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We present a comprehensive analysis of the anomalous Goos–Hänchen (GH) displacement that occurs during the reflection of light beams at an interface between air and an anisotropic medium. This analysis also applies to the Imbert–Fedorov effect. Our study suggests that the anomalous GH displacement is primarily caused by polarization-dependent abnormal interference effects between the direct and cross-reflected light fields. Using the interface between air and a type II Weyl semimetal as an example, we provide a clear physical explanation for the relationship between spin-dependent abnormal interference effects and anomalous GH displacement. We demonstrate that spin-dependent constructive interference leads to a reduction in the GH displacement of the total reflected light field, while spin-dependent destructive interference results in an increase in the GH displacement of the total reflected light field.

A novel and efficient modeling approach has been developed for simulating the responses of triaxial induction logging (TIL) in layered biaxial anisotropic (BA) formations. The core of this innovative technique lies in analytically calculating the primary fields within a homogeneous medium and approximating the scattered fields within layered formations. The former involves employing a two-level subtraction technique. Initially, the first-level subtraction entails altering the direction of the Fourier transform to mitigate the integral singularity of the spectral fields, particularly in high-angle and horizontal wells. Conversely, the second-level subtraction aims to further optimize integral convergence by creating an equivalent unbounded transverse isotropic (TI) formation and eliminating the corresponding spectral fields. With the two-level subtractions, the convergence of the spectral field has been enhanced by more than six orders of magnitude. Additionally, a strict recursive algorithm and approximation method are developed to compute the scattered fields in layered biaxial anisotropic media. The rigorous algorithm is based on a modified amplitude propagator matrix (MAPM) approach and serves as the benchmark for the approximation method. In contrast, the approximation method exploits the similarity between the spectral scattered field of the TI medium and the BA medium, establishing corresponding equivalent layered TI models for each magnetic component. Since the scattered field in TI models only involves a one-dimensional semi-infinite integral, the computational complexity is significantly reduced. Numerical simulation examples demonstrate that the new simulation method is at least two orders of magnitude faster than the current modeling approach while maintaining computational precision error within 0.5%. This significantly improved simulation efficiency provides a solid foundation for expediting the logging data processing.

Fractures significantly impact oil and gas production by altering the mechanical properties of subsurface rocks. This study investigates the forward and inverse problems associated with monoclinic anisotropic media, which arise from the presence of two sets of inclined fractures within an isotropic matrix. We first derive the stiffness matrix for this monoclinic medium by applying Schoenberg's linear slip theory. Subsequently, we employ the point spread function and the steady phase method to derive and simplify the reflection coefficient equation for seismic waves interacting with this complex medium. Finally, we verify the rationality of the method by using synthetic seismic records and actual seismic records. The proposed equations and inversion techniques offer a more accurate framework for characterizing monoclinic anisotropic media affected by inclined fractures, thereby enhancing the interpretation of subsurface structures in hydrocarbon exploration.

For model reduction techniques, there have been relatively few studies performed regarding the forward modeling of anisotropic media in comparison to transient electromagnetic (TEM) forward modeling of isotropic media. The transient electromagnetic method (TEM) responses after the current has been turned off can be represented as a homogeneous ordinary differential equation (ODE) with an initial value, and the ODE can be solved using a matrix exponential function. However, the order of the matrix exponential function is large and solving it directly is challenging, thus this study employs the Shift-and-Invert (SAI-Krylov) subspace algorithm. The SAI-Krylov subspace technique is classified as a single-pole approach compared to the multi-pole rational Krylov subspace approach. It only takes one LU factorization of the coefficient matrix, along with hundreds of backward substitutions. The research in this paper shows that the anisotropic medium has little effect on the optimal shift${\\gamma _{opt}}$and subspace order m. Furthermore, as compared to the mimetic finite volume method (SAI-MFV) of the SAI-Krylov subspace technique, the method proposed in this paper (SAI-FEM) can further improve the computing efficiency by roughly 13%. In contrast to the standard implicit time step iterative technique, the SAI-FEM method does not require discretization in time, and the TEM response at any moment within the off-time period can be easily computed. Next, the accuracy of the SAI-FEM algorithm was verified by 1D solutions for an anisotropic layer model and a 3D anisotropic model. Finally, the electromagnetic characteristics of the anisotropic anomalous body of the center loop device and separated device of the airborne transient electromagnetic method were analyzed, and it was found that horizontal conductivity has a considerable influence on the TEM response of the anisotropic medium.

We consider the interior transmission eigenvalue problem corresponding to the scattering for an anisotropic medium of the scalar Helmholtz equation in the case where the boundary ∂Ω is split into two disjoint parts and possesses different transmission conditions. Using the variational method, we obtain the well posedness of the interior transmission problem, which plays an important role in the proof of the discreteness of eigenvalues. Then we achieve the existence of an infinite discrete set of transmission eigenvalues provided that n ≡ 1, where a fourth order differential operator is applied. In the case of n ≢ 1, we show the discreteness of the transmission eigenvalues under restrictive assumptions by the analytic Fredholm theory and the T-coercive method.

Determining the microseismic event location is crucial in various fields of science such as hazard mitigation, exploration of new fossil energy sources, and others. However, in determining the source location, several problems arise, namely the determination of the source location that is not appropriate due to limited data. To determine the exact location of the event requires a lot of microseismic recording data. We developed a time reverse modeling method for elastic waves. The data used is synthetic data that is generated from forward modeling which seems to originate a source that is located in subsurface at 1,300 m depth. The seismic velocity model used is a layered seismic velocity model with the assumption that every layers is unabsorbed layers. Data from the wavefield recording on the surface is propagated back to the source. From the study, this was found that the microseismic event was at a depth of 1,300 m.

Fraunhofer diffraction of an electromagnetic wave by a slit on an opaque screen located between a vacuum and an anisotropic medium with an open wave vector surface (WVS) is considered. A formula for diffraction in a material medium is obtained, including the case of diffraction into a uniaxial anisotropic medium in the absence of absorption, and diffraction minima and maxima formation is studied.

The two-dimensional system of Navier–Stokes equations in a medium with anisotropic variable viscosity and periodic small obstacles is considered. It is proved that the trajectory attractors of the system tend in a certain weak topology to the trajectory attractors of the homogenized system of Navier–Stokes equations with an additional potential in a medium without obstacles.

Currently, new methods of energy transformation are of interest. The results of the researches in the directions of anisotropic transformation of electric energy show that the existing devices are characterized by a small coefficient of transformation. This is primarily due to the use of unipolar anisotropic materials and the appearance, as a result, in their volumes the electric vortices that are characterized by a laminar flow. This article proposes a new method of energy transformation, which is carried out by means of vortices with turbulent character of flow. According to our studies, such vortices appear in devices working on the basis of bipolar anisotropic materials. Depending on the characteristics of such material, they can be used as generators of electricity, heat, and cold. Their work is based on the transformation of the electric current by the bipolar anisotropic electric conductive medium and their further interaction with the external energy environment. These anisotropic materials in selected crystallographic directions are characterized by different p‐ and n‐types of conductivity providing the presence of ohmic contact between the layers. The leakage of the external electrical current of the sinusoidal form through this rectangular plate causes the appearance in its volume the electric current vortices with turbulent character of the current. For the first time it is shown that such a method of energy transformation is an effective mechanism that transfers energy between the external environment and the anisotropic plate. The results of the researches in the directions of anisotropic transformation of electric energy show that the existing devices are characterized by a small coefficient of transformation. This article proposes a new method of energy transformation which is carried out by means of vortices with turbulent character of flow. According to our studies, such vortices appear in devices working on the basis of bipolar anisotropic materials. Their work is based on the transformation of the electric current by the bipolar anisotropic electric conductive medium and their further interaction with the external energy environment.

A new numerical approach is presented to calculate the Green’s function for an anisotropic multi-layered half space. The formulation is explicit and unconditionally stable. It imposes no limit to the thickness of the layered medium and the magnitude of the frequency. In the analysis, the Fourier transform and the precise integration method (PIM) are employed. Here, the Fourier transform is employed to transform the wave motion equation from the spatial domain to the wavenumber domain. A second order ordinary differential equation (ODE) is observed. Then, the dual vector representation of the wave motion equation is used to reduce the second order ODE to first order. It is solved by the PIM. Finally, the Green’s function in the wavenumber domain is obtained. For the evaluation of the Green’s function in the spatial domain, the double inverse Fourier transform over the wavenumber is employed to derive the solutions. Especially, for the transversely isotropic medium, the double inverse Fourier transform can be further reduced to a single integral by the cylindrical polar coordinate transform. Numerical examples are provided. Comparisons with other methods are done. Very promising results are obtained.