Vector Domain P and S Decomposition in 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 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

High order finite volume approximations of differential operators on nonuniform grids

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.

The effect of fillers on the flow curves of polymeric liquid crystals is investigated. Suspensions of polystyrene particles in liquid crystalline solutions of hydroxypropylcellulose (HPC) in water are used. By reducing the HPC concentration an isotropic solution can be prepared. It serves as a reference to isolate the effect of the isotropic/anisotropic structure of the suspending medium on the rheological behaviour. Suspensions in the isotropic solution behave as expected for filled viscoelastic matrices in general. In the anisotropic medium the shear rate rather than the shear stress seems to govern the changes in the relative viscosity. This behaviour is clearly different from isotropic viscoelastic media. The most dramatic effect however is that even small amounts of particles eliminate or drastically shift the region of negative normal stress differences. As far as the structure is concerned, microscopic observations show that particles align in anisotropic as well as in isotropic media. At rest or at relatively low shear stresses the liquid crystalline structure is, in the present case, hardly affected by the presence of the particles. If anything, it becomes more homogeneous.

Due to coal beds existing in the middle of higher velocity wall rocks, seismic wave interferencing with each other generates guide wave propagating within coal beds and surrounding wall rocks (Liu,1994). Three dimensional channel wave propagating within homogeneous isotropic viscoelastic medium containing coal bed is simulated with an explicit, time-domain, high order staggered finite difference algorithm (Carcione, 1998). General standard linear solid (GSLS) system, representing dispersion/attenuation characteristic at both low frequency band and high frequency band, is used to reflect viscoelastic property. Considering massive computational time-consuming, a parallel computational implementation, utilizing OpenMP strategy, allows investigation of large-scale three dimensional models and/or broadband wave propagation within reasonable execution times.

In seismic numerical simulations of wave propagation, it is very important for us to consider surface topography and attenuation, which both have large effects (e.g., wave diffractions, conversion, amplitude/phase change) on seismic imaging and inversion. An irregular free surface provides significant information for interpreting the characteristics of seismic wave propagation in areas with rugged or rapidly varying topography, and viscoelastic media are a better representation of the earth’s properties than acoustic/elastic media. In this study, we develop an approach for seismic wavefield simulation in 2D viscoelastic isotropic media with an irregular free surface. Based on the boundary-conforming grid method, the 2D time-domain second-order viscoelastic isotropic equations and irregular free surface boundary conditions are transferred from a Cartesian coordinate system to a curvilinear coordinate system. Finite difference operators with second-order accuracy are applied to discretize the viscoelastic wave equations and the irregular free surface in the curvilinear coordinate system. In addition, we select the convolutional perfectly matched layer boundary condition in order to effectively suppress artificial reflections from the edges of the model. The snapshot and seismogram results from numerical tests show that our algorithm successfully simulates seismic wavefields (e.g., P-wave, Rayleigh wave and converted waves) in viscoelastic isotropic media with an irregular free surface.

Oscillatory numerical experiments in finely layered anisotropic viscoelastic media

This is the most important chapter in the entire book. A new method, designated the symbolic vector method, in treating vector analysis will be introduced. The main advantages of this method are the differential expressions of the three key functions in vector analysis are derived based on one basic formula, all of the integral theorems in vector analysis are deduced from one generalized theorem, the commonly used vector identities are found by an algebraic method without pedorming any differentiation, and two differential operators in the curvilinear coordinate system are introduced. They are designated as the divergence operator and the curl operator, which are distinct from the operator used for the gradient. The technical meaning of the words “divergence,” “curl”, and “gradient” will be explained shortly. It should be pointed out that the nomenclature for some technical terms introduced in this chapter is different from the original one used in [18).

An arbitrary-order method for magnetostatics on polyhedral meshes based on a discrete de Rham sequence

The separation of wave modes for isotropic elastic wavefields is typically done using Helmholtz decomposition. However, Helmholtz decomposition using conventional divergence and curl operators is not satisfactory for anisotropic media and leaves the different wave modes only partially separated. The separation of anisotropic waveflelds requires more sophisticated operators which depend on local material parameters. Wavefleld separation operators for TI (transversely isotropic) models can be constructed based on the polarization vectors evaluated at each point of the medium by solving the Christoffel equation using local medium parameters. These polarization vectors can be represented in the space domain as localized filters, which resemble conventional derivative operators. The spatially‐variable “pseudo” derivative operators perform well in 2D heterogeneous TI media even at places of rapid variation. Wave separation for 3D TI media can be performed in a similar way. In 3D TI media, P and SV waves are polarized only in symmetry planes, and SH waves are polarized orthogonal to symmetry planes. Using the mutual orthogonality property between these modes, we only need to solve for the P wave polarization vectors from the Christoffel equation, and SV and SH wave polarizations can be constructed using the relationship between these three modes. Synthetic results indicate that the operators can be used to separate wavefields for TI media with arbitrary strength of anisotropy.

The divergence and curl operators are fundamental mathematical tools introduced in university-level electromagnetism courses. Although students generally understand the operational use of these concepts, developing a clear physical intuition of what divergence and curl are can be challenging. This difficulty often arises due to the superficial or confusing treatment found in many standard textbooks. The goal of this article is to provide a clear demonstration, based on examples, that two-dimensional visualizations can be misleading when used to interpret these operators. A combination of graphical and analytical (symbolic) representations was found to be essential for developing an accurate physical intuition about divergence and curl. A deeper comprehension of these operators not only improves proficiency in electromagnetism, but also benefits a range of other fields, from fluid dynamics to meteorology and to technical applications.

The paper considered the signal processing of simplicial complexes. Hodge decomposition formula for discrete fields is given, which is similar to the Hodge decomposition formula for smooth vector fields. The construction and the estimation of the gradient, divergence and curl operators and Laplace matrices for discrete vector fields are considered.

Maxwell’s equations are usually formulated as differential equations (1.1)–(1.4), where we have derivatives with respect to space (represented by the curl and divergence operators) and derivatives with respect to time. The sought electromagnetic field is a function of space and time that must satisfy Maxwell’s equations. A rather simple way to pursue a numerical solution is to represent the electromagnetic field by its function values at a (finite) set of discrete grid points, where we would normally use a uniform Cartesian grid with respect to space and time. Then, it is quite natural to approximate the derivatives of the electromagnetic field in Maxwell’s equations by finite-difference approximations that involve the fields at neighboring points on the grid. This type of approach is referred to as a finite-difference method.KeywordsGrid PointStagger GridFinite Difference ApproximationComplex ExponentialDifference FormulaThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Atrial Fibrillation (AF) is the most common supraventricular arrhythmia and has different underlying activation mechanism, including functional rotors (FR) and ectopic foci (EF).In this work we propose an approach for locating FR and EF in potential maps, which were tested with mathematical phantoms.12 phantoms were created (128x128 array, 4 s, Fs 500 Hz), simulating the motion of: FR (4 maps), EF (4 maps) and superpositions of these (4 maps).These were downsampled to different grids from 16x16 to 8x8, simulating electrode acquisition.Noise (SNR from 2 to 60) was added.To locate the mechanism, the signals were filtered and interpolated.Farneback optical flow was applied to compute the motion vector field (MVF).The MVF was normalized and its temporal average was calculated.Finally, we computed curl and divergence, by using a x and y oriented 11 11 Sobel filter as a estimation of the of the partial derivatives.The location of extrema in the curl and divergence maps were used for locating FR and EF respectively.Method robustness was tested by comparing the algorithm performance on different grid sizes and SNR.The mechanism was considered to be detected accurately if the position was within a normalized error of 5% from its respective phantom location.The results showed that our approach was able to locate both mechanisms, but revealed a dependency on spatial resolution.