Comparison of methods for computing interference elastic waves in thin-layered media. 2

The paper is an immediate continuation of the paper where the solution of the problem on the propagation of low-frequency waves in thin-layered media by the dispersion equation method was considered in detail. In the present article, the solution of a similar problem is given for an elastic layer and a half-space, which are in rigid contact, by the method of superposition of complex plane waves. Bibliography: 17 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 342, 2007, pp. 217–232.     

An algorithm for determining wave fields in multilayer elastic media

An algorithm for computing wave fields propagating in elastic media with plane-parallel interfaces is proposed. This algorithm is based on summation of kinematically equivalent waves. All kinds of multiple waves of the exchange type arriving to an observer at the fixed interval of time (t₁, t₂) are taken into account. Dynamic analogues are determined within each kinematic group. Bibliography: 12 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 186, pp. 154–171, 1990. Translated by N. S. Zabavnikova.     

Quantitative study of nonstationary interference wave fields in layered homogeneous elastic media with plane-parallel interfaces. I. Statements of the problems and efficient methods for their solution

A new efficient approach to quantitatively studying nonstationary interference wave fields in layered homogeneous isotopic elastic media with plane-parallel interfaces is presented. Adequate statements of problems and efficient methods for their solution are discussed. Quantitative evaluations of fields in different regions of a medium are based on extensive application of contour integration methods, on modern numerical methods for studying the roots of the dispersion equations of problems, and on methods for constructing appropriate stationary contours of phase functions and for computing spectral functions of fields by methods of numerical integration along such contours. Bibligraphy: 10 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 214, 1994, pp. 7–186. Translated by N. S. Zabavnikova and T. N. Surkova.     

Elastic wave damping in thin-layered saturated porous media

Monochromatic wave propagation in thin-layered saturated porous media is examined by averaging differential equations with rapidly oscillating coefficients. Particular attention is given to the transformation mechanism for the damping of such waves. Existing results in this area /1/ are extended and refined.     

Simulation of elastic wave propagation in geological media: Intercomparison of three numerical methods

For wave propagation in heterogeneous media, we compare numerical results produced by grid-characteristic methods on structured rectangular and unstructured triangular meshes and by a discontinuous Galerkin method on unstructured triangular meshes as applied to the linear system of elasticity equations in the context of direct seismic exploration with an anticlinal trap model. It is shown that the resulting synthetic seismograms are in reasonable quantitative agreement. The grid-characteristic method on structured meshes requires more nodes for approximating curved boundaries, but it has a higher computation speed, which makes it preferable for the given class of problems.     

A new method of computing wave fields in the high-frequency approximation

In the paper a new method of computing wave fields in the high-frequency approximation is proposed. The method is based on the summation of Gaussian pencils at the point of observation; each pencil is connected with a ray passing in some neighborhood of this point. The idea of describing the high-frequency asymptotics of the wave field as an integral over Gaussian pencils was first proposed in the work of V. M. Babich and T. F. Pankratova of 1973 and was used in mathematical investigations of discontinuities of the Green function of a mixed problem for the wave equation.Translated from Zapiski Nauchnykh Seminarov Leningr ads kogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 104, pp. 195–216, 1981.     

On the calculation of wave fields propagating in multilayer media

A program for evaluating elastic waves in multilayer media is composed on the basis of the algorithm previously stated. The waves under consideration undergo all possible acts of reflection and refraction on the interfaces of the media. Exchanges on these interfaces are also taken into account. Two examples illustrating the calculation of theoretical seismograms of elastic waves are presented. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 203, 1992, pp. 156–165. Translated by N. S. Zabavnikova     

Nonstationary interference wave fields of SV type in a free elastic layer and the problem of ultrasonic simulation of plane seismic fields by using plate models

The problem of quantitatively studying wave fields in a free elastic layer excited by a nonstationary point source is discussed. Much attention is given to the low-frequency part of the Fourier spectrum. The roots of the dispersion equation of the layer are studied comprehensively, and on this basis efficient methods of quantitatively estimating interference wave modes are developed. Particular attention is given to the ultralow-frequency band of the spectrum. This allows us to make progress toward the mathematical justification of methods of seismic modeling of plane wave fields on plate models. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 225, 1997, pp. 121–239. Translated by N. S. Zabavnikova.     

Actual time integration methods for elastic wave propagation analysis

For the simulation of the interaction of elastic waves in CFRP plates with inhomogeneities and defects the spectral finite element method (SEM) is under investigation. The SEM uses high-order shape functions which are composed of Lagrange polynomials with nodes at the Gauss-Lobatto quadrature (GLq) points. In this way we obtain a diagonal mass matrix which makes an explicit time scheme more efficient. In this paper we analyse how actual time integration methods perform in combination with the SEM. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Algorithms for evaluating basic components of spectral functions of interference wave fields

Simple algorithms for finding a numerical solution on a complex plane of dispersion equations dependent on a real parameter are described. Algorithms for numerical integration along stationary contours on a complex plane in the case where the integrand possesses singularities of the pole and branch-point type are also presented. These algorithms are applied to compute the spectral functions of wave fields propagating in layered media. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 214, 1994, pp. 187–199. Translated by N. S. Zabavnikova.     

A boundary element procedure for transient wave propagations in two-dimensional isotropic elastic media

An integral formulation of the elastodynamic equations is presented and discretized to develope a numerical solution procedure. Constant space and linear time dependent interpolation functions are implemented. The expression obtained, the boundary element equations can be solved using a time-stepping scheme. An example is given for comparison with known solutions.     

Dispersion analysis of triangle-based spectral element methods for elastic wave propagation

We study the numerical dispersion/dissipation of Triangle-based Spectral Element Methods (TSEM) of order N????1 when coupled with the Leap-Frog (LF) finite difference scheme to simulate the elastic wave propagation over a structured triangulation of the 2D physical domain. The analysis relies on the discrete eigenvalue problem resulting from the approximation of the dispersion relation. First, we present semi-discrete dispersion graphs by varying the approximation polynomial degree and the number of discrete points per wavelength. The fully-discrete ones are then obtained by varying also the time step. Numerical results for the TSEM, resp. TSEM-LF, are compared with those of the classical Quadrangle-based Spectral Element Method (QSEM), resp. QSEM-LF.     

Superconvergence results for Galerkin methods for wave propagation in various porous media

Finite-element methods are considered for numerically solving the equations describing wave propagation in various porous media such as inhomogeneous elastic media, fluid saturated media, composite isotropic inhomogeneous elastic media, composite anisotropic media, etc. Quasi-projection analyses based on an asymptotic expansion to high order of finite-element solutions are given to obtain error estimates in Sobolev spaces of nonpositive index for the approximate solution. Superconvergence phenomena for the finite-element methods under consideration are also investigated. © 1996 John Wiley & Sons, Inc.     

The concentration of a wave field at the interface of elastic media

The steady symmetrical oscillations of a transversely non-uniform elastic rectangular region, consisting of three bonded uniformisotropic rectangles, are considered, where the elastic characteristics of the inner rectangle are assumed to be different from those of the external rectangles. Using methods developed previously [1,2], the dependence of the order of the singularity of the stress field at the interface on the combinations of the constants of elasticity of the joined media and on their wave impedances is investigated.     

Mathematical simulation of wave fields in media with arbitrary curvilinear boundaries

The method of calculation of wave fields in media with curvilinear interfaces is proposed. After integral transformations, the solution in the explicit form is obtained. For solving the problem, we introduce the notion of the generalized thickness of a layer. This value becomes real in the case of a plane boundary and coincides with an “ordinary” thickness of a layer. If the boundary is curvilinear, the generalized thickness of a layer becomes a complex value. The use of the generalized thickness of a layer brings about the fact that conditions on infinity (Sommerfeld’s conditions) are automatically fulfilled. The conditions of appearance of loops on the travel-time curves for different boundaries have been elucidated. Some examples of calculations are given.     

DFT modal analysis of spectral element methods for the 2D elastic wave equation

The DFT modal analysis is a dispersion analysis technique that transforms the equations of a numerical scheme to the discrete Fourier transform domain sampled in the mesh nodes. This technique provides a natural matching of exact and approximate modes of propagation. We extend this technique to spectral element methods for the 2D isotropic elastic wave equation, by using a Rayleigh quotient approximation of the eigenvalue problem that characterizes the dispersion relation, taking full advantage of the tensor product representation of the spectral element matrices. Numerical experiments illustrate the dependence of dispersion errors on the grid resolution, polynomial degree, and discretization in time. We consider spectral element methods with Chebyshev and Legendre collocation points.     

The elastic fields and interactions of point defects in isotropic and cubic media

Zusammenfassung Die Formel vonEshelby für die elastische Wechselwirkung zwischen Punktdefekten wird neu hergeleiter. Die elastischen Eigenschaften der Punktdefekte werden behandelt durch Anwendung der mathematischen Methoden der Quantenmechanik des Drehimpulses auf die Gleichungen der linearen Elastizitätslehre des Kontinuums. Für isotrope Medien werden die Multipole beliebiger Ordnung der elastischen Felder exakt berechnet, und für Medien kubischer Symmetrie wird eine Näherung für das elastische Dipolfeld gegeben. Die Wechselwirkungsenergie zweier elastischer Dipole wird für isotrope Medien exakt und für kubische Medien näherungsweise berechnet. Die Formel wird angewendet auf Paare vonO ₂ ^– Zentren und Paare vonOH ^– Zentren inKCl. Numerische Resultate sind gegeben.

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This work is part of a research program suported by the Swiss National Science Foundation under contract No. 4212.     

On specific features of the Lebedev scheme in simulating elastic wave propagation in anisotropic media

This paper presents the Lebedev scheme on staggered grids for the numerical simulation of wave propagation in anisotropic elastic media. Primary attention is given to the approximation of the elastic wave equation by the Lebedev scheme. Based on the differential approach, it is shown that the Lebedev scheme approximates a system of equations, which differs from the original equation. It is proved that the approximated system has a set of 24 characteristics, six of them coincide with those of the elastic wave equation and the rest ones are “artifacts.” Requiring the artificial solutions to be equal to zero and the true ones to coincide with those of the elastic wave equation, one comes to the classical definition of the approximation of the initial system on a sufficiently smooth solution. The results obtained and the knowledge of the complete set of characteristics are important for constructing reflectionless boundary conditions during approximation of point sources, etc.     

On the Rayleigh wave on a curvilinear boundary between elastic and fluid media

Along the boundary between elastic and fluid media, the surface Rayleigh wave propagates. The velocity of this wave v _R0 in the case of a plane boundary is less than the velocity of the Rayleigh wave v _R on a free plane boundary of an elastic medium and less than the velocity v _P0 in a fluid medium. To investigate the velocity v _R0 in the case of curvilinear boundaries, the propagation of Rayleigh waves under consideration along cylindrical and spherical surfaces is studied. The velocity of the Rayleigh wave depends on the curvature of the wave trajectory and the curvature in the direction perpendicular to the trajectory. Furthermore this velocity depends on the presence or absence of a fluid medium. Bibliography: 5 titles.