Displacement Eigenfunctions Research Articles

An eigenfunction representation of deep waveguides with application to unconventional reservoirs

Guided waves that propagate in deep low-velocity zones can be described using the displacement-stress eigenfunction theory. For a layered subsurface, these eigenfunctions provide a framework to calculate guided-wave properties at a fraction of the time required for fully numerical approaches for wave-equation modeling, such as the finite-difference approach. Using a 1D velocity model representing the low-velocity Eagle Ford Shale, an unconventional hydrocarbon reservoir, we verify the accuracy of the displacement eigenfunctions by comparing with finite-difference modeling. We use the amplitude portion of the Green’s function for source-receiver eigenfunction pairs as a proxy for expected guided-wave amplitude. These response functions are used to investigate the impact of the velocity contrast, reservoir thickness, and receiver depth on guided-wave amplitudes for discrete frequencies. We find that receivers located within the low-velocity zone record larger guided-wave amplitudes. This property may be used to infer the location of the recording array in relation to the low-velocity reservoir. We also study guided-wave energy distribution between the different layers of the Eagle Ford model and find that most of the high-frequency energy is confined to the low-velocity reservoir. We corroborate this measurement with field microseismic data recorded by distributed acoustic sensing fiber installed outside of the Eagle Ford. The data contain high-frequency body-wave energy, but the guided waves are confined to low frequencies because the recording array is outside the waveguide. We also study the energy distribution between the fundamental and first guided-wave modes as a function of the frequency and source depth and find a nodal point in the first mode for source depths originating in the middle of the low-velocity zone, which we validate with the same field data. The varying modal energy distribution can provide useful constraints for microseismic event depth estimation.

Rotational modes of Poincaré Earth models

ABSTRACT We study the following rotational modes of Poincaré Earth models: the tilt-over mode (TOM), the spin-over mode (SOM) and free core nutation (FCN), using first a simple Earth model with a homogeneous and incompressible liquid core (LC) and a rigid mantle (MT). We obtain analytical solutions for the periods of these modes as well as that of the Chandler wobble (CW). We show analytically the distinction between the TOM and the SOM and that the FCN is indeed the same mode as the SOM of a wobbling Earth. The reduced pressure, in terms of which the vector momentum equation is known to reduce to a scalar second-order partial differential equation called the Poincaré equation, is used as the independent variable. Analytical solutions are then found for the displacement eigenfunctions in a meridional plane of the liquid core for the aforementioned modes. We next consider a three-layer Earth model similar to above which also includes a rigid inner core (IC). We first show that analytical solutions exist for the period and eigenfunctions of the CW if the IC is locked to the MT, i.e. they have the same wobbling motion. We show that this is significant as it shows that the CW manifests itself for a Poincaré (incompressible and inviscid LC) wobbling Earth model. We further allow for the inner core to wobble independently and compute numerically the periods and displacement eigenfunctions of the TOM, SOM and FCN, as well as those for still another rotational mode, the inner-core wobble (ICW). Next we show that the presence of the characteristic surfaces intercepted by the inner-core, when computing the period and eigenfunctions of the free inner-core nutation (FICN), may be the reason for the slow (or lack of the) convergence of this mode. Finally, we show that even though the wobbling motion of the mantle is ignored when solving for the frequencies of the ICW and the FICN when Sasao's approximation is used, the analytical solutions for both these modes yield periods nearly identical to those in the literature for a similar Earth model with mantle allowed to wobble as well. We infer that the Sasao's approximation, or the severe truncation of the series solution of the field variables, the pressure, the gravitational potential and the components of the displacement vector, may not be adequate to accurately describe the motion in the liquid core during the excitation of the FICN.

Asymptotic Solutions of Plastic Stress and Displacement at V-Notch Tips Under Anti-Plane Shear

An efficient method was developed to determine the first- and high-order terms of asymptotic solutions of plastic stress and displacement near V-notch tips under anti-plane shear in power-law hardening materials. Through introduction of the asymptotic series expansions of stress and displacement fields around the V-notch tip into the fundamental equations of the elastoplastic theory, the governing ordinary differential equations (ODEs) with the stress and displacement eigen-functions were established. Then the interpolating matrix method was employed to solve the resulting nonlinear and linear ODEs. Consequently, the high-order stress exponents and the associated eigen-solutions were obtained. The presented method, being capable of dealing with the V-notches with arbitrary opening angles and strain hardening indexes under anti-plane shear, has the advantages of great versatility and high accuracy. Typical examples were given to demonstrate the accuracy and effectiveness of this method.

Complete elastic–plastic stress asymptotic solutions near general plane V-notch tips

An efficient method is developed to determine the multiple term eigen-solutions of the elastic–plastic stress fields at the plane V-notch tip in power-law hardening materials. By introducing the asymptotic expansions of stress and displacement fields around the V-notch tip into the fundamental equations of elastic–plastic theory, the governing ordinary differential equations (ODEs) with the stress and displacement eigen-functions are established. Then the interpolating matrix method is employed to solve the resulting nonlinear and linear ODEs. Consequently, the first four and even more terms of the stress exponents and the associated eigen-solutions are obtained. The present method has the advantages of greater versatility and high accuracy, which is capable of dealing with the V-notches with arbitrary opening angle under plane strain and plane stress. In the present analysis, both the elastic and the plastic deformations are considered, thus the complete elastic and plastic stress asymptotic solutions are evaluated. Numerical examples are shown to demonstrate the accuracy and effectiveness of the present method.

Symplectic Analysis of Wrinkles in Elastic Layers With Graded Stiffnesses

Wrinkles in layered neo-Hookean structures were recently formulated as a Hamiltonian system by taking the thickness direction as a pseudo-time variable. This enabled an efficient and accurate numerical method to solve the eigenvalue problem for onset wrinkles. Here, we show that wrinkles in graded elastic layers can also be described as a time-varying Hamiltonian system. The connection between wrinkles and the Hamiltonian system is established through an energy method. Within the Hamiltonian framework, the eigenvalue problem of predicting wrinkles is defined by a series of ordinary differential equations with varying coefficients. By modifying the boundary conditions at the top surface, the eigenvalue problem can be efficiently and accurately solved with numerical solvers of boundary value problems. We demonstrated the accuracy of the symplectic analysis by comparing the theoretically predicted displacement eigenfunctions, critical strains, and wavelengths of wrinkles in two typical graded structures with finite element simulations.

Resistive interchange mode destabilized by helically trapped energetic ions and its effects on energetic ions and bulk plasma in a helical plasma

A resistive interchange mode of the structure (, : poloidal and toroidal mode numbers, respectively) with a bursting character and rapid frequency chirping in the range less than 10 kHz is observed for the first time in the edge region of the net current-free, low beta LHD (Large Helical Device) plasmas during high power injection of perpendicular neutral beams. The mode resonates with the precession motion of helically trapped energetic ions (EPs), following the resonant condition. The radial mode structure is recognized to be similar to that of the pressure-driven resistive interchange mode, of which radial displacement eigenfunction quite localizes around the mode rational surface, and evolves into an odd-type (or island-type) during the late of frequency chirping phase. This beam driven mode is excited when the beta value of helically trapped EPs exceeds a certain threshold. This instability is thought to be a new branch of resistive interchange mode destabilized by the trapped energetic ions. The radial transport, i.e. redistribution and losses, of helically trapped energetic ions induced by the mode transiently generates significant radial electric field near the plasma peripheral region. The large shear of thus generated radial electric field is thought to contribute to the observed suppression of micro-turbulence and transient increases of the temperature of fully ionized carbon impurity ions and electron density, suggesting improvement of bulk plasma confinement.

Complex modal analysis of rods with viscous damping devices

Abstract The complex modal analysis of rods equipped with an arbitrary number of viscous damping devices is addressed. The following types of damping devices are considered: external (grounded) spring-damper, attached mass-spring-damper and internal spring-damper. Within a standard 1D formulation of the vibration problem, the theory of generalized functions is used to model axial stress and displacement discontinuities at the locations of the damping devices. By using the separate variable approach, a simple solution procedure of the motion equation leads to exact closed-form expressions of the characteristic equation and eigenfunctions, which inherently fulfill the required matching conditions at the locations of the damping devices. Based on the characteristic equation, a closed-form sensitivity analysis of the eigensolution is implemented. The displacement eigenfunctions exhibit orthogonality conditions. They can be used with the complex mode superposition principle to tackle forced vibration problems and, in conjunction with the stress eigenfunctions, to build the exact dynamic stiffness matrix of the rod for complex modal analysis of truss structures. Numerical results are discussed for a variety of parameters.

Dynamics of the Earth’s fluid core: Implementation of a Clairaut coordinate system

If the reference state of a rotating and self gravitating fluid body is one of hydrostatic equilibrium then the figure of the body is a spheroid such that a cross sectional area parallel to the equatorial plane of the body is a circle while that parallel to a meridional plane is an ellipse. The effects of the fluid body’s flattened (spheroidal) figure is small on the frequencies of the body’s short-period (shorter than a few hours in the case of the Earth) normal modes. For the long-period normal modes, however, these effects must be considered. Furthermore, the body’s wobble and nutation modes owe their existence to its ellipsoidal figure. In the conventional approach to computing these frequencies, an orthogonal coordinate system is usually considered. It is then necessary to have the knowledge of the derivatives of the material properties of the body, such as the density and Lamé parameters, in order to include the effects of the ellipticity in the dynamical equations. In the available Earth models, however, these derivatives are not well defined. In order to minimize the effects of these derivatives in the treatment of the dynamical problems we use a non-orthogonal (Clairaut) coordinate system. Using this approach, we compute the frequencies and displacement eigenfunctions for some of the inertial modes of a realistic spheroidal model of the Earth’s fluid core and compare them to the known results for an Earth model with a homogeneous and incompressible fluid core.

Ground motion prediction of realistic earthquake sources using the ambient seismic field

Predicting accurate ground motion is critical for earthquake hazard analysis, particularly in situations where sedimentary basins trap and amplify seismic waves. We exploit the information carried by the ambient seismic field to extract surface‐wave Green's functions between seismic stations and to predict long‐period ground motion from earthquakes. To do so, we modify the surface impulse response to correct for the source depth and for the double‐couple focal mechanism. These corrections are derived under the assumption that material properties in the immediate vicinity of the source depend only on depth. Using this local 1‐D assumption, we solve the surface‐wave eigenproblem and compute the fundamental‐mode displacement eigenfunctions to express the surface‐wave excitation at the source. We validate this technique, which we call the virtual earthquake approach, by comparing computed seismograms with earthquake waveforms from four moderate earthquakes that occur near broadband stations in southern California. The depth and mechanism corrections show clear improvements of the predicted ground motion relative to the surface impulse response.

Redistribution of high energy alpha particles due to sawteeth with partial reconnection

The redistribution of high energy alpha particles due to internal kink modes is studied in plasmas with ITER-like parameters. The exact particle trajectories in the total fields, equilibrium plus perturbation, are calculated. The equilibrium magnetic field is obtained by analytically solving the Grad–Shafranov equation and the perturbed electric and magnetic fields are reconstructed using ideal MHD and the experimental information about the displacement eigenfunction. The (1, 1), (2, 2) and (2, 1) modes are included and the effect of changing their amplitude and frequency is determined. The results show that if the conditions are similar to those reported in Igochine et al (2007 Nucl. Fusion 47 23), the peak density of counter-passing particles decreases between 25% and 40% (depending on the energy); the peak of the trapped particles density shifts outwards by approximately 10% of the minor radius and the total on axis density decreases by more than 25%. This redistribution occurs inside the q = 1 surface. The addition of a (2, 1) mode, which can produce the stochastization of the magnetic field, significantly increases particle redistribution and allows particles to spread beyond the q = 1 surface. Different groups of particles (co-passing, counter-passing, trapped) respond differently to the perturbations.

Interface corners in linear anisotropic viscoelastic materials

In this study, an extended Stroh formalism for two-dimensional linear anisotropic viscoelasticity is developed for the problems of interface corners between two dissimilar viscoelastic materials. In this formalism, the solutions for the displacements and stress functions in the time domain can be written in the form of a matrix function using complex variables. The correspondence relations for viscoelastic analysis are then obtained and verified for material eigenvectors, displacement and stress eigenfunctions, singularity orders of stresses, and stress intensity factors. Explicit solutions for the material eigenvector matrices in the Laplace domain are also obtained for standard linear and isotropic linear viscoelastic solids. To calculate the singularity orders and stress intensity factors of the interface corners, four different approaches are proposed. Through numerical examples on cracks, interface cracks, and interface corners, an approach using the path-independent H-integral in the Laplace domain with an elastic near-tip solution, which takes the correspondence relations for singularity orders and stress intensity factors, is demonstrated to be better than the other three approaches.

Alpha particle redistribution due to experimentally reconstructed internal kink modes

The redistribution of alpha particles due to internal kink modes is calculated. The exact particle trajectories in the total, equilibrium plus perturbation, fields are calculated. The equilibrium magnetic field is obtained by analytically solving the Grad–Shafranov equation. The perturbed electric and magnetic fields are reconstructed using the experimental information about the displacement eigenfunction. An effective diffusion coefficient is introduced to quantify the magnitude of the particle redistribution produced by the perturbations.

The Excitation of Guided-waves by Underground Point Source: an Investigation with Theoretical Seismograms

Near-Source scattering of Rg into S appears to be the primary contributor to the low-frequency Lg. The authors further suggest that the prominent low-frequency spectral null in Lg is due to Rg from a compensated linear vector dipole (CLVD) source, and the low-frequency null in Rg excitation is due to a zero-crossing of the horizontal displacement eigenfunctions. In this study, the mechanism of the excitation of Lg from explosions in layered earth structures are analyzed with theoretical seismograms. Our result shows that the CLVD source generates prominent Lg waves,and the null in the Lg spectra showing remarkably good agreement with those expected from Rg due to a CLVD source. We conclude that the derivative of displacement eigenfunction also takes a key role in the excitation of the null, only zero-crossing of the horizantall displacement eigenfunction can not fully explain it.

A fluid-saturated poroelastic model of the vocal folds with hydrated tissue

The purpose of this study is to develop a continuous model to describe the vibration of the vocal fold with hydrated tissue. This model is unique because it is based on the fluid-saturated porous solid theory. Therefore, this new model can be used to study some vocal fold characteristics that would be difficult to predict using previous models. Numerical simulations show that this model can generate self-oscillation and that its phonation threshold pressure (PTP) is 0.555 kPa. The basic outputs of this model, including fundamental frequency, maximum lateral displacement, surface dynamics, and empirical eigenfunctions, agree with previous models and experimental studies, which validates this new model. The ability to simulate the flow of liquid through the tissue is one of the important advantages of this new model. It was found that the liquid in the vocal fold tissue could be accumulated at the anterior–posterior midpoint during phonation, which could cause a pressure increase in the liquid. The liquid pressure increased from 0.033 to 0.150 kPa when the subglottal pressure increased from 0.555 kPa (PTP) to 0.7 kPa. It was believed that the liquid dynamics in the tissue during phonation could be related to the development of some vocal diseases, such as vocal nodules, edema, and so on. Therefore, we expect that this model might not only provide a more appropriate description of the vocal fold vibration, but that it could also have clinical value in investigating certain vocal fold diseases.

Gravity and inertial modes of rotating stars

We use the three potential description (3PD) to study the oscillatory dynamics of rotating stars. The 3PD scheme describes the exact linearized dynamics of rotating, self-gravitating, stratified, compressible and inviscid fluids. We compute the frequencies and the displacement eigenfunctions for some of the inertial and gravity modes of polytropic star models. The Coriolis force is the restoring force for the inertial modes while the gravity modes are induced by the buoyancy forces. We show that the frequencies of the inertial modes of these star models may be significantly different from those of a homogeneous and incompressible model, an approximation usually made to compute the frequencies of the inertial modes of rapidly rotating stars. We will also investigate the effects of rotation on the frequencies of the gravity modes of these models. In order to test the validity and the accuracy of the results from our approach, we compare the eigenfrequencies of some of the gravito-inertial modes of these models, computed using the 3PD, to those in the literature.

Inertial modes of a compressible fluid core model

Computational methods are used to investigate the effects of fluid compressibility on the frequencies of the inertial modes of the Earth's fluid core. The 3PD (the three potential description) is applied to two models of compressible fluid spheres, for one of which the dilatation vanishes, in order to study these modes. We show that compressibility may have a significant effect on some of the modal frequencies. Using the shape of the displacement eigenfunctions of a fluid sphere we also infer which modes of a sphere may have counterparts in a spherical shell. We then give a few examples of the inertial modes of a compressible fluid shell proportional to the Earth's fluid core.

Generation of Null in Rg Wave by Underground Explosions

It was widely accepted that the near-source scattering of explosion-generated Rg into S is the primary contributor to the low-frequency Lg from nuclear explosions, because the prominent low-frequency spectral null in Lg coincides with that in Rg from a compensated linear vector dipole (CLVD) source. In this study, the mechanism of the excitation of the spectral null of Rg generated by CLVD source is analyzed in detail with normal mode method. The most important result in this study is that the spectral null is due to a combination of displacement eigenfunction and its derivative of the fundamental-mode Rayleigh wave controlled by CLVD source, the zero-crossing alone can not fully explain the excitation of the spectral null.

Sharp Corner Functions for Mindlin Plates

Transverse displacement and rotation eigenfunctions for the bending of moderately thick plates are derived for the Mindlin plate theory so as to satisfy exactly the differential equations of equilibrium and the boundary conditions along two intersecting straight edges. These eigenfunctions are in some ways similar to those derived by Max Williams for thin plates a half century ago. The eigenfunctions are called “corner functions,” for they represent the state of stress currently in sharp corners, demonstrating the singularities that arise there for larger angles. The corner functions, together with others, may be used with energy approaches to obtain accurate results for global behavior of moderately thick plates, such as static deflections, free vibration frequencies, buckling loads, and mode shapes. Comparisons of Mindlin corner functions with those of thin-plate theory are made in this work, and remarkable differences are found.

Solution of the Rayleigh Eigenproblem in Viscoelastic Media

We present a technique for the solution of the complex-valued eigenproblem associated with the propagation of surface waves in general linear viscoelastic media. The new technique permits simultaneous determination of the Rayleigh dispersion and attenuation curves and the displacement and stress eigenfunctions for vertically heterogeneous, linear viscoelastic media with arbitrary values of material damping ratio. The technique is based on the Cauchy residue theorem of complex analysis that takes full advantage of the holomorphic properties of the Rayleigh secular function, which is viewed as an analytic mapping of the complex-valued Rayleigh phase velocity. Because the eigenvalue problem is solved directly in the complex domain with no simplifying assumptions, the algorithm implicitly accounts for the inherent coupling between phase velocity and attenuation of seismic waves as a result of material dispersion. The technique overcomes the limitations of previous algorithms that often break up the complex structure of the problem and/or require a priori information about the number of eigenvalues and their approximate value. The algorithm is validated via comparisons with closed-form solutions for a uniform half-space. Examples are also used to compare solutions obtained with the proposed technique and one based on the assumption of weak dissipation in strongly and weakly dissipative layered media.

Is there an observable lack of reciprocity inPKP(DF) traveltimes?

On average, traveltimes of PKPDF for equatorial ray paths through the quasi-eastern hemisphere of the inner core are around 0.5 s faster than equivalent ray paths through its quasi-western hemisphere. In these observations, the eastern hemisphere is sampled primarily by westward- and the western hemisphere by eastward-propagating waves. Noting that westward propagation is faster than eastward propagation inside a rotating earth, I estimate the expected traveltime difference from Coriolis splitting of the displacement eigenfunctions of the PKPDF-equivalent modes. It turns out that Coriolis effects are too small to give rise to residuals of the required magnitude. Thus, the observations must be primarily due to velocity heterogeneities.