3D Elastic Media Research Articles

On seismic gradiometric wave equation inversion for density

SUMMARY Material density remains poorly constrained in seismic imaging problems, yet knowledge of density would provide important insight into physical material properties for the interpretation of subsurface structures. We test the sensitivity to subsurface density contrasts of spatial and temporal gradients of seismic ambient noise wavefields, using wave equation inversion (WEI), a form of seismic gradiometry. Synthetic results for 3-D acoustic media suggest that it is possible to estimate relative density structure with WEI by using a full acoustic formulation for wave propagation and gradiometry. We show that imposing a constant density assumption on the medium can be detrimental to subsurface seismic velocity images. By contrast, the full acoustic formulation allows us to estimate density as an additional material parameter, as well as to improve phase velocity estimates. In 3-D elastic media, severe approximations in the governing wave physics are necessary in order to invert for density using only an array of receivers on the Earth's free surface. It is then not straightforward to isolate the comparatively weak density signal from the influence of phase velocity using gradiometric WEI. However, by using receivers both at the surface and in the shallow subsurface we show that it is possible to estimate density using fully elastic volumetric WEI.

Efficient design of helical higher-order topological insulators in 3D elastic medium

Topological materials (TMs) are well-known for their topological protected properties. Phononic system has the advantage of direct observation and engineering of topological phenomena on the macroscopic scale. For the inverse design of 3D TMs in continuum medium, however, it would be extremely difficult to classify the topological properties, tackle the computational complexity, and search solutions in an infinite parameter space. This work proposed a systematic design framework for the 3D mechanical higher-order topological insulators (HOTIs) by combining the symmetry indicators (SI) method and the moving morphable components (MMC) method. The 3D unit cells are described by the MMC method with only tens of design variables. By evaluating the inherent singularity properties in the 3D mechanical system, the classic formulas of topological invariants are modified accordingly for elastic waves. Then a mathematical formulation is proposed for designing the helical multipole topological insulators (MTIs) featured corner states and helical energy fluxes, by constraining the corresponding topological invariants and maximizing the width of band gap. Both valley-based and spin-based mechanical helical HOTIs are obtained by this method and verified by full wave simulations. This design paradigm can be further extended to design 3D TMs among different symmetry classes and space groups, and different physical systems.

Numerical Simulation of Wave Propagation in 3D Elastic Media with Viscoelastic Formations

Numerical Simulation of Wave Propagation in 3D Elastic Media with Viscoelastic Formations

To the Investigation of Plane Wave Propagation in an Elastic Anisotropic Media by a Recursive Operator Method

A solution of the equations of motion of a 3D anisotropic elastic medium without determining the roots of the determinant (secular) equation is obtained by a recursive operator method. A relationship between such solutions and classical solutions is established. The possibility of solving initial–boundary value problems for plane waves is considered. An example and comparative graphs of the solutions are given.

A computationally efficient approach to modeling contact problems and fracture closure using superposition method

The shape of fractures in an elastic medium under different stress distributions have been well studied in the literature, however, the fracture closure process during unloading has not been investigated thoroughly. The fracture surface is normally assumed to be perfectly smooth so that only the width and stress intensity at the fracture tip are critical in the analysis. In reality, the creation of a fracture in a rock seldom produces smooth surfaces and the resulting asperities on the fracture surfaces can impact fracture closure. Correctly modeling this fracture closure behavior has numerous applications in structural engineering and earth sciences. In this article, we present an approach to model fracture closure behavior in a 2D and 3D elastic media. The fracture surface displacements under arbitrary normal load are derived using superposition method. The contact stress, deformation, and fracture volume evolution are estimated in a computationally efficient manner for various fracture surface properties. When compared with the traditional integral transform methods used to model fracture closure, the superposition method presented in this study produces comparable results with significant less computation time. In addition, with the aid of parallel computation, large scale fracture closure and contact problems can be successfully simulated using our proposed dynamic fracture closure model (DFCM) with very modest computation times.

Numerical observation of the equipartition regime in a 3D random elastic medium, and discussion of the limiting parameters

At long lapse times in the weakly scattering regime, the energy of the coda in a randomly fluctuating isotropic medium is equipartitioned between P and S modes. This behavior is well understood mathematically and physically for full spaces. For realistic domains, analytical results are more scarce and numerical simulations become a valuable tool. This paper discusses, based on numerical simulations of wave propagation in a 3D randomly heterogeneous elastic medium, the transition to an equipartitioned regime of the wave field. Both the time to transition and the value of the ratio of energies after transition are evaluated. Several influencing parameters are discussed, either physical (ratio of background P- and S- velocities, propagation length, variance of the heterogeneities) or numerical (influence of Perfectly Matched Layers). Setting up of a localization regime, inefficient mixture of body waves and small propagation length compared to the transport mean free paths are identified as constraining for the transition toward an equipartition regime.

Fast 3D elastic micro-seismic source location using new GPU features

In this paper, we describe new GPU features and their applications in passive seismic – micro-seismic location. Locating micro-seismic events is quite important in seismic exploration, especially when searching for unconventional oil and gas resources. Different from the traditional ray-based methods, the wave equation method, such as the method we use in our paper, has a remarkable advantage in adapting to low signal-to-noise ratio conditions and does not need a person to select the data. However, because it has a conspicuous deficiency due to its computation cost, these methods are not widely used in industrial fields. To make the method useful, we implement imaging-like wave equation micro-seismic location in a 3D elastic media and use GPU to accelerate our algorithm. We also introduce some new GPU features into the implementation to solve the data transfer and GPU utilization problems. Numerical and field data experiments show that our method can achieve a more than 30% performance improvement in GPU implementation just by using these new features.

Analysis of planar cracks in 3D elastic media with consideration of surface elasticity

This paper presents an efficient numerical procedure for analysis of arbitrary shaped, planar cracks in a three-dimensional, linear elastic infinite medium by taking the surface elasticity into account. The concept of surface stresses via Gurtin–Murdoch surface theory is integrated with the classical theory of linear elasticity to form a mathematical model capable of simulating cracks under general loading conditions and nano-scale influences. The key governing equations for the bulk material are established in terms of weakly singular displacement and traction boundary integral equations involving only unknowns on the crack face, whereas that for a zero-thickness material layer on each crack face is established in a weak form following the weighted residual technique. The final governing equations for the bulk material and the two crack-face layers are fully coupled via the continuity condition along the interface. The solution of the resulting coupled system is determined numerically by the coupling of a standard finite element procedure and a symmetric Galerkin boundary element method. Owing to the weakly singular feature of all involved boundary integrals, C\(^{0}\) basis functions are adopted everywhere in the approximation of crack-face data and only the special quadrature for evaluating nearly singular and weakly singular double surface integrals is required. Once the implemented numerical scheme is fully tested with existing reference solutions, it is then applied to investigate the role and influence of surface parameters such as the residual surface tension and the in-plane modulus on the near-front field and the size dependency of predicted solutions. Results of several examples under various scenarios are reported not only to demonstrate the capability and robustness of the proposed method but also to indicate the significance of the surface stresses in enhancing the near-surface stiffness and reducing stresses in a local region ahead of the crack front.

State space solution to 3D multilayered elastic soils based on order reduction method

Starting with the governing equations in terms of displacements of 3D elastic media, the solutions to displacement components and their first derivatives are obtained by the application of a double Fourier transform and an order reduction method based on the Cayley-Hamilton theorem. Combining the solutions and the constitutive equations which connect the displacements and stresses, the transfer matrix of a single soil layer is acquired. Then, the state space solution to multilayered elastic soils is further obtained by introducing the boundary conditions and continuity conditions between adjacent soil layers. The numerical analysis based on the present theory is carried out, and the vertical displacements of multilayered foundation with a weak and a hard underlying stratums are compared and discussed.

Time domain topological gradient and time reversal analogy: an inverse method for ultrasonic target detection

To find the shape and position of one or several obstacles embedded in a 2D or 3D elastic medium given ultrasonic measurements performed with a linear array of sensors, we use a topological gradient method. It consists in minimizing a cost function which evaluates the adequation between the measurements obtained for the inspected medium and measurements performed on a reference medium known as safe. The topological gradient is a function of both the solutions of the forward and adjoint problems. The mathematical solution of such an adjoint problem corresponds to a physical time reversal operation. This inverse method is therefore physically justified and inheritance of the refocusing properties of the time reversal phenomenon is expected.

Optimization of the reinforcement of a 3D medium with thin composite plates

The reinforcement with a thin composite plate of a 3D linear elastic medium on its external boundary or inside is considered. A linear analysis of the 3D problem leads to a variational formulation in which the reinforcement is modelled by a Kirchhoff–Love plate. Considering the sum of the compliance and a cost as the design objective, a numerical example of the optimization of this reinforcement is performed taking into account the in-plane membrane rigidity only (i.e. the bending aspects are not treated numerically).

Finite element mapping for spring network representations of the mechanics of solids.

We present a general finite element mapping procedure for defining spring network representations of solid mechanics. The procedure is rigorous and equally suitable for setting regular and unstructured spring network models of generally anisotropic solids. We use the procedure to define close-packed triangular and simple cubic lattice spring models of isotropic 2D and 3D elastic media, respectively. We extend the study to heterogeneous solids and show that the mapped spring network approach constitutes an appealing route for incorporating subelement level constitutive equations.

High-order behavior of fully bonded and delaminated circular sandwich plates with laminated face sheets and a “soft” core

The bending behavior of circular, fully or partially bonded (delaminated), sandwich plates with a “soft” core and composite laminated face sheets of general layup is investigated. The analytical model derived follows the principles of the High-Order Sandwich Plate Theory and it is based on equilibrium and compatibility requirements. The variational principle of virtual work is employed for the derivation of the field equations along with the boundary and continuity conditions. The delamination considered forms a penny shaped disbonded zone, in which the delaminated faces can slip horizontally with respect one to another, yet they may be in contact accommodating vertical normal compressive stresses. The core is assumed a 3D elastic medium and the composite face sheets are modeled using the classical lamination theory, which yields coordinate dependent constitutive relations. The governing equations take the form of partial differential equations with variable coefficients. The solution procedure adopts the Galerkin method with truncated Fourier series in the circumferential direction and the multiple-points shooting method in the radial direction. Numerical results concerning typical fully and partially bonded sandwich plates are presented and discussed. The results reveal the effect of the anisotropy of the face sheets on the response of the structure and the influence of the delaminated area on the behavior of the sandwich plate.

Uniform asymptotic approximations for accurate modeling of cracks in layered elastic media

We present uniform asymptotic solutions (UAS) for displacement discontinuities (DD) that lie within the middle layer of a three layer elastic medium. The DDs are assumed to be normal to the two parallel interfaces between the leastic media, and solutions will be presented for both 2D and 3D elastic media. Using the Fourier transform (FT) method we construct the leading term in the asymptotic expansion for the spectral coefficient functions for a DD in a three layer medium. Although a closed form solution will require an infinite series solution, we demonstrate how this UAS can be used to construct highly efficient and accurate solutions even in the case in which the DD actually touches the interface. We present an explicit UAS for elements in which the DD fields are assumed to be piecewise constant throughout a line segment in 2D and a rectangular element in 3D. We demonstrate the usefulness of this UAS by providing a number of examples in which the UAS is used to solve problems in which cracks just touch or cross an interface. The accuracy and efficiency of the algorithm is demonstrated and compared with other numerical methods such as the finite element method and the boudary integral method.

3D Elastic finite-difference modeling of seismic motion using staggered grids with nonuniform spacing

AbstractThis article provides a technique to model seismic motions in 3D elastic media using fourth-order staggered-grid finite-difference (FD) operators implemented on a mesh with nonuniform grid spacing. The accuracy of the proposed technique has been tested through comparisons with analytical solutions, conventional 3D staggered-grid FD with uniform grid spacing, and reflectivity methods for a variety of velocity models. Numerical tests with nonuniform grids suggest that the method allows sufficiently accurate modeling when the grid sampling rate is at least 6 grid points per shortest shear wavelength. The applicability for a finite fault with non-uniform distribution of point sources is also confirmed. The use of nonuniform spacing improves the efficiency of the FD methods when applied to large-scale structures by partially avoiding the spatial oversampling introduced by the uniform spacing in zones with high velocity. The significant reduction in computer memory that can be obtained by the new technique improves the efficiency of the 3D-FD method at handling shorter wavelengths, larger areas, or more realistic 3D velocity structures.

Rupture dynamics of a planar fault in a 3D elastic medium: Rate- and slip-weakening friction

Abstract We propose a new boundary integral equation method (BIEM) to model the spontaneous propagation of rupture on a planar fault embedded in a homogeneous elastic medium. The BIEM formulation is very compact and fast in computation, so that we are able to study the effect of different slip- and rate-dependent friction laws on dynamic shear fault propagation. We simulated a spontaneous rupture by a sudden break of asperity under two friction laws, rate and/or slip weakening. We examined both long and circular asperities. The long asperity model corresponds to the 2D anti-plane or in-plane problem. The obtained result shows that slip-weakening friction is important at the crack tip, and rate-weakening friction plays an important role in the healing stage. If slip-weakening friction is strong stress drops gradually at the crack tip. On the other hand, if rate-weakening friction is strong stress drops abruptly, but it stops suddenly and sometimes stress recovers. This sudden stop of rupture produces a heterogeneous stress distribution, which will in turn produce aftershocks. Finally, we studied a realistic asperity model in which rupture starts from a small patch and then propagates with finite rupture speed. Depending on the asperity size healing can occur even without rate-weakening friction. This is due to very strong healing waves produced by the edges of the asperity.

Simulating seismic wave propagation in 3D elastic media using staggered-grid finite differences

Abstract This article provides an overview of the application of the staggered-grid finite-difference technique to model wave propagation problems in 3D elastic media. In addition to presenting generalized, discrete representations of the differential equations of motion using the staggered-grid approach, we also provide detailed formulations that describe the incorporation of moment-tensor sources, the implementation of a stable and accurate representation of a planar free-surface boundary for 3D models, and the derivation and implementation of an approximate technique to model spatially variable anelastic attenuation within time-domain finite-difference computations. The comparison of results obtained using the staggered-grid technique with those obtained using a frequency-wavenumber algorithm shows excellent agreement between the two methods for a variety of models. In addition, this article also introduces a memory optimization procedure that allows large-scale 3D finite-difference problems to be computed on a conventional, single-processor desktop workstation. With this technique, model storage is accommodated using both external (hard-disk) and internal (core) memory. To reduce system overhead, a cascaded time update procedure is utilized to maximize the number of computations performed between I/O operations. This formulation greatly expands the applicability of the 3D finite-difference technique by providing an efficient and practical algorithm for implementation on commonly available workstation platforms.

Integral equation method for plane crack with arbitrary shape in 3D elastic medium

AbstractWe have derived stress integral equations for an arbitrary shaped plane crack in a three-dimensional (3D) infinite homogeneous isotropic elastic medium. First we obtained the integral equation for static crack, which is consistent with previous works. Then we derived a new boundary integral equation for the study of dynamic tensile and shear cracks. We removed the hypersingularity that appears in the usual formulation of the stress-boundary integral equation method by a regularization technique. The dynamic stress integral equation consists of three terms which can be physically explained. The first term is a modification of the static solution, the second one represents the diffraction of waves at slip discontinuities, and the last term is related to the effect of an instantaneous change in slip velocity. The first term controls the overall shape of the stress drop inside the crack. This term is important when we consider the final stress drop on the fault. However, the second and the third terms are dominant at a moving crack tip, and are responsible for the radiation of P and S waves. They are also very important for the application of a fracture criterion at the rupture front. Using this boundary integral equation we have calculated several numerical examples for both the static and dynamic fault models.

Hermite-Gaussian beams in inhomogeneous elastic media

Hermite-Gaussian beams in a 3D elastic inhomogeneous medium are obtained as high-frequency asymptotic solutions of equations of motion concentrated in a vicinity of P- and S-wave rays. Equations of motion are transformed into the parabolic equation (Schroedinger equation) in this case. Explicit expressions for a complete set of linearly independent solutions of a parabolic equation are derived. The method of creation and anihilation operators known from quantum mechanics are used in the derivation.