General Anisotropic Media Research Articles

ELECTROMAGNETIC FIELD SOLUTIONS IN AN ISOTROPIC MEDIUM WITH WEAKLY-RANDOM FLUCTUATIONS IN TIME AND SOME APPLICATIONS IN THE ELECTRODYNAMICS OF THE IONOSPHERE

Stochastic wave equations are derived to describe electromagnetic wave propagation in an isotropic medium in which the electric permittivity and the magnetic permeability are weaklyrandom functions of time. Approximate analytical solutions are obtained using separation of variables and the WKB method for some configurations that can be used to model the electromagnetic field in the ionosphere. The form of the initial and boundary conditions determines whether the solution takes a form representing a direct current electric field or continuous pulsation electromagnetic waves. The temporal variation of the calculated induced electromotive force (EMF) is in agreement with observations.

3D forward modeling of magnetotelluric fields in general anisotropic media and its numerical implementation in Julia

We have developed a finite volume (FV) algorithm for magnetotelluric (MT) forward modeling in 3D conductivity structures with general anisotropy. The electric and magnetic fields are discretized on a conventional staggered grid, which cannot directly address the full-tensor conductivity. To overcome this difficulty, an interpolation scheme is used to average different components of the electric field to the same position. We formulate the algorithm in pure matrix form and implement it in a new language, Julia, making the programming process highly efficient and leading to a code with excellent readability, maintainability, and extendability. The validity of the FV Julia code is demonstrated using a layered 1D anisotropic model. For this model, the FV code provides accurate results, and the computational cost is reasonable. Being preconditioned with the electromagnetic potential ([Formula: see text]) system, the iterative solvers including quasi-minimal residual and biconjugate gradient stabilized exhibit a good convergence rate for a wide range of periods. The direct solvers MUMPS and PARDISO are highly efficient for small model sizes. For a relatively large model size with 2.18 millions unknowns, the linear system of one period can be solved by MUMPS within 360 s with multiple threads involved in the computation, and the memory usage is only 11.6 GB in the “out-of-core” mode. We further calculated MT responses of a 3D model with dipping and horizontal anisotropy, respectively. The results suggest that the electrical anisotropy can have significant influence on the MT response.

Computation of Tensor Green’s Functions in Uniaxial Planar-Stratified Media With a Rescaled Equivalent Boundary Approach

An equivalent boundary approach is developed to compute electric- and magnetic-type tensor Green’s functions in general anisotropic media. With this approach, a succinct and robust representation is provided of the tensor Green’s functions for planar-stratified media with uniaxial anisotropic layers based on the propagator matrix method in cylindrical coordinates. In deriving the tensor Green’s functions in the spectral domain, the point electric- or magnetic-current source is replaced by appropriate boundary conditions. In the cylindrical coordinate system, a cylindrical Hankel integral transformation is adopted to expedite the integration of the tensor Green’s functions from the spectral domain to the spatial domain. A Levin-type collocation method for the fast integration of rapidly oscillatory functions is applied to effectively compute the spectral integrals with oscillatory and slowly decaying behavior along the contour integral path.

Microseismic Full Waveform Modeling in Anisotropic Media with Moment Tensor Implementation

Seismic anisotropy which is common in shale and fractured rocks will cause travel-time and amplitude discrepancy in different propagation directions. For microseismic monitoring which is often implemented in shale or fractured rocks, seismic anisotropy needs to be carefully accounted for in source location and mechanism determination. We have developed an efficient finite-difference full waveform modeling tool with an arbitrary moment tensor source. The modeling tool is suitable for simulating wave propagation in anisotropic media for microseismic monitoring. As both dislocation and non-double-couple source are often observed in microseismic monitoring, an arbitrary moment tensor source is implemented in our forward modeling tool. The increments of shear stress are equally distributed on the staggered grid to implement an accurate and symmetric moment tensor source. Our modeling tool provides an efficient way to obtain the Green’s function in anisotropic media, which is the key of anisotropic moment tensor inversion and source mechanism characterization in microseismic monitoring. In our research, wavefields in anisotropic media have been carefully simulated and analyzed in both surface array and downhole array. The variation characteristics of travel-time and amplitude of direct P- and S-wave in vertical transverse isotropic media and horizontal transverse isotropic media are distinct, thus providing a feasible way to distinguish and identify the anisotropic type of the subsurface. Analyzing the travel-times and amplitudes of the microseismic data is a feasible way to estimate the orientation and density of the induced cracks in hydraulic fracturing. Our anisotropic modeling tool can be used to generate and analyze microseismic full wavefield with full moment tensor source in anisotropic media, which can help promote the anisotropic interpretation and inversion of field data.

The stress state around an elliptical borehole in anisotropy medium

Under the assumption of generalized plane strain condition, this paper presents an analytical solution for stresses near an inclined elliptical wellbore drilled in anisotropic formation based on the linear theory of anisotropic elasticity and complex variable method. The solution is suitable for calculating the stresses around an elliptical borehole drilled with any inclination and orientation angles in a linear elastic, continuous, homogeneous and general anisotropic medium. The solution for the stresses can also show the effects of the liquid pressure exerted by drilling mud, the ratio and the direction of the in-situ stresses. For engineering utilization the transformation of compliance matrix for the transversely isotropic medium is formulated. An example is calculated and shown by contours of stress components around an inclined elliptical wellbore in a transversely isotropic formation. The stress contours can be utilized to some extent to predict the location, the range, the depth and even the outlines of fractured or collapsed well wall. Then a range of parametric analysis is carried out on the impacts of borehole inclination, orientation and ellipticity on the first principal stress and maximum shearing stress. The results indicate that for various boreholes with different degree of formation anisotropy the algebraic values of the first principal stress and maximum shearing stress go down gradually with the increase of borehole inclination, and these two stresses vary in the pattern of half a sinusoidal wave with the variation of orientation angles. Furthermore, the stresses go up with the increase of ellipticity. The solution presented in this paper might supply a more realistic technique for assessing the stability of borehole and optimizing the well path in anisotropic medium.

Rayleigh wave at the surface of a general anisotropic poroelastic medium: derivation of real secular equation

A secular equation governs the propagation of Rayleigh wave at the surface of an anisotropic poroelastic medium. In the case of anisotropy with symmetry, this equation is obtained as a real irrational equation. But, in the absence of anisotropic symmetries, this secular equation is obtained as a complex irrational equation. True surface waves in non-dissipative materials decay only with depth. That means, propagation of Rayleigh wave in anisotropic poroelastic solid should be represented by a real phase velocity. In this study, the determinantal system leading to the complex secular equation is manipulated to obtain a transformed equation. Even for arbitrary (triclinic) anisotropy, this transformed equation remains real for the propagation of true surface waves. Such a real secular equation is obtained with the option of boundary pores being opened or sealed. A numerical example is solved to study the existence and propagation of Rayleigh waves in porous media for the top three (i.e. triclinic, monoclinic and orthorhombic) anisotropies.

Application of transformation optics for the purpose of cloaking

The advent of transformation optics has lead to the initiation of designing devices and applications associated to electromagnetic wave propagation in anisotropic media. Here, a method is suggested using a coordinate transformation with spherical coordinates for the purpose of designing a three dimensional cloak for a body having an arbitrary convex geometry. For the purpose of verification of the algorithm, a ray tracing process is carried out for an ellipsoid having axial symmetry.

Adaptive finite element modelling of three-dimensional magnetotelluric fields in general anisotropic media

We present a goal-oriented adaptive finite element (FE) modelling algorithm for 3-D magnetotelluric fields in generally anisotropic conductivity media. The model consists of a background layered structure, containing anisotropic blocks. Each block and layer might be anisotropic by assigning to them 3×3 conductivity tensors. The second-order partial differential equations are solved using the adaptive finite element method (FEM). The computational domain is subdivided into unstructured tetrahedral elements, which allow for complex geometries including bathymetry and dipping interfaces. The grid refinement process is guided by a global posteriori error estimator and is performed iteratively.The system of linear FE equations for electric field E is solved with a direct solver MUMPS. Then the magnetic field H can be found, in which the required derivatives are computed numerically using cubic spline interpolation. The 3-D FE algorithm has been validated by comparisons with both the 3-D finite-difference solution and 2-D FE results. Two model types are used to demonstrate the effects of anisotropy upon 3-D magnetotelluric responses: horizontal and dipping anisotropy. Finally, a 3D sea hill model is modelled to study the effect of oblique interfaces and the dipping anisotropy.

The Brezis–Nirenberg problem for the curl–curl operator

We look for solutions E:Ω→R3 of the problem{∇×(∇×E)+λE=|E|p−2Ein Ων×E=0on ∂Ω on a bounded Lipschitz domain Ω⊂R3, where ∇× denotes the curl operator in R3. The equation describes the propagation of the time-harmonic electric field ℜ{E(x)eiωt} in a nonlinear isotropic material Ω with λ=−μεω2≤0, where μ and ε stand for the permeability and the linear part of the permittivity of the material. The nonlinear term |E|p−2E with p>2 is responsible for the nonlinear polarisation of Ω and the boundary conditions are those for Ω surrounded by a perfect conductor. The problem has a variational structure and we deal with the critical value p, for instance, in convex domains Ω or in domains with C1,1 boundary, p=6=2⁎ is the Sobolev critical exponent and we get the quintic nonlinearity in the equation. We show that there exist a cylindrically symmetric ground state solution and a finite number of cylindrically symmetric bound states depending on λ≤0. We develop a new critical point theory which allows to solve the problem, and which enables us to treat more general anisotropic media as well as other variational problems.

Generation of transparent boundary conditions for modeling wave propagation in anisotropic media

Generation of transparent boundary conditions for modeling wave propagation in anisotropic media

The Plane Wave Methods Combined with Local Spectral Finite Elements for the Wave Propagation in Anisotropic Media

The Plane Wave Methods Combined with Local Spectral Finite Elements for the Wave Propagation in Anisotropic Media

An efficient Helmholtz solver for acoustic transversely isotropic media

The acoustic approximation, even for anisotropic media, is widely used in current industry imaging and inversion algorithms mainly because P-waves constitute most of the energy recorded in seismic exploration. The resulting acoustic formulas tend to be simpler, resulting in more efficient implementations, and they depend on fewer medium parameters. However, conventional solutions of the acoustic-wave equation with higher-order derivatives suffer from S-wave artifacts. Thus, we separate the quasi-P-wave propagation in anisotropic media into the elliptic anisotropic operator (free of the artifacts) and the nonelliptic anisotropic components, which form a pseudodifferential operator. We then develop a separable approximation of the dispersion relation of nonelliptic-anisotropic components, specifically for transversely isotropic media. Finally, we iteratively solve the simpler lower-order elliptical wave equation for a modified source function that includes the nonelliptical terms represented in the Fourier domain. A frequency-domain Helmholtz formulation of the approach renders the iterative implementation efficient because the cost is dominated by the lower-upper decomposition of the impedance matrix for the simpler elliptical anisotropic model. In addition, the resulting wavefield is free of S-wave artifacts and has a balanced amplitude. Numerical examples indicate that the method is reasonably accurate and efficient.

A meshless average source boundary node method for steady-state heat conduction in general anisotropic media

The average source boundary node method (ASBNM) is a recent boundary-type meshless method, which uses only the boundary nodes in the solution procedure without involving any element or integration notion, that is truly meshless and easy to implement. This paper documents the first attempt to extend the ASBNM for solving the steady-state heat conduction problems in general anisotropic media. Noteworthily, for boundary-type meshless/meshfree methods which depend on the boundary integral equations, whatever their forms are, a key but difficult issue is to accurately and efficiently determine the diagonal coefficients of influence matrices. In this study, we develop a new scheme to evaluate the diagonal coefficients via the pure boundary node implementation based on coupling a new regularized boundary integral equation with direct unknowns of considered problems and the average source technique (AST). Seven two- and three-dimensional benchmark examples are tested in comparison with some existing methods. Numerical results demonstrate that the present ASBNM is superior in the light of overall accuracy, efficiency, stability and convergence rates, especially for the solution of the boundary quantities.

Three-dimensional magnetotelluric modeling in anisotropic media using edge-based finite element method

It is important to understand how magnetotelluric (MT) modeling can most effectively be performed in general anisotropic media. However, previous studies in this area have mainly focused on the use of one-dimensional (1D) and two-dimensional (2D) algorithms. Thus, building on earlier work, it is important to study the performance of three-dimensional (3D) modeling in arbitrary conductivity media; therefore, an edge-based finite element (FE) method has been developed for 3D MT modeling in arbitrary conductivity media. This approach is based on the initial derivation of a series of equivalent variational equations that are based on Maxwell equations, generated using the weighted residual method. Specific values were then obtained for coefficient matrixes of this edge-based FE method using hexahedral meshes, and the algorithm was verified by comparing its results with finite difference (FD) solutions generated using a 2D anisotropic model. Finally, the results of a 3D anisotropic model were analyzed detailed for three conditions; another 3D anisotropic model was designed and its results were compared with two isotropic models'.

Optimal damping profile ratios for stabilization of perfectly matched layers in general anisotropic media

Conventional perfectly matched layers (PMLs) can be unstable for certain kinds of anisotropic media. The multiaxial PML removes such instability using nonzero damping coefficients in the directions tangential with the PML interface. Although using nonzero damping profile ratios can stabilize PMLs, it is important to obtain the smallest possible damping profile ratios to minimize artificial reflections caused by these nonzero ratios, particularly for 3D general anisotropic media. Using the eigenvectors of the PML system matrix, we have developed a straightforward and efficient numerical algorithm to determine the optimal damping profile ratios to stabilize PMLs in 2D and 3D general anisotropic media. Numerical examples indicate that our algorithm provides optimal damping profile ratios to ensure the stability of PMLs and complex-frequency-shifted PMLs for elastic-wave modeling in 2D and 3D general anisotropic media.

Local Interaction Simulation Approach for Efficient Modeling of Linear and Nonlinear Ultrasonic Guided Wave Active Sensing of Complex Structures

This paper presents the local interaction simulation approach (LISA) for efficient modeling of linear and nonlinear ultrasonic guided wave active sensing of complex structures. Three major modeling challenges are considered: material anisotropy with damping effects, nonlinear interactions between guided waves and structural damage, as well as geometric complexity of waveguides. To demonstrate LISA's prowess in addressing such challenges, carefully designed numerical case studies are presented. First, guided wave propagation and attenuation in carbon fiber composite panels are simulated. The numerical results are compared with experimental measurements obtained from scanning laser Doppler vibrometry (SLDV) to illustrate LISA's capability in modeling damped wave propagation in anisotropic medium. Second, nonlinear interactions between guided waves and structural damage are modeled by integrating contact dynamics into the LISA formulations. Comparison with commercial finite element software reveals that LISA can accurately simulate nonlinear ultrasonics but with much higher efficiency. Finally, guided wave propagation in geometrically complex waveguides is studied. The numerical example of multimodal guided wave propagation in a rail track structure with a fatigue crack is presented, demonstrating LISA's versatility to model complex waveguides and arbitrary damage profiles. This article serves as a comprehensive, systematic showcase of LISA's superb capability for efficient modeling of transient dynamic guided wave phenomena in structural health monitoring (SHM).

Material point method for crack propagation in anisotropic media: a phase field approach

A novel phase field formulation implemented within a material point method setting is developed to address brittle fracture simulation in anisotropic media. The case of strong anisotropy in the crack surface energy is treated by considering an appropriate variational, i.e., phase field approach. Material point method is utilized to efficiently treat the resulting coupled governing equations. The brittle fracture governing equations are defined at a set of Lagrangian material points and subsequently interpolated at the nodes of a fixed Eulerian mesh where solution is performed. As a result, the quality of the solution does not depend on the quality of the underlying finite element mesh and is relieved from mesh-distortion errors. The efficiency and validity of the proposed method is assessed through a set of benchmark problems.

Ultrasonic field modeling in anisotropic materials by distributed point source method

DPSM (distributed point source method) is a modeling technique which is based on the concept of Green’s function. First, a collection of source and target points are distributed over the solution domain based on the problem description and solution requirements. Then, the effects from all source points are superimposed at the location of every individual target point. Therefore, a successful implementation of DPSM entails an effective evaluation of Green’s function between many pairs of source and target points. For homogeneous and isotropic media, the Green’s function is available as a closed-form analytical expression. But for anisotropic solids, the evaluation of Green’s function is more complicated and needs to be done numerically. Nevertheless, important applications such as defect detection in composite materials require anisotropic analysis. In this paper, the DPSM is used for ultrasonic field modeling in anisotropic materials. Considering the prohibitive computational cost of evaluating Green’s function numerically for a large number of points, a technique called “windowing” is suggested which employs the repetitive pattern of points in DPSM in order to considerably reduce the number of evaluations of Green’s function. In addition, different resolutions of numerical integration are used for computing Green’s function corresponding to different distances in order to achieve a good balance between time and accuracy. The developed anisotropic DPSM model equipped with windowing technique and multi-resolution numerical integration is then applied to the problem of ultrasonic wave modeling in a plate immersed in a fluid. The transducers are placed in the fluid on both sides of the plate. First an isotropic plate is considered for the sake of verification and rough calibration of numerical integration. Then a composite plate is considered to demonstrate applicability and effectiveness of the developed model for simulating ultrasonic wave propagation in anisotropic media.

Local normal vector field formulation for periodic scattering problems formulated in the spectral domain.

We present two adapted formulations, one tailored to isotropic media and one for general anisotropic media, of the normal vector field framework previously introduced to improve convergence near arbitrarily shaped material interfaces in spectral simulation methods for periodic scattering geometries. The adapted formulations enable the definition and generation of the normal vector fields to be confined to a region of prolongation that includes the material interfaces but is otherwise limited. This allows for a more flexible application of geometrical transformations like rotation and translation per scattering object in the unit cell. Moreover, these geometrical transformations enable a cut-and-connect strategy to compose general geometries from elementary building blocks. The entire framework gives rise to continuously parameterized geometries.

On the determination of defect dipoles from atomistic simulations using periodic boundary conditions

By solving the problem of a periodic distribution of point defects in general anisotropic media, we give an alternative, more direct proof, of the relatively recent procedure that extracts dipole tensors from the stress acting on the cell of atomistic simulations performed under periodic boundary conditions. Moreover, we show that naive superposition of individual defect fields is not a solution of the problem, though correction terms can be identified; as a byproduct, analysis of the latter allows us to reveal a spurious contribution to the elastic interaction energy as calculated in current literature procedures, that therefore must be subtracted in order to obtain correct results.