Anisotropic Elastodynamics Research Articles

Modeling 3-D anisotropic elastodynamics using mimetic finite differences and fully staggered grids

Modeling 3-D anisotropic elastodynamics using mimetic finite differences and fully staggered grids

深層学習を用いた異方性弾性体中の欠陥形状再構成

In this study, the shape reconstruction method of a cavity is developed for 2-D pure antiplane anisotropic elastodynamics. Elastic wave scattering by a cavity is implemented by using the convolution quadrature time-domain boundary element method. Scattered waves obtained at receiver points are prepared and utilized for the deep learning. The created deep learning model estimates position and size of a cavity in 2-D pure antiplane anisotropic solids. Numerical example shows that deep learning can be useful for inverse analysis of a defect in anisotropic elastic materials.

Tensorial elastodynamics for coupled acoustic/elastic anisotropic media: incorporating bathymetry

SUMMARYCorrectly implementing the fluid/solid boundary conditions at the seafloor is important for accurate full-wavefield imaging and inversion of marine seismic data. Because bathymetric profiles are rarely flat, the associated undulations influence wave modes interacting with the seafloor and, therefore, the ensuing imaging and inversion results. The conventional method of using single-domain elastic finite-difference (FD) solutions assumes that the fluid/solid contact is welded, which leads to incorrect handling of the boundary conditions and introduces modelling errors. We present a mimetic finite-difference (MFD) approach to solve the equations of anisotropic elastodynamics in a non-orthogonal coordinate system conformal to the bathymetric interface. The vertically deformed coordinate mapping transforms the irregular Cartesian (physical) domain into a regularly sampled generalized computational domain. We partition the medium into the acoustic and elastic subdomains and explicitly satisfy the fluid/solid boundary conditions with a split-node approach involving high-order one-sided MFD operators that achieve uniform spatial accuracy throughout the computational domain. Fully staggered grids (FSGs) are used to solve the velocity–pressure and velocity–stress formulations of the acoustic and anisotropic elastic wave equations, respectively. Numerical examples demonstrate that the proposed MFD+FSG algorithm accurately simulates wavefields even for strongly undulating bathymetric surfaces overlying structurally complex anisotropic media, and produces no spurious numerical artefacts (e.g. staircasing) or unphysical wave modes often caused by improper handling of the strong-contrast bathymetric interface. The wavefields generated by the tensorial MFD scheme closely match those from the more computationally expensive spectral-element method used to validate our implementation. The developed MFD+FSG technique can be effectively employed as the modelling kernel in a variety of coupled acoustic/elastic imaging and inversion applications.

Optimized first-order absorbing boundary conditions for anisotropic elastodynamics

We propose new forms of low-order absorbing boundary conditions (ABC) for time-dependent elastic waves in isotropic and anisotropic media. The configuration considered is that of a two-dimensional elastic waveguide. The ABC shares with the widely-known Lysmer–Kuhlemeyer (LK) boundary condition the ease of implementation, especially in a finite-element (FE) setting, while offering gains in accuracy over the LK-like condition (or the LK condition in the isotropic case). The new conditions are proved to be energy-stable, even in the case when inverse modes are present, for which the normal components of the phase and group velocities have an opposite direction. The developed ABCs are particularly convenient to use in a Finite Element setting as they preserve the symmetry and positivity of the formulation and are simple to implement. The ABC features several parameters, which are optimized for accuracy. The optimization criterion is based on the notion of the energy-rate reflection coefficient. We test the performance of the scheme via a numerical example. While the gain in accuracy is found to be moderate (around 20% relative to the LK condition), the main result of this investigation lies in the development of ABCs for anisotropic elastodynamics that are proved to be stable from the outset, and are amenable for accuracy optimization. We plan to design in a future study analogous stable and optimized high-order ABCs using the methodology developed here.

Equations of anisotropic elastodynamics in 3D quasicrystals as a symmetric hyperbolic system: Deriving the time-dependent fundamental solutions

The fundamental solution (FS) of the time-dependent differential equations of anisotropic elasticity in 3D quasicrystals are studied in the paper. Equations of the time-dependent differential equations of anisotropic elasticity in 3D quasicrystals are written in the form of a symmetric hyperbolic system of the first order. Using the Fourier transform with respect to the space variables and matrix transformations we obtain explicit formulae for Fourier images of the FS columns; finally, the FS is computed by the inverse Fourier transform. As a computational example applying the suggested approach FS components are computed for icosahedral QCs.

Computation of the time-dependent fundamental solution for equations of elastodynamics in general anisotropic media

A new method for computation of the fundamental solution (FS) of anisotropic elastodynamics is presented in the paper. This method consists of the following steps: equations for each column of the FS are reduced to a symmetric hyperbolic system; using the Fourier transform with respect to the space variables and matrix transformations the explicit formulae for Fourier images of the FS columns are obtained; finally, the FS is computed by the inverse Fourier transform. Applying the suggested approach, FS components are computed in trigonal, monoclinic and triclinic anisotropic media. Computational examples confirm the robustness of the suggested method.

Equations of anisotropic elastodynamics as a symmetric hyperbolic system: Deriving the time-dependent fundamental solution

The dynamic system of anisotropic elasticity from three second order partial differential equations is written in the form of the time-dependent first order symmetric hyperbolic system with respect to displacement velocity and stress components. A new method of deriving the time-dependent fundamental solution of the obtained system is suggested in this paper. This method consists of the following. The Fourier transform image of the fundamental solution with respect to a space variable is presented as a power series expansion relative to the Fourier parameters. Then explicit formulae for the coefficients of these power series are derived successively. Using these formulae the computer calculation of fundamental solution components (displacement velocity and stress components arising from pulse point forces) has been made for general anisotropic media (orthorhombic and monoclinic) and the simulation of elastic waves has been obtained. These computational examples confirm the robustness of the suggested method.

Anisotropic elastodynamics in a half space: An analytic method for polynomial data

An initial boundary value problem for the dynamic system of anisotropic elasticity in a half space is studied in the paper. A novel method of finding an exact solution of this problem for a special polynomial form of initial data and inhomogeneous term of the system is described. On the base of this method the simulation of elastic waves in different homogeneous anisotropic half spaces is implemented.

On the theory of wave propagation in anisotropic plates

A theoretical framework is developed which describes wave propagation in infinite homogeneous elastic plates of unrestricted anisotropy. The approach exploits the propagator matrix which is the exponential of the fundamental elasticity matrix underlying Stroh9s formalism of anisotropic elastodynamics. The matrices of plate impedance and admittance are introduced, and their analytical properties are established, which appear fruitful for computing and analysing the plate wave spectra. On this basis, the dispersion equation can be cast into the form of a real equation involving the monotonic function, whose zeros and poles are the wave velocities in a given plate subjected to different boundary conditions (traction–free or clamped faces). It is proved that three fundamental wave branches exist for any orientation of wave propagation in an anisotropic plate with traction–free faces, and that those branches are missing if one or both faces are clamped. The intrinsic symmetry of the wave motion in an arbitrary plate is revealed. The general formalism is applied to elaborate the long–wavelength low–frequency approximation for a (thin) free plate of unrestricted anisotropy. The frequency–dispersive wave velocities, displacements and tractions at the onset of the fundamental wave branches are derived explicitly. The conditions for the extreme velocity values and some other useful universal connections are determined for an arbitrary thin plate, and exemplified for specific anisotropy cases.

Some contact problems of anisotropic elastodynamics: integral characteristics and exact solutions

The author's method of integral characteristics of solutions to boundary-initial value problems of contact is applied to solve some new problems of anisotropic elastodynamics. For both anisotropic elastic and elastic with initial stresses media, solutions are obtained to impact problems which were solved earlier for only an isotropic linear elastic medium. It is also found the resultant contact force in the problems of both frictional and frictionless pressing normal to the boundary plane of the anisotropic continuous both an elastic with initial stresses and a linear viscoelastic orthotropic half-spaces. Some general properties of dynamical contact problems are also formulated. In particular, conditions are provided when problems of vertical pressing are self-similar.

Integral properties of solutions to some boundary initial-value problems in linear anisotropic elastodynamics

We study some integral properties of weak solutions to some boundary initial-value problems in the linearized dynamical theory of elasticity. These problems arise during the indentation of an anisotropic half-space by a convex punch having an arbitrary indenting velocity and shape. In contrast to previous studies of the problem, we consider various boundary conditions, e. g., adhesive or frictional, in the case when this non-frictionless boundary initial-contact problem has two orthogonal planes of symmetry, which are both orthogonal to the boundary of half-space. We show that if the non-frictionless boundary initial-contact problem has such a symmetry, then the problem for integral characteristics of the solutions is equivalent to a problem of plane-wave propagation in the same medium. A proof is given that the instantaneous value of the force required to indent the punch during the first supersonic stage of contact is directly proportional to the product of the velocity of indentation and the area of contact at that instant, and is independent of the boundary conditions in the contact region.

Stress characterization of elastodynamics with continuously distributed defects

A pure stress formulation of linear nonhomogeneous anisotropic elastodynamics with continuously distributed defects (EcDD) is proposed, and well known solutions of ECDD corresponding to stationary and moving concentrated plastic fields in an infinite body are recovered by direct integration of the stress field equation. A new interpretation of the solutions is given.

A tensorial classification of elastic waves

Pure stress equations of motion of linear homogeneous anisotropic elastodynamics are used to present a classification of the plane stress progressive waves in an infinite body. A plane stress wave is decomposed into ‘‘normal’’ and ‘‘tangential’’ parts with respect to the wave front in such a way that: (i) both parts can propagate with the same velocity, (ii) the total stress energy of the wave is represented by the positive difference between ‘‘normal’’ and ‘‘tangential’’ stress energies, and (iii) pure ‘‘tangential’’ stress wave cannot propagate. The classification is consistent with and complementary to that of the displacement progressive waves, at least for homogeneous isotropic and transversely isotropic elastic media, and provides a number of new properties of the plane progressive waves in a general anisotropic elastic body.

Three-dimensional inverse scattering in anisotropic elastic media

The three-dimensional inverse scattering problem is considered for localised scatterers in an anisotropic elastic solid. Isotropic elastic solids are automatically included in the analysis as a special case. The authors obtain three results: (1) a near-field version of the Newton-Marchenko equation for anisotropic elastodynamics, (2) a set of self-consistent equations for the exact spatial reconstruction of the inhomogeneous density function, and (3) a new generalised completeness relation for the elastodynamic Green function corresponding to an inhomogeneous anisotropic media.

On the structure and invariance of the Barnett-Lothe tensors

The three real tensors introduced by Barnett and Lothe into the theory of steady plane motions of an anisotropic elastic body are shown to have algebraic representations the structure of which is largely independent of material symmetry. The allied form of the complex impedance tensor central to the analyses of surface and interfacial waves in anisotropic elastodynamics is also obtained. A detailed study of the representations yields alternative routes to known results and a variety of new relations. The paper concludes with a discussion of the invariance properties of quantities appearing in the representations under rotations of the reference frame about the direction in which the deformation is uniform.

Some stability results in the linear theory of anisotropic elastodynamics

Conditions are established to ensure continuous data dependence for classical solutions of certain initial-boundary value problems in linearised anisotropic elastodynamics. Two uniqueness theorems are also presented.

On some extremum principles in linear elastodynamics

New extremum principles are established for linear anisotropic elastodynamics in terms of polarisation tensors. The new principles provide upper and lower bounds on the action integral and are thus suitable for the derivation of bounds on effective elastic moduli and wave velocities in heterogeneous materials. It is shown that when certain constraints are removed the present principles reduce to previous polarisation principles given by the author in [3].