Elastic Wave Modeling Research Articles

A mechanistic analysis of stone comminution in lithotripsy

In vitro experiments and an elastic wave model were employed to isolate and assess the importance of individual mechanisms in stone comminution in lithotripsy. Cylindrical U-30 cement stones were treated in an HM-3-style research lithotripter. Baffles were used to block specific waves responsible for spallation, squeezing, or shear. Surface cracks were added to stones to simulate the effect of cavitation, then tested in water and glycerol (a cavitation suppressive medium). Each case was simulated using the elasticity equations for an isotropic medium. The calculated location of maximum stress compared well with the experimental observations of where cracks naturally formed. Shear waves from the shock wave in the fluid traveling along the stone surface (a kind of dynamic squeezing) led to the largest stresses in the cylindrical stones and the fewest SWs to fracture. Reflection of the longitudinal wave from the back of the stone—spallation—and bubble-jet impact on the proximal and distal faces of the stone produced lower stresses and required more SWs to break stones. Surface cracks accelerated fragmentation when created near the location where the maximum stress was predicted. [Work supported by NIH DK43881, NIH-Fogarty, NSBRI SMS00203, RFBR, and ONRIFO.]

Elastic wave modelling in heterogeneous media with high velocity contrasts

SUMMARY In this paper a new numerical technique is proposed for modelling elastic wave propagation in heterogeneous media with high velocity contrasts, as for instance in the presence of weathered layers or soft marine sediments. The scheme allows a sharp jump in grid spacing across the interface with high velocity contrasts. That is, the number of nodes along the interface with high velocity contrasts may be different for two meshes beside the interface. Sharp jumps of grid spacing can occur along a curved interface. Therefore, the proposed scheme can use a fine mesh in regions with low velocities and a coarser mesh in regions with higher velocities by modelling the interface topography with a piecewise linear function, so both spatial and temporal oversampling is avoided. This leads to a distinct reduction in the storage requirements and in the computational cost. The scheme is developed by formulating the problem in terms of stresses (at nodes) in regions with low S-wave velocities and in terms of velocities (at nodes) in regions with higher velocities, and then defining an explicit boundary between them. The boundary conditions on this explicit boundary with a complex geometry are implemented by introducing an integral approach to the equations on the explicit boundary. The outcome of this is that no nodes need to be added through the interpolations of the field variables. Moreover, no smoothing of the material parameters is needed on the interface with high velocity contrasts. A stress-grid scheme is developed to solve the problem inside regions with low S-wave velocities, and the (velocity) grid method is used to solve the problem inside regions with higher velocities. The algorithm is a second-order approximation for an irregular mesh in comparison to conventional finite differences. Implementation of the free-surface conditions of a complex geometrical surface is obtained for the proposed stress-grid scheme when the surface topography is approximated to be piecewise linear. Thus, the new algorithm can model elastic wave propagation even when the regions of low S-wave velocities reach the surface. Numerical examples demonstrate the efficiency of the proposed numerical technique.

An Acoustic Method for Soil Moisture Measurement

The paper deals with the problem of measuring the moisture of agricultural soils by an accurate, on-site, real-time method. The idea is to estimate the moisture by measuring the speed of sound in the medium: the main issue is therefore to determine a precise relationship between the two quantities. To this purpose, the Brutsaert's model for elastic waves in porous media is applied, taking into consideration different kinds of soil of common interest in agriculture. The authors have derived the speed-moisture curves, the conditions for the actual validity of the curves, and the suitable sound frequency for performing the measurement, for a wide range of agricultural soils in different physical conditions. The paper results are therefore the basis to realize an inexpensive and accurate moisture sensor for farmers.

The numerical modeling of 3-D elastic wave equation using a high-order, staggered-grid, finite difference scheme

This article provides the application of the high-order, staggered-grid, finite-difference scheme to model elastic wave propagation in 3-D isotropic media. Here, we use second-order, temporal- and high-order spatial finite-difference formulations with a staggered grid for discretization of the 3-D elastic wave equations of motion. The set of absorbing boundary conditions based on paraxial approximations of 3-D elastic wave equations are applied to the numerical boundaries. The trial results for the salt model show that the numerical dispersion is decreased to a minimum extent, the accuracy high and diffracted waves abundant. It also shows that this method can be used for modeling wave propagation in complex media with the lateral variation of velocity.

Delay Boundary Conditions for Elastic Wave Modeling

AbstractIn numerical modeling elastic wave equations, an artificial boundary must be given around the area in which the wave propagation is interesting. Unlike the wave filed within the area, the wave filed on the boundary cannot be calculated using the center difference method, thus the numerical calculation precision is decreased greatly, and as a result reflection is generated. For avoiding the reflection generating on the boundary, the delay boundary conditions method is proposed by this paper. On the basis of the fact that there are the same phase‐delay and the same coefficient of amplitude attenuation on equidistant particles in the direction of wave propagation, the phase‐delay and the coefficient of amplitude attenuation on the boundary can be calculated using the wave field within the boundary. Thus the precision of the wave field calculation on the boundary is improved and the reflection is removed.

Free-Surface Boundary Condition in Finite-Difference Elastic Wave Modeling

We design a new frequency-domain, finite-difference approach, based on a displacement formulation, which correctly describes the stress-free conditions at a free surface. In the conventional, displacement-based finite-difference method, we assign both displacements and material properties such as density and Lame constants to nodal points (a node-based grid set), whereas in our new finite-difference method, displacements are still defined at nodal points but material properties within cells (a cell-based grid set). In our new finite-difference technique using the cell-based grid set, stress-free conditions at the free surface are described by the changes in the material properties without any additional stress-free boundary condition. By conducting accuracy tests, we confirmed that the new second-order finite differences formulated in the cell-based grid set generate numerical solutions compatible with analytic solutions within the range of errors determined by dispersion analysis; the new, cell-based, weighted-averaging finite-difference method also yields better solutions than the old, node-based, weighted-averaging finite-difference method. The cell-based finite-difference method does not violate the reciprocity.

A numerical study of 1-D nonlinear P-wave propagation in solid

Two central schemes of finite difference (FD) up to different accuracy orders of space sampling step Δx (Fourth order and Sixth order respectively) were used to study the 1-D nonlinear P-wave propagation in the nonlinear solid media by the numerical method. Distinctly different from the case of numerical modeling of linear elastic wave, there may be several difficulties in the numerical treatment to the nonlinear partial differential equation, such as the steep gradients, shocks and unphysical oscillations. All of them are the great obstacles to the stability and convergence of numerical calculation. Fortunately, the comparative study on the modeling of nonlinear wave by the two FD schemes presented in the paper can provide us with an easy method to keep the stability and convergence in the calculation field when the product of the absolute value of nonlinear coefficient and the value of ϖu/ϖx are small enough, namely, the value of βϖu/ϖx is much smaller than 1. Several results are founded in the numerical study of nonlinear P-wave propagation, such as the waveform aberration, the generation and growth of harmonic wave and the energy redistribution among different frequency components. All of them will be more violent when the initial amplitudeA 0 is larger or the nonlinearity of medium is stronger. Correspondingly, we have found that the nonlinear P-wave propagation velocity will change with different initial frequencyf of source wave or the wave velocityc (equal to the P-wave velocity in the same medium without considering nonlinearity).

N-Times Absorbing Boundary Conditions for Compact Finite-Difference Modeling of Acoustic and Elastic Wave Propagation in the 2D TI Medium

This article presents decoupling n-times absorbing boundary conditions designed to model acoustic and elastic wave propagation in a 2D transversely iso- tropic (TI) medium. More general n-times boundary conditions with absorbing pa- rameters are also obtained by cascading first-order differential operators with param- eters. These boundary conditions are approximated with simple finite-difference schemes for numerical simulations. The numerical results show that the absorbing for the reflection waves strengthens with increasing the absorbing times n and the discretization boundary formulas are stable. Specially, the n-times absorbing bound- ary condition with absorbing parameters is better than that without the absorbing parameters under the case of same absorbing order. Elastic wave fields and three- component synthetic seismograms, generated by using the compact finite-difference and the decoupling n-times absorbing boundary, also illustrate that the n-times ab- sorbing boundary condition can eliminate effectively the spurious numerical reflec- tions in the acoustic and elastic wave modeling for the TI medium case.

Weighted-Averaging Finite-Element Method for 2D Elastic Wave Equations in the Frequency Domain

We present a weighted-averaging frequency-domain finite-element method for an accurate and efficient 2D elastic wave modeling technique. Our method introduces three kinds of supplementary element sets in addition to a basic element set that is used in the standard finite-element method. By constructing global stiffness and mass matrices for four kinds of element sets and then averaging them with weighting coefficients, we obtain a new global stiffness and mass matrix. With optimal weighting coefficients determined by a Marquardt–Levenberg method to minimize grid dispersion and grid anisotropy, we can reduce the number of nodal points per shear wavelength from 33.3 (using the standard finite-element method) and 20 (using the eclectic method) to 5, with the errors of group velocities no larger than 1%. By reducing the number of grid points per wavelength, we achieve a 97% and 75% reduction of computer memory required to store the complex impedance matrix for a band-type matrix solver and a nested dissection method, respectively, compared with those of the eclectic method. Our method gives approximate solutions compatible with exact solutions for an infinite homogeneous, a semi-infinite homogeneous (Lamb9s problem), and a horizontal two-layer model with fewer grid points than the standard and the eclectic method. A major advantage of the weighted-averaging finite-element method for the elastic wave equation is that it provides solutions very close to correct solutions for Lamb9s problem economically, unlike most of the displacement approaches. In addition, our scheme makes the complex impedance matrix symmetric, which satisfies reciprocity. Seismic forward modeling techniques that satisfy reciprocity are of critical importance in seismic imaging and inversion because we can economically calculate a Jacobian matrix using the reciprocity. Successful simulation of a large-size model shows that our method can be used for the simulation of wave propagation in the geological model needed in the reverse-time migration or seismic inversion.

Solving elastodynamics in a fluid–solid heterogeneous sphere: a parallel spectral element approximation on non-conforming grids

We present a spectral element approach for modeling elastic wave propagation in a solid–fluid sphere where the local effects of gravity are taken into account. The equations are discretized in terms of the displacement in the solid and the velocity potential in a neutrally stratified fluid. The spatial approximation is based upon a spherical mesh of hexahedra in which local refinement allows for adapting the discretization to the variation of elastic parameters in both the solid and the fluid regions. Continuity constraints across the non-conforming interfaces are introduced through Lagrange multipliers which are further discretized by the mortar element method. Due to the spherical nature of the non-conforming interfaces the mortar method turns out to be functionally conforming and allows for an equal-order interpolation of the primal variables and the Lagrange multipliers. The method is shown to provide an accurate solution when compared to analytical calculations obtained for radial models of elastic parameters. Its parallel implementation is based upon a simple domain decomposition strategy which makes it efficient to solve large problems as those imposed by planetary scales.

Elastic wave modeling with free surfaces: Stability of long simulations

Three‐dimensional (3D) elastic wave propagation modeling in the velocity‐stress formulation using finite differences (FDs) have been investigated for a homogeneous medium covered by a representative, relatively steep surface topography consisting of a 1D square root function. This scenario using various numerical implementations is explored. The behavior with regard to stability of long simulations is expected to be indicative of each numerical implementation's robustness for other types of topographies/media. Employing various combinations of the FD order was found only to change the time of the first incidence of instability. On the other hand, nonequidistant grids in the horizontal and vertical directions are found to be extremely useful for long‐term stability of 3D wave propagation modeling with our free‐surface boundary condition for single‐valued topographies. In particular dz ≥ (3/2) dx is found completely stable for all tested vP/vSratios. Such relationships of using dz > dx are also favorable for more accurate Rayleigh‐wave modeling. Setting the density equal to 1/10 of its interior value at one layer only at the surface is another simple means of achieving stability.

Local shear wave observations in João Câmara, northeast Brazil

We present shear wave splitting data from a 10 km aperture, eight‐station digital three‐component seismograph network, which operated from December 1992 to August 1994 near João Câmara (5°33′S, 35°51′W), to record shear waves from continuing microearthquake activity persisting 6 years after a main shock in November 1986. Previous aftershock recording had shown remarkably impulsive simple seismograms and hypocentral errors of typically 200 m in a structure of near‐uniform wave speed, but only one station was three component. The pattern of polarization directions shown is consistent with the trend of Precambrian foliation observed in the field. It is therefore concluded that these shear wave observations are controlled by the seismic anisotropy associated with the Precambrian foliation and hence bear the signature of the “paleo‐stress field,” i.e., the Precambrian deformation regime. This conclusion has important implications for the interpretation of shear wave splitting observations made in crustal crystalline rock, because other authors have asserted that shear wave splitting normally results from the effect of stress‐aligned fluid‐filled cracks (extensive dilatancy anisotropy or EDA). EDA is, by contrast, a signature of the present‐day stress field. In this region, there is firm evidence that the present‐day maximum compressive stress is E‐W. This evidence comes from earthquake focal mechanisms and borehole breakout data and is summarized in the World Stress Map. We therefore conclude that our present data are in maximum conflict with an interpretation in terms of EDA. Our interpretation is supported by the observation that delay times are consistent with those expected from modeling of elastic waves in the crust.

Elastic wave modelling in 3D heterogeneous media: 3D grid method

SUMMARY We present a new numerical technique for elastic wave modelling in 3D heterogeneous media with surface topography, which is called the 3D grid method in this paper. This work is an extension of the 2D grid method that models P-SV wave propagation in 2D heterogeneous media. Similar to the finite-element method in the discretization of a numerical mesh, the proposed scheme is flexible in incorporating surface topography and curved interfaces; moreover it satisfies the free-surface boundary conditions of 3D topography naturally. The algorithm, developed from a parsimonious staggered-grid scheme, solves the problem using integral equilibrium around each node, instead of satisfying elastodynamic differential equations at each node as in the conventional finite-difference method. The computational cost and memory requirements for the proposed scheme are approximately the same as those used by the same order finite-difference method. In this paper, a mixed tetrahedral and parallelepiped grid method is presented; and the numerical dispersion and stability criteria on the tetrahedral grid method and parallelepiped grid method are discussed in detail. The proposed scheme is successfully tested against an analytical solution for the 3D Lamb problem and a solution of the boundary method for the diffraction of a hemispherical crater. Moreover, examples of surface-wave propagation in an elastic half-space with a semi-cylindrical trench on the surface and 3D plane-layered model are presented.

Elastic waves interacting with buried land mines: a study using the FDTD method

A three-dimensional (3-D) finite-difference model for elastic waves in the ground has been developed and implemented. The model has been created to supplement the development of a sensor that uses elastic waves to detect buried land mines. The model is used to investigate the propagation characteristics of elastic waves in the ground and to explore the interaction of elastic waves with buried land mines. When elastic waves interact with a buried mine, a strong resonance occurs at the mine location. The resonance can be used to enhance the mine's signature and to distinguish the mine from clutter. Results presented in this paper explain the features of elastic wave propagation in the ground and show the interaction of elastic waves with both an anti-personnel mine and an anti-tank mine.

Effective velocities in fractured media: a numerical study using the rotated staggered finite‐difference grid

ABSTRACTThe modelling of elastic waves in fractured media with an explicit finite‐difference scheme causes instability problems on a staggered grid when the medium possesses high‐contrast discontinuities (strong heterogeneities). For the present study we apply the rotated staggered grid. Using this modified grid it is possible to simulate the propagation of elastic waves in a 2D or 3D medium containing cracks, pores or free surfaces without hard‐coded boundary conditions. Therefore it allows an efficient and precise numerical study of effective velocities in fractured structures. We model the propagation of plane waves through a set of different, randomly cracked media. In these numerical experiments we vary the wavelength of the plane waves, the crack porosity and the crack density. The synthetic results are compared with several static theories that predict the effective P‐ and S‐wave velocities in fractured materials in the long wavelength limit. For randomly distributed and randomly orientated, rectilinear, non‐intersecting, thin, dry cracks, the numerical simulations of velocities of P‐, SV‐ and SH‐waves are in excellent agreement with the results of the modified (or differential) self‐consistent theory. On the other hand for intersecting cracks, the critical crack‐density (porosity) concept must be taken into account. To describe the wave velocities in media with intersecting cracks, we propose introducing the critical crack‐density concept into the modified self‐consistent theory. Numerical simulations show that this new formulation predicts effective elastic properties accurately for such a case.

Use of coordinate-free numerics in elastic wave simulation

The modeling of elastic waves in solids and fluids is widely applied in physics and geophysics and often requires large computer codes for its realization. Despite the fact that different wave propagation problems have many similarities from the point of view of the abstract mathematical formulation, such codes are usually hard to adapt to new problems and changing requirements. In this paper we discuss a so-called coordinate-free approach for the general computer implementation of tensor-field equations. We show how it applies to the modeling of seismic waves in isotropic and anisotropic structures, sonic waves in boreholes and ultrasonic waves in poro-elastic fluid-filled materials. The results clearly indicate that the coordinate-free approach is very flexible for the implementation of various wave propagation problems.

TIME DOMAIN MODELING OF AXISYMMETRIC WAVE PROPAGATION IN ISOTROPIC ELASTIC MEDIA WITH CEFIT — CYLINDRICAL ELASTODYNAMIC FINITE INTEGRATION TECHNIQUE

The Elastodynamic Finite Integration Technique (EFIT), originally developed by Fellinger et al.,1–3 represents a stable and efficient numerical code to model elastic wave propagation in linearly-elastic isotropic and anisotropic, homogeneous and heterogeneous as well as dissipative and nondissipative media. In previous works, the FIT discretization of the basic equations of linear elasticity, Hooke's law and Cauchy's equation of motion, was exclusively carried out in Cartesian coordinates. For problems in cylindrical geometries it is more suitable to use cylindrical coordinates. By that, axisymmetric problems can be treated in a two-dimensional staggered grid in the r,z-plane. The paper presents an EFIT version for axisymmetric problems in cylindrical coordinates called Cylindrical EFIT (CEFIT). After demonstrating the accuracy of the numerical code by a comparison between simulation results and analytical solutions, different examples of application are given. These examples include modeling of sound fields of ultrasonic transducers, thermoelastic laser sources, geophysical borehole probes, impact-echo measurements in layered media, and load simulations of the European Spallation Source (ESS) mercury target.

Time Domain Modeling of Axisymmetric Wave Propagation in Isotropic Elastic Media With Cefit - Cylindrical Elastodynamic Finite Integration Technique

The Elastodynamic Finite Integration Technique (EFIT), originally developed by Fellinger et al.,1–3 represents a stable and efficient numerical code to model elastic wave propagation in linearly-elastic isotropic and anisotropic, homogeneous and heterogeneous as well as dissipative and nondissipative media. In previous works, the FIT discretization of the basic equations of linear elasticity, Hooke's law and Cauchy's equation of motion, was exclusively carried out in Cartesian coordinates. For problems in cylindrical geometries it is more suitable to use cylindrical coordinates. By that, axisymmetric problems can be treated in a two-dimensional staggered grid in the r,z-plane. The paper presents an EFIT version for axisymmetric problems in cylindrical coordinates called Cylindrical EFIT (CEFIT). After demonstrating the accuracy of the numerical code by a comparison between simulation results and analytical solutions, different examples of application are given. These examples include modeling of sound fields of ultrasonic transducers, thermoelastic laser sources, geophysical borehole probes, impact-echo measurements in layered media, and load simulations of the European Spallation Source (ESS) mercury target.

Acoustic detection of buried objects in 3-D fluid saturated porous media: numerical modeling

Acoustic waves can be a viable tool for the detection and identification of land mines, unexplored ordnance (UXO), and other buried objects. Design of acoustic instruments and interpretation and processing of acoustic measurements call for accurate numerical models to simulate acoustic wave propagation in a heterogeneous soil with buried objects. Compared with the traditional seismic exploration, high attenuation is unfortunately ubiquitous for shallow surface acoustic measurements because of the loose soil and the fluid in its pore space. To adequately model such acoustic attenuation, we propose a comprehensive multidimensional finite-difference time-domain (FDTD) model to simulate the acoustic wave interactions with land mines and soils based on the Biot theory for poroelastic media. For the truncation of the computational domain, we use the perfectly matched layer (PML). The method is validated by comparison with analytical solutions. Unlike the pure elastic wave model, this efficient PML-FDTD model for poroelastic media incorporates the interactions of waves and the fluid-saturated pore space. Several typical land mine detection measurements are simulated to illustrate the application.

A multidomain approach of the Fourier pseudospectral method using discontinuous grid for elastic wave modeling

Pseudospectral method is an efficient and high-accuracy numerical modeling technique that has been applied to the simulation of seismic wavefields in realistic models with complex structures such as sedimentary basin. However, if the sedimentary layers are included in the model, high velocity contrasts arise because of the very low S-wave velocities of the sediments as compared with the bedrock. The grid spacing then has to be very small, which leads to spatial oversampling in regions with higher velocities. As a result, we also have to use very small time interval for time marching, so that the computation requires much CPU time. A treatment of this problem is use of a multidomain approach that can employ different grid spacings in different regions, i.e. “discontinuous grid”. In this study, we propose a scheme for the Fourier pseudospectral method using discontinuous grid, which uses an interpolation by the FFT. This scheme can reduce the computation time and the required computer memory as compared with the conventional one. The accuracy and efficiency of this approach are demonstrated in the present paper through examples of 2-D elastic wave modeling.