Elliptic Cone Research Articles

A Novel Breast-Volume Self-Measurement Method with Improved Convenient and Accuracy

Breast volume is crucial for ensuring proper bra fit and comfort, significantly influencing women’s physiological and psychological well-being. This study aims to develop a novel method for breast-volume self-measurement, allowing women to accurately assess their breast volume without specialized equipment. We employed a geometric approximation of the breast as a combination of a partial elliptical cone and an irregular partial ellipsoid, leading to the formulation of a new volume equation. The method was validated against established standards, including the specimen drainage method and 3D scanning techniques. The findings revealed that our self-measurement approach achieved a relative error of only 3.8%, outperforming the 4.8% of 3D scanning and the 86.3% associated with traditional breast-volume equations. This innovative self-measurement technique enhances accuracy and serves as a practical solution for health and nutritional assessments, alongside body image evaluations. Its user-friendly nature positions it as a valuable tool for women’s health, particularly in personal fitness and ergonomic design.

Study on thermal penetration process of oil casing in shallow water environment by plasma arc based on heat source simulation and experiments

Abandoned offshore oil wellheads must be removed and plugged to protect the marine environment. Underwater plasma arc cutting holds the promise of playing a significant role in this work. However, much remains unknown about the thermal burn-off process of underwater plasma arcs for oil well casing. This work investigated the thermal penetration process of oil casing in a shallow water environment through experiments and heat source modeling simulations. A workpiece made of mild steel plate and cement block was considered as a small segment of the oil casing to be penetrated. The hole diameters, depths, and removal volumes of the workpieces were investigated experimentally. A combined heat source consisting of an elliptical cone heat source and a semi-ellipsoidal heat source was established based on the hole shape. The simulated and experimental hole profiles matched well. The results showed that the evolution of the hole shape of the workpiece was related to the motion state of the plasma arc and could be reflected by the variation of the arc voltage. Compared to steel plates, cement materials were easily removed by heat. These findings are favorable to promote the development of underwater plasma arc cutting in oil casing removal.

MATHEMATICAL MODEL OF A LARGE RESERVOIR, ESTABLISHING AND FINDING ITS MAIN ENERGY CHARACTERISTICS

Water plays an important role in the normal life of a person as a member of society. The presence or absence of fresh water reserves is one of the most important economic and social factors for the successful existence and development of any country in modern conditions. It is difficult to overestimate the importance of having large fresh water reserves for every sector of the economy. Ukraine has one of the largest such reserves in the world. Over time, as established by research and forecasts, the value and cost of fresh water will only increase. Therefore, preserving, protecting, properly utilizing and maintaining large, man-made water reservoirs is an important and always relevant task for the people of Ukraine. This strategy seems especially important in the current state of full-scale military aggression by the neighboring enemy. Thus, it seems reasonable to care about the preservation and proper use of fresh water in the context of the current geopolitical conflict. Water resources indeed play a critical role in all areas of the economy and daily life. Fresh water supplies are essential for agriculture, industry, drinking water for the population, as well as for energy and other industries. Ukraine, as a country with large freshwater reserves, has the opportunity to use this resource to ensure sustainable development and reduce dependence on imports. Water reservoirs can also play a strategic role in the face of possible threats and military conflicts. The storage and maintenance of water reservoirs is a necessary precautionary measure in case of any possible threats to infrastructure and access to fresh water. It is also important to develop and implement strategies for water conservation and efficient use to guarantee long-term resilience and sustainability in the water supply sector. The context of armed conflict can pose additional challenges to water management. It is therefore important to have contingency plans in place and to ensure the safety of water bodies in such circumstances. The development and improvement of water supply systems, as well as the implementation of effective water management strategies, will contribute to the sustainable development and conservation of an important natural resource, which is fresh water. Taking into account the sufficient importance and relevance of all these issues, in the following studies, a certain mathematical model of a large modern reservoir is proposed and investigated in the form of such a volumetric geometric structure as a spatially horizontally oriented elliptical half cylinder with the corresponding parameters that correspond to the specific dimensions of a real reservoir, such as the length, width and depth of filling such a structure with water.It is believed that the height (depth) of the filling can change, and this fact is essentially the most important factor in the corresponding change in the current volume of stored water and the potential energy of the accumulated liquid.The main results of this work are the possibility of determining the value of the current volume and the accumulated potential energy of such a reservoir. The paper presents a geometric view of the model of such a structure, formulas and their results for finding the values of the mentioned energy characteristics of a reservoir of the corresponding geometry, draws conclusions, points out the advantages and possible disadvantages of the given mathematical model, and provides opportunities for the future creation of an alternative mathematical more adequate model of a modern reservoir in the form of a horizontally oriented elliptical half cone.

Volume-Preserving Shear Transformation of an Elliptical Slant Cone to a Right Cone

One nappe of a right circular cone, cut by a transverse plane, splits the cone into an infinite frustum and a cone with an elliptical section of finite volume. There is a standard way of computing this finite volume, which involves finding the parameters of the so-called shadow ellipse, the characteristics of the oblique ellipse (the cut) and, finally, the projection of the vertex of the cone onto the oblique ellipse. This paper shows that it is possible to compute that volume just by using the information of the shadow ellipse and the height of the cone. Indeed, the finite slant cone has the same volume of an elliptic right cone, with the base being the shadow ellipse of the cut portion and with the height being the distance between the vertex of the cone and the intersection of the height of the original cone with the cutting plane. This is proved by introducing a volume-preserving shear transformation of the elliptical slant cone to a right cone, so that the standard volume formula for a cone can be straightforwardly applied. This implies a simplification in the procedure for computing the volume, since the oblique ellipse—i.e., the difficult part—can be neglected because only the shadow ellipse needs to be determined.

Ruled Shells of Conical Type on Elliptical Base

The information about main results on geometry of developable surfaces with an edge of regression which have a directrix ellipse in the base is gathered. These surfaces constitute a group called “Ruled surfaces of conical type on elliptical base”. This group includes elliptical cones, torses with two ellipses defined in the parallel planes, equal slope surfaces, and ruled surfaces with the main frame of three superellipses that are ellipses in one coordinate plane and broken straight lines in the other two coordinate planes. The paper presents a method for developing torses onto a plane, approximation of torses by folded surfaces, and parabolic ending of a thin sheet from elastic material into a torse shell. A brief review of the methods of stress-strain and buckling analysis of the considered ruled shells is given, including the displacement-based finite element method and variational energy method. It is shown that analytical methods can be used only in the case of applying the momentless shell theory for ruled thin shells of conical type. The analytical formulae for determining the normal and tangent internal forces in any momentless conic shell with a superellipse in the base are derived. References to forty four scientific articles of other authors, working or having worked on the subject of the paper are given. These references confirm the conclusions of the author and the perspectives of investigations of the considered ruled surfaces and shells.

Inertial wave super-attractor in a truncated elliptic cone

We consider inertial waves propagating in a fluid contained in a non-axisymmetric three-dimensional rotating cavity. We focus on the particular case of a fluid enclosed inside a truncated cone or frustum, which is the volume that lies between two horizontal parallel planes cutting an upright cone. While this geometry has been studied in the past, we generalise it by breaking its axisymmetry and consider the case of a truncated elliptic cone for which the horizontal sections are elliptic instead of circular. The problem is first tackled using ray tracing, where local wave packets are geometrically propagated and reflected within the closed volume without attenuation. We complement these results with a local asymptotic analysis and numerical simulations of the original linear viscous problem. We show that the attractors, well known in two dimensional or axisymmetric domains, can be trapped in a particular plane in three dimensions provided that the axisymmetry of the domain is broken. Contrary to previous examples of attractors in three-dimensional domains, all rays converge towards the same limit cycle regardless of initial conditions, and it is localised in the bulk of the fluid.

Prediction of Transition Locations in Regions of the Minor Axis of Hypersonic Elliptic Cones using the BiGlobal-$e^N$ Method

Prediction of Transition Locations in Regions of the Minor Axis of Hypersonic Elliptic Cones using the BiGlobal-$e^N$ Method

Supersonic Flow over Elliptic Cone with Different Ellipticity Ratio

This study investigates supersonic flow characteristics over circular and elliptic cones at various angles of attack. Simulations were conducted on the cones with the same base area and length-to-diameter ratio. The elliptic cones considered had axis ratios of 1.5 and 3. The angle of attack varied from 0o to 50o, with two different Mach numbers (1.97 and 2.94) employed for the analysis. The numerical results were compared with the experimental and theoretical findings from existing literature. The results revealed that increasing the ellipticity ratio of the cones resulted in higher lift generation. The pressure distributions on the windward and leeward sides of the cones were also examined. The results demonstrated that elliptic cones outperformed circular cones in terms of lift production, and this advantage increased with higher ellipticity ratios. Specifically, when the ellipticity ratio was increased from 1 to 3, the maximum increase in lift coefficient was 96% and 100% at Mach numbers 2.94 and 1.97, respectively. Additionally, by changing the ellipticity ratio from 1 to 1.5, the maximum gain in the lift-drag ratio was 16% and 22% at Mach numbers 1.97 and 2.94, respectively. Notably, an elliptic cone with an ellipticity ratio of 3 achieved a remarkable 46% gain in lift-to-drag ratio compared to a circular cone. However, as the angle of attack increased, a primary bow shock formed on the windward side of the cone, with an embedded shock appearing on the leeward side.

An In-depth Comparative Analysis of Conical Curve Variations: Exploring the Distinctive Properties and Applications of Elliptic and Hyperbolic Cones

An In-depth Comparative Analysis of Conical Curve Variations: Exploring the Distinctive Properties and Applications of Elliptic and Hyperbolic Cones

The general method of the tanker car mouth pose measurement

PurposeThe coupling process between the loading mechanism and the tank car mouth is a crucial step in the tank car loading process. The purpose of this paper is to design a method to accurately measure the pose of the tanker car.Design/methodology/approachThe collected image is first subjected to a gray enhancement operation, and the black parts of the image are extracted using Otsu’s threshold segmentation and morphological processing. The edge pixels are then filtered to remove outliers and noise, and the remaining effective points are used to fit the contour information of the tank car mouth. Using the successfully extracted contour information, the pose information of the tank car mouth in the camera coordinate system is obtained by establishing a binocular projection elliptical cone model, and the pixel position of the real circle center is obtained through the projection section. Finally, the binocular triangulation method is used to determine the position information of the tank car mouth in space.FindingsExperimental results have shown that this method for measuring the position and orientation of the tank car mouth is highly accurate and can meet the requirements for industrial loading accuracy.Originality/valueA method for extracting the contours of various types of complex tanker mouth is proposed. This method can accurately extract the contour of the tanker mouth when the contour is occluded or disturbed. Based on the binocular elliptic conical model and perspective projection theory, an innovative method for measuring the pose of the tanker mouth is proposed, and according to the space characteristics of the tanker mouth itself, the ambiguity of understanding is removed. This provides a new idea for the automatic loading of ash tank cars.

Dirac cones in bipartite square-octagon lattice: A theoretical approach.

Dirac cones are difficult to achieve in a square lattice with full symmetry. Here, we have theoretically investigated a bipartite tetragonal lattice composed of tetragons and octagons using both Tight-Binding (TB) model and density functional theory (DFT) calculations. The TB model predicts that the system exhibits nodal line semi-metallic properties when the on-site energies of all atoms are identical. When the on-site energies differ, the formation of an elliptical Dirac cone is predicted. Its physical properties (anisotropy, tilting, merging, and emerging) can be regulated by the hopping energies. An exact analytical formula is derived to determine the position of the Dirac point by the TB parameters, and a criterion for the existence of Dirac cones is obtained. The "divide-and-coupling" method is applied to understand the origin of the Dirac cone, which involves dividing the bands into several groups and examining the couplings among inter-groups and intra-groups. Various practical systems computed by DFT methods, e.g., t-BN, t-Si, 4,12,2-graphyne, and t-SiC, are also examined, and they all possess nodal lines or Dirac cones as predicted by the TB model. The results provide theoretical foundation for designing novel Dirac materials with tetragonal symmetry.

AXOIDS OF HOOKE’S JOINT

The analysis of literary sources devoted to the research and construction of cardan transmissions using the Hooke's joint showed that the peculiarities of the spherical movement of the cardan joint crosspiece are insufficiently covered. This is explained by the relatively small weight of the crosspiece, due to which it almost does not affect the dynamics of the cardan transmission. The need to take into account the mass-geometric parameters of the crosspiece and to analyze comprehensively its movement may arise during accurate calculations of high-speed cardan transmissions. In this case, it is convenient to consider the movement of the crosspiece as rolling without sliding of the moving axoid on the stationary axoid. This approach makes it possible to more fully investigate the peculiarities of the spherical movement of the crosspiece and gives a visual representation of the orientation of its instantaneous axis of rotation in fixed and moving coordinate systems. Based on this, this work is devoted to the construction of a kinematic model of the Hooke's joint crosspiece, the movement of which is considered as rolling without sliding of a moving axoid on a stationary one.
 As a result of the kinematic analysis of the Hooke's joint, the equations of the hodographs of the vector of the absolute angular velocity of the crosspiece in the fixed and moving coordinate systems were obtained, and the parameters of the fixed and moving axoids of the crosspiece of the cardan joint were determined. It is proved that the stationary axoid of the crosspiece of the Hooke's joint is an oblique elliptic cone with the apex in the center of the crosspiece, and its opposite generatrices, crossing the major axis of the guide ellipse, coincide with the axes of the drive and driven shafts of the joint. The moving axoid of the crosspiece is a closed conical surface with a variable opening angle, the maximum value of which is equal to twice the value of the interaxial transmission angle. With small values of the interaxial angle, which occur in the practice of using a single Hooke’s joint, a moving axoid can be approximately considered as a straight circular cone with a vertex in the center of the crosspiece and an opening angle equal to twice the value of the interaxial transmission angle. The error of this assumption increases rapidly with the increase of the interaxial transmission angle. The correctness of the obtained mathematical model was confirmed by computer modeling in the Autodesk Inventor environment.
 The obtained results can be used in the study of dynamic processes that occur in cardan transmissions using Hooke's joint.

Design of Quadruple Offset Butterfly Valve Used in Power Plants

The elliptical disc is a specific characteristic of a triple offset butterfly valve (TOBV), but the structure of the elliptical disc leads to a non-uniform distribution of contact at the closing position with non-uniform wear. On the other hand, the continuous processing of elliptical sections to finish the entire disc contour is expensive compared to the processing of a circular section. In the present study, to improve the above disadvantages of an elliptical disc, an elliptic cone function was established to describe the quadruple butterfly valve (QOBV) disc shape, which has a circular section disc. In addition, some designs of a fixed thickness (7 mm) laminated seal for good sealing were proved. Sealing performance simulations were carried out using the thermal–structural coupling analysis. The best design of the QOBV showed that the maximum contact pressure was 37.2% lower than the TOBV using the seven-way laminated seal layers (thickness of A240–316 stainless and graphite was 1.3 mm and 0.6 mm, respectively). Compared to TOBV, the QOBV disc operation during the opening and closing process was faster and smoother. The disc can be applied widely to various industrial fields, engine exhaust systems, and turbine and power plants under severe environments.

Research on prevention and control methods of land subsidence induced by groundwater overexploitation based on three-dimensional fluid solid coupling model—a case study of Guangrao County

Land subsidence is an environmental geological phenomenon with slowly decreasing ground elevation, The North China Plain is one of the areas with the most serious land subsidence in China, and Guangrao County is one of the subsidence centers. This paper is based on the hydrogeological and engineering geological data of Guangrao County, the groundwater monitoring data for many years and the land subsidence monitoring data, systematically analyzes the dynamic characteristics of groundwater, the distribution and evolution of land subsidence, and the correlation between groundwater exploitation and land subsidence development in different layers of this area. Based on Biot porous medium consolidation theory, establishes a three-dimensional fluid solid coupling numerical model of land subsidence in Guangrao County, restores the development process of land subsidence, predicts and analyzes the subsidence evolution law under different groundwater exploitation schemes, and proposes targeted prevention and control measures. The research results show that: the shallow groundwater forms a cone of depression with Guangbei Salt Field as the center, and the deep groundwater forms an elliptical regional cone of depression with the urban area as the center. The ground is gradually formed two small settlement areas with the urban area of Guangrao County and Guangbei Salt Field as the settlement center, and there is a trend of interrelated expansion. The three-dimensional fluid solid coupling model of land subsidence accurately restored the development process of land subsidence in the study area, predicted that under the current groundwater exploitation conditions, by 2040, the settlement of Guangrao urban settlement center will increase to 1,350 mm, forming a large regional funnel centered around the urban area, and gradually developing and expanding around. Prohibition of groundwater exploitation in the main funnel area is a more reasonable and effective exploitation plan to prevent the development of land subsidence.

In‐Plane Hyperbolic Phonon‐Polaritons in van der Waals Nanocrystals

AbstractAnisotropic hyperbolic phonon‐polaritons (HPhPs) in a van der Waals material, orthorhombic‐phase molybdenum trioxide (α‐MoO3), provide strategies in multiple dimensions to manipulate photons at the nanoscale. However, the nano‐imaging studies of the in‐plane HPhPs along orthogonal crystal orientations are still rare. In this work, the launching and propagating properties of HPhPs are studied in α‐MoO3 upon a holey silicon nitride microcavity through a photon‐induced force microscope (PiFM) over a broad tunable mid‐infrared region of 780–1010 cm−1. Near‐field patterns stemming from the orthogonal polariton momenta are observed in this spectral band. A simple model based on the geometrical optics of elliptic resonance cones is used to explain the near‐field phonon‐polariton features. These near‐field optical properties indicate abundantly possible operations and applications for nanophotonics.

Underwater 3D Scanner to Counteract Refraction: Calibration and Experimental Results

Underwater 3-D laser scanners are an essential type of sensors used by unmanned underwater vehicle (UUVs) for operations such as inspection, navigation, and object recognition and manipulation. This article presents a novel 3-D laser scanner, which uses a 2-axis mirror to project straight lines into the water by compensating for refraction-related distortions. This is achieved by projecting optimally curved lines, so that the refraction when they enter the water transforms them into straight lines. The relevance of this approach lies in the fact that 3-D triangulation using planes is noticeably faster than using elliptic cones. The goal of this work is twofold: first, to prove that refraction-related distortions can in practice be compensated for by using a 2-axis mirror, and second, to present a simple calibration algorithm that only needs to compute the coefficients of polynomial functions. To the best of the authors’ knowledge, the prototype presented in this article is the first laser line scanner that actively counteracts the refraction of the projected light in the context of underwater robotics.

Numerical investigation of erosion characteristics of coupling separators with different conical profiles

Solid particles exist widely in liquid–liquid systems. Although a separator coupled centrifugal and electric fields can achieve liquid–liquid separation efficiently, the solid particles are usually not considered, and the erosion characteristics are unclear. Thus, in this study, a numerical model was established to investigate the erosion characteristics of coupling devices. Two conical profiles, namely bi-cone and elliptical, were examined. The effect of structural parameters on erosion was investigated, and the relationships between the erosion rate and the inlet velocity, initial drop size and solid loading were explored. The results showed that maximum erosion occurred in the interaction regions of the two cones for the bi-cone, and that the cone with an elliptical profile had a lower erosion rate. The maximum erosion rate depended on the centrifugal force acting on the solid particles and the inertial effect of particles in the mixture. Furthermore, the precessing vortex core was the dominant factor influencing the position, including maximum erosion rate, which depended on the centrifugal force and the vortex core radial deviation. Moreover, the potential reason for the effect of geometric parameters on wall erosion is different. From the perspective of erosion resistance, the optimal values of straight and underflow pipe section diameters and conical structure length for the device with an elliptical profile cone were 35, 400, and 430 mm. Additionally, increasing inlet velocity, initial drop size and solid loading can increase average erosion rate on the walls, and strengthen of breakage effect can indirectly reflect that the increase in erosion rate.

Resolvents and complex powers of semiclassical cone operators

We give a uniform description of resolvents and complex powers of elliptic semiclassical cone differential operators as the semiclassical parameter h tends to 0. An example of such an operator is the shifted semiclassical Laplacian h 2 Δ g + 1 $h^2\Delta _g+1$ on a manifold ( X , g ) $(X,g)$ of dimension n ≥ 3 $n\ge 3$ with conic singularities. Our approach is constructive and based on techniques from geometric microlocal analysis: we construct the Schwartz kernels of resolvents and complex powers as conormal distributions on a suitable resolution of the space [ 0 , 1 ) h × X × X $[0,1)_h\times X\times X$ of h-dependent integral kernels; the construction of complex powers relies on a calculus with a second semiclassical parameter. As an application, we characterize the domains of ( h 2 Δ g + 1 ) w / 2 ${\big (h^2\Delta _g+1\big )}^{w/2}$ for Re w ∈ − n 2 , n 2 $\operatorname{Re}w\in \left(-\tfrac{n}{2},\tfrac{n}{2}\right)$ and use this to prove the propagation of semiclassical regularity through a cone point on a range of weighted semiclassical function spaces.

Stress concentration of perforated cement sheath and the effect of cement sheath elastic parameters on its integrity failure during shale gas fracturing

Due to the extremely low porosity and permeability of shale, fracturing is often used to develop shale gas reservoirs. During shale fracturing, extremely high fracturing pressure may invalidate the integrity of the cement sheath and bring hidden dangers to the safe development of shale gas. This paper compares the stress state of the three-dimensional unperforated and perforated finite element models of casing-cement sheath-formation to obtain the influence of perforation on the stress concentration and failure area of the cement sheath. The stress state comparison incorporates three sets of perforation models with different hole densities and diameter verifies the stress concentration law of perforation on the cement sheath. By studying the effects of the elastic modulus and Poisson’s ratio of the cement sheath on the maximum tensile and compressive stresses of the cement sheath under fracturing pressure, the integrity failure form of the cement sheath and the measures for integrity failure are obtained. The results show that the peak stress concentration of the perforated cement sheath is about twice the normal value of the stress. The failure area is the two ends of the perforation, the shape is similar to the bottom surface of the elliptical cone, and the thickness is up to half of the wall thickness of the cement sheath. The line length of the largest failure area is 4.5 times the diameter of the hole. Under fracturing conditions, the cement sheath generally undergoes tensile failure. The lower the elastic modulus and Poisson’s ratio of the cement sheath, the smaller the tensile stress of the cement sheath, and the less likely it is to cause tensile failure.

Location of singular points in orthorhombic media

The dependence of the location of singular points of orthorhombic (ORT) media on the stiffness coefficients , and phase velocity at the singularity point is studied under the assumption that are larger than and . In this case, singular points appear only at the intersection of slowness surfaces of S1- and S2-waves. To simplify the presentation of the results, the values , are fixed and changed within the limits at which the stiffness matrix remains positive definite. We define the parameters, , , , which result in 0, 1, or 2 singular points in the symmetry planes of the ORT medium. The types of these singular points and their location on the unit circle are described. It is shown that, by selecting parameters , any singular point in the symmetry plane 13 can be combined with the limiting position of the singularity point in-between the symmetry planes, or this point can be included in the singular curve of the degenerate ORT medium. We derive equations for the semi-axes of an ellipse of conical refraction, which is the image of a singular point from plane 13 in the group domain. Conditions for degeneration of the ellipse of conical refraction into a segment or a point are defined. It is shown that there exists only one ORT medium with a fixed phase velocity of S1- and S2-waves in a given singular direction n. If we present the ORT media as projections of these vectors n onto the plane 12 and mark the value of the Poincaré index of the S1- or S2-wave at the point n, we get 2 regions with indices 1/2 and –1/2 separated by projection of the singular curve in the form of an ellipse or hyperbola. We compute equations for of a degenerate ORT medium in terms of the values, , and velocity of S1-, S2-waves on a singular curve. The singular curve is defined by the intersection of a unit sphere with an elliptic cone. It is proved that a degenerate ORT medium for or is, respectively, a transversally isotropic medium with a vertical or horizontal axis of symmetry. The results are illustrated in several examples.