An Analytical Elastic Solution for Right-Angle Trapezoidal Opening in Steeply Inclined Coal Seam

In the process of underground mining, steeply inclined rocks or coal seams are often encountered, forming the openings of right-angle trapezoid. According to the geological conditions of a mining project in China, an analytical elastic solution of stress and displacement around right-angle trapezoidal opening in a homogeneous, isotropic, and linear elastic geomaterial is presented, which is based on the evaluation of the conformal mapping representation by an appropriate numerical calculation and the complex potential functions. The different results from other shaped openings are shown as follows. In a right-angle trapezoidal opening, the maximum displacements of roof falling occur on the low side, while the most horizontal displacements on the low side are around the roof and the most horizontal displacements on the high side are around the middle of the high side in this opening. These results are also compared with the numerical calculations in FLAC software, illustrating that the solution may be easily applied to rock mechanics or rock engineering for understanding the deformation of floor heave and roof falling down. The solution is also suitable for optimum design of bolt supporting in a right-angle trapezoidal opening, which is different from the traditional concept of symmetrical bolt supporting. Finally, a methodology is proposed for the estimation of conformal mapping coefficients for a given cross-sectional shape of an opening without symmetrical axis.

Automatic Registration of the Nascence and Propagation of Cracks in Electrically Conductive Materials

A correlation is established between the length of the propagating crack in electrically conductive flat specimens and the change in the field of electric potentials over the surface of the specimen when an electric current is passed through it. The experimental data correspond to the analytical solution in a two-dimensional formulation obtained by means of a conformal mapping. Work can be attributed to the field of converting mechanical values into electrical ones. This transformation simplifies the process of recording mechanical processes and provides a convenient form of their control. The work is aimed at automatic control of the process of fracture of electrically conductive materials.

2D Ship-Like Section Slamming Pressure Numerical Analysis by Conformal Mapping

In this paper, a method to predict slamming pressures and pressure distribution at the time of water entry for 2D sections is presented. The mathematical model is based on the Schwarz-Christoffel conformal mapping method. This conformal mapping technique has been used to calculate slamming loads during water entry. The pile-up of water during motion is also considered and an alternative pile-up coefficient is assumed against Wagner’s generalized method. A simplified and accurate method is presented, which does not include non-linear terms and jet flow in the calculated pressure distribution on monotonically increasing 2D sections like wedge shapes. Comparison with real ship sections has been done to show accuracy of the results. Finally, a simple yet powerful method is obtained to aid the initial design stage of ships.

Conformal Mapping-Based Discrete Vortex Method for Simulating 2-D Flows around Arbitrary Cylinders

A novel technique based on conformal mapping and the circle theorem has been developed to tackle the boundary penetration issue, in which vortex blobs leak into structures in two-dimensional discrete vortex simulations, as an alternative to the traditional method in which the blobs crossing the boundary are simply removed from the fluid field or reflected back to their mirror-image positions outside the structure. The present algorithm introduces an identical vortex blob outside the body using the mapping method to avoid circulation loss caused by the vortex blob penetrating the body. This can keep the body surface streamlined and guarantees that the total circulation will be constant at any time step. The model was validated using cases of viscous incompressible flow passing elliptic cylinders with various thickness-to-chord ratios at Reynolds numbers greater than Re = 1 × 105. The force and velocity fields revealed that this boundary scheme converged, and the resultant time-averaged surface pressure distributions were all in excellent agreement with wind tunnel tests. Furthermore, a flow around a symmetrical Joukowski foil at Reynolds number Re = 4.62 × 104, without considering the trailing cusp, was investigated, and a close agreement with the experimental data was obtained.

Near Centimeter-scale FT-MIR Spectrometer based on NZIM Perfect Absorption using Inverted TanCirc Conformal Mapping Geometry

A long sought objective of MEMS research within the oil & gas industry has been the realization of “FT-IR on a chip,” which could hold the potential to migrate laboratory grade chemical spectroscopy into downhole sensor suites. A fundamental obstacle to this research has been the cooling demands of conventional technologies which conflict with the miniaturized sensor volumes required in downhole logging systems and environmental conditions that routinely exceed 125 • C. Near centimeter scale spectroscopic devices are required by a majority of downhole sensor suites, which stands in stark contrast to multiple-decimeter size conventional devices. Here we report a near-centimeter scale FT-MIR ATR spectrometer compatible with downhole volumetric and temperature constraints. The spectrometer is based upon a high-temperature broadband mid-infrared metasurface detector/source combination derived from a geometric inversion of a set of conformal mapping contours. The metasurface elemental structure is derived from a geometric inversion of the canonical TanCirc conformal mapping contours and was found to exhibit a near zero index metamaterial (NZIM) behavior over a spectral range of interest for downhole chemical spectroscopy. The NZIM properties of the metasurface lead to an absorption phenomenon characterized by surface plasmon resonances which confine the absorption mechanism within the ultrathin (λ /300) metasurface plane and make the absorption properties of the microbolometer design relatively insensitive to the material properties of the remaining laminae. This unusual feature allows the metasurface to be integrated on a single VO 2 material thermometric layer and operated at elevated downhole temperatures despite corresponding to the VO2 metal-insulator-transition (MIT) region. Within this transition region however the VO2 layer exhibits a substantially beneficial property in that the VO 2 layer exhibits more than an order of magnitude enhancement in its ambient thermometric properties, leading to an uncooled microbolometer design with predicted maximum detectivity D * = 1.5 × 10 10 cm √ Hz/W and noise equivalent difference temperature NEDT of 1 mK at a modulation frequency of 500 Hz. A sub-millimeter scale thermal infrared source with intrinsic mid-infrared band-limited emission is formed from the same cellular geometric building block, enabling the spectrometer miniaturization. These performance parameters compare well with lower tier laboratory grade FT-MIR spectroscopic instruments and could represent a significant step in the effort towards deploying miniaturized high-temperature mid-infrared spectroscopy into oilfield downhole logging applications.

Wavefront detection method for orbital angular momentum modes based on conformal mapping-spatial phase shifting interferometry

The orbital angular momentum (OAM) of light has garnered significant interest in recent years owing to its various applications, and extensive creative research has been conducted to generate OAM. However, the particular helical phase structure of an optical vortex leads to non-smooth and discontinuous phase profiles and hinders the accurate recovery of the phase distribution of the vortex beam. Significantly, the existence of a wavefront dislocation leads to the failure of the traditional phase unwrapping algorithm. At the same time, it is essential to detect the wavefront of OAM modes in real-time for free-space optical communication and optical precision measurement. Therefore, we designed conformal mapping-spatial phase-shifting interferometry and achieved rapid and high-precision wavefront measurements for the OAM modes. The wavefront of the OAM modes with a topological charge of 1,2,4 and 6 were measured, respectively. The results were significantly consistent with the anticipated results based on simulations. This study revealed the mathematical mechanism behind the forked fringe patterns and presented a method for demodulating the helical wavefront from the forked fringe patterns.

On conformally reducible pseudo-Riemannian spaces

The present paper studies the main type of conformal reducible conformally flat spaces. We prove that these spaces are subprojective spaces of Kagan, while Riemann tensor is defined by a vector defining the conformal mapping. This allows to carry out the complete classification of these spaces. The obtained results can be effectively applied in further research in mechanics, geometry, and general theory of relativity. Under certain conditions the obtained equations describe the state of an ideal fluid and represent quasi-Einstein spaces. Research is carried out locally in tensor shape.