersion criteria and numerical examples. The methods studied here include: the classical finite-difference method (FDM) based on the second-order displacement formulation of the elastic wave equation (DFDM), the staggered-grid finite difference method (SGFDM), the velocity-stress FDM with a standard grid (VSFDM) and the spectral-element method (SEM). We observe that among these, DFDM and the first-order SEM have a large amount of grid dispersion in the fluid region which renders them impractical for this application. On the other hand, SGFDM, VSFDM and SEM of order greater or equal to 2 yield accurate results for the body waves in the fluid and solid regions if a sufficient number of nodes per wavelength is used. All of the considered methods yield limited accuracy for the surface waves because the proper boundary conditions are not incorporated into the numerical scheme. Overall, we demonstrate both by analytic treatment and numerical experiments, that a first-order velocity-stress formulation can, in general, be used in dealing with fluid-solid interfaces without using staggered grids necessarily.

Numerical simulation of wave propagation in anisotropic media

Numerical simulation of wave propagation in anisotropic media

Workflow to numerically reproduce laboratory ultrasonic datasets

The risks and uncertainties related to the storage of high-level radioactive waste (HLRW) can be reduced thanks to focused studies and investigations. HLRWs are going to be placed in deep geological repositories, enveloped in an engineered bentonite barrier, whose physical conditions are subjected to change throughout the lifespan of the infrastructure. Seismic tomography can be employed to monitor its physical state and integrity. The design of the seismic monitoring system can be optimized via conducting and analyzing numerical simulations of wave propagation in representative repository geometry. However, the quality of the numerical results relies on their initial calibration. The main aim of this paper is to provide a workflow to calibrate numerical tools employing laboratory ultrasonic datasets. The finite difference code SOFI2D was employed to model ultrasonic waves propagating through a laboratory sample. Specifically, the input velocity model was calibrated to achieve a best match between experimental and numerical ultrasonic traces. Likely due to the imperfections of the contact surfaces, the resultant velocities of P- and S-wave propagation tend to be noticeably lower than those a priori assigned. Then, the calibrated model was employed to estimate the attenuation in a montmorillonite sample. The obtained low quality factors (Q) suggest that pronounced inelastic behavior of the clay has to be taken into account in geophysical modeling and analysis. Consequently, this contribution should be considered as a first step towards the creation of a numerical tool to evaluate wave propagation in nuclear waste repositories.

Local time–space mesh refinement for simulation of elastic wave propagation in multi-scale media

This paper presents an original approach to local time–space grid refinement for the numerical simulation of wave propagation in models with localized clusters of micro-heterogeneities. The main features of the algorithm are–the application of temporal and spatial refinement on two different surfaces;–the use of the embedded-stencil technique for the refinement of grid step with respect to time;–the use of the Fast Fourier Transform (FFT)-based interpolation to couple variables for spatial mesh refinement. The latter makes it possible to perform filtration of high spatial frequencies, which provides stability in the proposed finite-difference schemes.In the present work, the technique is implemented for the finite-difference simulation of seismic wave propagation and the interaction of such waves with fluid-filled fractures and cavities of carbonate reservoirs. However, this approach is easy to adapt and/or combine with other numerical techniques, such as finite elements, discontinuous Galerkin method, or finite volumes used for approximation of various types of linear and nonlinear hyperbolic equations.

SYNTHETIC OBSERVATIONS OF WAVE PROPAGATION IN A SUNSPOT UMBRA.

Spectropolarimetric temporal series from Fe i λ6301.5 Å and Ca ii infrared triplet lines are obtained by applying the Stokes synthesis code NICOLE to a numerical simulation of wave propagation in a sunspot umbra from MANCHA code. The analysis of the phase difference between Doppler velocity and intensity core oscillations of the Fe i λ6301.5 Å line reveals that variations in the intensity are produced by opacity fluctuations rather than intrinsic temperature oscillations, except for frequencies between 5 and 6.5 mHz. On the other hand, the photospheric magnetic field retrieved from the weak field approximation provides the intrinsic magnetic field oscillations associated to wave propagation. Our results suggest that this is due to the low magnetic field gradient of our sunspot model. The Stokes parameters of the chromospheric Ca ii infrared triplet lines show striking variations as shock waves travel through the formation height of the lines, including emission self-reversals in the line core and highly abnormal Stokes V profiles. Magnetic field oscillations inferred from the Ca ii infrared lines using the weak field approximation appear to be related with the magnetic field strength variation between the photosphere and the chromosphere.

Numerical study of the interface errors of finite-difference simulations of seismic waves

Numerical simulations of wave propagation produce different errors and the most well known is numerical dispersion, which is only valid for homogeneous media. However, there is a lack of error studies for heterogeneous media or even for the canonical case of media that have two constant velocity layers. The error associated with media that have two layers is called an interface error, and it typically converges to zero with a lower order of convergence compared to the theoretical convergence rate of the finite-difference schemes (FDS) for homogeneous media. We evaluated a detailed numerical study of the interface error for three staggered-grid FDS that are commonly used in the simulation of seismic-wave propagation. We determined that a standard staggered-grid scheme (SSGS) (also known as the Virieux scheme), a rotated staggered-grid scheme (RSGS), and a Lebedev scheme (LS) preserve the second order of convergence at horizontal/vertical solid-solid interfaces when the medium parameters have been properly modified, such as by harmonic averaging of finely layered media for the stiffness tensor and arithmetic mean for the density. However, for a fluid-solid interface aligned with the grid line, a second-order convergence can only be achieved by an SSGS. In addition, the presence of a fluid-solid interface reduces the order of convergence for the LS and the RSGS to a first order of convergence. The presence of inclined interfaces makes high-order (second and more) convergence impossible.

Feasibility Study on Evaluating Surface Opening Cracks in Concrete by Multi-Channel Instrumented Surface Rayleigh Wave Propagation

In the present paper, numerical simulations of wave propagation were conducted for concrete containing surface opening crack. The purpose is to examine the behavior of surface Rayleigh wave (R-wave) components when impinged by the crack and to identify wave parameters that could be utilized to characterize the crack, including depth (length) and degree of inclination. In the study, cracks with varying inclination angle were investigated. By changing the frequencies of source, the change in amplitudes and velocities of R-waves that propagate from one side to another side of the crack were extracted from the observed waveforms. The change of R-wave parameters (velocity, amplitude and frequency) with respect to crack depth as well as the inclination angle was analyzed. The simulation results revealed that promising relationships can be established between velocity or amplitude with the crack characteristics, leading to the suggestion that simple methodology based on R-waves could be developed for assessing surface opening cracks in concrete structures.

Decorrelation and phase-shift of coda waves induced by local changes: multiple scattering approach and numerical validation

We report on theoretical predictions of the decorrelation and phase-shift of coda waves induced by local changes in multiple scattering media. Using the multiple scattering formalism, we show that both expressions (decorrelation and phase-shift) involve a same sensitivity kernel based on the intensity transport in the medium. We compare the kernels based on the diffusion approximation with the ones based on the radiative transfer approximation, showing that the latter is more accurate at short times or for changes located close to the source or the receiver. We also perform a series of numerical simulations of wave propagation (finite differences) to validate our models in different configurations.

2D and 3D modeling of wave propagation in cold magnetized plasma near the Tore Supra ICRH antenna relying on the perfecly matched layer technique

A novel method to simulate ion cyclotron wave coupling in the edge of a tokamak plasma with the finite element technique is presented. It is applied in the commercial software COMSOL Multiphysics. Its main features include the perfectly matched layer (PML) technique to emulate radiating boundary conditions beyond a critical cutoff layer for the fast wave (FW), full-wave propagation across the inhomogeneous cold peripheral plasma and a detailed description of the wave launcher geometry. The PML technique, while widely used in numerical simulations of wave propagation, has scarcely been used for magnetized plasmas, due to specificities of this gyrotropic material. A versatile PML formulation, valid for full dielectric tensors, is summarized and interpreted as wave propagation in an artificial medium. The behavior of this technique has been checked for plane waves on homogeneous plasmas. Wave reflection has been quantified and compared to analytical predictions. An incompatibility issue for adapting the PML for forward (FW) and backward (slow wave (SW)) propagating waves simultaneously has been evidenced. In a tokamak plasma, this critical issue is overcome by taking advantage of the inhomogeneous density profile to reflect the SW before it reaches the PML. The simulated coupling properties of a Tore Supra ion cyclotron resonance heating (ICRH) antenna have been compared to experimental values in a situation of good single-pass absorption. The necessary antenna elements to include in the geometry to recover the coupling properties obtained experimentally are also discussed.

Weak Discontinuities in Solutions of Long‐Wave Equations for Viscous Flow

Nonlinear integrodifferential equations describing the propagation of disturbances in a thin layer of viscous liquid with free surface are studied. These equations admit solutions with weak discontinuities, which are located on the characteristics. The possibility of an unbounded increase in the amplitude of the weak discontinuity and the formation of the shock in the process of flow evolution is established. Differential balance laws approximating the integrodifferential model are proposed. These laws are used to perform numerical simulation of wave propagation in a fluid.

Efficient implementation of power-law attenuation of elastic waves in time-domain numerical simulations

Numerical simulation of wave propagation in the time domain is easily parallelizable on high performance computing systems due to the spatially local nature of the governing equations. The disadvantage of working in the time domain arises when lossy media must be modeled which generally gives rise to convolution-type loss terms in the governing time-domain equations. Computation of these convolutions usually requires the storage of several solution fields at thousands of previous time steps. This requirement can be memory prohibitive in three-dimensions. In this talk we present a recursive convolution approach to computing lossy (power-law) elastic wave propagation that is an extension of the one-way, one-dimensional acoustic wave equation work done by Liebler [Liebler et al., J. Acoust. Soc. Am. 116 (2004)] in order to handle multiple dimensions and shear waves. Convolutions are computed recursively by first using a nonlinear least-squares technique to fit the kernel of the convolution with a series of decaying exponentials. We demonstrate how graphical processing units (GPUs) can be used to obtain speed-up factors as high as 35 on a test computation of time-domain scattering from a highly resonant but lossy elastic cylinder. [Work sponsored by the Office of Naval Research.]

Numerical Simulation the Stress Uniformity in Split Hopkinson Pressure Bar Testing

The assumption of uniform stress in a test specimen is fundamental to SHPB test technique. In the present paper, a numerical simulation of wave propagation in SHPB is performed to validate the assumption. A one-dimensional model based on CSPM is firstly developed. Then the wave propagations in SHPB with various area ratios of bar/specimen are simulated. The results show that the condition of stress uniformity is not satisfied, especially at the beginning of wave propagation. For the large area specimen, the stress tends to be uniform. While for the small area specimen, the non-uniformity of stress is more apparent.

Parallel Simulation of 3D Wave Propagation by Domain Decomposition

In order to perform large scale numerical simulation of wave propagation in 3D heterogeneous multiscale viscoelastic media, Finite Difference technique and its parallel implementation based on domain decomposition is used. A couple of typical statements of borehole geophysics are dealt with—sonic log and cross well measurements. Both of them are essentially multiscales, which claims to take into account heterogeneities of very different sizes in order to provide reliable results of simulations. Locally refined spatial grids help us to avoid the use of redundantly tiny grid cells in a target area, but cause some troubles with uniform load of Processor Units involved in computations. We present results of scalability tests together with results of numerical simulations for both statements performed for some realistic models.

Numerical simulation of ambient seismic wavefield modification caused by pore-fluid effects in an oil reservoir

We have modeled numerically the seismic response of a poroelastic inclusion with properties applicable to an oil reservoir that interacts with an ambient wavefield. The model includes wave-induced fluid flow caused by pressure differences between mesoscopic-scale (i.e., in the order of centimeters to meters) heterogeneities. We used a viscoelastic approximation on the macroscopic scale to implement the attenuation and dispersion resulting from this mesoscopic-scale theory in numerical simulations of wave propagation on the kilometer scale. This upscaling method includes finite-element modeling of wave-induced fluid flow to determine effective seismic properties of the poroelastic media, such as attenuation of P- and S-waves. The fitted, equivalent, viscoelastic behavior is implemented in finite-difference wave propagation simulations. With this two-stage process, we model numerically the quasi-poroelastic wave-propagation on the kilometer scale and study the impact of fluid properties and fluid saturation on the modeled seismic amplitudes. In particular, we addressed the question of whether poroelastic effects within an oil reservoir may be a plausible explanation for low-frequency ambient wavefield modifications observed at oil fields in recent years. Our results indicate that ambient wavefield modification is expected to occur for oil reservoirs exhibiting high attenuation. Whether or not such modifications can be detected in surface recordings, however, will depend on acquisition design and noise mitigation processing as well as site-specific conditions, such as the geologic complexity of the subsurface, the nature of the ambient wavefield, and the amount of surface noise.

Helioseismic Holography of an Artificial Submerged Sound Speed Perturbation and Implications for the Detection of Pre-emergence Signatures of Active Regions

We use a publicly available numerical wave-propagation simulation of Hartlep et al. 2011 to test the ability of helioseismic holography to detect signatures of a compact, fully submerged, 5% sound-speed perturbation placed at a depth of 50 Mm within a solar model. We find that helioseismic holography as employed in a nominal "lateral-vantage" or "deep-focus" geometry employing quadrants of an annular pupil is capable of detecting and characterizing the perturbation. A number of tests of the methodology, including the use of a plane-parallel approximation, the definition of travel-time shifts, the use of different phase-speed filters, and changes to the pupils, are also performed. It is found that travel-time shifts made using Gabor-wavelet fitting are essentially identical to those derived from the phase of the Fourier transform of the cross-covariance functions. The errors in travel-time shifts caused by the plane-parallel approximation can be minimized to less than a second for the depths and fields of view considered here. Based on the measured strength of the mean travel-time signal of the perturbation, no substantial improvement in sensitivity is produced by varying the analysis procedure from the nominal methodology in conformance with expectations. The measured travel-time shifts are essentially unchanged by varying the profile of the phase-speed filter or omitting the filter entirely. The method remains maximally sensitive when applied with pupils that are wide quadrants, as opposed to narrower quadrants or with pupils composed of smaller arcs. We discuss the significance of these results for the recent controversy regarding suspected pre-emergence signatures of active regions.

Seismic modeling to monitor CO2 geological storage: The Atzbach‐Schwanenstadt gas field

We develop a petro‐elastical numerical methodology to compute realistic synthetic seismograms and analyze the sensitivity of the seismic response when injecting carbon dioxide (CO2) in a depleted gas reservoir. The petro‐elastical model describes the seismic properties of the reservoir rock saturated with CO2, methane and brine, and allows us to estimate the distribution and saturation of CO2 during the injection process. The gas properties, as a function of the in‐situ pressure and temperature conditions, are computed with the Peng‐Robinson equation of state, taking into account the absorption of gas by brine. Wave attenuation and velocity dispersion are based on the mesoscopic loss mechanism, which is simulated by an upscaling procedure to obtain an equivalent viscoelastic medium corresponding to partial saturation at the mesoscopic scale. Having the equivalent complex and frequency‐dependent bulk (dilatational) modulus, we include shear attenuation and perform numerical simulations of wave propagation at the macroscale by solving the viscoelastic differential equations using the memory‐variable approach. The pseudo‐spectral modeling method allows general material variability and provides a complete and accurate characterization of the reservoir. The methodology is used to assess the sensitivity of the seismic method for monitoring the CO2 geological storage at the Atzbach‐Schwanestadt depleted gas‐field in Austria. The objective of monitoring is the detection of the CO2 plume in the reservoir and possible leakages of CO2. The leakages are located at different depths, where the CO2 is present as gaseous, liquid and supercritical phases. Even though the differences can be very subtle, this work shows that seismic monitoring of CO2 from the surface is possible. While the identification of shallow leakages is feasible, the detection of the plume and deep leakages, located in the caprock just above the injection formation, is more difficult, but possible by using repeatability metrics, such as the normalized RMS (NRMS) images. Considering real‐data conditions, affected by random noise, a reference detection threshold for deep leakages and the CO2 plume in the reservoir corresponds to a signal‐to‐noise ratio of about 10 dB.