Elastic Equation Research Articles

Stochastic electromechanical bidomain model *

We analyse a system of nonlinear stochastic partial differential equations (SPDEs) of mixed elliptic-parabolic type that models the propagation of electric signals and their effect on the deformation of cardiac tissue. The system governs the dynamics of ionic quantities, intra and extra-cellular potentials, and linearised elasticity equations. We introduce a framework called the active strain decomposition, which factors the material gradient of deformation into an active (electrophysiology-dependent) part and an elastic (passive) part, to capture the coupling between muscle contraction, biochemical reactions, and electric activity. Under the assumption of linearised elastic behaviour and a truncation of the nonlinear diffusivities, we propose a stochastic electromechanical bidomain model, and establish the existence of weak solutions for this model. To prove existence through the convergence of approximate solutions, we employ a stochastic compactness method in tandem with an auxiliary non-degenerate system and the Faedo–Galerkin method. We utilise a stochastic adaptation of de Rham’s theorem to deduce the weak convergence of the pressure approximations.

Effect of Indentation at Rolling Body Surfaces on the Film Formation in Elastohydrodynamic Lubrication Contact under Dynamic Loads

Vibration is one of the phenomena that can occur during the operation of roller bearing which can lead to dynamic loads being subjected to the contact and thus, affect the film formation in elastohydrodynamic (EHD) contacts. On the other hand, surface textures such as artificial indentations and grooves greatly affect the film formation as well as the corresponding pressure distribution in the contact area, since the depths of the textures are usually much greater than the film thickness in the contact. This paper investigates the influence of an artificial indentation located on the rolling element, which passes through a point contact under dynamic loads by means of numerical simulations. Solutions to the Reynolds equation are performed and calculations of the elastic deformation equation, force balance equation and lubricant properties equations to show the effects of both indentation passing through the contact and a sinusoidal dynamic load on the film thickness as well as pressure profile of EHD point contact. Moreover, the effect of frequency and amplitude excitation of the dynamic load on the film formation was investigated. The results revealed that the artificial indentation under sinusoidal dynamic load of EHD point contact induced a significant effect on the thickness of the film formation and pressure distribution.

Space‐Time Monitoring of Seafloor Velocity Changes Using Seismic Ambient Noise

AbstractWe use seismic ambient noise recorded by dense ocean bottom nodes (OBNs) in the Gorgon gas field, Western Australia, to compute time‐lapse seafloor models of shear‐wave velocity. The extracted hourly cross‐correlation (CC) functions in the frequency band 0.1–1 Hz contain mainly Scholte waves with very high signal‐to‐noise ratio. We observe temporal velocity variations (dv/v) at the order of 0.1% with a peak velocity change of 0.8% averaged from all station pairs, from the conventional time‐lapse analysis with the assumption of a spatially homogeneous dv/v. With a high‐resolution reference (baseline) model from full waveform inversion of Scholte waves, we present an elastic wave equation based double‐difference inversion (EW‐DD) method, using arrival time differences between the reference and time‐lapsed Scholte waves, for mapping temporally varying dv/v in the heterogeneous subsurface. The time‐lapse velocity models reveal increasing/decreasing patterns of shear‐wave velocity in agreement with those from the conventional analysis. The velocity variation exhibits a ∼24‐hr cycling pattern, which appears to be inversely correlated with the diurnal variations in sea level height, possibly associated with dilatant effects for porous, low‐velocity shallow seafloor and rising pore pressure with higher sea level. This study demonstrates the feasibility of using dense passive seismic surveys and wave‐equation time‐lapse inversion for quantitative monitoring of subsurface property changes in the horizontal and depth domain.

Study on the structure, elasticity, and thermal conductivity of orthocarbonate Sr2CO4

Orthocarbonate Sr2CO4 is a recently discovered Sr-carbonate that plays a crucial role in understanding the global long-term carbon cycle. In this work, the structure, equation of state, elasticity, and thermal conductivity of Sr2CO4 were studied by first-principles calculations. The results show that the surfaces of the SrO9 and SrO11 polyhedrons, which make up Sr2CO4, are composed of triangles and quadrangles. By comparison with other alkaline earth metal orthocarbonates, SrCO3 polymorphs, and the Preliminary Reference Earth Model (PREM), Sr2CO4 is found to be a carbonate with high density, low wave velocity, low anisotropy, and low thermal conductivity in the entire Earth’s mantle. This study may help to understand the high-pressure behavior of alkaline earth metal orthocarbonates in the Earth’s interior.

Deformation of rectangular orthotropic laminate resting on thin elastic foundation with nonlinear change of displacements over thickness under edge loading

Deformation of rectangular orthotropic laminate resting on a thin elastic orthotropic foundation with fully fixed bottom plane is studied in this paper. The laminate is subjected to an in-plane uniaxial load uniformly distributed over one of its edges. Models of joint deformation of the laminate and foundation are created based on the equations of plane elasticity for an orthotropic body. The applicability of such equations is justified by the uniformity of edge loading. A laminate model allows for the transverse shear strain, whereas a model created for the foundation considers a nonlinear character of variation of the transverse and axial displacements. A governing system of differential equationsmodelling the joined deformation of the laminate and foundation is derived using static and kinematic contact conditions. The corresponding boundary value problem is solved using the sweep method. Based on this solution, effects of laminate and foundation thicknesses, and elasticity characteristics on the stress–strain state of the jointed structure are investigated. The model created and the method of its analysis enable the strong oscillating behaviour of stresses and strains to be studied. The results obtained can be utilized for verification of solutions of similar problems found by other numerical methods, including finite-element analyses.

Comparative Analysis and Unified Derivation of Reissner’s Equations For 2D Bending of Thick Plates and Timoshenko’s Equations for Bending of Beams

Currently, calculations of flexural deformations of lithospheric plates are carried out on the basis of Kirchhoff’s theory of bending of thin plates formulated about 170 years ago. The paper examines the possibility of refining these calculations based on the theory of bending of thick plates by Timoshenko and Reissner. A new unified derivation is presented of the Timoshenko equations for 2D bending of beams and the Reissner equations for bending of slabs by direct transformation of the general elasticity equations with a simple approximate replacement of power cubic functions with effective linear ones. This derivation offers a simpler and more detailed understanding of the difference between the equations and the meaning of the simplifications made in these theories. By comparing the analytical solutions of the Timoshenko and Reissner equations with the existing test analytical solutions of the exact elasticity equations, quantitative estimates of the accuracy of these theories are presented.

Electromechanical coupling characteristics of multilayered piezoelectric quasicrystal plates in an elastic medium

AbstractQuasicrystals (QCs) have attracted tremendous attention of researchers for their unusual properties. In this paper, an exact electric‐elastic solution of the simply supported and multilayered three‐dimensional (3D) cubic piezoelectric quasicrystal (PQC) nanoplate with the nonlocal effect is derived. Based on the basic elasticity equation of 3D QCs, we construct the linear eigenvalue system in terms of the pseudo‐Stroh formalism, from which the general solutions of the extended displacements and stresses in any homogeneous layer can be obtained. The two‐parameter foundation model is utilized to simulate the interaction between the nanoplate and elastic medium. The propagator matrices are employed to connect the field variables at the upper interface to those at the lower interface of each layer. Based on the boundary conditions of the upper and lower surfaces of the laminate and foundation model, the solutions are employed to derive from the global propagator matrix. Compared with the conventional propagator matrix method, a new propagator method is reestablished to deal with numerical instabilities of the case of large aspect ratio and high‐order frequencies for QC laminates. Finally, typical numerical examples are presented to illustrate the influence of nonlocal parameters and elastic medium coefficients on phonon, phason, and electric variables of 3D PQC nanoplates.

Acoustic full waveform inversion with elastic wave equation forward modeling

Full-waveform inversion (FWI) is a high-resolution velocity inversion method. Currently, acoustic full waveform inversion (AFWI) has achieved promising results in marine data inversion, but it still faces significant challenges in land data inversion. One of the reasons for the failure of AFWI in land data inversion is that the acoustic wave equation does not account for the elastic effects of seismic waves, such as surface waves and converted waves. This results in errors between the synthesized acoustic waveform and the observed elastic waveform, particularly in the amplitude and phase of the P-wave. To reduce the error caused by the elastic effect, we have improved the inversion process of AFWI. During the forward modeling stage, we replace the original acoustic wave equation with the elastic wave equation to simulate the wavefield. This allows us to obtain synthetic waveforms that incorporate elasticity, thereby reducing the corresponding error. When computing model gradients, we still base on the acoustic wave equation and only update the P-wave velocity model. This approach helps maintain a lower computational cost. We tested the method on the Marmousi model and field land data, and the results demonstrate that the method successfully reduces the error caused by the elastic effect, thereby improving the clarity of the inversion results.

A finite element framework for fluid–structure interaction of turbulent cavitating flows with flexible structures

We present a finite element framework for the numerical prediction of cavitating turbulent flows interacting with flexible structures. The vapor–fluid phases are captured through a homogeneous mixture model, with a scalar transport equation governing the spatio-temporal evolution of cavitation dynamics. High-density gradients in the two-phase cavitating flow motivate the use of a positivity-preserving Petrov–Galerkin stabilization method in the variational framework. A mass transfer source term introduces local compressibility effects arising as a consequence of phase change. The turbulent fluid flow is modeled through a dynamic subgrid-scale method for large eddy simulations. The flexible structure is represented by a set of eigenmodes, obtained through a modal decomposition of the linear elasticity equations. While a partitioned iterative approach is adopted to couple the structural dynamics and cavitating fluid flow, the deforming flow domain is described by an arbitrary Lagrangian–Eulerian frame of reference. We establish the fidelity of the proposed framework by comparing it against experimental and numerical studies for both rigid and flexible hydrofoils in cavitating flows. Under unstable partial cavitating conditions, we identify specific vortical structures leading to cloud cavity collapse. We further explore features of cavitating flow past a rigid body such as re-entrant jet and turbulence-cavity interactions during cloud cavity collapse. Based on the validation study conducted over a flexible NACA66 rectangular hydrofoil, we elucidate the role of cavity and vortex shedding in governing the structural dynamics. Subsequently, we identify a broad spectrum frequency band whose central peak does not correlate to the frequency content of the cavitation dynamics or the natural frequencies of the structure, indicating the induction of unsteady flow patterns around the hydrofoil. Finally, we discuss the coupled fluid–structure dynamics during a cavitation cycle and the underlying mechanism associated with the promotion and mitigation of cavitation.

Polytopal discontinuous Galerkin discretization of brain multiphysics flow dynamics

A comprehensive mathematical model of the multiphysics flow of blood and Cerebrospinal Fluid (CSF) in the brain can be expressed as the coupling of a poromechanics system and Stokes' equations: the first describes fluids filtration through the cerebral tissue and the tissue's elastic response, while the latter models the flow of the CSF in the brain ventricles. This model describes the functioning of the brain's waste clearance mechanism, which has been recently discovered to play an essential role in the progress of neurodegenerative diseases. To model the interactions between different scales in the porous medium, we propose a physically consistent coupling between Multi-compartment Poroelasticity (MPE) equations and Stokes' equations. In this work, we introduce a numerical scheme for the discretization of such coupled MPE-Stokes system, employing a high-order discontinuous Galerkin method on polytopal grids to efficiently account for the geometric complexity of the domain. We analyze the stability and convergence of the space semidiscretized formulation, we prove a-priori error estimates, and we present a temporal discretization based on a combination of Newmark's β-method for the elastic wave equation and the θ-method for the other equations of the model. Numerical simulations carried out on test cases with manufactured solutions validate the theoretical error estimates. We also present numerical results on a two-dimensional slice of a patient-specific brain geometry reconstructed from diagnostic images, to test in practice the advantages of the proposed approach.

Reconstruction of elastic inclusions in layered medium

We consider the recovery of small inclusions in a two-layered and an arbitrary multi-layered medium for the elastic equation, respectively. We use layer potential techniques and asymptotic analysis to obtain asymptotic expansions of the perturbed elastic field in a two-layered and an arbitrary multi-layered medium, respectively. Furthermore, we show the uniqueness of the recovery of the locations and Lamé constants of small inclusions through a single measurement.

Multi-method examination of contact mechanics in orthotropic layers under gravity

Analyzing the behavior of intelligent unconventional materials under diverse contact scenarios, in comparison to conventional materials, is a critical step in the initial design of engineering systems. This paper presents the development of analytical and numerical approaches for analyzing contact mechanics in a system comprising an orthotropic layer resting on a rigid foundation. Parametric analyses include the consideration of cylindrical and flat punch profiles. Analytical formulation utilizes integral transform methods to convert planar elasticity equations into a Cauchy-type singular integral equation of the second kind. A detailed description of the solution technique for the integral equation is provided, encompassing both the analytical formulation and the required discretization for obtaining the solution. Subsequently, a finite element approach (FEA) is employed to approximate contact bodies using a collection of finite elements, while contact boundaries are approximated by utilizing a set of polygons. Alternatively, the problem is addressed using the Multilayer Perceptron approach, a form of artificial neural network frequently applied in diverse machine learning applications, including scenarios involving contact problems. Finally, the resolution to the problem is achieved by employing the multilayer perceptron (MLP) method. The study yields determinations of contact stresses under the punch, the critical separation load resulting in the detachment of the orthotropic layer from the rigid foundation, and the corresponding separation distance. This analysis considers a range of dimensionless parameters and explored the behavior of diverse orthotropic materials. Comparisons between the results of the analytical method and computations from FEA and MLP reveal the exceptional accuracy attained by all three approaches. As the pioneer study employing three distinct approaches to examine continuous contact mechanics within an orthotropic layer under the influence of gravity, this research constitutes a valuable point of reference for upcoming scholars in the field.

A global space–time Trefftz DG scheme for the time-dependent isotropic elastic wave equations

In this paper we are concerned with Trefftz discretizations of the time-dependent linear elastic wave equation in arbitrary space dimensional domains Ω⊂Rd(d∈N). We propose a Trefftz discontinuous Galerkin method, construct Trefftz basis functions and prove that the corresponding approximate solutions possess optimal-order error estimates with respect to the meshwidth. A space–time domain decomposition preconditioner is introduced for the system generated by the proposed variational formula. Besides, we propose the global Trefftz DG method combined with local DG methods to solve the nonhomogeneous model. The numerical results verify the validity of the theoretical results, and show that the resulting approximate solutions possess high accuracy, and the proposed preconditioner is effective.

Existence Theorems for Parameter Dependent Weakly Continuous Operators with Applications

The paper presents results on the solvability and parameter dependence for problems driven by weakly continuous potential operators with continuously differentiable and coercive potential. We provide a parametric version on the existence result to nonlinear equations involving coercive and weakly continuous operators. Applications address a variant of elastic beam equation.

Gradient preserving Operator Inference: Data-driven reduced-order models for equations with gradient structure

Hamiltonian Operator Inference has been introduced in Sharma et al. (2022) to learn structure-preserving reduced-order models (ROMs) for Hamiltonian systems. This approach constructs a low-dimensional model using only data and knowledge of the Hamiltonian function. Such ROMs can keep the intrinsic structure of the system, allowing them to capture the physics described by the governing equations. In this work, we extend this approach to more general systems that are either conservative or dissipative in energy, and which possess a gradient structure. We derive the optimization problems for inferring structure-preserving ROMs that preserve the gradient structure. We further derive an a priori error estimate for the reduced-order approximation. To test the algorithms, we consider semi-discretized partial differential equations with gradient structure, such as the parameterized wave and Korteweg–de-Vries equations, and equations of three-dimensional linear elasticity in the conservative case and the one- and two-dimensional Allen-Cahn equations in the dissipative case. The numerical results illustrate the accuracy, structure-preservation properties, and predictive capabilities of the gradient-preserving Operator Inference ROMs.

Shear wave anisotropy response characteristics of laminated shale and its correction methods

Laminated shale exhibits strong anisotropic characteristics in horizontal wells, significantly affecting the application of shear wave travel time data. Using equivalent medium theory and the elastic wave equation, we constructed a physical model for laminated shale, simulating the impact of lamination angle and lamination density on shear wave velocity and shear wave anisotropy coefficient. We established the relationship between the shear wave anisotropy coefficient and the cosine of the lamination angle. The simulation results indicate that shear wave velocity decreases with an increase in lamination angle and density, while the shear wave anisotropy coefficient gradually decreases with an increase in lamination angle and initially increases and then decreases with an increase in lamination density. Moreover, it exhibits a power exponential relationship with the cosine of the lamination angle. After correction, the minimum horizontal principal stress and fracture pressure calculated from shear wave data show an average relative error reduction of 6.72% and 7.77%, respectively, compared to measured data. This demonstrates that the correction method effectively eliminates the impact of laminated shale anisotropy on shear waves.

A quantum computing concept for 1-D elastic wave simulation with exponential speedup

SUMMARY Quantum computing has attracted considerable attention in recent years because it promises speedups that conventional supercomputers cannot offer, at least for some applications. Though existing quantum computers (QCs) are, in most cases, still too small to solve significant problems, their future impact on domain sciences is already being explored now. Within this context, we present a quantum computing concept for 1-D elastic wave propagation in heterogeneous media with two components: a theoretical formulation and an implementation on a real QC. The method rests on a finite-difference approximation, followed by a sparsity-preserving transformation of the discrete elastic wave equation to a Schrödinger equation, which can be simulated directly on a gate-based QC. An implementation on an error-free quantum simulator verifies our approach and forms the basis of numerical experiments with small problems on the real QC IBM Brisbane. The latter produce simulation results that qualitatively agree with the error-free version but are contaminated by quantum decoherence and noise effects. Complementing the discrete transformation to the Schrödinger equation by a continuous version allows the replacement of finite differences by other spatial discretization schemes, such as the spectral-element method. Anticipating the emergence of error-corrected quantum chips, we analyse the computational complexity of the best quantum simulation algorithms for future QCs. This analysis suggests that our quantum computing approach may lead to wavefield simulations that run exponentially faster than simulations on classical computers.

Angular and modal equipartitioning of elastic waves in scattering media: An illustration based on energy transport.

We illustrate the angular and modal equipartitioning of elastic waves in scattering media using two-dimensional elastic radiative transfer equations . To solve these equations, we decompose the P and S specific intensities into direct and scattered components. We handle the direct component analytically, and derive integral equations for the scattered components of the P and S specific intensities. We construct a time-stepping algorithm with which we evolve the scattered components of the specific intensities numerically in time. We handle the advection of P and S energy analytically at the computational grid points and use numerical interpolation to deal with advection terms which do not lie on the grid points. We test this algorithm for a pure P source and a double couple, which radiates both P and S energy. We compare our numerical solutions against known approximations and find good agreement. We use this algorithm to illustrate the local behavior of equipartitioning over wave modes and angular directions. We find that both types of equipartitioning are a function of space and time, depending on the extent of scattering. This local behavior must be taken into account when studying diffusion and equipartitioning of elastic waves.

Numerical Simulation of Proppant Transport in Transverse Fractures of Horizontal Wells

Proppant transport and distribution law in hydraulic fractures has important theoretical and field guidance significance for the optimization design of hydraulic fracturing schemes and accurate production prediction. Many studies aim to understand proppant transportation in complex fracture systems. Few studies, however, have addressed the flow path mechanism between the transverse fracture and horizontal well, which is often neglected in practical design. In this paper, a series of mathematical equations, including the rock elastic deformation equation, fracturing fluid continuity equation, fracturing fluid flow equation, and proppant continuity equation for the proppant transport, were established for the transverse fracture of a horizontal well, while the finite element method was used for the solution. Moreover, the two-dimensional radial flow was considered in the proppant transport modeling. The results show that proppant breakage, embedding, and particle migration are harmful to fracture conductivity. The proppant concentration and fracture wall roughness effect can slow down the proppant settling rate, but at the same time, it can also block the horizontal transportation of the proppant and shorten the effective proppant seam length. Increasing the fracturing fluid viscosity and construction displacement, reducing the proppant density and particle size, and adopting appropriate sanding procedures can all lead to better proppant placement and, thus, better fracturing and remodeling results. This paper can serve as a reference for the future study of proppant design for horizontal wells.

Petrophysical parameter estimation using Biot-poroelastic full-waveform inversion

SUMMARY The increasing need for petrophysical parameter inversion has spurred the advancement of full-waveform inversion (FWI). This study introduces a new inversion method that can directly estimate petrophysical parameters from seismic waveform data. Specifically, we modified the regular elastic wave equation by including Biot poroelastic parameters and designed an inversion workflow based on the framework of elastic FWI. The newly introduced inversion algorithms have demonstrated their effectiveness and accuracy using the Marmousi model and a synthetic model related to the gas hydrate deposits based on the Shenhu area of the South China Sea. Our work demonstrates the feasibility of directly inverting petrophysical parameters and assessing if a deposit contains gas hydrate.