Wave Propagation In Elastic Media Research Articles

A residual perfectly matched layer for wave propagation in elastic media

A residual perfectly matched layer for wave propagation in elastic media

Nonlinear convolution finite element method for solution of large deformation elastodynamics

A new algorithm based on the convolution finite element method (CFEM) is proposed for the nonlinear wave propagation in elastic media. The formulation is developed in the context of the total Lagrangian framework, encompassing contributions due to both geometrical and material nonlinearities. As a basis, a counterpart of equations of motion – namely, the alternative field equations – is first established. The satisfaction of the alternative field equations is then realized in a weak sense. Next, the Newton–Raphson procedure and the consistent tangential matrix are applied to the weak formulation, where the CFEM is used as the linear solver in each iteration. Finally, several examples are carried out to examine the theoretical aspects and the feasibility of the proposed algorithm. In particular, problems of free vibration of Neo-Hookean and Saint Venant–Kirchhoff plates are explored. Also, a cantilever beam of the Neo-Hookean material is simulated for the case of forced vibrations. Conspicuously, in contrast to the existing time-step methods with finite order of accuracy, the new solution procedure obtains the accurate solution when the time-step size is increased.

Digitally tunable non-reciprocal wave propagation using piezoelectric metamaterials with synthetic impedance circuits

One common approach to enable non-reciprocal wave propagation in elastic media relies on introducing a directional bias by parameter modulation (e.g., stiffness modulation) in space and time. Researchers have successfully demonstrated non-reciprocal wave propagation by coupling elastic waveguides with magnetic electrical coil elements, piezoelectric elements shunted to capacitive analog circuits, or rotating mechanical resonators. The existing efforts are often limited by narrow operation frequency ranges and cumbersome implementations that lack smooth modulation (e.g., with analog circuit elements and switches). In this work, we demonstrate precise and smooth non-reciprocal wave propagation in a piezoelectric metamaterial by connecting each unit cell to a digitally controlled synthetic impedance circuit. The effective impedance of each unit cell can be externally controlled according to a desired space-time modulation scheme over a wide frequency range. We present numerical and experimental investigations on the dynamics of a piezoelectric metamaterial beam with 30 bimorph unit cells whose impedances can be varied smoothly. Specifically, we employ the spatiotemporal modulation on synthetic inductance circuits rather than capacitive. Experimental results are presented for various space-time modulation profiles, investigating the effects of the inductance value (target frequency), as well as the modulation frequency and amplitude on digitally tunable non-reciprocal wave propagation.

ASSESSMENT OF GROUND SURFACE FLUCTUATIONS DURING SHIELD TUNNELING OF SUBWAY TUNNELS

T The problems of evaluation of ground surface vibrations created by underground vibration sources are quite acute, especially in the conditions of urban development. During the construction of subway tunnels by the shield method there is a dynamic impact during the work of working mechanisms. The oscillations during the excavation of the soil by cutting mechanisms of the shield complex, during the ground bailing and pumping of mortar behind the lining can reach the day surface and cause significant deformations. When designing long underground tunnels built next to existing buildings, it is necessary to estimate the parameters of dynamic impact on the buildings caused by the operation of shielding mechanisms. The paper deals with the estimation of dynamic effects of various devices and mechanisms used in underground works. At this stage, the problems of oscillations of the surface of an elastic half-space from various point sources of waves located at different depths, taking into account the waves reflected from the free surface, are solved. The distribution of stresses and displacements during wave propagation in elastic media is a complex picture. In confined media the picture of wave propagation becomes even more complicated due to reflection of waves from boundaries. The reciprocity theorem is used to estimate the levels of ground surface oscillations created by dynamic effects of various devices and mechanisms during shield tunneling. The dynamic effects of working mechanisms are modeled by: the force directed along the axis of the shield, a pair of forces, two pairs of forces without momentum, a combination of a pair of forces and a longitudinal force, a combination of two pairs without momentum and longitudinal force.

Elastodynamic wave propagation modelling in geological structures considering fully-adaptive explicit time-marching procedures

In this paper, two fully-adaptive explicit time-marching procedures are discussed for the time-domain solution of wave propagation in elastic media. In these procedures, the time integration parameters of the methods are adaptively locally evaluated, following the properties of the discretized model and the values of the calculated responses, engendering more accurate solution procedures. In addition, automated domain decomposition and sub-cycling procedures are also performed, providing more efficient analyses. In this case, a domain decomposition procedure divides the domain of the model into different time-marching subdomains according to the properties of the discretized problem and the stability limit of the adopted time-marching technique, so that a different time-step value is applied to each subdomain, leading to more efficient (and yet stable) explicit time-domain computations. As it is indicated in this work, the adoption of multi-time-steps/sub-cycling splitting procedures further improves the accuracy of the discussed adaptive time-marching formulations. At the end of the paper, benchmark analyses are performed to illustrate the effectiveness of the proposed techniques, followed by synthetic case analyses, with degrees of complexity equivalent to that of real geophysical applications in the OIL & GAS industry, demonstrating the robustness of the proposed explicit approaches.

First steps on modelling wave propagation in isotropic-heterogeneous media: numerical simulation of P–SV waves

In geophysics, wave propagation in elastic media is a crucial subject. In this context, seismology has made significant progress as a result of numerous advances, among these stands out the advancement of numerical methods such as the finite-difference one. Usually, seismic wave propagation in elastic media results in complex systems of partial differential equations, whose solutions cannot be obtained in an analytical way, especially when dealing with heterogeneous media. In consequence, there exists a necessity to implement numerical methods. However, the available information about the construction of a numerical solution of these equations is not as explicit as it should be. Our main goal is to provide pedagogical instructions for undergraduate students who want to improve their computational skills as well as their knowledge of the subject. The last through current teaching methods involving challenging problems and transversal tools. Particularly, we provide a brief description of the equations and a detailed description of the numerical solution of the seismic wave equation. Furthermore, we model two different seismic explosive sources in both homogeneous and heterogeneous media as an illustrative example. In the results, we include velocity maps showing wave propagation in the x–z plane, and the z-velocity as a function of time measured in a series of detectors distributed in the numerical domain.

The influence of heterogeneity and initial stress on the propagation of Love-type wave in a transversely isotropic layer subjected to rotation.

In this paper, a mathematical model of Love-type wave propagation in a heterogeneous transversely isotropic elastic layer subjected to initial stress and rotation of the resting on a rigid foundation. Frequency equation of Love-type wave is obtained in closed form. The material constants and initial stress have been taken as space dependent and arbitrary functions of depth in the respective media. The dispersion equation is determined to study the effect of different types of parameters such as inhomogeneity, initial stress, rotation, wave number, the phase velocity on the Love-type wave propagation. The analytical solution has been obtained, we have used the separation of variables, method and the numerical solution using the bisection method implemented in MATLAB. We present a general dispersion relation to describe the impacts as the propagation of Love-type waves in the structures. Numerical results analyzing the dispersion equation are discussed and presented graphically. Moreover, the obtained dispersion relation is found in well agreement with the classical case in isotropic and transversely isotropic layer resting on a rigid foundation. Finally, some graphical presentations have been made to assess the effects of various parameters in the plane wave propagation in elastic media of different nature.

An overset-grid finite-difference algorithm for simulating elastic wave propagation in media with complex free-surface topography

We have developed an overset-grid algorithm to simplify the difficulty of curvilinear grid (CG) generation and increase the computational efficiency for seismic wavefield modeling by using the finite-difference method in areas with complex surface topography. The overset grid comprises a Cartesian grid block and an approximately orthogonal CG block. The Cartesian grid covers most of the simulation domain, whereas the CG discretizes the near-surface topography. The Cartesian grid and the CG overlap each other arbitrarily. We use sixth-order explicit Lagrangian interpolation to exchange data between the Cartesian grid and the CG, which is shown to be sufficiently accurate. We also find that spatially smoothing the source term is important for reducing strong artificial reflections when the source is near the overlapping zone. Finally, numerical tests are performed to verify that the proposed overset grid is well suited for effective numerical simulation of seismic wave propagation.

A level-set approach based on reaction–diffusion equation applied to inversion problems in acoustic wave propagation

In this work, a topology optimization procedure based on the level-set method is applied to the solution of inverse problems for acoustic wave propagation in the time-domain. In this class of inverse problems the presence of obstacles in a background medium must be identified. Obstacles and background are defined by means of a level-set function that evolves by following the solution of a reaction–diffusion equation. Within this approach, no initial guess for the topology nor level-set reinitialization procedures are necessary, contrary to what is commonly observed when the Hamilton–Jacobi equation is used. The objective function is defined as the domain and time integration of the squared difference between experimental and simulation pressure signals. The finite element method is used for the spatial and level-set function discretizations and a time-marching procedure (Newmark scheme) is used to solve the wave propagation problem, as well as the adjoint problem for the sensitivity analysis. Both procedures provide the information needed to define the velocity field for the level set evolution. Results show that the proposed technique is capable to find the location and shape of obstacles within a background medium. Systematic tests show that, as expected, the distribution of sources and receivers shows to have a major influence on the final solution. Results also reproduce known difficulties; when the so called inverse crime is avoided, the identification procedure worsens its performance. Filters and smoothing are among different features that deserve further investigation. Although the formulation presented here focuses on the acoustic wave propagation problem, its extension to wave propagation in elastic media is straightforward.

Application of comprehensive advanced geological prediction technology in Da-puling tunnel

For the Da-puling tunnel of Puqing Expressway in Guangxi, the advanced geological prediction is carried out by combining TSP long-distance forecast method with short distance geological radar method. This paper describes the principle of seismic wave propagation in elastic medium, as well as the key points of data processing and analysis, some requirements that should be paid attention to the field test and scientific way of image interpretation put forward to improve the accuracy of prediction; When TSP is deployed, it should be sharp angle with potential joint surface. P-wave reacts surrounding rock properties, the shear wave is closely related to the transverse skeleton of medium. In data interpretation, it is necessary to focus on the analysis of the characteristics of P-wave and S-wave, weakening Poisson’s ratio and Young’s modulus. TSP and GPR can achieve the mutual complement and improve the detection accuracy.

2D multi-parameter waveform inversion of land reflection seismic data obtained from the particle-motion response from the vertical geophone

Onshore seismic exploration analyzes seismic wave propagation in elastic media, which includes the conversion between P- and S-waves. The development of multi-wave and multi-component seismic exploration methods provides data that enable onshore elastic wave full-waveform inversion. However, most data sets of onshore exploration are single component obtained from the particle-motion response from the vertical geophone. When the aiming area has a low-velocity zone, the ray path of reflected wave that propagates to the detector is nearly perpendicular to the ground surface, so that we call it P-wave data. In this paper, we focus on multi-parameter waveform inversion using P-wave reflection seismic data. Although only P-wave data are received, it still contains the converted P-wave information, and the converted P-wave energy gradually increases as the offset increases. As seismic acquisition technology, observation systems and science develop, the folds and acquisition offset increase significantly, and the seismic data contain important converted P-wave information. In this paper, the first-order elastic velocity–stress equation is decomposed to obtain the scalar-P-wave equation from which the S-wave velocity is included firstly. Then we present the theoretical framework for onshore multi-parameter full-waveform inversion using P-wave data. In order to explore the inversion potential of the P-wave data (extracting the S-wave velocity from the converted P-wave information) and accuracy and stability of the P- and S-wave velocity inverted by our method, we carry out numerical tests via different inversion strategies, by using the P-wave data regarded as containing converted P-wave information, and get successful results.

Modified LSM for size-dependent wave propagation: comparison with modified couple stress theory

A modified LSM is proposed by introducing an independent micro-rotational inertia, which may help characterize the scale-dependent effect and avoid the Poisson’s ratio limitation of regular triangle lattices in two dimensions. For this method, some factors may affect data pickup and modeling accuracy, but the ‘optimal’ inputs, like stiffness ratio, numerical damping, and micro-rotational inertia, could be obtained from parameter identification by the Dakota toolkit (Adams et al., Tech Rep SAND2010–2183, 2009), when a suitable excitation source function and lattice spacing are set up. By comparing with the modified couple stress theory, we analyze the dispersion relationship of elastic waves for the estimation of the characteristic material length. It shows that this modified LSM may provide an alternative and promising way to investigate the size-dependent wave propagation in elastic media numerically.

New acoustic approximation for transversely isotropic media with a vertical symmetry axis

Seismic data processing in the elastic anisotropic model is complicated due to multiparameter dependency. Approximations to the P-wave kinematics are necessary for practical purposes. The acoustic approximation for P-waves in a transversely isotropic medium with a vertical symmetry axis (VTI) simplifies the description of wave propagation in elastic media, and as a result, it is widely adopted in seismic data processing and analysis. However, finite-difference implementations of that approximation are plagued with S-wave artifacts. Specifically, the resulting wavefield also includes artificial diamond-shaped S-waves resulting in a redundant signal for many applications that require pure P-wave data. To derive a totally S-wave-free acoustic approximation, we have developed a new acoustic approximation for pure P-waves that is totally free of S-wave artifacts in the homogeneous VTI model. To keep the S-wave velocity equal to zero, we formulate the vertical S-wave velocity to be a function of the model parameters, rather than setting it to zero. Then, the corresponding P-wave phase and group velocities for the new acoustic approximation are derived. For this new acoustic approximation, the kinematics is described by a new eikonal equation for pure P-wave propagation, which defines the new vertical slowness for the P-waves. The corresponding perturbation-based approximation for our new eikonal equation is used to compare the new equation with the original acoustic eikonal. The accuracy of our new P-wave acoustic approximation is tested on numerical examples for homogeneous and multilayered VTI models. We find that the accuracy of our new acoustic approximation is as good as the original one for the phase velocity, group velocity, and the kinematic parameters such as vertical slowness, traveltime, and relative geometric spreading. Therefore, the S-wave-free acoustic approximation could be further applied in seismic processing that requires pure P-wave data.

Frequency-domain elastic wavefield simulation with hybrid absorbing boundary conditions

AbstractThe frequency-domain seismic modeling has advantages over the time-domain modeling, including the efficient implementation of multiple sources and straightforward extension for adding attenuation factors. One of the most persistent challenges in the frequency domain as well as in the time domain is how to effectively suppress the unwanted seismic reflections from the truncated boundaries of the model. Here, we propose a 2D frequency-domain finite-difference wavefield simulation in elastic media with hybrid absorbing boundary conditions, which combine the perfectly matched layer (PML) boundary condition with the Clayton absorbing boundary conditions (first and second orders). The PML boundary condition is implemented in the damping zones of the model, while the Clayton absorbing boundary conditions are applied to the outer boundaries of the damping zones. To improve the absorbing performance of the hybrid absorbing boundary conditions in the frequency domain, we apply the complex coordinate stretching method to the spatial partial derivatives in the Clayton absorbing boundary conditions. To testify the validity of our proposed algorithm, we compare the calculated seismograms with an analytical solution. Numerical tests show the hybrid absorbing boundary condition (PML plus the stretched second-order Clayton absorbing condition) has the best absorbing performance over the other absorbing boundary conditions. In the model tests, we also successfully apply the complex coordinate stretching method to the free surface boundary condition when simulating seismic wave propagation in elastic media with a free surface.

Perturbation-based prediction of vibration phase shift along fluid-conveying pipes due to Coriolis forces, nonuniformity, and nonlinearity

Flexural vibrations of a fluid-conveying pipe are investigated theoretically, with special consideration to the spatial shift in vibration phase caused by fluid flow and various imperfections. The latter includes small nonuniformity or asymmetry in stiffness, mass, or damping, and weak stiffness and damping nonlinearity. Besides contributing general understanding of wave propagation in elastic media with gyroscopic forces, this is relevant for the design, control, and troubleshooting of phase shift measuring devices like Coriolis mass flowmeters. A multiple time-scaling perturbation analysis is employed with a simple model of a fluid-conveying pipe with relevant imperfections, resulting in simple analytical expressions for the prediction of phase shift. For applications like Coriolis flowmetering, this allows for readily examining effects of a variety of relevant features, like small sensors and actuators, production inaccuracies, mounting conditions, wear, contamination, and corrosion. To second order of accuracy, only mass flow and asymmetrically distributed damping are predicted to introduce spatial phase shift, while nonuniformly distributed linear mass and stiffness, symmetrically distributed linear damping, and uniformly nonlinear stiffness and damping are all negligible in comparison. The analytical predictions are illustrated by examples and validated with excellent agreement against numerical analysis for realistic magnitudes of parameters.

An algorithm for fast elastic wave simulation using a vectorized finite difference operator

Modern geophysical imaging techniques exploit the full wavefield information which can be simulated numerically. These numerical simulations are computationally expensive due to several factors, such as a large number of time steps and nodes, big size of the derivative stencil and huge model size. Besides these constraints, it is also important to reformulate the numerical derivative operator for improved efficiency. In this paper, we have introduced a vectorized derivative operator over the staggered grid with shifted coordinate systems. The operator increases the efficiency of simulation by exploiting the fact that each variable can be represented in the form of a matrix. This operator allows updating all nodes of a variable defined on the staggered grid, in a manner similar to the collocated grid scheme and thereby reducing the computational run-time considerably. Here we demonstrate an application of this operator to simulate the seismic wave propagation in elastic media (Marmousi model), by discretizing the equations on a staggered grid. We have compared the performance of this operator on three programming languages, which reveals that it can increase the execution speed by a factor of at least 2–3 times for FORTRAN and MATLAB; and nearly 100 times for Python. We have further carried out various tests in MATLAB to analyze the effect of model size and the number of time steps on total simulation run-time. We find that there is an additional, though small, computational overhead for each step and it depends on total number of time steps used in the simulation. A MATLAB code package, ’FDwave’, for the proposed simulation scheme is available upon request.

A rectangular-grid lattice spring model for modeling elastic waves in Poisson’s solids

The lattice spring model (LSM) combined with the velocity Verlet algorithm is a newly developed scheme for modeling elastic wave propagation in solid media. Unlike conventional wave equations, LSM is established on the basis of micromechanics of the subsurface media, which enjoys better dynamic characteristics of elastic systems. We develop a rectangular-grid LSM scheme for elastic waves simulation in Poisson’s solids, and the direction-dependent elastic constants are deduced to keep the isotropy of the discrete grid. The stability condition and numerical dispersion properties of LSM are discussed and compared with other numerical methods. The 2D and 3D numerical simulations are carried out using the rectangular-grid LSM, as well as the second- and fourth-order accuracy staggered finite-difference method (FDM). Wavefields obtained by LSM are fairly similar with those by analytical solution and FDM, which demonstrates the correctness of the proposed scheme and its capability of modeling elastic wave propagation in heterogeneous media. Moreover, we perform plane P-wave simulation through a semi-infinite cavity model via LSM and FDM, the recorded wavefield snapshots indicate that our proposed rectangular-grid LSM obtains more reasonable wavefield details compared with those of FDM, especially in media with high compliance and structure complexity. Our main contribution lies in offering an alternative simulation scheme for modeling elastic wave propagation in media with some kinds of complexities, which conventional FDM may fail to simulate.

The Propagation of Thermoelastic Waves in Anisotropic Media of Orthorhombic, Hexagonal, and Tetragonal Syngonies

The investigation of wave propagation in elastic medium with thermomechanical effects is bound to have important economic implications in the field of composite materials, seismology, geophysics, and so on. In this article, thermoelastic wave propagation in anisotropic mediums of orthorhombic and hexagonal syngony having heterogeneity along z-axis is studied. Such medium has second-order axis symmetry. By using analytical matriciant method, a set of equations of motions in thermoelastic medium are reduced to an equivalent set of the first-order differential equations. In the general case, for the given set of equations, structures of fundamental solutions are made and dispersion relations are obtained.

The Analytical Form of the Dispersion Equation of Elastic Waves in Periodically Inhomogeneous Medium of Different Classes of Crystals

The investigation of thermoelastic wave propagation in elastic media is bound to have much influence in the fields of material science, geophysics, seismology, and so on. The heat conduction equations and bound equations of motions differ by the difficulty level and presence of many physical and mechanical parameters in them. Therefore thermoelasticity is being extensively studied and developed. In this paper by using analytical matrizant method set of equation of motions in elastic media are reduced to equivalent set of first-order differential equations. Moreover, for given set of equations, the structure of fundamental solutions for the general case has been derived and also dispersion relations are obtained.

Discontinuous Galerkin method for wave propagation in elastic media with inhomogeneous inclusions

AbstractA discontinuous Galerkin method on unstructured grids is adapted and implemented for simulation of wave response of subvertical fractured systems in carbonate rocks for numerical solution of direct problems of seismic exploration. The present paper compares seismic responses for several mechanical-mathematical models of a fractured reservoir. The models of collectors differ in the presence and location of media interfaces relative to the collector and also in a way of its specification, namely, an explicit selection of fractures with the parameters of the medium in the domain of collector coinciding with the host medium, or differing from it. We indicate the ability to take into account inter-fracture interactions with the use of the model of a fractured layer presented in the paper and study wave processes formed as the result of interaction of seismic pulses with fractured reservoirs.