Anisotropic Equation Research Articles

Optimal control of anisotropic Allen-Cahn equations

This paper aims at solving an optimal control problem governed by an anisotropic Allen-Cahn equation numerically. Therefor we first prove the Frechet differentiability of an in time discretized parabolic control problem under certain assumptions on the involved quasilinearity and formulate the first order necessary conditions. As a next step, since the anisotropies are in general not smooth enough, the convergence behavior of the optimal controls is studied for a sequence of (smooth) approximations of the former quasilinear term. In addition the simultaneous limit in the approximation and the time step size is considered. For a class covering a large variety of anisotropies we introduce a certain regularization and show the previously formulated requirements. Finally, a trust region Newton solver is applied to various anisotropies and configurations, and numerical evidence for mesh independent behavior and convergence with respect to regularization is presented.

INITIAL-BOUNDARY VALUE PROBLEM FOR HIGHER-ORDERS NONLINEAR PARABOLIC EQUATIONS WITH VARIABLE EXPONENTS OF THE NONLINEARITY IN UNBOUNDED DOMAINS WITHOUT CONDITIONS AT INFINITY

Initial-boundary value problems for parabolic equations in unbounded domains with respect to the spatial variables were studied by many authors. As is well known, to guarantee the uniqueness of the solution of the initial-boundary value problems for linear and some nonlinear parabolic equations in unbounded domains we need some restrictions on solution's behavior as $|x|\to +\infty$ (for example, solution's growth restriction as $|x|\to +\infty$, or belonging of solution to some functional spaces). Note that we need some restrictions on the data-in behavior as $|x|\to +\infty$ to solvability of the initial-boundary value problems for parabolic equations considered above. However, there are nonlinear parabolic equations for which the corresponding initial-boundary value problems are unique solvable without any conditions at infinity. Nonlinear differential equations with variable exponents of the nonlinearity appear as mathematical models in various physical processes. In particular, these equations describe electroreological substance flows, image recovering processes, electric current in the conductor with changing temperature field. Nonlinear differential equations with variable exponents of the nonlinearity were intensively studied in many works. The corresponding generalizations of Lebesgue and Sobolev spaces were used in these investigations. In this paper we prove the unique solvability of the initial--boundary value problem without conditions at infinity for some of the higher-orders anisotropic parabolic equations with variable exponents of the nonlinearity. An a priori estimate of the generalized solutions of this problem was also obtained.

Anisotropic Poroelasticity and AVAZ Inversion for in Situ Stress Estimate in Fractured Shale-Gas Reservoirs

Knowledge of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in situ</i> stress is of great significance for hydraulic fracture stimulating of unconventional reservoirs. Quantitative estimation of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in situ</i> stress, especially in the far field, is a major challenge. This research mainly focuses on developing a novel Bayesian AVAZ inversion approach to estimate <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in situ</i> stress of fractured shale-gas reservoirs with horizontal transversely isotropic (HTI) symmetry. Using the generalized Hooke’s law and Schoenberg’s linear slip model, we first deduce the anisotropic horizontal stress equation in an HTI medium formed by a single set of vertically aligned fractures embedded in an isotropic background rock. Based on the critical porosity model, we then obtain the saturated stiffness matrix with vertical effective stress-sensitive parameters and dry fracture weaknesses using the relationship between vertical effective stress and porosity. Combining the scattering function and the perturbed stiffness matrix, we deduce a linearized PP-wave reflection coefficient as a function of fluid bulk modulus, vertical effective stress-sensitive parameter, dry-rock P- and S-wave moduli, density, and fracture density. Finally, we propose a novel Bayesian amplitude variation with azimuth (AVAZ) inversion constrained by the Cauchy regularization and low-frequency regularization to estimate these model parameters, which are further used to calculate <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in situ</i> stress. The analysis of synthetic data demonstrates that <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in situ</i> stress can be reasonably estimated even with moderate noise. A field data set test reveals that the inversion results agree well with the well log interpretation, and the proposed approach can generate meaningful results that are useful for seismic identification of potential fracturing barriers during fracturing stimulation of shale-gas reservoirs.

Initial mixed-boundary value problem for anisotropic fractional degenerate parabolic equations

Initial mixed-boundary value problem for anisotropic fractional degenerate parabolic equations

Spheroidal magnetic stars rotating in vacuum

Context. Gravity shapes stars to become almost spherical because of the isotropic nature of gravitational attraction in Newton’s theory. However, several mechanisms break this isotropy, such as their rotation generating a centrifugal force, magnetic pressure, or anisotropic equations of state. The stellar surface therefore slightly or significantly deviates from a sphere depending on the strength of these anisotropic perturbations. Aims. In this paper, we compute analytical and numerical solutions of the electromagnetic field produced by a rotating spheroidal star of oblate or prolate nature. This study is particularly relevant for millisecond pulsars for which strong deformations are produced by rotation or a strong magnetic field, leading to indirect observational signatures of the polar cap thermal X-ray emission. Methods. First we solve the time harmonic Maxwell equations in vacuum by using oblate and prolate spheroidal coordinates adapted to the stellar boundary conditions. The solutions are expanded in series of radial and angular spheroidal wave functions. Particular emphasis is put on the magnetic dipole radiation. Second, we compute approximate solutions by integrating the time-dependent Maxwell equations in spheroidal coordinates numerically. Results. We show that the spin-down luminosity corrections compared to a perfect sphere are, to leading order, given by terms involving (a/rL)2 and (a/R)2 where a is the stellar oblateness or prolateness, R the smallest star radius, and rL the light-cylinder radius. The corresponding perturbations in the electromagnetic field are only perceptible close to the surface, deforming the polar cap rims. At large distances r ≫ a, the solution tends asymptotically to the perfect spherical case of a rotating dipole.

Bounded nonnegative weak solutions to anisotropic parabolic double phase problems with variable growth

We study the homogeneous Dirichlet problem for a class of nonlinear anisotropic parabolic double phase equations with nonstandard growth conditions. We prove the existence of a unique nonnegative weak solution in suitable Orlicz-Sobolev spaces, and derive the global boundedness of the weak solution.

Multi-stencils fast marching method for factored eikonal equations with quadratic anisotropy

In this study, we propose a multi-stencils fast marching method to solve the factored eikonal equations for media with quadratic anisotropy, also known as Riemannian anisotropy. Compared with the conventional fast marching method, the new method has an improved accuracy in tracing moving wavefronts via computing the solution at each grid point along several stencils. To deal with the source singularity, we solve for the viscosity solution iteratively by factorize the unknown traveltime function into a function that captures the singularity and a multiplicative factor which is differentiable at the source location. By leveraging the strengths of multi-stencils and factorisation, the newly developed algorithm is highly accurate in solving anisotropic eikonal equations. Through a series of numerical examples, we explicitly show that the algorithm (1) has an obvious first (second) order convergence rate with the first (second) order finite-difference scheme, and (2) is unconditionally stable as long as the initial time table around the source is specified. Finally, we design an eikonal-based anisotropic ray tracing technique. We test the anisotropic ray tracing technique on both radial and azimuthal anisotropy cases and the numerical results demonstrate that the usage of multi-stencils scheme increase the accuracy of the anisotropic ray paths especially for azimuthal anisotropy case. This high-accurate eikonal solver and the associated ray tracing technique can be used in high-resolution seismic anisotropic tomography and other researches in the future.

Prediction of an unusual trigonal phase of superconducting LaH10 stable at high pressures

Based on evolutionary crystal structure searches in combination with ab initio calculations, we predict an unusual structural phase of the superconducting LaH$_{10}$ that is stable from about 250 GPa to 425 GPa pressure. This new phase belongs to a trigonal $R\bar{3}m$ crystal lattice with an atypical cell angle, $\alpha_{rhom}$ $\sim$ 24.56$^{\circ}$. We find that the new structure contains three units of LaH$_{10}$ in its primitive cell, unlike the previously known trigonal phase, where primitive cell contains only one LaH$_{10}$ unit. In this phase, a 32-H atoms cage encapsulates La atoms, analogous to the lower pressure face centred cubic phase. However, the hydrogen cages of the trigonal phase consist of quadrilaterals and hexagons, in contrast to the cubic phase, that exhibits squares and regular hexagons. Surprisingly, the shortest H-H distance in the new phase is shorter than that of the lower pressure cubic phase and of atomic hydrogen metal. We find a structural phase transition from trigonal to hexagonal at 425 GPa, where the hexagonal crystal lattice coincides with earlier predictions. Solving the anisotropic Migdal-Eliashberg equations we obtain that the predicted trigonal phase (for standard values of the Coulomb pseudopotential) is expected to become superconducting at a critical temperature of about 175 K, which is less than $T_c \sim$250 K measured for cubic LaH$_{10}$.

The primitive equations approximation of the anisotropic horizontally viscous 3D Navier-Stokes equations

In this paper, we provide rigorous justification of the hydrostatic approximation and the derivation of primitive equations as the small aspect ratio limit of the incompressible three-dimensional Navier-Stokes equations in the anisotropic horizontal viscosity regime. Setting ε>0 to be the small aspect ratio of the vertical to the horizontal scales of the domain, we investigate the case when the horizontal and vertical viscosities in the incompressible three-dimensional Navier-Stokes equations are of orders O(1) and O(εα), respectively, with α>2, for which the limiting system is the primitive equations with only horizontal viscosity as ε tends to zero. In particular we show that for “well prepared” initial data the solutions of the scaled incompressible three-dimensional Navier-Stokes equations converge strongly, in any finite interval of time, to the corresponding solutions of the anisotropic primitive equations with only horizontal viscosities, as ε tends to zero, and that the convergence rate is of order O(εβ2), where β=min⁡{α−2,2}. Note that this result is different from the case α=2 studied in Li and Titi (2019) [38], where the limiting system is the primitive equations with full viscosities and the convergence is globally in time and its rate of order O(ε).

Automatic Recognition of Financial Instruments Based on Anisotropic Partial Differential Equations

In this paper, anisotropic partial differential equations are used to conduct an indepth study and analysis of automatic recognition of financial bills. Firstly, it obtains the invoices of the group enterprise, uses scanning technology and related image recognition technology to capture, process, compress and slice the paper bill content, and then carries out data identification and verification of the image. It classifies the obtained electronic data information into bills, converts it into electronic information related to bills according to the corresponding categories of the bill template, and stores it in the bill table of the database to achieve the management operation of formatted electronic files. After categorizing the bills according to the electronic information of bills to match the business scenarios, financial journal vouchers can be generated according to the preconfigured voucher templates of the corresponding business scenarios, and the financial journal vouchers are converted into voucher messages using XML technology. Finally, we use agent technology to design middleware for heterogeneous financial systems to realize the function of communicating voucher messages to each other in different business systems. The system automatically extracts the key information of invoices through OCR technology and performs real-time verification and cyclical feedback to the verification results to the suppliers. The system has realized the intelligent management of the power company’s VAT invoices, thus greatly enhancing the efficiency of VAT invoice verification and settlement. The automatic tax invoice recognition system adopts a network structured tax invoice recognition model, which eliminates the cumbersome steps of character decomposition and character classification in traditional OCR character recognition. After several trials, it has obtained better experimental results in terms of recognition accuracy, with an accuracy rate of over 93% in the recognition of tax invoice data set.

Unconventional superconductivity mediated solely by isotropic electron-phonon interaction

Unconventional superconductivity is commonly linked to electronic pairing mechanisms, since it is believed that the conventional electron-phonon interaction (EPI) cannot cause sign-changing superconducting gap symmetries. Here, we show that this common understanding needs to be revised when one considers a more elaborate theory of electron-phonon superconductivity beyond standard approximations. We self-consistently solve the full-bandwidth, anisotropic Eliashberg equations including vertex corrections beyond Migdal's approximation assuming the usual isotropic EPI for cuprate, Fe-based, and heavy-fermion superconductors with nested Fermi surfaces. In the case of the high-${T}_{c}$ cuprates we find a $d$-wave order parameter, as well as a nematic state upon increased doping. For Fe-based superconductors, we obtain ${s}_{\ifmmode\pm\else\textpm\fi{}}$ gap symmetry, while for heavy-fermion ${\mathrm{CeCoIn}}_{5}$ we find unconventional $d$-wave pairing. These results provide a proof of concept that EPI cannot be excluded as a mediator of unconventional and of high-${T}_{c}$ superconductivity.

A fixed grid based accurate phase-field method for dendritic solidification in complex geometries

In this paper, an easily implementable immersed interface method based high-resolution phase-field model is proposed for simulating dendritic growth related problems in complicated geometries. At first, resolution issues of different Fourier Pseudo-spectral based dealiasing schemes are shown for the anisotropic phase field equations at high thermal undercooling conditions using just adequate spatial grid resolution. The problems of aliasing error and spectral leakage are studied using high-order based exponential smoothing spectral filters, which generally occur at low sampling rate situations. A fully dealiased zero padding scheme based anisotropic phase-field method is also developed in conjunction with an optimal third order strong stability preserving Runge–Kutta (SSPRK3) time integration scheme for obtaining more accurate and stable numerical solutions. In the later part, a modified phase-field method is implemented with the indicator function which simulates different crystal growth problems in complex geometries without using non-trivial mesh refinement algorithm. Effect of non-interacting obstacles present in the computational domain can also be readily incorporated in the present numerical model by using the single order parameter based phase-field equations.

A Fast Computational Method for the Optimal Thermal Design of Anisotropic Multilayer Structures with Discrete Heat Sources for Electrified Propulsion Systems

This work investigates the effect of the anisotropy on the heat transfer properties of layered composite materials with discrete heat sources, that are used in the power electronics of hybrid-electric propulsion systems. An analytical method, based on closed-form expressions structured on Fourier expansion series, is proposed for the solution of the Laplace’s steady-state anisotropic heat equation. The physical presence of electrical components is replaced by externally applied power sources. The method provides an accurate prediction of the temperature distribution and of the heat transfer across perfect layer-to-layer adhesion or finite conductance interfaces; its use is therefore encouraged for the optimization of composite substrates. Code verification has been employed on test cases for which the analytical solution was available.

Adjoint‐State Traveltime Tomography for Azimuthally Anisotropic Media and Insight Into the Crustal Structure of Central California Near Parkfield

AbstractSeismic anisotropy provides crucial information on the stress state and geodynamic processes inside the Earth. We develop a novel adjoint‐state traveltime tomography method using P‐wave traveltime data to simultaneously determine velocity heterogeneity and azimuthal anisotropy of the subsurface. First, an anisotropic eikonal equation is derived to model first‐arrival traveltimes in azimuthally anisotropic media. Traveltime tomography is then formulated as an optimization problem constrained by the anisotropic eikonal equation, which is subsequently solved by the adjoint‐state method. Ray tracing is not required. Its high accuracy is achieved by solving the anisotropic eikonal equation and the associated adjoint equation with efficient numerical solvers. In addition, an eikonal equation‐based earthquake location method for azimuthally anisotropic media is developed to solve the coupled hypocenter‐velocity problem. The tomography and earthquake location methods are applied to central California near Parkfield to test their performance in practice. A total of 1,068,850 first P‐wave traveltimes clearly maps the velocity heterogeneity and azimuthal anisotropy in the upper and middle crust. The average P‐wave velocity model shows a striking velocity contrast across the San Andreas Fault (SAF). In the upper crust, we find structural anisotropy in the SAF zone and stress‐induced anisotropy off the SAF zone. In the middle crust, the fast P‐wave velocity directions are generally fault‐parallel due to the decreased effect of the maximum horizontal compressive stress. In all, the real‐data application suggests that the new adjoint‐state traveltime tomography method can be reliably used to investigate anisotropic seismic structures.

Infinitely many solutions for anisotropic elliptic equations with variable exponent

In this article, we study the existence and multiplicity of solutions for a class of anisotropic elliptic equations First we establisch that anisotropic space is separable and by using the Fountain theorem, and dual Fountain theorem we prove, under suitable conditions, that the problem (P) admits two sequences of weak solutions.

Effect of Fluids on the Elastic Properties of 3D-Printed Anisotropic Rock Models

Laboratory physical models play an important role in understanding rock properties and wave propagation, both theoretically and at the field scale. In some cases, 3D-printing technology can be adopted to construct complex rock models faster, more inexpensively, and with more specific features than previous model-building techniques. In this study, we use 3D-printed rock models to assist in understanding the effects of various fluids (air, water, engine oil, crude oil, and glycerol) on the models’ elastic properties. We first used a 3D-printed, 1-in. cube-shaped layered model. This model was created with a 6% primary porosity and a bulk density of 0.98 g/cc with VTI anisotropy. We next employed a similar cube but with horizontal inclusions embedded in the layered background, which contributed to its total 24% porosity (including primary porosity). For air to liquid saturation, P-velocities increased for all liquids in both models, with the highest increase being with glycerol (57%) and an approximately 45% increase for other fluids in the inclusion model. For the inclusion model (dry and saturated), we observed a greater difference between two orthogonally polarized S-wave velocities (Vs1 and Vs2) than between two P-wave velocities (VP0 and VP90). We attribute this to the S2-wave (polarized normal to both the layering and the plane of horizontal inclusions), which appears more sensitive to horizontal inclusions than the P-wave. For the inclusion model, Thomsen’s P-wave anisotropic parameter (ɛ) decreased from 26% for the air case to 4% for the water-saturated cube and to 1% for glycerol saturation. The small difference between the bulk modulus of the frame and the pore fluid significantly reduces the velocity anisotropy of the medium, making it almost isotropic. We compared our experimental results with theory and found that predictions using Schoenberg’s linear slip theory combined with Gassmann’s anisotropic equation were closer to actual measurements than Hudson’s isotropic calculations. This work provides insights into the usefulness of 3D-printed models to understand elastic rock properties and wave propagation under various fluid saturations.

Solving anisotropic heat equations by exponential shift-and-invert and polynomial Krylov subspace methods

We assess performance of the exponential Krylov subspace methods for solving a class of parabolic problems with a strong anisotropy in coefficients. Different boundary conditions are considered, which have a direct impact on the smallest eigenvalue of the discretized operator and, hence, on the convergence behavior of the exponential Krylov subspace solvers. Restarted polynomial Krylov subspace methods and shift-and-invert Krylov subspace methods combined with algebraic multigrid are considered.

Second-order maximum principle preserving Strang’s splitting schemes for anisotropic fractional Allen-Cahn equations

Second-order maximum principle preserving Strang’s splitting schemes for anisotropic fractional Allen-Cahn equations

Tensorial elastodynamics for coupled acoustic/elastic anisotropic media: incorporating bathymetry

SUMMARYCorrectly implementing the fluid/solid boundary conditions at the seafloor is important for accurate full-wavefield imaging and inversion of marine seismic data. Because bathymetric profiles are rarely flat, the associated undulations influence wave modes interacting with the seafloor and, therefore, the ensuing imaging and inversion results. The conventional method of using single-domain elastic finite-difference (FD) solutions assumes that the fluid/solid contact is welded, which leads to incorrect handling of the boundary conditions and introduces modelling errors. We present a mimetic finite-difference (MFD) approach to solve the equations of anisotropic elastodynamics in a non-orthogonal coordinate system conformal to the bathymetric interface. The vertically deformed coordinate mapping transforms the irregular Cartesian (physical) domain into a regularly sampled generalized computational domain. We partition the medium into the acoustic and elastic subdomains and explicitly satisfy the fluid/solid boundary conditions with a split-node approach involving high-order one-sided MFD operators that achieve uniform spatial accuracy throughout the computational domain. Fully staggered grids (FSGs) are used to solve the velocity–pressure and velocity–stress formulations of the acoustic and anisotropic elastic wave equations, respectively. Numerical examples demonstrate that the proposed MFD+FSG algorithm accurately simulates wavefields even for strongly undulating bathymetric surfaces overlying structurally complex anisotropic media, and produces no spurious numerical artefacts (e.g. staircasing) or unphysical wave modes often caused by improper handling of the strong-contrast bathymetric interface. The wavefields generated by the tensorial MFD scheme closely match those from the more computationally expensive spectral-element method used to validate our implementation. The developed MFD+FSG technique can be effectively employed as the modelling kernel in a variety of coupled acoustic/elastic imaging and inversion applications.

Phonon-mediated high-temperature superconductivity in the ternary borohydride KB2H8 under pressure near 12 GPa

The discovery of high-temperature superconductivity in hydrogen-rich compounds has fueled the enthusiasm for finding materials with more promising superconducting properties among hydrides. However, the ultrahigh pressure needed to synthesize and maintain high-temperature hydrogen-rich superconductors hinders the experimental investigation of these materials. For practical applications, it is also highly desired to find more hydrogen-rich materials that superconduct at high temperatures but under relatively lower pressures. Based on first-principles density functional theory, we calculate the electronic and phonon band structures for a ternary borohydride formed by intercalating ${\mathrm{BH}}_{4}$ tetrahedrons into a fcc potassium lattice, ${\mathrm{KB}}_{2}{\mathrm{H}}_{8}$. Remarkably, we find that this material is dynamically stable and one of its $s{p}^{3}$-hybridized $\ensuremath{\sigma}$-bonding bands is metallized (i.e., partially filled) above a moderate high pressure. This metallized $\ensuremath{\sigma}$-bonding band couples strongly with phonons, giving rise to a strong superconducting pairing potential. By solving the anisotropic Eliashberg equations, we predict that the superconducting transition temperature of this compound is 134--146 K around 12 GPa.