Christoffel Matrix Research Articles

Anisotropic media with singular slowness surfaces

It is proved that if an anisotropic medium has an open set of singular directions, then this medium has two slowness surfaces that completely coincide. The coinciding slowness surfaces form one double singular slowness surface. The corresponding anisotropic medium is an elliptical orthorhombic (ORT) medium with equal stiffness coefficients c44=c55=c66 rotated to an arbitrary coordinate system. Based on the representation of the Christoffel matrix as a uniaxial tensor and considering that the elements of the Christoffel matrix are quadratic forms in the components of the slowness vector, a system of homogeneous polynomial equations was derived. Then, the identical equalities between homogeneous polynomials are replaced by the equalities between their coefficients. As a result, a new system of equations is obtained, the solution of which is the values of the reduced (density normalized) stiffness coefficients in a medium with a singular surface. Conditions for the positive definite of the obtained stiffness matrix are studied. For the defined medium, the Christoffel equations and equations of group velocity surfaces are derived. The orthogonal rotation matrix that transforms the medium with a singular surface into an elliptic ORT medium in the canonical coordinate system is determined. In the canonical coordinate system, the slowness surfaces S1 and S2 waves coincide and are given by a sphere with a radius . The slowness surface of qP waves in the canonical coordinate system is an ellipsoid with semi-axes , , . The polarization vectors of S1 and S2 waves can be arbitrarily selected in the plane orthogonal to the polarization vector of the qP wave. However, the qP wave polarization vector can be significantly different from the wave vector. This feature should be taken into account in the joint processing and modelling of S and qP waves. The results are illustrated in one example of an elliptical ORT medium.

Negative refraction of elastic waves on a metamaterial with anisotropic local resonance

Wave propagation through a locally resonant metamaterial characterized by anisotropic dynamic mass density is studied. The polarizations and velocities of waves are obtained from an extended Christoffel matrix. The set of waves induced by a body wave coming through the interface with an elastic material is described. Reflexion and refraction coefficients corresponding to the ratio of real Poynting vectors are obtained, showing that some of these coefficients are singular. In this case, the Poynting vector of the set of coupled waves must be used. This Poynting vector can display a negative refraction. The conditions leading to negative refraction are described and several examples of negative refraction are displayed.

Two S-wave eigenvectors of the Christoffel matrix need not exist in anisotropic viscoelastic media

Two S-wave eigenvectors of the Christoffel matrix need not exist in anisotropic viscoelastic media

Computation of velocities and polarization vectors in weakly anisotropic media

To compute the phase velocities in the weakly anisotropic media, we propose to transform the Christoffel matrix K into an adapted coordinate system, and, then, apply the perturbation theory to the resulting matrix X. For a weakly anisotropic medium, the off-diagonal elements of the matrix X are small compared to the diagonal ones, and two of them are equal to 0. The diagonal elements of the matrix X are initial approximations of the phase velocities squared. To refine them, it is proposed to use either iterative schemes or Taylor series expansions. The initial terms of the series and the formulas of iterative schemes are expressed through the elements of the matrix X and have a compact analytical representation. The odd-order terms in the series are equal to 0. To approximate the phase velocities of the S1 and S2 waves, a stable method is proposed based on solving a quadratic equation with the coefficients being expressed in terms of the matrix elements and the precomputed value of the qP wave phase velocity squared. For all iterative schemes and series, the convergence conditions are derived. The polarization vector of the wave with the square of the phase velocity is defined as the column with maximum modulus of cofactor of the matrix K-I. The group velocities vectors are computed based on the known components of the polarization vector, the directional vector, and the density-normalized stiffness coefficients. The computational accuracy is demonstrated for the standard orthorhombic model. It is shown how the perturbation theory can be applied to media with strong anisotropy. To do this, first we need to apply several QR transforms or Jacobi rotations of the Christoffel matrix, and then use the perturbation theory. This method with four Jacobi rotations is applied to the calculation of the phase velocities squared for a triclinic medium with a maximum number (32) of singularity points. In this case, the phase velocities are computed with a relative error less than 0,004 %.

OUP accepted manuscript

Wave motion in two- and three-dimensional periodic lattices of beam members supporting longitudinal and flexural waves is considered. An analytic method for solving the Bloch wave spectrum is developed, characterized by a generalized eigenvalue equation obtained by enforcing the Floquet condition. The dynamic stiffness matrix is shown to be explicitly Hermitian and to admit positive eigenvalues. Lattices with hexagonal, rectangular, tetrahedral and cubic unit cells are analyzed. The semi-analytical method can be asymptotically expanded for low frequency yielding explicit forms for the Christoffel matrix describing wave motion in the quasistatic limit.

Weak-anisotropy approximations of P-wave phase and ray velocities for anisotropy of arbitrary symmetry

We test the accuracy of the previously derived weak-anisotropy approximations of the P-wave phase and ray (group) velocities. The formulae are of varying accuracy, and are applicable to weak or moderate anisotropy of arbitrary symmetry and orientation. They have a form of expressions for squares of phase velocity depending on the phase-velocity direction and of ray velocity depending on the ray-velocity direction. Both velocities are expressed in terms of three elements of the rotated Christoffel matrix, which depend linearly on the parameters specifying the medium, the weak-anisotropy parameters. The least accurate formulae are fully independent of the choice of a reference isotropic medium and depend linearly on weak-anisotropy parameters. The most accurate formulae depend only slightly on the ratio of S- and P-wave velocities of a reference medium, their dependence on weak-anisotropy parameters being quadratic. We show that the accuracy of the formulae is quite high, in fact, in some cases, it is close to the accuracy of the socalled anelliptic approximations, very accurate approximations based on a perturbation of the elliptical anisotropy.

Hyperbolicity of Velocity-Stress Equations for Waves in Anisotropic Elastic Solids

This paper reports mathematical properties of the three-dimensional, first-order, velocity-stress equations for propagating waves in anisotropic, linear elastic solids. The velocity-stress equations are useful for numerical solution. The original equations include the equation of motion and the elasticity relation differentiated by time. The result is a set of nine, first-order partial differential equations (PDEs) of which the velocity and stress components are the unknowns. Cast into a vector-matrix form, the equations can be characterized by three Jacobian matrices. Hyperbolicity of the equations is formally proved by analyzing (i) the spectrum of a linear combination of the three Jacobian matrices, and (ii) the eigenvector matrix for diagonalizing the linearly combined Jacobian matrices. In the three-dimensional space, linearly combined Jacobian matrices are shown to be connected to the classic Christoffel matrix, leading to a simpler derivation for the eigenvalues and eigenvectors. The results in the present paper provide critical information for applying modern numerical methods, originally developed for solving conservation laws, to elastodynamics.

Eigenstructure of First-Order Velocity-Stress Equations for Waves in Elastic Solids of Trigonal 32 Symmetry

AbstractThis paper reports the eigenstructure of a set of first-order hyperbolic partial differential equations for modeling waves in solids with a trigonal 32 symmetry. The governing equations include the equation of motion and partial differentiation of the elastic constitutive relation with respect to time. The result is a set of nine, first-order, fully coupled, hyperbolic partial differential equations with velocity and stress components as the unknowns. Shown in the vector form, the model equations have three 9×9 coefficient matrices. The wave physics are fully described by the eigenvalues and eigenvectors of these matrices; i.e., the nontrivial eigenvalues are the wave speeds, and a part of the corresponding left eigenvectors represents wave polarization. For a wave moving in a certain direction, three wave speeds can be identified by calculating the eigenvalues of the coefficient matrix in a rotated coordinate system. In this process, without using the plane-wave solution, we recover the Christoffel matrix and thus validate the formulation. To demonstrate this approach, two- and three-dimensional slowness profiles of quartz are calculated. Wave polarization vectors for wave propagation in several compression directions as well as noncompression directions are discussed.

First-order ray computations of coupledSwaves in inhomogeneous weakly anisotropic media

SUMMARY We propose an approximate procedure for computing coupled S waves in inhomogeneous weakly anisotropic media. The procedure is based on the first-order ray tracing (FORT) and dynamic ray tracing (FODRT), which was originally developed for P waves. We use the so-called common ray tracing concept to derive approximate ray tracing and dynamic ray tracing equations, and an approximate solution of the transport equation for coupled S waves propagating in laterally varying, weakly anisotropic media. In our common ray tracing, ray equations are governed by the first-order Hamiltonian formed by the average of first-order eigenvalues of the Christoffel matrix, corresponding to the two S-wave modes propagating in anisotropic media. The solution of the transport equation for the coupled S waves leads to a system of two coupled frequency-dependent, linear ordinary differential equations for amplitude coefficients, which is evaluated along the S-wave common ray. For derivation of the FORT and FODRT equations, we use the perturbation theory in which deviations of anisotropy from isotropy are considered to be first-order perturbations. To derive the coupled differential equations for S-wave amplitudes, we assume that the first-order perturbations are of order O(ω −1 ), where ω is the circular frequency. This makes it possible to express the amplitude coefficients in the coupled differential equations in terms of geometrical spreading and other quantities related to the common ray. The proposed procedure removes problems of most currently available ray tracers, which yield distorted results or even collapse when shear waves propagating in weakly anisotropic media are computed. The first-order approximation leads to simpler ray tracing and dynamic ray tracing equations than in the exact case. For anisotropic media of higher-symmetry than monoclinic, all equations involved differ only slightly from the corresponding equations for isotropic media. If the anisotropy vanishes, the equations reduce to standard, exact ray tracing and dynamic ray tracing equations for S waves propagating in isotropic media. The proposed ray tracing and dynamic ray tracing equations, corresponding traveltimes and geometrical spreading are all given to the first order. The accuracy of the traveltimes along S-wave first-order common rays can be increased by calculating a secondorder traveltime correction.

Dispersion feature in arbitrary direction of surface acoustic wave applied to property characterization of ultra-large-scale integrated circuit interconnect films

The ultrasonic surface acoustic wave (SAW) technique is becoming attractive for accurate and nondestructive property determination for thin low-k films employed in the modern ultra-large-scale integrated circuit (ULSI). In this paper, the dispersive characteristics of SAW propagating along arbitrary directions on the layered films deposited on Si(100) were studied in detail. The computation time was reduced greatly by appling coodinate transformation. The calculation of 9×9 Christoffel matrix was simplified to 6×6 matrix for one film layer structure, and from 15×15 matrix to 10×10 matrix for two-layered structure. This work will benefit the on-line test of low-k film properties during ULSI fabrication. Furthermore, the limitation on the propagating direction of SAW can be eliminated in the measurement.

Weak elastic anisotropy by perturbation theory

We demonstrate the advantages of adopting a wave-vector-based coordinate system (WCS) for the application of perturbation theory to derive and display approximate expressions for qP- and qS-wave polarization vectors, phase velocities, and group velocities in general weakly anisotropic media. The advantages stem from two important properties of the Christoffel equation when expressed in the WCS: (1) Each element of the Christoffel matrix is identical to a specific stiffness component in the WCS, and (2) the Christoffel matrix of an isotropic medium is diagonal in the WCS. Using these properties, one can easily identify the small components of the Christoffel matrix in the WCS for a weakly anisotropic medium. Approximate solutions to the Christoffel equation are then obtained by straightforward algebraic manipulations, which make our perturbation theory solution considerably simpler than previously published methods. We compare and contrast our solutions with those discussed by other workers. Numerical comparisons between the exact, first-order, and zero-order qS-wave polarization vectors illustrate the accuracy of our approximate formulas. The form of the WCS phase-velocity expressions facilitates the derivation of closed-form, first-order expressions for qP- and qS-wave group-velocity vectors, providing explicit formulas for the direction of propagation of seismic energy in general weakly anisotropic media. Numerical evaluation of our group-velocity expressions demonstrates their accuracy. We discuss problems with the approximate qS-wave group velocities and polarizations in neighboring directions of singularities. Standard methods are used to transform our solutions from the WCS to the acquisition coordinates, as illustrated by application to orthorhombic symmetry.

On the Calculation of the Eshelby Tensor and the Beauty of our Nature

The calculation of the Eshelby tensor occupies the dominant part of time in modern selfconsistent schemes modeling texture developments. A further time-reduction can be got representing the denominator in the explicit expression of the integrand of the Eshelby integral by finite series, leading to known library integrals. The possibility to get fast converging series depends on the degree of anisotropy of the stiffness tensor C. As universal (any symmetries) parameter, in order to classify the degree of anisotropy of a given C-data set, the variance s0 of the normalized determinant D0(C,r) of the Christoffel matrix is suggested. s0 can exactly be determined without great effort. For typical C-sets time reducing factors in the order of 2 – 8 have been got. Beyond this special result the anisotropy parameter 0 £ s0 < 1 seems to be of general importance. 3d-figures of D0(C,r) show characteristic (and sometimes unexpected ‘beautiful’) individual faces of the elastic properties of monocrystalline or textured polycrystalline substances.

Plane Inhomogeneous Elastic Waves in Transversely Isotropic Media: Explicit Analysis of Different Types of Degeneracy

An intrinsic trait of the plane inhomogeneous waves in elastic media is a particular parametrization duality, stemming from two components characterizing the complex wave vector (bivector) k with respect to the reference frame of orthonormal vectors. Correspondingly, the characteristic condition associated with the equation of motion involves two parameters p and λ, where p is the eigenvalue of the Stroh matrix N (λ) and λ is the eigenvalue of the complex Christoffel matrix λ (p). One of the consequences is a rather elaborate layout of the complex-valued degeneracy types implying repeated p, or λ, or both p and λ. Different types of degeneracy are characterized by dissimilar analytical, algebraic and topological features. The general theory, which has been established for an arbitrary anisotropic medium, is developed in the present paper into an explicit form for a transversely isotropic continuum. By reducing the number of geometrical parameters and enabling factorization of the characteristic polynomial, transverse isotropy fosters explicit resolving of the equations which define different degeneracy types. A particular thrust of the paper is concerned with the algebraic analysis of the conditions, under which the degenerate inhomogeneous modes, having linearly varying or/and circularly polarized amplitude, can be excited on a traction-free boundary by means of reflection or within the Rayleigh wave for, respectively, two- and one-parameter manifolds in the space of orientations of the free surface and propagation direction. Various settings are identified which ensure reflected and surface-localized wave packets comprising a degenerate inhomogeneous mode.

Reformulated Kinematic and Dynamic Ray Tracing Systems for Arbitrarily Anisotropic Media

I reformulate well-established systems for kinematic and dynamic ray tracing in 3D heterogeneous media with arbitrary anisotropy. Matrices of size 3 × 3, e.g. the Christoffel matrix, are substituted by six-component vectors, and the Christoffel matrix elements are expressed explicitly in terms of the elements of the 6 × 6 matrix of elastic coefficients, written in Voigt notation. Thereby, I find it easier to see the effects on the ray tracing systems of vanishing elastic coefficients and Christoffel matrix elements and of vanishing derivatives with respect to spatial coordinates and slowness vector components. The eigenvalue of the current wave and its derivatives with respect to the ray parameters are included explicitly, which may be favorable for ray tracing processes optimized with respect to speed. With the ANRAY program as a reference, I show that the new formulation requires less optimization than the conventional one.

In-Plane Ray Tracing with Calculation of Out-of-Plane Geometrical Spreading in Anisotropic Media

The objective is to provide, in one single paper, a complete collection of equations governing kinematic and dynamic ray tracing related to a symmetry plane of an anisotropic medium. Well known systems for kinematic ray tracing and in-plane dynamic ray tracing are reformulated for the purpose of clarity, by taking advantage of a vector representation of the Christoffel matrix elements and related quantities. A generalized formula is derived for the integrand in out-of-plane dynamic ray tracing, pertaining to a monoclinic medium. Integrands corresponding to non-tilted orthorhombic and transversely isotropic media are obtained as special cases.

On Semi-tensor Product of Matrices and Its Applications

The left semi-tensor product of matrices was proposed in [2]. In this paper the right semi-tensor product is introduced first. Some basic properties are presented and compared with those of the left semi-tensor product. Then two new applications are investigated. Firstly, its applications to connection, an important concept in differential geometry, is considered. The structure matrix and the Christoffel matrix are introduced. The transfer formulas under coordinate transformation are expressed in matrix form. Certain new results are obtained. Secondly, the structure of finite dimensional Lie algebra, etc. are investigated under the matrix expression. These applications demonstrate the usefulness of the new matrix products.

Relation of the Wave-Propagation Metric Tensor to the Curvatures of the Slowness and Ray-Velocity Surfaces

The contravariant components of the wave-propagation metric tensor equal half the second-order partial derivatives of the selected eigenvalue of the Christoffel matrix with respect to the slowness-vector components. The relations of the wave-propagation metric tensor to the curvature matrix and Gaussian curvature of the slowness surface and to the curvature matrix and Gaussian curvature of the ray-velocity surface are demonstrated with the help of ray-centred coordinates.

On the theory of plane inhomogeneous waves in anisotropic elastic media

In the theory of plane inhomogeneous elastic waves, the complex wave vector constituted by two real vectors in a given plane may be described with the aid of two complex scalar parameters. Either of those parameters may be taken as a free one in the characteristic condition assigned to the wave equation. This alternative underlies the two fundamental approaches in the theory, namely, one associated with the Stroh eigenvalue problem and the other with the generalized Christoffel eigenvalue problem. The two approaches are identical insofar as a partial nondegenerate wave solution (partial mode) is concerned, but they differ in the fundamental solution (wave packet) assembling, and their dissimilarity is also revealed in the presence of degeneracies, which may involve either of the two governing parameters or both of them. Therefore, use of both approaches is essential for studying the degeneracy phenomenon in the theory of inhomogeneous waves. The criteria for different types of degeneracy, related to a double eigenvalue of the Stroh matrix or the Christoffel matrix and at the same time to a repeated root of the characteristic condition, are formulated by appeal to the matrix algebra and to the theory of polynomial equations. On this basis, dimensions of the manifolds, associated with degeneracy of different types in the space of variables, are established for elastic media of unrestricted anisotropy. The relation to the boundary-value problems is discussed.

Elastic wave propagation in inhomogeneous anisotropic media

Plane wave decomposition is a convenient and effective method in the study of wave field. Various complex wave fields can be obtained by using plane wave composition. In this paper, the method of plane wave is used to study elastic wave propagation in inhomogeneous and anisotropic media. The f-k transformation is applied to time-space domain wave equation in inhomogeneous anisotropic media. As a result, the frequency-wave number domain wave equation (Christoffel equation) can be obtained. Using the relation between elastic parameters or its spatial derivatives and Christoffel matrix elements, a method for solving the Christoffel matrix in inhomogeneous anisotropic media is formulated and applied to inhomogeneous TIV as well as inhomogeneous EDA media. The results obtained show that directional derivative of wave amplitude in continuum is negative, so amplitude is reduced, when propagating direction directs to velocity increasing direction; and directional derivative of wave amplitude is positive that means amplitude enhanced when propagating direction directs to velocity decreasing direction. Thus, wave amplitude depends on propagation direction (even in isotropic medium), but does not always attenuate. The conclusion that in continuum wave amplitude attenuates does not apply to all propagation direction.

Theory of anisotropic dynamic ray tracing in ray-centred coordinates

A 4×4-propagator matrix formalism is presented for anisotropic dynamic ray tracing, including the propagation across curved interfaces. The computations are organised in the same way as in Cervený's well-known isotropic propagator matrix formalism. Attention is paid to cases where double eigenvalues of the Christoffel matrix result in unstable expressions in the dynamic ray tracing system, but where geometrical spreading is well-behaved.