Ellipsoidal Surface Research Articles

Modeling the Approach of Non-mated Rock Fracture Surfaces Under Quasi-static Normal Load Cycles

This paper examines the quasi-static behaviour of a non-mated rock fracture during the first and second loading cycles. Asperity deformation, substrate deformation, asperity interaction and gouge production are included in the modelling. The composite topography assumption is revisited, and the irregular fracture topography is idealized by representing the asperities as ellipsoidal surfaces. It is shown that the modelling can capture the closure behaviour and hysteresis response of rock fractures on both the first and second loading–unloading cycles with an acceptable accuracy. The model is then used to examine sensitivity of deformation behaviours to mechanical properties. It is shown that Young’s modulus of the rock is the most sensitive parameter with regard to the estimation of the closure behaviour of joints, followed by the compressive strength of the material, while the deformation is independent of the Poisson’s ratio variations in the range of ± 0.1. Moreover, while the closure behaviour is sensitive to variations in the joint roughness coefficient on the first loading cycle, this sensitivity reduces considerably on the second cycle.

Electrical conductivity of a thin film in the case of an arbitrarily oriented ellipsoidal isoenergetic surface of a conductor

The static electrical conductivity of a thin conducting film is calculated using the kinetic approach. The model of diffuse boundary conditions is used as boundary conditions for the charge carrier distribution function. The isoenergetic surface of the conducting film material is an ellipsoid of rotation that is arbitrarily oriented in the pulse space. The volume of the conductor is dominated by scattering on impurity atoms and on dislocations of electrons (holes). The electrical conductivity of the film is analyzed depending on the angles that determine the direction of the ellipsoid axis of rotation, and a comparison with the results for the case of a macroscopic sample is presented. A comparison is made with experimental data for the electrical conductivity of Bi and Si thin films.

Geoid undulation prediction using Gaussian Processes Regression: A case study in a local region in Turkey

In order to convert ellipsoidal heights obtained by the Global Navigation Satellite System (GNSS) to orthometric heights, it is necessary to know the distance between the ellipsoidal and geoid surface, called the geoid undulation. The geoid undulation can be predicted using emerging mathematics tools and algorithms. The objective of this study was to develop a model for predicting the geoid undulation using Gaussian process regression (GPR), one of the soft machine learning algorithms having different covariance functions. This method was then compared with the radial basis function neural network (RBFNN), generalized regression neural network (GRNN), and the interpolation method of inverse distance to a power (IDP) with the power of 1, 2, 3, 4, and 5. First, 70 % of GNSS/leveling data (422 points) were used in the training phase. The remaining 185 points were used as testing data to check the effectiveness of the constructed model. In the GPR modeling, ten covariance functions (Materniso d= 1, 3, 5; Maternard d= 1, 3, 5; SEiso; SEard; RQiso; and RQard) were tested for prediction on this dataset. The GPR based on the Materniso (d = 1) covariance function model was introduced as an effective method for predicting geoid undulation and provided the best results (RMSE = 8.32 cm, MAE = 5.51 cm, R2 = 0.98968) when compared with the other developed GPR models. In addition, the statistical findings showed that the accuracy of all the GPR models was also better in predicting geoid undulation than the RBFNN, GRNN, and IDP with the power of 1, 2, 3, 4, and 5.

A new approach for calculating the slope length factor in the Revised Universal Soil Loss Equation

This paper proposes a flow-path-network-triangular-facet-network (FPN_TFN) method based on a flow path network (FPN) and triangular facet network (TFN) constructed from a digital elevation model (DEM) to enhance the calculation of the slope length factor (<i>L</i>) in the Revised Universal Soil Loss Equation (RUSLE). First, the slope length was derived from a FPN, which is traced by the flow path network model. Then, a TFN was used to estimate the slope, and finally, <i>L</i> was obtained by combining slope length with slope using the equation in RUSLE. Experiments were conducted to (1) quantitatively validate the accuracy of the FPN_TFN method using 5 m DEMs that were generated from four mathematical surface models; and (2) qualitatively evaluate the error by using a 30 m DEM of a real-world mountainous region located in central Ganzi Tibetan Autonomous Prefecture of Sichuan Province, China. The performance of the four algorithms proposed by Moore and Burch (1986a), Böhner and Selige (2006), Desmet and Govers (1996), and Hickey (2000) were compared with that of the new approach. In the former two algorithms, two single flow direction and four multiple flow direction algorithms were selected for calculating the upslope contributing area (CA) and specific catchment area (SCA), such that 14 algorithms would be used for the comparison. The <i>L</i> estimates computed with the proposed algorithm were closer to the theoretical value. The root mean standard errors (RMSE) of the proposed algorithm are 79%, 96%, 69%, and 90% less than the same errors generated by the best of 14 comparison algorithms on the ellipsoid, inverse ellipsoid, saddle, and plane, respectively. The mean absolute errors (MEs) of the proposed algorithm are 0.83, 0.14, 1.00, and 0.39 m using the aforementioned order of surfaces, obviously lower than that of the comparison algorithms. The spatial error of the proposed algorithm is distributed between –1 and 1 for the inverse ellipsoid and plane surfaces, and falls mainly between –1 and 1 on the ellipsoid and saddle surfaces. They are still lower than that for the other algorithms. The new approach also produces better spatial distributions of <i>L</i> on the real-world mountainous region.

Antlers Are Shaped as Elements of Concave Surfaces that Reflect and Concentrate Incoming Sounds

In 1917 D’Arcy Thompson observed that the pair of antlers of a red deer, a sambur or a wapiti Make up between them a single surface that is more or less spherical or occasionally an ellipsoidal surface. The palmated antlers of the moose are explored to provide an exam-ple of how antlers can be instrumental in lowering the threshold sensitivity of hearing.

Mathematical modeling of triode system on the basis of field emitter with ellipsoid shape

In this paper the mathematical modeling of the triode emission axially symmetric system on the basis of field emitter is considered. Emitter is an ellipsoid of revolution, anode is a confocal ellipsoidal surface of revolution. Modulator is a part of the ellipsoidal surface of revolution, confocal with the cathode and anode surfaces. The boundary-value problem for the Laplace's equation in the prolate spheroidal coordinates with the boundary conditions of the first kind is solved. The variable separation method is applied to calculate the axisymmetrical electrostatic potential distribution. The potential distribution is represented as the Legendre functions expansion. The expansion coefficients are the solution of the system of linear equations. All geometrical dimensions of the system are the parameters of the problem.

Accuracy analysis and verification of the method for calculation of geodetic problem on earth ellipsoid surface

The method for calculation of geodetic problem on earth ellipsoid surface is the basic premise for the realization of high-precision marine delimitation technology. Based on the analysis of the construction principle of the existing solution of geodetic problem method, the algorithm for direct geodetic problem solution(DGPS) based on nested coefficient method and Bessel’s inverse geodetic problem solution(IGPS), which are suitable for various distance solutions, are optimized. Then, the experimental verification scheme of the solution of geodetic problem is designed. Finally, the crossing and reducing accuracy of different calculation examples are verified by different geodetic problem solution methods, and the experiment shows that: the verified accuracy of point-position plotting, distance calculation and azimuth calculation meet the requirements of the indexes of high-precision marine delimitation.

An RBF-Based h-Adaptive Cartesian Grid Refinement Method for Arbitrary Single/Multi-Body Hull Modeling and Reconstruction

Complex single/multi-body structures are generally found in ship and ocean engineering. They have the smooth, sharp, concave, and convex surface features in common. Precise modeling of the structures is the basis of numerical simulation. However, the most widely used explicit modeling method requires considerable manual operations. The result is also difficult to reproduce. Therefore, this paper presents a Radial basis function (RBF) based hierarchical (h-) adaptive Cartesian grid method. The RBF is introduced for arbitrary implicit modeling over the Cartesian framework. Thanks to its natural properties, the RBF is easy to use, highly automated, and only needs a set of scatter points for modeling. The block-based h-adaptive mesh refinement (AMR) combined with the RBF aims to enhance the local grid resolution. It accelerates the calculation of intersecting points compared with the uniform Cartesian grid. The accuracy, efficiency, and robustness of the proposed method are validated by the simulation of the 3D analytical ellipsoidal surface and the unclosed conic surface. To select suitable parameters, we thoroughly investigated the uncertainty factors including sample points, RBF functions, and h-AMR grids. The simulation results of the single/multi-body Wigley hull and KCS hull forms verified the proper selection of the factors and the feasibility of our method to solve practical problems.

Метод розрахунків значень геодезичної дальності, азимута і висоти точки в задачах зовнішньої балістики

У статті розглянуто метод розрахунку значень геодезичної дальності (по поверхні земного еліпсоїда), азимута і висоти точки, координати якої задані в стартовій системі координат (ССК) при вирішенні завдань зовнішньої балістики. Початок ССК визначено в геодезичній системі координат геодезичною широтою, довготою і висотою. Розв'язана задача перетворення значень координат точки з ССК в геодезичну систему координат (ГСК) шляхом проведення послідовних перетворень з ССК в прямокутну геоцентричну систему координат (ПГЦ СК), а потім з ПГЦ СК в ГСК. Особливістю перетворення координат з ССК в ПГЦ СК є додатковий облік складових повного ухилення прямовисній лінії в точці початку ССК. Представлений вивід матриці перетворення координат із ССК в ПГЦ СК. Проведено аналіз впливу неврахування ухилення прямовисній лінії в точці початку ССК на точність обчислень значень геодезичної дальності, азимута і висоти точки. Представлені графіки залежності відносних похибок обчислення значень зазначених величин від величини повного ухилення прямовисній лінії в точці початку ССК. Представлені оцінки абсолютних похибок. Зроблено висновок про необхідність додаткового обліку складових повного ухилення прямовисній лінії в точці початку ССК при проведенні точних обчислень значень геодезичної дальності, азимута і висоти точки, координати якої задані в ССК.

A three-dimensional slope stability analysis method based on finite element method stress analysis

A new three-dimensional slope stability analysis method based on stress calculation by the finite element method is proposed. This method conforms to the uniform assumption of the sliding direction of the rigid limit equilibrium method and can easily be used calculate the three-dimensional sliding direction. By using this method, both the local and global safety factors of the slip surface can be calculated according to the integral mean value theorem. Additionally, an improved ellipsoidal slip surface construction method is proposed for roughly searching for the minimum safety factor and the critical slip surface. A two-stage particle swarm optimization is used to search for the critical slip surface more accurately. Two numerical examples of fixed slip surfaces are provided to confirm the effectiveness of the proposed method and its applicability to nonspherical slip surfaces. The influence of the slip surface mesh grid size, boundary conditions, Poisson's ratio, and dilatancy angle on the calculation results is revealed. Another two numerical examples of searching for the critical slip surface are then provided, and the results of the proposed method are compared with the results of the strength reduction method to confirm the validity of the proposed method.

Regionalized tissue fluidization is required for epithelial gap closure during insect gastrulation

Many animal embryos pull and close an epithelial sheet around the ellipsoidal egg surface during a gastrulation process known as epiboly. The ovoidal geometry dictates that the epithelial sheet first expands and subsequently compacts. Moreover, the spreading epithelium is mechanically stressed and this stress needs to be released. Here we show that during extraembryonic tissue (serosa) epiboly in the insect Tribolium castaneum, the non-proliferative serosa becomes regionalized into a solid-like dorsal region with larger non-rearranging cells, and a more fluid-like ventral region surrounding the leading edge with smaller cells undergoing intercalations. Our results suggest that a heterogeneous actomyosin cable contributes to the fluidization of the leading edge by driving sequential eviction and intercalation of individual cells away from the serosa margin. Since this developmental solution utilized during epiboly resembles the mechanism of wound healing, we propose actomyosin cable-driven local tissue fluidization as a conserved morphogenetic module for closure of epithelial gaps.

Problem of the Vertical Deflection in High-Precision Inertial Navigation

The paper addresses the systematic error of an inertial navigation system, caused by the discrepancy between the plumb line and the normal to the reference ellipsoid surface. The methods of this discrepancy estimation, and their use for correcting the output data of inertial navigation systems are studied.

Magnetotransport properties and topological phase transition in NaCd4As3

We report the magnetotransport properties and the electronic structure of $\mathrm{Na}{\mathrm{Cd}}_{4}{\mathrm{As}}_{3}$ single crystal. The electric resistivity of $\mathrm{Na}{\mathrm{Cd}}_{4}{\mathrm{As}}_{3}$ exhibits a metallic behavior. Clear Shubnikov--de Haas (SdH) oscillations have been observed at low temperatures and high magnetic fields. A single fundamental frequency and small cyclotron effective mass are obtained from the analysis of SdH oscillations, which originate from an ellipsoidal Fermi surface. The Hall resistivity reveals the multiband nature at low temperature and the electron carriers dominate the transport properties. First-principles calculations show that the electronic structure of $\mathrm{Na}{\mathrm{Cd}}_{4}{\mathrm{As}}_{3}$ undergoes a topological phase transition from topological crystalline insulator (TCI) to topological insulator (TI) induced by the structural phase transition. The breaking of mirror symmetry is considered to play a key role in the topological phase transition. Our findings show that $\mathrm{Na}{\mathrm{Cd}}_{4}{\mathrm{As}}_{3}$ is a suitable platform to study the effect of mirror symmetry on the band topology of TCI.

Profile error compensation in precision grinding of ellipsoid optical surface

Optical glass elements with the ellipsoidal surface are used in many critical aerospace instruments, such as satellites, telescope and cameras. Their optical performance is mainly affected by profile accuracy and surface quality. In this paper, a rectangular ellipsoid surface is precisely ground on a BK7 optical glass blank by grating scanning grinding path with a three-axis CNC precision surface grinder. A profile error compensation procedure for ellipsoidal grinding is proposed based on the error analysis about the primary error sources in the XY and YZ projection planes during the grinding process. The mathematical prediction models of the wheel arc profile error, the measurement and wear error of the grinding wheel radiuses are established. By applying the proposed error compensation procedure, the profile accuracy of the ellipsoidal surface was improved from 4 μm to 2 μm in the XY plane, and improved from 15 μm to 5 μm in the YZ plane.

Mitigation of Geostationary Lightning Mapper Geolocation Errors

AbstractThe geolocation of lightning flashes observed by spaceborne optical sensors depends upon a priori assumptions of the cloud-top height (or, more generally, the height of the radiant emitter) as observed by the satellite. Lightning observations from the Geostationary Lightning Mappers (GLMs) on Geostationary Operational Environmental Satellite 16 (GOES-16) and GOES-17 were originally geolocated by assuming that the global cloud-top height can be modeled as an ellipsoidal surface with an altitude of 16 km at the equator and sloping down to 6 km at the poles. This method produced parallax errors of 20–30 km or more near the limb, where GLM can detect side-cloud illumination or below-cloud lightning channels at lower altitudes than assumed by the ellipsoid. Based on analysis of GLM location accuracy using a suite of alternate lightning ellipsoids, a lower ellipsoid (14 km at the equator, 6 km at the poles) was implemented in October and December 2018 for GLM-16 and GLM-17, respectively. While the lower ellipsoid slightly improves overall GLM location accuracy, parallax-related errors remain, particularly near the limb. This study describes the identification of optimized assumed emitter heights, defined as those that produce the closest agreement with the ground-based reference networks. Derived using the first year of observations from GOES-East position, the optimal emitter height varies geographically and seasonally in a manner consistent with known meteorological regimes. Application of the optimal emitter height approximately doubles the fraction of area near the limb for which peak location errors are less than half a GLM pixel.

A New Algorithm for Calculating the Flow Path Curvature (C) from the Square-Grid Digital Elevation Model (DEM)

This paper proposes a flow-path-network-based (FPN-based) algorithm, constructed from a square-grid digital elevation model (DEM) to improve the simulation of the flow path curvature (C). First, the flow-path network model was utilized to obtain an FPN. Then, a flow-path-network-flow-path-curvature (FPN-C) algorithm was proposed to estimate C from the FPN. The experiments consisted of two sections: (1) quantitatively evaluating the accuracy using 5 m DEMs generated from the mathematical ellipsoid and Gauss models, and (2) qualitatively assessing the accuracy using a 30 m DEM of a real-world complex region. The three algorithms proposed by Evans (1980), Zevenbergen and Throne (1987), and Shary (1995) were used to validate the accuracy of the new algorithm. The results demonstrate that the C value of the proposed algorithm was generally closer to the theoretical C value derived from two mathematical surfaces. The root mean standard error (RMSE) and mean absolute error (MAE) of the new method are 0.0014 and 0.0002 m, reduced by 42% and 82% of that of the third algorithm on the ellipsoid surface, respectively. The RMSE and MAE of the presented method are 0.0043 and 0.0025 m at best, reduced by up to 35% and 14% of that of the former two algorithms on the Gauss surface, respectively. The proposed algorithm generally produces better spatial distributions of C on different terrain surfaces.

DETERMINATION OF AN APPROPRIATE PROJECTION SYSTEM FOR FOREST AREAS IN TURKEY

Geometrical surfaces such as sphere and ellipsoid are considered as reference surfaces since there is no geometric shape that perfectly represents the earth when translating the earth into a map plane. Hence, on 3D reference surfaces, it is almost impossible to perfectly preserve the angle, direction and area properties and transfer them to a map plane without any deformations. The scaled topographic maps produced in our country under provision of map production regulations are conformal projections that do not preserve area properties but angle and shape properties. Area values calculated by projection coordinates cannot be considered the exact area values therefore, an area reduction is needed. Area values calculated by ignoring this situation in GIS based software do not represent the accurate area values on reference surfaces. The aim of this study is to determine the best area preserving projection for GIS applications in which area values are important. In this study, the real area values of 25 large-extent forest parcels are determined by employing the Danielsen method with geographical coordinates on ellipsoid surface. These parcels are also calculated by using the area-preserving projections available in ArcGIS software and are compared to their real area values.

Design and Analysis of Conformal Antenna for Future Public Safety Communications: Enabling Future Public Safety Communication Infrastructure

Future 4G wireless communication systems include, in their capabilities portfolio, emergency-specific needs, such as data support, broadband communication, and extremely high reliability. An emergency situation can be addressed with undoubtedly more chances of success if augmented information is enabled within the public safety communication novel capabilities. In this article, for a fully augmented information provision based on broadband transmission, a user-end (UE) communication-capabilities enhancement is addressed by deploying multiple antennas without compromising the portability and light weight of first-responder equipment. With this aim, we propose the design of a 4.9-GHz conformal antenna array at the rescuer side (integrated in a helmet) and evaluate its performance in terms of relative data rate gain. The conformal array design is based on traditional patch antennas that consider the need for deployment over an ellipsoidal surface. The antenna array is simulated and then built, and several parameter characterizations (bandwidth, radiation pattern, reflection coefficient, and MC) and measurements are undertaken to ensure the suitability of the design. Furthermore, an analysis of the specific absorption rate (SAR) is performed to guarantee that the exposure to electromagnetic fields is below the standardized levels.

Development of Lee’s exact method for Gauss–Krüger projection

Lee’s exact method was developed to enable the Gauss–Kruger (GK) projection to be implemented without iterative procedures via the expansion of the intermediate mapping (Thompson projection) into series approximations in terms of isothermal coordinates $$\left( \psi , \lambda \right) $$ for the forward mapping and GK coordinates $$\left( x, y \right) $$ for the reverse mapping. The straightforward procedures expressed by new formulas for both forward and reverse mapping of the GK projection were composed by three sequential steps which essentially reveal the mapping procedures and intrinsic properties of the GK projection: The first step of deriving the isothermal coordinates $$\left( \psi , \lambda \right) $$ from the geodetic coordinates $$\left( \varphi , \lambda \right) $$ specifies a conformal mapping (i.e., the Normal Mercator projection) of the Earth ellipsoid surface into the Euclidean plane excluding the South and North poles, and the subsequent two steps allow the GK projection to be expressed analytically via the elliptic functions and integrals. Based on the three-step procedure, the conformality and singularities over the entire ellipsoid of the Normal Mercator, Thompson and GK projections were analyzed and the fundamental domains of them were determined. With respect to the precision and efficiency, it was verified that the new algorithm and the complex latitude method had equivalent precision levels for the same orders of the third flatting n with the Kruger-n series from $$n^2$$ to $$n^{12}$$ but slower about 0.19 to $${0.21}\,\upmu \hbox {s}$$ than the Kruger-n series for a GK coordinates calculating. However, the new formulas provide series approximations for the forward mapping of the Thompson projection and projective transformations for the Normal Mercator, Thompson and GK projections.

Continuos Parameterization of the Median Surface of an Ellipsoidal Shell and Its Geometric Parameters

When analyzing the stress-strain state of thin-walled structural elements that have the shape of an ellipsoid, it becomes necessary to calculate the geometric characteristics of the ellipsoidal surface. When using the canonical ellipsoid equation, regions of uncertainty appear in the Cartesian coordinate system at the intersection points of the ellipsoid surface with the horizontal coordinate plane. To exclude these areas of uncertainty, we propose an expression of the radius vector of an ellipsoidal surface whose projections are functions of two parametric representations in mutually perpendicular planes. One of the planes is the vertical plane XOZ, and the other plane is the plane perpendicular to the axis O at the point with the x coordinate. The parameter T of the ellipse obtained from the intersection of the ellipsoid with the XOZ plane was chosen as the argument of the first parametric function. The argument of the second parametric function t is the parameter of an ellipse formed as a result of the intersection of an ellipsoidal surface with a plane perpendicular to the abscissa axis at a distance of x from the origin. The proposed representation of the ellipsoidal surface allowed us to exclude uncertainties at the intersection points of the ellipsoid with the HOWE coordinate plane. By differentiating the proposed radius-vector expression at an arbitrary point on an ellipsoidal surface, we obtain relations for the basis vectors of an arbitrary point and their derivatives represented by components in the same local basis. These relations are necessary for the development of algorithms for numerical analysis of deformation processes of engineering structures that have ellipsoidal surfaces.