Inhomogeneous Anisotropic Media Research Articles

Mapping capabilities of geometric EM induction sounding in southern Yakutia permafrost

The purpose of the article is to consider the possibilities of the geophysical method in solving the problem of quantitative estimation of the strength of sedimentary rocks that form the basis of engineering structures at the Kyurgellakh station of the Amur-Yakutsk Railway located 572 km away from the town of Neryungri, which is an administrative center of southern Yakutia (Russia). The problem is solved using the method of geometric electromagnetic induction sounding and a new technique that explored the attenuation of the harmonic field induced by a vertical magnetic dipole in the inhomogeneous anisotropic geological medium at the frequencies of 1.125 and 0.281 MHz in the intermediate separation zone of 5–100 m. A comparative analysis has determined a good agreement in changes of geological and geophysical estimates of the average strength of laboratory water-saturated samples and sedimentary rock mass predicted in the same state at the comparable depth of 6–12 m. The measure of agreement when using the most adequate equation of the power function is high and equals 0.815 according to the normalized coefficient of multiple determination. This means that strength is the most important factor among the frozen ground characteristics affecting the electromagnetic field attenuation at the frequencies and spacing specified, contributing no less than 80 %. The tested geophysical method can thus correctly map the distribution boundaries of different strength sedimentary rocks. The crushed and fissured rocks with the strengths below 40–35 MPa are confined to the tectonic fractures of varying directions with a polygon-like structure. Geophysical data clearly delineate the structure at the depths of 12.3–27.5 m where predominate the rocks with high strengths (50–120 MPa). The geometric EM induction sounding is recommended to use at all stages of geotechnical investigations to map the development areas of southern Yakutia by rock strength classes.

Volume integral equations for electromagnetic scattering by an orthotropic infinite cylinder

We investigate time-harmonic electromagnetic wave scattering by an inhomogeneous anisotropic medium. For the case of an orthotropic cylindrical obstacle and fields invariant along the axis of the cylinder, the problem can be reduced to a two-dimensional one. This 2D problem can be written equivalently in terms of two different strongly singular volume integral equations (VIEs), one is vector-valued and the other is scalar-valued. The aim of this study is to analyze the essential spectrum of the integral operators that describe these VIEs in the case where the physical parameters are piecewise valued matrix functions with some bounds on their eigenvalues. For Lipschitz interfaces, we show that the spectrum is contained in some subsets which depend on the spectral properties of the scattering parameters. The results on the spectrum will then be used to derive sufficient conditions which ensure the uniqueness of solutions to the diffraction problem.

A point-to-point convolutional neural network for reconstructing electromagnetic parameters of multiple cavities scattering with inhomogeneous anisotropic media

We propose a point-to-point convolutional neural network (CNN) model to reconstruct the electromagnetic parameters of multiple cavities filled with inhomogeneous anisotropic media. The input of the model is a grid matrix of electromagnetic field magnitude solved by Petrov–Galerkin finite element interface method, and the output is a distribution matrix of the electromagnetic parameters of the cavities. The uniform non-body-fitted mesh is set in the computational domain, which does not need to be updated repeatedly at the complex interfaces of different media. Compared with the standard finite element, this mesh reduction method effectively saves the cost of mesh generation. Level set functions are applied to capture these arbitrarily shaped interfaces. The matrix coefficients caused by the dielectric constant and permeability tensors of anisotropic media can also be handled with this method. The point-to-point CNN model effectively reconstructs the value and the distribution of electromagnetic parameters in the cavities, also predicting the shape and the position of the interfaces. Three optimized schemes are then proposed to improve reconstruction accuracy. Numerical results show that the size of the convolution kernel and the number of fully connected layers are the crucial parameters in determining interfaces and reducing the number of artefacts.

The electromagnetic scattering from multiple arbitrarily shaped cavities with inhomogeneous anisotropic media

We study the electromagnetic scattering problem from multiple arbitrarily shaped cavities filled with inhomogeneous anisotropic media. A local-refined Petrov-Galerkin finite element interface (PGFEI) method is developed for solving the scattering problem. By introducing the transparent boundary condition, the unbounded scattering problem can be reduced into bounded domain problem with coupled boundary conditions on the apertures of multiple cavities. The multiple cavity scattering in transverse magnetic (TM) and transverse electric (TE) polarizations are both investigated based on uniform non-body-fitted meshes. We adopt the level-set functions to capture the irregular cavity wall and the interfaces of inhomogeneous media. The special basis is constructed to satisfy the jump conditions across the interfaces of different materials further. The local-refined algorithm is proposed to enhance the accuracy of the solutions around the discontinuous interfaces. The generation of the grid points facilitates a special numbering scheme which enables a stepwise Schur complement decomposition of the discrete linear system derived by the PGFEI method. It could be an alternative method for solving the scattering problems which have similar structures. The proposed method can handle matrix coefficients caused by the permittivity and permeability tensor of anisotropic media. Numerical experiments illustrate the validity and efficiency of the method for solving the scattering from multiple arbitrarily shaped cavities with inhomogeneous anisotropic media in TM and TE polarization.

Numerical Solutions of the Electromagnetic Scattering by Overfilled Cavities with Inhomogeneous Anisotropic Media

Numerical Solutions of the Electromagnetic Scattering by Overfilled Cavities with Inhomogeneous Anisotropic Media

Novel applications of local optimization semi-Cartesian grid for the complex band structure analysis of phononic crystals

In this paper, we propose a finite element method based on the Floquet transform and local optimization semi-Cartesian grid to compute the complex band structure of phononic crystals. Floquet transform is applied to transform the elastic wave equation into a quadratic eigenvalue equation. Value-periodic test function independent of the interface is defined to derive the weak formulation. The local optimization semi-Cartesian grid is proposed to improve the quality of triangular elements near the interface. Our results are in good agreement with ones obtained by other mature methods, and the purely propagative mode coincides with the traditional band diagram. In addition to simple scatterer shapes such as circle or square, the proposed method is able to calculate the complex band structures of phononic crystals with complicated scatterer shapes. Numerical experiments demonstrated the effectiveness of this method in the computation of anisotropic inhomogeneous medium. The complex band structures under different lattice types, scatterer shapes, material properties, rotation angle and modes are obtained, and their effects on the width and number of band gaps are analyzed.

A mixed element scheme for the Helmholtz transmission eigenvalue problem for anisotropic media

In this paper, we study the Helmholtz transmission eigenvalue problem for inhomogeneous anisotropic media with the index of refraction in two and three dimensions. Starting with the nonlinear fourth-order formulation established by Cakoni et al 2009 J. Integral Equ. Appl. 21 203–27, we present an equivalent mixed formulation for this problem with auxiliary variables, followed by finite element discretization. Using the proposed scheme, we rigorously show that the optimal convergence rate for the transmission eigenvalues on both convex and nonconvex domains can be expected. With this scheme, we obtain a sparse generalized eigenvalue problem whose size is too demanding, even with a coarse mesh that its smallest few real eigenvalues fail to be solved by the shift and invert method. We partially overcome this critical issue by deflating nearly all of the eigenvalues with huge multiplicity, resulting in a marked reduction in the matrix size without deteriorating the sparsity. Extensive numerical examples are reported to demonstrate the effectiveness and efficiency of the proposed scheme.

On the Regularity of Non-scattering Anisotropic Inhomogeneities

In this paper we examine the necessary conditions for an anisotropic inhomogeneous medium to be non-scattering at a single wave number and for a single incident field. These conditions are expressed in terms of the regularity of the boundary of the inhomogeneity. We assume that the coefficients, characterizing the constitutive material properties of the medium, are sufficiently smooth, and that the incident wave is appropriately non-degenerate. Our analysis utilizes the Hodograph transform as well as regularity results for nonlinear elliptic partial differential equations. Our approach requires that the boundary a-priori is of class $$C^{1,\alpha }$$ for some $$0<\alpha <1$$ .

Identification and focusing of edge diffractions using wavefront attributes

Edge diffractions are more common in 3D seismic data than point diffractions; however, most current studies on processing and imaging diffractions consider point diffractions. We find that the 3D kinematic response of an edge diffractor manifests itself locally as a 2D diffraction for a specific azimuth, which can be uniquely determined using the local characteristics of the wavefronts. This applies to arbitrarily oriented curvilinear edges embedded in inhomogeneous anisotropic media. We further establish a novel 3D time-domain event classification, generalizing the existing 2D technique. A new criterion based on data-derived wavefront curvature matrices allows the unambiguous zero-offset identification of the wave scatterer types: point diffractors, edge diffractors, and reflecting boundaries. We further develop a new data-driven technique of receiver grouping to focus edge diffractions, which contributes to diffraction imaging and tomography. Both proposed methods do not rely on a priori knowledge of the subsurface. Synthetic examples validate these findings.

Optics in anisotropic inhomogeneous media and Lagrange geometry

In this paper, the equations of motion associated with a Lagrangian inspired by relativistic optics in inhomogeneous moving medium are considered. The model describes optical effects in the inhomogeneous moving medium with special optical properties given by the self-consistent system. When using the metric restricted to the Minkowski manifold, we have established the Euler–Lagrange equations for corresponding geodesics. We have specified the general model to the special case when the metric coefficient [Formula: see text] linearly increases along the direction [Formula: see text]. The exact analytical solutions of the Euler–Lagrange equations have been constructed. The study of the obtained solutions shows that the light ray bends to the axis [Formula: see text] along which the effective refractive index increases.

NUMERICAL SOLUTIONS FOR 2D UNSTEADY LAPLACE-TYPE PROBLEMS OF ANISOTROPIC FUNCTIONALLY GRADED MATERIALS

The time-dependent Laplace-type equation of variable coefficients for anisotropic inhomogeneous media is discussed in this paper. Numerical solutions to problems which are governed by the equation are sought by using a combined Laplace transform and boundary element method. The variable coefficients equation is transformed to a constant coefficients equation. The constant coefficients equation after being Laplace transformed is then written in a boundary-only integral equation involving a time-free fundamental solution. The boundary integral equation is therefore employed to find the numerical solutions using a standard boundary element method. Finally the numerical results are inversely transformed numerically using the Stehfest formula to obtain solutions in the time variable. Some problems of anisotropic functionally graded media are considered. The results show that the combined Laplace transform and boundary element method is accurate and easy to implement.

Effective Medium Theory for Embedded Obstacles in Elasticity with Applications to Inverse Problems

We consider the time-harmonic elastic wave scattering from a general (possibly anisotropic) inhomogeneous medium with an embedded impenetrable obstacle. We show that the impenetrable obstacle can be effectively approximated by an isotropic elastic medium with a particular choice of material parameters. We derive sharp estimates to rigorously verify such an effective approximation. Our study is strongly motivated by the related studies of two challenging inverse elastic problems including the inverse boundary problem with partial data and the inverse scattering problem of recovering mediums with buried obstacles. The proposed effective medium theory readily yields some interesting applications of practical significance to these inverse problems.

Поверхностные электромагнитные волны Дьяконова на границе раздела анизотропных двуосных кристаллов

A general approach to analysis of Dyakonov surface waves at the interfaces of anisotropic biaxial crystals is proposed, taking into account arbitrary spatial orientation of the media tensors principal axes. This approach is based on the operator representation of macroscopic Maxwell equations corresponding to the quantum-mechanical equations for photon states in an inhomogeneous anisotropic media. Surface wave dispersion law is investigated in the most general case. It is established that the interface eigenmode dispersion is closely related to the dispersion of bulk waves in the partnering media, which is a specific feature of Dyakonov waves. The electromagnetic field spatial distribution is investigated in the direction orthogonal to the boundary plane. The surface wave angular existence domain is determined. It’s dependencies on the rotation angles of the media optical axes are studied as well.

Dyakonov Surface Waves at the Boundary of Anisotropic Biaxial Crystals

A general approach to analysis of Dyakonov surface waves at the interfaces of anisotropic biaxial crystals is proposed, taking into account arbitrary spatial orientation of the media tensors principal axes. This approach is based on the operator representation of macroscopic Maxwell equations corresponding to the quantum-mechanical equations for photon states in an inhomogeneous anisotropic media. Surface wave dispersion law is investigated in the most general case. It is established that the interface eigenmode dispersion is closely related to the dispersion of bulk waves in the partnering media, which is a specific feature of Dyakonov waves. The electromagnetic field spatial distribution is investigated in the direction orthogonal to the boundary plane. The surface wave angular existence domain is determined. It's dependencies on the rotation angles of the media optical axes are studied as well. Keywords: electromagnetic surface waves, interface photon states, anisotropic media, biaxial materials.

3-D Numerical Mode Matching Method for Off- Centered Electromagnetic Well Logging Tools in Noncircular Vertical Borehole and Invasion Zones in Multilayered Media

In electromagnetic (EM) well logging for petroleum exploration, off-centered tools within a noncircular borehole and invasion zones in multilayered media represent a challenging 3-D problem for traditional numerical methods. The 3-D finite-element numerical mode-matching (FNMM) method and the spectral numerical mode-matching (SNMM) method are developed to address this problem in this work. The numerical mode-matching (NMM) methods reduce the original 3-D well logging problem into a series of 2-D open waveguide eigenvalue problems plus a 1-D layered medium problem, which can be analytically solved by a recursion procedure. These waveguide eigenvalue problems with anisotropic inhomogeneous media in the horizontal directions are solved numerically by the mixed finite-element method (MFEM) with the flexibility for the complex geometry and the mixed spectral element method (MSEM) with the exponential convergence. A general operator form of Maxwell’s equations is also derived to obtain the source excitation vector independent of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$z$ </tex-math></inline-formula> -axis. Therefore, the NMM solvers can effectively and accurately simulate the response of off-centered EM well logging tools in an irregular borehole in complex formations. The NMM methods are applied to simulate three formations including a five-layer isotropic homogeneous formation, an off-centered logging tool in a noncircular borehole with anisotropic invasion zones in a five-layer anisotropic formation, and the Oklahoma formation. Numerical experiments indicate that the NMM methods are highly efficient and accurate for the EM well logging models.

Spatial dynamics of light quasispin vector in an anisotropic medium with twisting

On the Poincar's sphere the spatial evolution of the light's polarization state has been studied quantitatively in the smoothly inhomogeneous anisotropic media with gyrotropy. The main features of the dependence on the sample?s thickness, the gyration vector and the azimuth of the polarization of the incident linearly polarized wave were revealed. An analytical solution of the system of coupled equations relative to Cartesian components of light's wave electric component obtained by transition to rotating coordinate system has been used for analysis. Keywords: Poincar' sphere, polarization evolution, medium with gyrotropy, rotating coordinate system.

Пространственная динамика вектора квазиспина света в анизотропной среде с кручением

On the Poincaré sphere the spatial evolution of the light’s polarization state has been studied quantitatively in the smoothly inhomogeneous anisotropic media with gyrotropy. The main features of the dependence on the sample’s thickness, the gyration vector and the azimuth of the polarization of the incident linearly polarized wave were revealed. An analytical solution of the system of coupled equations relative to Cartesian components of light's wave electric component obtained by transition to rotating coordinate system has been used for analysis.

Propagation of electromagnetic waves in stationary anisotropic media

In this paper we have investigated the fundamental properties of the solutions of Maxwell’s equations describing free electromagnetic waves in stationary anisotropic media, with tensor characteristics along the Z axis. A matrix of coefficients, the structure of the matrix of Maxwell’s equations, dispersion equations and indicatrix curves for a transparent anisotropic one-dimensional inhomogeneous conductive medium have been obtained. Exact analytical solutions in case of homogeneous anisotropic media are constructed on the basis of the matrix structure.

Surface waves in a Goubau line filled with nonlinear anisotropic inhomogeneous medium

The propagation of monochromatic nonlinear symmetric hybrid surface waves in a cylindrical nonlinear anisotropic inhomogeneous metal-dielectric waveguide is considered. The physical problem is reduced to solving a transmission eigenvalue problem for a system of ordinary differential equations. Spectral parameters of the problem are propagation constants of the waveguide. The setting is reduced to a new type of nonlinear eigenvalue problem and an analytical method of its solution is developed. The propagation modes are found that have not been previously reported in the literature. For the numerical solution, a method is proposed based on solving an auxiliary Cauchy problem (a version of the shooting method). As a result of comprehensive numerical modeling, new propagation regimes are discovered.

Collective Synchronous Spiking in a Brain Network of Coupled Nonlinear Oscillators.

A network of propagating nonlinear oscillatory modes (waves) in the human brain is shown to generate collectively synchronized spiking activity (hypersynchronous spiking) when both amplitude and phase coupling between modes are taken into account. The nonlinear behavior of the modes participating in the network are the result of the nonresonant dynamics of weakly evanescent cortical waves that, as shown recently, adhere to an inverse frequency-wave number dispersion relation when propagating through an inhomogeneous anisotropic media characteristic of the brain cortex. This description provides a missing link between simplistic models of synchronization in networks of small amplitude phase coupled oscillators and in networks built with various empirically fitted models of pulse or amplitude coupled spiking neurons. Overall the phase-amplitude coupling mechanism presented in the Letter shows significantly more efficient synchronization compared to current standard approaches and demonstrates an emergence of collective synchronized spiking from subthreshold oscillations that neither phase nor amplitude coupling alone are capable of explaining.