Elastic Equation Research Articles

Elastic reverse time migration using an efficient and accurate adaptive variable grid discretization method

Abstract The conventional reverse time migration utilizes regularly sampled computational grids to simulate wave propagation. Selecting the appropriate grid sampling is important for computational accuracy and efficiency. In general, the uniform-size grid cannot represent the complexity of the geology well. The grid may appear sparse in the low-velocity zone, especially in shallow depths where dispersion may occur. Conversely, it may appear excessively dense in the high-velocity zone, such as at greater depths or within a salt body, which results in higher computational memory and time consumption. To overcome these issues, we developed an efficient and accurate adaptive variable grid discretization method that automatically selects the vertical grid size based on the velocity, depth, and dominant frequency of the wavelet in elastic medium. Then we reformulated the elastic equations based on the adaptive variable grid by introducing a mapping relationship. To test the effectiveness, accuracy, and efficiency of the equation, we implemented it to both the forward propagation and migration of elastic wavefield. Synthetic numerical examples demonstrate that our proposed method can achieve elastic wavefield separation and no significant dispersion phenomenon. The multi-component imaging accuracy of reverse time migration is nearly equivalent to the traditional method, while significantly improving computational efficiency and saving storage space.

Advancing river cruise product design: insights from cross-quality elasticity of river cruises to ocean cruises

ABSTRACT This paper puts forward the concept of cross-quality elasticity, leveraging the commonalities of river and ocean cruise tourism, and develops a cross-quality elasticity equation with price, quality, and popularity as independent variables, while market share as dependent variable, calculating the cross-quality elasticity for river cruise tourism relative to ocean one. A model is developed for optimising river cruise itinerary and ship choice based on the cross-quality elasticity and the constraints of river channel capacity and itinerary type, to maximise the increased profits for river cruise operators. In the case study on the segments of Chongqing-Wuhan, Wuhan-Shanghai and Nanjing-Zhoushan along the Yangtze River, the home ports, ports of call, order of call, days of cruise travel, adoptions of cruise ships, and the number of tourists shifting from ocean cruise market have been determined. The results indicate that well-designed river cruises can expand the market of river cruise tourism; popularity exerts a moderating influence on the relationship between cruise quality and consumer demand; river cruise of high quality may serve as a viable alternative for consumers accustomed to ocean cruises.

On Hybrid and Non-Hybrid Discrete Fractional Difference Inclusion Problems for the Elastic Beam Equation

On Hybrid and Non-Hybrid Discrete Fractional Difference Inclusion Problems for the Elastic Beam Equation

Numerical Analysis of the Cylindrical Shell Pipe with Preformed Holes Subjected to a Compressive Load Using Non-Uniform Rational B-Splines and T-Splines for an Isogeometric Analysis Approach

In this paper, we implement the finite detail technique primarily based on T-Splines for approximating solutions to the linear elasticity equations in the connected and bounded Lipschitz domain. Both theoretical and numerical analyses of the Dirichlet and Neumann boundary problems are presented. The Reissner–Mindlin (RM) hypothesis is considered for the investigation of the mechanical performance of a 3D cylindrical shell pipe without and with preformed hole problems under concentrated and compression loading in the linear elastic behavior for trimmed and untrimmed surfaces in structural engineering problems. Bézier extraction from T-Splines is integrated for an isogeometric analysis (IGA) approach. The numerical results obtained, particularly for the displacement and von Mises stress, are compared with and validated against the literature results, particularly with those for Non-Uniform Rational B-Spline (NURBS) IGA and the finite element method (FEM) Abaqus methods. The obtained results show that the computation time of the IGA based on the T-Spline method is shorter than that of the IGA NURBS and FEM Abaqus/CAE (computer-aided engineering) methods. Furthermore, the highlighted results confirm that the IGA approach based on the T-Spline method shows more success than numerical reference methods. We observed that the NURBS IGA method is very limited for studying trimmed surfaces. The T-Spline method shows its power and capability in computing trimmed and untrimmed surfaces.

Graviton mass due to dark energy as a superconducting medium-theoretical and phenomenological aspects

It is well known that the cosmological constant term in the Einstein field equations can be interpreted as a stress tensor for dark energy. This stress tensor is formally analogous to an elastic constitutive equation in continuum mechanics. As a result, the cosmological constant leads to a “shear modulus” and “bulk modulus” affecting all gravitational fields in the universe. The form of the constitutive equation is also analogous to the London constitutive equation for a superconductor. Treating dark energy as a type of superconducting medium for gravitational waves leads to a Yukawa-like gravitational potential and a massive graviton within standard General Relativity. We discuss a number of resulting phenomenological aspects such as a screening length scale that can also be used to describe the effects generally attributed to dark matter. In addition, we find a gravitational wave plasma frequency, index of refraction, and impedance. The expansion of the universe is interpreted as a Meissner-like effect as dark energy causes an outward “expulsion” of space-time similar to a superconductor expelling a magnetic field. The fundamental cause of these effects is interpreted as a type of spontaneous symmetry breaking of a scalar field. There is an associated chemical potential, critical temperature, and an Unruh-Hawking effect associated with the formulation.

Nonlinear elastic general integral equations in micromechanics of random structure composites

We consider static problems for composite materials (CMs) with either locally elastic or peridynamic constitutive properties. The general integral equation (GIE) is the exact integral equationWe consider static problems for composite materials connecting the random fields at the point being considered and the surrounding points. There is a very long and colored history of the development of GIE which goes back to Lord Rayleigh. Owing to the new GIE (forming the second background of micromechanics also called the computational analytical micromechanics, CAM), one proved that local micromechanics (LM) and peridynamic micromechanics (PM) are formally similar to each other for CM of random structures. By now, the GIEs are generalized to CMs (of statistically homogeneous and inhomogeneous structures) with the phases described by local models, strongly nonlocal models (strain type and displacement type, peridynamics), and weakly nonlocal models (strain-gradient theories, stress-gradient theories, and higher-order models). However, a fundamental restriction of all mentioned GIEs is their linearity with respect to a primary unknown variable. The goal of this study is obtaining nonlinear GIEs for PM, and, in a particular case, for LM. For the presentation of PM as a unified theory, we describe PM as the formalized schemes of blocked (or modular) structures so that the experts developing one block need not be experts in the underlying another block (this is a good background for effective collaborations of different teams in so many multidisciplinary areas as PM). The opportunity for the creation of this blocked structure of the PM is supported by a critical generalization of CAM which is extremely flexible, robust, and general.

Two-Dimensional Linear Elasticity Equations of Thermo-Piezoelectric Semiconductor Thin-Film Devices and Their Application in Static Characteristic Analysis

Two-Dimensional Linear Elasticity Equations of Thermo-Piezoelectric Semiconductor Thin-Film Devices and Their Application in Static Characteristic Analysis

A discontinuous Galerkin based multiscale method for heterogeneous elastic wave equations

In this paper, we develop a local multiscale model reduction strategy for the elastic wave equation in strongly heterogeneous media, which is achieved by solving the problem in a coarse mesh with multiscale basis functions. We use the interior penalty discontinuous Galerkin (IPDG) to couple the multiscale basis functions that contain important heterogeneous media information. The construction of efficient multiscale basis functions starts with extracting dominant modes of carefully defined spectral problems to represent important media feature, which is followed by solving a constraint energy minimization problem. Then a Petrov-Galerkin projection and systematization onto the coarse grid are applied. As a result, an explicit and energy-conserving scheme is obtained for fast online simulation. The method exhibits both coarse-mesh and spectral convergence as long as one appropriately chooses the oversampling size. We rigorously analyze the stability and convergence of the proposed method. Numerical results are provided to show the performance of the multiscale method and confirm the theoretical results.

Use of the integration by part to derive beam/plate equilibrium equations from elasticity with applications to beams

This article proposes a rigorous mathematical justification of the derivation of the governing equations of any beam and plate theory by the three three-dimensional (3D) elasticity equations. This is done by the application of the integration by part theorem via weight residual-like methods. The Fis polynomials (subscripts i and s denote the i-direction and the generic s-term of the displacement expansion uis, respectively) related to a given beam/plate theory are used as weight functions for each of the displacement unknown. To show how the method works a number of simple problems are discussed. These are related to the equilibrium equations and boundary conditions of bending beams according to shear deformation theory. Three methods for obtaining these equations are compared in some detail. The first two are the well-known methods used in mechanics of deformable bodies: (1) use of the Principle of Virtual Displacements; (2) use of the six cardinal ‘Newton’ equations of statics. The third method coincides with the one proposed and rigorously justified in this paper which is (3) the direct integration of the equations of 3D elasticity. Beam theories based on Lagrange polynomials are used as well. The latter makes it possible to immediately develop theories for layered beams and obtain so-called ‘layer-wise’ models, which are only applied here to the case of sandwich structures. Three appendixes discuss: A – The problem in which the derivative of one of the unknowns appears among the unknowns, as is the case with the Euler–Bernoulli theory; B – The extension to dynamics; C-Simple derivation of constitutive equations and related closed-form solution via Navier method.

A method for extracting P‐SV‐converted wave angle‐domain common‐image gathers based on elastic‐wave reverse‐time migration

AbstractMulticomponent seismic technology utilizes the kinematic and dynamic characteristics of reflected P‐waves and converted S‐waves to reduce ambiguity in seismic exploration. The imaging and inversion accuracy of P‐SV‐converted waves are important in determining whether multicomponent seismic exploration can achieve higher exploration accuracy than conventional P‐wave exploration. Pre‐stack inversion of P‐SV‐converted waves requires precise input of P‐SV‐converted wave angle‐domain common‐image gathers. Consequently, the P‐SV‐converted wave angle‐domain common‐image gather extraction accuracy will significantly affect the P‐SV‐converted wave inversion accuracy. However, existing methods for extracting P‐SV‐converted wave angle‐domain common‐image gathers are constrained by issues such as the P‐ and S‐wave crosstalk artefacts, low‐frequency noises and inaccurate calculation of P‐wave incident angles, leading to poor imaging accuracy. We study an angle‐domain cross‐correlation imaging condition and address three key issues based on this condition: the decoupling of P‐ and S‐waves, the separation of up‐going and down‐going waves and the precise calculation of P‐wave incident angles. Our strategies facilitate high‐precision extraction of P‐SV‐converted wave angle‐domain common‐image gathers using elastic wave reverse‐time migration. In this paper, first, we employ the first‐order velocity‐dilatation‐rotation elastic wave equations to decouple P‐ and S‐waves automatically during source and receiver wavefield extrapolations. Second, we calculate the optical flow vectors of P‐ and S‐waves to ensure stable calculations of wave propagation directions. Based on this, we obtain up‐going and down‐going waves of P‐ and S‐waves. Meanwhile, we calculate the incident angle of the source P‐wave using geometric relations. Lastly, we apply the angle‐domain imaging condition to achieve high‐precision extraction of P‐SV‐converted wave angle‐domain common‐image gathers. Model examples demonstrate the effectiveness and advantages of the proposed method.

Simulate the elastic wavefields in media with an irregular surface topography based on staggered grid finite difference

Abstract Wave equation forward modeling is a useful method to study the propagation regulation of seismic wavefields. Finite difference (FD) is one of the most extensively employed numerical approaches for computing wavefields in earthquake and exploration seismology. However, the FD approach relying on regular grids often struggles to calculate wavefields in regions featuring surface topographies. The elastic wave equation can more accurately describe the propagation of seismic wavefields in elastic media compared to the acoustic wave equation. We introduce a new FD scheme to calculate the elastic wavefields in an isotropic model with a surface topography. The novel approach can use a conventional staggered grid FD (SGFD) approach based on regular grids. A new elastic model with a horizontal surface is first obtained from the nearby surface's elastic properties and the undulating terrain elevation. We subsequently employ a topography-related strategy to eliminate the effects of surface topographies on the seismic wavefields in models with irregular surface topographies. The merits of our proposed scheme lie in its ability to stable numerically compute wavefields in models with irregular surface topographies without altering the conventional SGFD relying on regular grids. To validate the effectiveness and practicality of our method, we utilize elastic models featuring complex surface topographies. Numerical experiments demonstrate that our approach efficiently calculates elastic wavefields in isotropic media with irregular topographies based on conventional SGFD.

On Higher-Order Nonlinear Fractional Elastic Equations with Dependence on Lower Order Derivatives in Nonlinearity

The paper studied high-order nonlinear fractional elastic equations that depend on low-order derivatives in nonlinearity and established the existence and uniqueness results by using the Leray–Schauder alternative theorem and Perov’s fixed point theorem on an appropriate space under mild assumptions. Examples are given to illustrate the key results.

Elastic reverse time migration based on first-order velocity-dilatation-rotation equations using the optical flow vector

The determination of the P- and S-wave propagation directions is crucial for wavefield decomposition, polarity reversal correction, and noise suppression in elastic reverse time migration (RTM). Compared with the conventional decoupled elastic wave equation, the first-order velocity-dilatation-rotation equations enable more accurate computation of propagation directions for P and S waves. Moreover, compared with the Poynting vector, the optical flow vector signifies the wavefield propagation directions more accurately. To effectively enhance the accuracy of elastic wave imaging, we develop an elastic RTM based on first-order velocity-dilatation-rotation equations using the optical flow vector. Numerical tests illustrate that our method, with or without noise, can better eliminate migration artifacts and improve the imaging accuracy of the elastic RTM than conventional methods, achieving more accurate wavefield decomposition and superior S-wave polarity reversal correction.

Elastic full-waveform inversion using tools of neural networks

In this paper, we investigate the full-waveform inversion (FWI) for elastic wave equation as training a neural network. The forward modeling of the elastic wave equation in the time domain by the staggered-grid difference schemes can be reformulated as a process of a recurrent neural network (RNN). As a result, the FWI problem is equivalent to neural network training, and the parameter of RNN coincides with the model parameter of inversion. Furthermore, a variety of stochastic optimizers including Adgrad, RMSprop, Adam, Nadam and Admax in neural networks can be applied in the training process. The gradient of the objective function with the model parameters is computed by the technique of automatic differentiation instead of the adjoint-state method in the traditional FWI. A new objective function of FWI is also proposed. Compared to the traditional FWI methods, the developed FWI using tools of neural networks has a relatively good robustness. Numerical computations and comparisons with Marmousi model for two and three parameters simultaneous inversion are completed. The results show that the algorithms except Adgrad can yield good inversion results. The FWI framework developed in this paper has potential applications for other complex partial equations.

Global existence of nonlinear elastic waves: Non-hyperelastic case

For the Cauchy problem of nonlinear elastic wave equations for three-dimensional isotropic, homogeneous and hyperelastic materials with null condition, global existence of classical solutions with small initial data was proved in Agemi (2000) [1] and Sideris (2000) [17] independently. How to treat the more general non-hyperelastic case is suggested as an open problem in Agemi (2005) [2]. The aim of this paper is to resolve this problem.

Hamiltonian/Stroh formalism for reversible poroelasticity (and thermoelasticity)

Stroh’s sextic formalism represents the equilibrium equations of anisotropic elasticity in a particularly attractive form, that is most suitable for studying interface-dominated multilayered solids, composite materials and time-harmonic problems. Taking advantage of the fact that the Stroh formalism really amounts to the canonical form of the equations in the Hamiltonian sense, the case of Biot’s reversible (i.e. no fluid dissipation) poroelasticity is here addressed, in the absence of a fluid pressure gradient. This framework is the same as thermoelasticity of perfect conductors. Two Hamiltonian formulations are developed: the first describes both the solid and the fluid phases and it exhibits, besides energy conservation, momentum conservation, as a result of pressure uniformity (perfectly drained conditions). The second is restricted to the solid skeleton and perfectly parallels anisotropic elasticity, where the Stroh matrices refer to the effective stress tensor. The case of weak fluid–solid coupling is also considered and it produces a perturbation from anisotropic elasticity with the same structure as incompressibility, although in an “opposing” manner. This comparison suggests that the incompressibility limit introduced by Biot should be revised. The energy conservation integral and the edge impedance matrix are also illustrated.

A high-order, localized-artificial-diffusivity method for Eulerian simulation of multi-material elastic-plastic deformation with strain hardening

A high-order method for Eulerian simulation of material undergoing large elastic–plastic deformation is developed. Thermodynamically consistent hyperelastic constitutive relations are assumed, facilitating the treatment of solids, liquids, and gases in a unified manner. The method enables the simulation of multi-material interactions using a diffuse interface approach. Numerical capturing of material interfaces, shock waves, contact surfaces, and elastic-plastic strain discontinuities using high-order compact-difference schemes is assisted by Localized Artificial Diffusivity (LAD). In the new setting involving elastic–plastic deformation, the previously established terms for the artificial properties are verified to effectively regularize normal shocks. Additional LAD terms are introduced to the elastic and plastic kinematic equations to regularize shear shocks and other strain discontinuities, improving solution stability. Other important features of the method that improve robustness include the numerical treatment of compatibility terms in the kinematic equations, and the treatment of rotation. Particular emphasis is focused toward new advancements of the methods for plastic-deformation integration and the associated strain hardening of the material, including rate-dependent plasticity. The method is demonstrated on a variety of test problems, including 1-D impacts, a variant of the Shu-Osher problem, a Taylor impact, and a Richtmyer-Meshkov instability between two elastic–plastic solids with strain hardening.

Experimental Investigation of Low-Frequency Distributed Acoustic Sensor Responses to Two Parallel Propagating Fractures.

Low-frequency distributed acoustic sensing (LF-DAS) is a diagnostic tool for hydraulic fracture propagation with far-field monitoring using fiber optic sensors. LF-DAS senses strain rate variation caused by stress field change due to fracture propagation. Fiber optic sensors are installed in the monitoring wells in the vicinity of a fractured well. From the strain responses, fracture propagation can be evaluated. To understand subsurface conditions with multiple propagating fractures, a laboratory-scale hydraulic fracture experiment was performed simulating the LF-DAS response to fracture propagation with embedded distributed optical fiber strain sensors under these conditions. The experiment was performed using a transparent cube of epoxy with two parallel radial initial flaws centered in the cube. Fluid was injected into the sample to generate fractures along the initial flaws. The experiment used distributed high-definition fiber optic strain sensors with tight spatial resolutions. The sensors were embedded at two different locations on opposite sides of the initial flaws, serving as observation/monitoring locations. We also employed finite element modeling to numerically solve the linear elastic equations of equilibrium continuity and stress-strain relationships. The measured strains from the experiment were compared to simulation results from the finite element model. The experimentally derived strain and strain-rate waterfall plots from this study show the responses to both fractures propagating, while the fracture at the lower position took most of the fluid during the experiment. Interestingly, a fracture first began propagating from the upper flaw of the two flaws, but once the lower fracture was initiated, it grew much faster than the upper fracture. Both fibers were intercepted by the lower fracture, further verifying the strain signature as a fracture is approaching and intersecting an offset fiber.

A reduced-dimension method for unknown Crank-Nicolson finite element solution coefficient vectors of elastic wave equation with singular source term

Herein, we mainly develop a new reduced-dimension method for the unknown finite element (FE) solution coefficient vectors to the elastic wave equation containing the known singular source term, the unknown displacement vector, and the unknown stress tensor matrix. For this end, we need first to develop the Crank-Nicolson (CN) FE (CNFE) method with unconditional stability, unconditional convergence, and second-order time accuracy for the elastic wave equation and analyze the existence, unconditional stability, and errors of the CNFE solutions. After that we employ a proper orthogonal decomposition to lower the dimensionality of the unknown FE solution coefficient vectors in the CNFE method for the elastic wave equation and to design a new reduced-dimension extrapolated CNFE (RDECNFE) method. Subsequently, we employ matrix analysis to discuss the existence, unconditional stability, unconditional convergence, and errors of the RDECNFE solutions. Finally, we use two numerical examples to show the superiority of the RDECNFE method and our theoretical results. It is also shown that the RDECNFE method is feasible and effective for solving the elastic wave equation.

An inf-sup approach to C0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_0$$\\end{document}-semigroup generation for an interactive composite structure-Stokes PDE dynamics

In this work, we investigate the existence and uniqueness properties of a composite structure (multilayered)–fluid interaction PDE system which arises in multi-physics problems, and particularly in biofluidic applications related to the mammalian blood transportation process. The PDE system under consideration consists of the interactive coupling of 3D Stokes flow and 3D elastic dynamics which gives rise to an additional 2D elastic equation on the boundary interface between these 3D PDE systems. By means of a nonstandard mixed variational formulation, we show that the PDE system generates a C0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_0$$\\end{document}-semigroup on the associated finite energy space of data. In this work, the presence of the pressure term in the 3D Stokes equation adds a great challenge to our analysis. To overcome this difficulty, we follow a methodology which is based on the necessarily non-Leray-based elimination of the associated pressure term, via appropriate nonlocal operators. Moreover, while we express the fluid solution variable via decoupling of the Stokes equation, we construct the elastic solution variables by solving a mixed variational formulation via a Babuska–Brezzi approach.