Problems In Anisotropic Media Research Articles

A local absorbing boundary condition for 3D seepage and heat transfer in unbounded domains

AbstractFor the parabolic problems in an infinite space, previous methods basically focused on the one‐ and two‐dimensional artificial boundary. Here, a high‐order local absorbing boundary condition (ABC) used for the fluid seepage and heat transfer in unbounded one‐ and two‐dimensional domains is extended to the relative three‐dimensional analysis. The local ABCs are first derived for the problem in an isotropic media and then stretched to the case in an orthotropic media. The function including time‐related variables in Laplace‐Fourier space is approximated through the Gauss‐Legendre quadrature formula. By using the inverse Laplace‐Fourier transformation, the local ABCs in Laplace‐Fourier space are inverted into the ones in time space. The numerical examples indicate that the local ABCs can provide satisfactory results with high computational efficiency, especially for the long‐term analysis. Moreover, the relationship among the diffusion coefficient, maximum simulation time and approximation order value is also investigated.

Spectral Element Method for the Elastic/Acoustic Waveguide Problem in Anisotropic Metamaterials

Waveguide problems are fundamental to elastic and acoustic wave propagation, where we are interested in finding the propagation constants and modal patterns of waveguide modes. As the waveguide is assumed uniform in one direction, the original 3-D problem can be converted into a so-called 2.5-D problem by using the Fourier transform in that direction. However, the introduction of elastic metamaterials (EMM) broadens the horizon of this subject, and new features are required in EMM waveguides that cannot be obtained by most traditional waveguide solvers. In this work, a spectral element method (SEM) is developed to simulate the elastic/acoustic waveguide problem in anisotropic media with anisotropic mass density and/or negative index parameters. We design an anisotropic density EMM waveguide with our SEM solver to demonstrate several intriguing phenomena. Comparisons with the traditional finite element method (FEM) for several examples show the significant advantages of the SEM in term of accuracy and efficiency.

On the singularity of the Yeh 4 × 4 transfer matrix formalism

The 4×4 matrix method proposed by Yeh is a robust technique to solve the wave propagation problem in anisotropic media. However, it fails in the isotropic limit. Although the singularity of this method has been widely noted, a comprehensive discussion on the reason behind such a failure has not been reported previously. In this paper, we review the Yeh's 4×4 transfer matrix formulation in multi-layered systems and evaluate it in the limit of isotropic media to obtain a physical meaning of the singularity of the formalism. We demonstrate that the method is built on a constrained polarization vector p, which is subject to the orientation of the wavevector k. When the constraint is released in the limit of isotropic media, p vanishes, and this entails the singularity of the method.

Spatial Analogues of Numerical Quasiconformal Mapping Methods for Solving Identification Problems in Anisotropic Media

The approach to solving the gradient problems of image reconstruction of spatial bodies using applied quasipotential tomographic data that is based on numerical complex analysis methods is extended to cases of anisotropic media. Here the distribution of eigen-directions of the conductivity tensor is considered a priori known. We propose to identify the parameters of the corresponding quasiideal stream by the way of minimizing the functional of the sum of squares of residuals which constructed using differential equations in partial derivatives that relate the quasipotential of velocity and the spatially quasicomplex conjugated stream functions

Spectral properties of exterior transmission problem for spherically stratified anisotropic media

ABSTRACT The exterior transmission eigenvalue problem for anisotropic media with spherical symmetry assumptions is considered. First, we discuss the existence and the asymptotic distribution of exterior transmission eigenvalues. Then, we investigate the inverse problem of determining the index of refraction from a knowledge of two sets of spectral data under appropriate conditions.

Anti-plane dynamic response of a non-circular tunnel with imperfect interface in anisotropic rock mass

In anisotropic rock mass, the dynamic response of a non-circular tunnel with imperfect interface is theoretically presented under an incidence of an anti-plane SH wave. For the anti-plane problem in anisotropic medium, three elastic constants can describe the characteristics of medium anisotropy according to the generalized Hooke’s Law. The wave function expansion method and the complex function are applied to express the wave fields with three unknown coefficients. By introducing the spring elastic model, the imperfect boundary condition around the tunnel lining can be satisfied to determine the coefficients. In numerical examples, effects of rock anisotropy and imperfect interface around the tunnel lining are discussed in detail, accompanied with incident angle and wave frequency. Analytical solutions show that both rock anisotropy and interface have considerable influence on the dynamic response of underground structures, which should be paid attention to for the design and construction of deep underground tunnels.

Gradient estimates for the perfect conductivity problem in anisotropic media

We study the perfect conductivity problem when two perfectly conducting inclusions are closely located to each other in an anisotropic background medium. We establish optimal upper and lower gradient bounds for the solution in any dimension which characterize the singular behavior of the electric field as the distance between the inclusions goes to zero.

Traveltime parameters in tilted orthorhombic medium

Traveltime parameters, defined through the series coefficients of the traveltime squared as a function of the horizontal offset projections, play an important role in moveout approximations and corrections, and in model parameter inversion. We evaluate an approach to derive the traveltime parameters in a single homogeneous anisotropic layer of tilted orthorhombic symmetry for one- and two-way traveling waves. The approach allows us to obtain the traveltime parameters of pure and converted modes. We use numerical models to illustrate the dependence of the high-order traveltime parameters on the Euler angles and the anisotropy parameters. The traveltime parameter inversion is a strongly ill-posed problem in anisotropic media, and improvements due to inclusion of the high-order traveltime parameters can sufficiently reduce the space of equivalent kinematic models. We perform a numerical model parameter inversion using the concept of artificial neural networks to demonstrate the accuracy improvements due to inclusion of the high-order traveltime parameters over the inversion of the second-order coefficient, conventionally known as normal moveout velocity, only. We demonstrate algebraically and numerically that the presented approach to calculate the traveltime parameters is easily extended to multilayered media. It can be used for Dix-type inversion to obtain the interval medium parameters.

P-wave ray velocities and the inverse acoustic problem for anisotropic media

The specific features of the calculation of ray velocities of quasi-longitudinal waves in anisotropic media have been considered. A technique for calculating elastic constants using P-wave ray velocities measured in an ultrasonic experiment on spherical samples is presented. It is shown by an example of tabular data that elastic constants С 11, С 22, and С 33 and combinations of constants (С 12 + 2С 66), (С 13 + 2С 55), (С 23 + 2С 44), (С 14 + 2С 56), (С 25 + 2С 46), and (С 36 + 2С 45) can be calculated most accurately for the general case of anisotropic media with elastic properties of arbitrary symmetry. Since the determining system of equations is illconditioned, the values of elastic constants entering these combinations depend on the choosed initial approximation.

Применение разрывного метода Галеркина для решения параболических задач в анизотропных средах на треугольных сетках

Применение разрывного метода Галеркина для решения параболических задач в анизотропных средах на треугольных сетках

Plane Problem of Magnetoelasticity for a Piezomagnetic Medium with Cracks

We solve a boundary-value problem of magnetoelasticity for a piezomagnetic plane weakened by cracks. For this purpose, we generalize the method aimed at the solution of similar problems for anisotropic media. The boundary-value problem is reduced to a matrix singular integral equation and the solution of this equation is found in the class of vector functions unbounded at the ends of the notches. The numerical solution of this equation is obtained by the method of mechanical quadratures. The proposed numerical-analytic algorithm is used to investigate the influence of magnetoelastic fields on the stress intensity factors in the vicinity of the crack tips.

A generalized multiscale finite element method for high-contrast single-phase flow problems in anisotropic media

In this paper, we describe a systematic framework called the Generalized Multiscale Finite Element Method (GMsFEM) for performing multiscale simulations, and we investigate the application of this framework to the high-contrast flow problems in anisotropic media. We present a multiscale model reduction technique for constructing a reduced-dimension space based on localized spectral decompositions. We discuss the multiscale coarse space both in parameter-independent and parameter-dependent cases. Furthermore, we highlight the relation between the solution convergence and the spectral behavior of the eigenvalue problems. A variety of numerical examples are presented to illustrate the performance of the proposed methodology.

An explicit form of the solution of the Cauchy problem for the Barenblatt–Zheltov–Kochina equation

A solution of the Cauchy problem in an isotropic medium and in an anisotropic medium with clearly expressed vertical or horizontal permeability is constructed for the Barenblatt–Zheltov–Kochina model representation of the seepage of a fluid in fissured porous rock by reducing the seepage problems under consideration to solving an abstract Cauchy problem in a Banach space.

MIXED FINITE ELEMENT METHOD FOR 2D VECTOR MAXWELL'S EIGENVALUE PROBLEM IN ANISOTROPIC MEDIA

It is well known that the conventional edge element method in solving vector Maxwell's eigenvalue problem will lead to the presence of nonphysical zero eigenvalues. This paper uses the mixed flnite element method to suppress the presence of these nonphysical zero eigenvalues for 2D vector Maxwell's eigenvalue problem in anisotropic media. We introduce a Lagrangian multiplier to deal with the constraint of divergence-free condition. Our method is based on employing the flrst-order edge element basis functions to expand the electric fleld and linear nodal element basis functions to expand the Lagrangian multiplier. Our numerical experiments show that this method can successfully remove all nonphysical zero and nonzero eigenvalues. We verify that when the cavity has a connected perfect electric boundary, then there is no physical zero eigenvalue. Otherwise, the number of physical zero eigenvalues is one less than the number of disconnected perfect electric boundaries.

Modeling anisotropic diffusion using a departure from isotropy approach

There are a large number of finite volume solvers available for solution of isotropic diffusion equation. This article presents an approach of adapting these solvers to solve anisotropic diffusion equations. The formulation works by decomposing the diffusive flux into a component associated with isotropic diffusion and another component associated with departure from isotropic diffusion. This results in an isotropic diffusion equation with additional terms to account for the anisotropic effect. These additional terms are treated using a deferred correction approach and coupled via an iterative procedure. The presented approach is validated against various diffusion problems in anisotropic media with known analytical or numerical solutions. Although demonstrated for two-dimensional problems, extension of the present approach to three-dimensional problems is straight forward. Other than the finite volume method, this approach can be applied to any discretization method.

A parallel finite-element method for three-dimensional controlled-source electromagnetic forward modelling

SUMMARY We present a nodal finite-element method that can be used to compute in parallel highly accurate solutions for 3-D controlled-source electromagnetic forward-modelling problems in anisotropic media. Secondary coupled-potential formulation of Maxwell’s equations allows to avoid the singularities introduced by the sources, while completely unstructured tetrahedral meshes and mesh refinement support an accurate representation of geological and bathymetric complexity and improve the solution accuracy. Different complex iterative solvers and an efficient pre-conditioner based on the sparse approximate inverse are used for solving the resulting large sparse linear system of equations. Results are compared with the ones of other researchers to check the accuracy of the method. We demonstrate the performance of the code in large problems with tens and even hundreds of millions of degrees of freedom. Scalability tests on massively parallel computers show that our code is highly scalable.

Ellipticity in the Interior Transmission Problem in Anisotropic Media

The paper concerns the discreteness of the eigenvalues and the solvability of the interior transmission problem for anisotropic media. Conditions for the ellipticity of the problem are written explicitly, and it is shown that they do not guarantee the discreteness of the eigenvalues. Some simple sufficient conditions for the discreteness and solvability are found. They are expressed in terms of the values of the anisotropy matrix and the refraction index at the boundary of the domain. The discreteness of the eigenvalues and the solvability of the interior transmission problem are shown if a small perturbation is applied to the refraction index.

A singular edge-based smoothed finite element method (ES-FEM) for crack analyses in anisotropic media

Abstract The recently developed edge-based smoothed finite element method (ES-FEM) is extended to fracture problems in anisotropic media using a specially designed five-node singular crack-tip (T5) element. In the formulation of singular ES-FEM, only the assumed displacement values (not the derivatives) on the boundaries of the smoothing domains are needed. Thus, a layer of T5 crack-tip element is devised to construct “singular” shape functions via a simple point interpolation with a fractional order basis, without mapping procedure. The effectiveness of the present singular ES-FEM is demonstrated by intensive examples for a wide range of degrees of anisotropy.

Influence of Media Properties on Fluid Flow in Faulted and Fractured Anisotropic Media: A Study Using a Modified Flux-Vector-Based Green Element Method

This article illustrates the use of the modified ‘q-based’ GEM for complex geological problems in anisotropic media involving faults/fractures, by adopting various values of the permeability for both the faults/fractures and the media. The bulk permeabilities are compared and illustrated for different cases. Examples based on generating randomly positioned faults for which the number of faults can vary, but the sum of their lengths is fixed, are investigated. Situations are presented for a number of different total fault lengths and for various physical properties of the faults within a fixed total fault length. For some of the cases, particle traces are shown to better illustrate the behaviour of the flow.

A meshless method for stress-wave propagation in anisotropic and cracked media

Several transient wave propagation problems in anisotropic media and isotropic media with cracks are numerically analyzed by using a new numerical algorithm based on meshless local Petrov–Galerking (MLPG) method. In this algorithm, a novel modified Moving Least Squares (MLS) approximation is introduced to simplify the treatment of essential boundary conditions. By using a variant type of MLPG1 methods, the stabilized scheme of the discretized elasto-dynamic equations is obtained. Explicit central difference method with lumped mass matrix is used to solve the coupled ODEs to increase the efficiency of present algorithm. Visibility criterion is used to present the cracks, and the path-independent dynamic J′ integral is adopted to evaluate the dynamic stress intensity factors. The availability and accuracy of the present algorithm in solving dynamic problems in isotropic or anisotropic media with cracks are tested through the comparison with the results obtained by the LS-DYNA and the method of characteristic. Finally, the transient stress wave interacting with a slanted crack under an impact loading is investigated in detail, in which the ability of extracting the different stress-wave components in a complex acoustic field is also proved.