Oscillatory numerical experiments in finely layered anisotropic viscoelastic media

ABSTRACTIn this study, the characteristics of Love waves in viscoelastic vertical transversely isotropic layered media are investigated by finite-difference numerical modeling. The accuracy of the modeling scheme is tested against the theoretical seismograms of isotropic-elastic and isotropic-viscoelastic media. The correctness of the modeling results is verified by the theoretical phase-velocity dispersion curves of Love waves in isotropic or anisotropic elastic or viscoelastic media. In two-layer half-space models, the effects of velocity anisotropy, viscoelasticity, and attenuation anisotropy of media on Love waves are studied in detail by comparing the modeling results obtained for anisotropic-elastic, isotropic-viscoelastic, and anisotropic-viscoelastic media with those obtained for isotropic-elastic media. Then, Love waves in three typical four-layer half-space models are simulated to further analyze the characteristics of Love waves in anisotropic-viscoelastic layered media. The results show that Love waves propagating in anisotropic-viscoelastic media are affected by both the anisotropy and viscoelasticity of media. The velocity anisotropy of media causes substantial changes in the values and distribution range of phase velocities of Love waves. The viscoelasticity of media leads to the amplitude attenuation and phase velocity dispersion of Love waves, and these effects increase with decreasing quality factors. The attenuation anisotropy of media indicates that the viscoelasticity degree of media is direction dependent. Comparisons of phase velocity ratios suggest that the change degree of Love-wave phase velocities due to viscoelasticity is much less than that caused by velocity anisotropy.

SUMMARY FracturesarecommonintheEarth’scrustduetodifferentfactors,forinstance,tectonicstresses and natural or artificial hydraulic fracturing caused by a pressurized fluid. A dense set of fractures behaves as an effective long-wavelength anisotropic medium, leading to azimuthally varying velocity and attenuation of seismic waves. Effective in this case means that the predominant wavelength is much longer than the fracture spacing. Here, fractures are represented by surface discontinuities in the displacement u and particle velocity v as [κ ·u + η ·v], where the brackets denote the discontinuity across the surface, κ is a fracture stiffness and η is a fracture viscosity. We consider an isotropic background medium, where a set of fractures are embedded. There exists an analytical solution—with five stiffness components—for equispaced plane fractures and an homogeneous background medium. The theory predicts that the equivalent medium is transversely isotropic and viscoelastic. We then perform harmonic numerical experiments to compute the stiffness components as a function of frequency, by using a Galerkin finite-element procedure, and obtain the complex velocities of the medium as a function of frequency and propagation direction, which provide the phase velocities, energy velocities (wavefronts) and quality factors. The algorithm is tested with the analytical solution and then used to obtain the stiffness components for general heterogeneous cases, where fractal variations of the fracture compliances and background stiffnesses are considered.

Seepage flow problems in layered soils solved by the finite element method

SUMMARY We study properties of the energy-flux vector and other related energy quantities of homogeneous and inhomogeneous time-harmonic P and S plane waves, propagating in unbounded viscoelastic anisotropic media, both analytically and numerically. We propose an algorithm for the computation of the energy-flux vector, which can be used for media of unrestricted anisotropy and viscoelasticity, and for arbitrary homogeneous or inhomogeneous plane waves. Basic part of the algorithm is determination of the slowness vector of a homogeneous or inhomogeneous wave, which satisfies certain constraints following from the equation of motion. Approaches for determination of a slowness vector commonly used in viscoelastic isotropic media are usually difficult to use in viscoelastic anisotropic media. Sometimes they may even lead to non-physical solutions. To avoid these problems, we use the so-called mixed specification of the slowness vector, which requires, in a general case, solution of a complex-valued algebraic equation of the sixth degree. For simpler cases, as for SH waves propagating in symmetry planes, the algorithm yields simple analytic solutions. Once the slowness vector is known, determination of energy flux and of other energy quantities is easy. We present numerical examples illustrating the behaviour of the energy-flux vector and other energy quantities, for homogeneous and inhomogeneous plane P, SV and SH waves. 1 I NTR O DUCTION In this paper, we study the properties of the energy flux of homogeneous and inhomogeneous time-harmonic plane waves propagating in an unbounded viscoelastic anisotropic medium. The complexvalued energy-flux vector F (also called the Poynting or Umov‐ Poynting vector) is used as the quantity which fully characterizes the energy flux and the dissipation of energy of the wave. The real part of the vector F represents the time-averaged energy flux S. Both real, Re F = S, and imaginary, Im F, parts are closely related to the density of the time-averaged dissipated energy W d : W d equals twice the scalar product of −Im F and propagation vector P (real part of the slowness vector, perpendicular to the wave front) or twice the scalar product of Re F and attenuation vector A (imaginary part of the slowness vector, perpendicular to the plane of constant amplitude). The aim of this paper is to present a concise derivation of the expression for the complex-valued energy-flux vector F of homogeneous and inhomogeneous plane waves propagating in unbounded media of unrestricted anisotropy and viscoelasticity, to propose an algorithm for its calculation, and to perform analytical and numerical studies necessary for correct understanding of energy propagation and dissipation of the relevant plane waves. Considerable attention in seismology has been devoted to the energy flux of plane waves propagating in elastic anisotropic and in viscoelastic isotropic media. For elastic anisotropic media, see, for

Thin inclusions of large or small conductivity in an anisotropic homogeneous medium

Elastic study on singularities interacting with interfaces using alternating technique

A hyper-lens with planar object/image surfaces and uniform magnification factor is designed based on directional projecting property of optic-null medium (ONM). The proposed planar hyper-lens only requires one homogeneous anisotropic medium, i.e. ONM. We use metallic plates inside ϵ-near-zero medium to realize the proposed hyper-lens. 0.191λ 0 resolution is observed (0.238λ 0 can be clearly resolved) in experimental measurement, which consists very well with 3D numerical simulations and verifies the performance of the proposed planar hyper-lens. The planar hyper-lens with uniform pre-designed magnification factor may have some applications in super-resolution imaging technology.

A numerical comparison of finite difference (FD) and finite element (FE) methods for a stochastic ordinary differential equation is made. The stochastic ordinary differential equation is turned into a set of ordinary differential equations by applying polynomial chaos, and the FD and FE methods are then implemented. The resulting numerical solutions are all non-negative. When orthogonal polynomials are used for either continuous or discrete processes, numerical experiments also show that the FE method is more accurate and efficient than the FD method.

We study behavior of attenuation (inhomogeneity) angles γ, i.e., angles between real and imaginary parts of the slowness vectors of inhomogeneous plane waves propagating in isotropic or anisotropic, perfectly elastic or viscoelastic, unbounded media. The angle γ never exceeds the boundary attenuation angle γ*. In isotropic viscoelastic media γ* = 90°; in anisotropic viscoelastic media γ* may be greater than, equal to, or less than 90°. Plane waves with γ > γ* do not exist. Because γ* in anisotropic viscoelastic media is usually not known a priori, the commonly used specification of an inhomogeneous plane wave by the attenuation angle γ may lead to serious problems. If γ is chosen close to γ* or even larger, indeterminate, unstable or even nonphysical results are obtained. We study properties of γ* and show that the approach based on the mixed specification of the slowness vector fully avoids the problems mentioned above. The approach allows exact determination of γ* and removes instabilities known from the use of the specification of the slowness vector by γ. For γ = γ*, the approach yields zero phase velocity, i.e., the corresponding wave is a nonpropagating wave mode. The use of the mixed specification leads to the explanation of the deviation of γ* from 90° as a consequence of different orientations of energy-flux and propagation vectors in anisotropic media. The approach is universal; it may be used for isotropic or anisotropic, perfectly elastic or viscoelastic media, and for homogeneous and inhomogeneous waves, including strongly inhomogeneous waves, like evanescent waves.

Deformable, rigid, and inviscid elliptical inclusions in a homogeneous incompressible anisotropic viscous fluid

Energy-preserving finite element methods for a class of nonlinear wave equations

Layered composite, bonded with two half planes, one of which contains a crack, is compared with homogeneous anisotropic media, bonded similar half planes. This problem is analyzed for the case of antiplane strain. Stress singularities are investigated, and numerical results are presented and compared for these models. Conclusions drawn concern thresholds within which the model of homogeneous media can serve to simulate a layered composite.

A 2-D problem of the electromagnetic wave scattering in a homogeneous anisotropic medium from an infinitely long cylinder of arbitrary cross-section shape made of inhomogeneous anisotropic material is considered. The problem is formulated in the form of a system of integro-differential equations with respect to the field components over the cross-section of a scatterer. These equations are solved numerically via the moment method using pulse functions as the basis functions and the point-matching technique.

SUMMARY The influence of the free surface on shear wavetrains in a homogeneous anisotropic medium is analysed. We use a numerical technique which allows one to compute full waveforms from a buried dislocation source in a model with arbitrary symmetry. Azimuthally isotropic (hexagonal) and orthorhombic media are considered. In a transversely isotropic model and in symmetry planes of an orthorhombic medium the SH-wave preserves its polarization at the surface and orthogonal particle motions of the incident shear waves are clearly seen even beyond the critical angle. If sagittal symmetry is absent, particle motions in the horizontal and sagittal planes are seriously distorted beyond critical angles by head waves and inhomogeneous reflected SP phases. However, in spite of the transverse components acquired by head and reflected waves, shear wave polarizations in the transverse plane remain close to the polarizations of the incident waves up to incidence angles of 55-60. Some complications in the detection of shear wave splitting caused by source parameters and anisotropic structure are also investigated. Distinctive shear wave splitting in the transverse plane for angles of incidence 35-60 indicates the possibility of extending the limits of the shear wave window. Investigation of particle motions beyond critical angles may help to avoid ambiguities in the interpretation of the limited data set obtained inside the 'classical' shear wave window.

P and S decomposition is an essential step in isotropic viscoelastic reverse time migration (VRTM). Separation using divergence and curl operators doesn’t preserve the phase or amplitude of the input viscoelastic wavefield, so an alternative decomposition that preserves the vector components of the Pand S-waves in the input wavefield is desired. With a decoupled assumption, we rearrange the memory variables in the elastodynamic equations based on the standard linear solid. The vector components of the propagating Pand S-wave can be can be decomposed in isotropic viscoelastic media without losing accuracy. Synthetic tests show that the Pand S-waves can be decomposed with no distortion of the amplitude or phase, and have the same vector components of particle velocity that exist in the viscoelastic wavefield before decomposition.