Elliptic Cone Research Articles

Sampling random directions within an elliptical cone

This work extends the spherical surface sampling algorithm in order to uniformly generate random directions within an elliptical cone. This has applications in Monte Carlo particle transport simulations, for example modeling asymmetric beam divergence or scattering interactions. Two methods are presented. The first obeys the strict boundary of the elliptical cone. The second relaxes this requirement, increasing the range of generated directions by up to 10% for elliptical cones of extreme eccentricity. However, the second method is able to generate directions beyond the equator.

Direct Simulation of Hypersonic Crossflow Instability on an Elliptic Cone

Direct numerical simulation is performed on a 38.1% scale Hypersonic International Flight Research Experimentation Program Flight 5 forebody to study stationary crossflow instability. Computations use the US3D Navier–Stokes solver to simulate Mach 6 flow at Reynolds numbers of and , which are conditions used by quiet-tunnel experiments at Purdue University. Distributed roughness with point-to-point height variation on the computational grid and maximum heights of is used with the intent to emulate smooth-body transition and excite the naturally occurring most unstable disturbance wavenumber. Cases at the low-Reynolds number condition use three grid sizes, and hence three different roughness patterns, and demonstrate that the exact flow solution is dependent on the particular roughness pattern. The same roughness pattern is interpolated onto each grid, which yields similar solutions, indicating grid convergence. A steady physical mechanism is introduced for the sharp increase in wall heat flux seen in both computations and experiment at the high-Reynolds number condition. Evolution of disturbance spanwise wavelength is computed, and is found to be more sensitive to Reynolds number than roughness, indicating that the disturbance wavelength is primarily the naturally occurring, flow-selected wavelength for these cases.

КОНСТРУИРОВАНИЕ КАНАЛОВЫХ ПОВЕРХНОСТЕЙ С ПЕРЕМЕННОЙ ОБРАЗУЮЩЕЙ И ПЛОСКОСТЬЮ ПАРАЛЛЕЛИЗМА НА ОСНОВЕ ЭКВИАФФИННЫХ ПРЕОБРАЗОВАНИЙ ПЛОСКОСТИ

A kinematic approach to the construction of surfaces with equiaffine invariant for a volume of a compartment bounded by a surface, and a square of cross sections, parallel to a parallelism plane on the basis of the plane’s equiaffine transformations has been proposed in this paper. In particular, in the paper have been considered parametric equations of canal surfaces with a parallelism plane xOy, which sections have been transformed by elliptic, parabolic and hyperbolic rotations. A curve-prototype lies in the section z = 0, and a curve-image lies in the section z = h after transformation with rotations set parameters. The parametric equations have been obtained in a general form with randomness of the choice both a curve-prototype for canal surface construction, and functions defining a rate of rotation parameters change. Some general properties of the obtained surfaces have been considered, as well as some special cases of surfaces are considered, and trajectories of generating lines have been defined for them. In the case of parameter’s linear variation along the z-axis the elliptical rotation’s surfaces represent spiral and straight helical surfaces with constant and variable step however the generatrix trajectory envelops elliptical cone and cylinder. In all cases, the generatrices of the elliptical rotation’s surfaces are space curves. In the case of all parameters’ linear variation the parabolic rotation’s surfaces represent ruled surfaces. One particular case of parabolic rotation’s surfaces is the surface with two parallelism planes: xOy for equiaffine changing curve-section and xOz for generatrix. In the cases related to non-linear variation of only one parameter, generatrices for surfaces of parabolic and hyperbolic rotations are plane curves. Considered surfaces require further investigation with a view to their possible use in mechanisms and structures.

Reversals of Least-Square Estimates and Model-Invariant Estimation for Directions of Unique Effects

ABSTRACTWhen a linear model is adjusted to control for additional explanatory variables, the sign of a fitted coefficient may reverse. Here, these reversals are studied using coefficients of determination. The resulting theory can be used to determine directions of unique effects in the presence of model uncertainty. This process is called model-invariant estimation when the estimates are invariant across changes to the model structure. When a single covariate is added, the reversal region can be understood geometrically as an elliptical cone of two nappes with an axis of symmetry relating to a best-possible condition for a reversal using a single coefficient of determination. When a set of covariates are added to a model with a single explanatory variable, model-invariant estimation can be implemented using subject matter knowledge. More general theory with partial coefficients is applicable to analysis of large datasets. Applications are demonstrated with dietary health data from the United Nations.

Nonadiabatic eigenfunctions can have conical nodes

It has been argued the total vibrational probability amplitude for a molecular nonadiabatic eigenfunction should not have nodes unless required by symmetry. For a model with two nonadiabatically coupled electronic states, exact factorization of nonadiabatic eigenfunctions into a total vibrational probability amplitude and a normalized electronic factor reveals zero-dimensional nodes in nonadiabatic eigenfunctions over a two-dimensional vibrational space. These conical nodes have the shape of a right elliptical cone with the node at the vertex. Low dimensionality nodes are likely when the number of nonadiabatically coupled electronic states is less than or equal to the dimensionality of the vibrational space.

小児腹部CT 検査における自動露出制御の一手法―年齢および腹部サイズを考慮した標準偏差(SD)値および管電流値の設定法―

We aimed to apply the pediatric abdominal CT protocol of Donnelly et al. in the United States to the pediatric abdominal CT-AEC. Examining CT images of 100 children, we found that the sectional area of the hepatic portal region (y) was strongly correlated with the body weight (x) as follows: y=7.14x + 84.39 (correlation coefficient=0.9574). We scanned an elliptical cone phantom that simulates the human body using a pediatric abdominal CT scanning method of Donnelly et al. in, and measured SD values. We further scanned the same phantom under the settings for adult CT-AEC scan and obtained the relationship between the sectional areas (y) and the SD values. Using these results, we obtained the following preset noise factors for CT-AEC at each body weight range: 6.90 at 4.5-8.9 kg, 8.40 at 9.0-17.9 kg, 8.68 at 18.0-26.9 kg, 9.89 at 27.0-35.9 kg, 12.22 at 36.0-45.0 kg, 13.52 at 45.1-70.0 kg, 15.29 at more than 70 kg. From the relation between age, weight and the distance of liver and tuber ischiadicum of 500 children, we obtained the CTDIvol values and DLP values under the scanning protocol of Donnelly et al. Almost all of DRL from these values turned out to be smaller than the DRL data of IAEA and various countries. Thus, by setting the maximum current values of CT-AEC to be the Donnelly et al.'s age-wise current values, and using our weight-wise noise factors, we think we can perform pediatric abdominal CT-AEC scans that are consistent with the same radiation safety and the image quality as those proposed by Donnelly et al.

Toward robust prediction of crossflow-wave instability in hypersonic boundary layers

Abstract Traveling and stationary crossflow-wave instabilities in a laminar Mach 6 boundary layer are investigated on a 38.1% scale model of the Fifth Hypersonic International Flight Research Experiment (HIFiRE-5) elliptic cone at zero angle of attack and yaw. The Langley Stability and Transition Analysis Code (LASTRAC) was used to analyze the crossflow dominated boundary layer in the mid-span region near the downstream end of the model. Disturbance growth rates, wave angles, and phase speeds are computed with LASTRAC using quasi-parallel Linear Stability Theory (LST), Linear Parabolized StabilityEquations (LPSE), and two-plane or surface marching LPSE (2pLPSE). The predicted wave angles and phase speeds are validated using experimental data, and are found to be in better agreement than previous computations. Further numerical analysis is conducted using the Spatial BiGlobal technique (SBG), which simultaneously accounts for wall-normal and spanwise gradients in the mean boundary layer at a particular axial station. For the first time in the literature, a comparison is made between crossflow wave growth rates computed using LST, LPSE, and 2pLPSE and those computed using SBG, accounting for curvature and geometric divergence of the elliptic cone. The agreement between LST, LPSE, and SBG is fair at best, but excellent agreement is realized between 2pLPSE and SBG. This result constitutes a co-verification of the LASTRAC and SBG stability codes, and provides evidence that 2pLPSE accurately models the physics of the traveling-crossflow instability.

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Linear modal instabilities of hypersonic flow over an elliptic cone

Steady laminar flow over a rounded-tip $2\,:\,1$ elliptic cone of 0.86 m length at zero angle of attack and yaw has been computed at Mach number $7.45$ and unit Reynolds number $Re^{\prime }=1.015\times 10^{7}~\text{m}^{-1}$. The flow conditions are selected to match the planned flight of the Hypersonic Flight Research Experimentation HIFiRE-5 test geometry at an altitude of 21.8 km. Spatial linear BiGlobal modal instability analysis of this flow has been performed at selected streamwise locations on planes normal to the cone symmetry axis, resolving the entire flow domain in a coupled manner while exploiting flow symmetries. Four amplified classes of linear eigenmodes have been unravelled. The shear layer formed near the cone minor-axis centreline gives rise to amplified symmetric and antisymmetric centreline instability modes, classified as shear-layer instabilities. At the attachment line formed along the major axis of the cone, both symmetric and antisymmetric instabilities are also discovered and identified as boundary-layer second Mack modes. In both cases of centreline and attachment-line modes, symmetric instabilities are found to be more unstable than their antisymmetric counterparts. Furthermore, spatial BiGlobal analysis is used for the first time to resolve oblique second modes and cross-flow instabilities in the boundary layer between the major- and minor-axis meridians. Contrary to predictions for the incompressible regime for swept infinite wing flow, the cross-flow instabilities are not found to be linked to the attachment-line instabilities. In fact, cross-flow modes peak along most of the surface of the cone, but vanish towards the attachment line. On the other hand, the leading oblique second modes peak near the leading edge and their associated frequencies are in the range of the attachment-line instability frequencies. Consequently, the attachment-line instabilities are observed to be related to oblique second modes at the major-axis meridian. The linear amplification of centreline and attachment-line instability modes is found to be strong enough to lead to laminar–turbulent flow transition within the length of the test object. The predictions of global linear theory are compared with those of local instability analysis, also performed here under the assumption of locally parallel flow, where use of this assumption is permissible. Fair agreement is obtained for symmetric centreline and symmetric attachment-line modes, while for all other classes of linear disturbances use of the proposed global analysis methodology is warranted for accurate linear instability predictions.

A three dimensional infinite wedge shaped solid block sliding into water along an inclined beach

The three dimensional (3D) problem of a solid block sliding into water along an inclined beach is investigated. The main part of the block is an infinite wedge cylinder and the front of the body is part of an elliptical cone. Incompressible velocity potential theory is used together with fully nonlinear boundary conditions. When gravity is ignored, it is found that self-similar solution is possible. The boundary element method is used to solve the problem. The free surface shape is updated together with the potential on the free surface until the flow has become self-similar. Convergence studies are taken with respect to marching step and element size. Simulations are made for different bodies and different beach angles. Extensive results are provided for the pressure as well as the free surface shape, and their implications in physics are discussed.

Combined complex-source beam and spherical-multipole analysis for the electromagnetic probing of conical structures

The paper addresses the combination of the spherical-multipole analysis in sphero-conal coordinates with a uniform complex-source beam (CSB) in order to analyze the scattering of a localized electromagnetic plane wave by any desired part of a perfectly conducting elliptic cone. The concept of uniform CSB is introduced and rigorously applied to the diffraction by a semi-infinite elliptic cone. The analysis takes into account the fact that the incident CSB does not satisfy the radiation condition. A new modal form of the Green's function for the elliptic cone is derived based on the principle that there is no energy loss to infinity. The numerical evaluation includes the scattered far fields of a CSB incident on the corner of a plane angular sector with different opening angles. Cet article présente la combinaison de l'analyse de multipôles sphériques en coordonnées sphéro-coniques avec une faisceau de source complexe ( Complex Source Beam , CSB) dans le but d'analyser la diffusion localisée par un cône elliptique parfaitement conducteur d'une onde plane électromagnétique. Le concept de CSB est introduit au travers de la diffraction par un cône elliptique semi-infini. L'analyse prend en compte le fait que l'onde CSB incidente ne satisfait pas les conditions de radiation. Un nouveau modèle de la fonction de Green pour un cône elliptique est développé en faisant l'hypothèse qu'il n'y a pas de pertes d'énergie à l'infini. Le modèle numérique inclut la diffusion en champ lointain d'une source CSB sur le coin d'un secteur angulaire avec différents angles d'ouverture.

Image Processing of Illuminated Ellipsoid

This study introduces novel algorithms and the underlying mathematics to process photographs of planetary illuminated bodies and use them for navigation purposes. The goal is to accurately estimate the observer-to-body relative position in inertial coordinates. The main motivation is to provide autonomous navigation capabilities to spacecrafts by observing a planet or a moon. This is needed, for example, in human-rated vehicles in order to provide navigation capabilities in a loss-of-communications scenario. The algorithm is derived for the general case of a triaxial ellipsoid that is observed bounded by an elliptical cone. The orientation of the elliptical cone reference frame is obtained by eigenanalysis, and the offset between the elliptical cone axis and the body center direction as well as the equation of the terminator are quantified. The main contribution of this paper is in the image-processing approach adopted to derive centroid and distance to the body. This is done by selecting a set of pixels around the body limb and modeling the smooth limb transition using two-dimensional circular and elliptical sigmoid functions. More accurate estimates of the centroid and distance are then obtained by iterative nonlinear least-squares using these models. A sensitivity analysis is performed, and numerical examples using a real moon photograph taken from Earth are provided to clarify the image-processing steps and to validate the proposed theory.

Tract Orientation and Angular Dispersion Deviation Indicator (TOADDI): A framework for single-subject analysis in diffusion tensor imaging

The purpose of this work is to develop a framework for single-subject analysis of diffusion tensor imaging (DTI) data. This framework is termed Tract Orientation and Angular Dispersion Deviation Indicator (TOADDI) because it is capable of testing whether an individual tract as represented by the major eigenvector of the diffusion tensor and its corresponding angular dispersion are significantly different from a group of tracts on a voxel-by-voxel basis. This work develops two complementary statistical tests based on the elliptical cone of uncertainty, which is a model of uncertainty or dispersion of the major eigenvector of the diffusion tensor. The orientation deviation test examines whether the major eigenvector from a single subject is within the average elliptical cone of uncertainty formed by a collection of elliptical cones of uncertainty. The shape deviation test is based on the two-tailed Wilcoxon–Mann–Whitney two-sample test between the normalized shape measures (area and circumference) of the elliptical cones of uncertainty of the single subject against a group of controls. The False Discovery Rate (FDR) and False Non-discovery Rate (FNR) were incorporated in the orientation deviation test. The shape deviation test uses FDR only. TOADDI was found to be numerically accurate and statistically effective. Clinical data from two Traumatic Brain Injury (TBI) patients and one non-TBI subject were tested against the data obtained from a group of 45 non-TBI controls to illustrate the application of the proposed framework in single-subject analysis. The frontal portion of the superior longitudinal fasciculus seemed to be implicated in both tests (orientation and shape) as significantly different from that of the control group. The TBI patients and the single non-TBI subject were well separated under the shape deviation test at the chosen FDR level of 0.0005. TOADDI is a simple but novel geometrically based statistical framework for analyzing DTI data. TOADDI may be found useful in single-subject, graph-theoretic and group analyses of DTI data or DTI-based tractography techniques.

Quantitative Comparison of Mathematical Models to Measure Surface Area of Canine Teeth Prepared to Receive Full Veneer Crowns in Dogs.

This study was performed in order to determine if mathematical modeling of the canine teeth in dogs could be utilized to provide an accurate and reliable estimation of crown surface area that could be used in both a research and a clinical setting. Actual surface area (aSA) calculations for 32 stone dies of clinical crown preparations were acquired utilizing a tridimensional (3D) laser scanner and 3D computer-aided design and manufacturing (CAD/CAM) software applications. These calculations were used as a control. Seventeen unique mathematical models from eight geometric shapes were used to calculate estimated surface area (eSA) of each stone die. Linear association and agreement between eSA and aSA calculations were assessed with multiple statistical methods. All methods of eSA showed a significant linear association with aSA. Five of the mathematical models [right elliptical frustum (H3), right elliptical cone (G3), right pyramidal cone (A3), right circular frustum (F2), and right circular cone (E1)] were superior to the other 12 models. The H3 mathematical model based on the right elliptical frustum provided the most accurate estimate of crown surface area of dog teeth. However, H3 requires the use of laser scans and a 3D CAD software program. As a result, this model would be recommended for research applications. The E1 mathematical model was similar in accuracy to H3 and, given it requires only two measurements and a comparatively simple equation for calculation, this method would be recommended for clinical chair-side use.

On the Conical Point Crossing: The Case of the Orthorhombic Lattice

It is shown that the conical point crossing, known to appear at the K point in the Brilloiun zone of the hexagonal lattice, may appear also in those of the orthorhombic and the monoclinic lattices. The feature of the conical point crossing in the orthorhombic lattice induced by the ferroelastic phase transition is concretely studied. Generally in the lattices with the symmetry lower than hexagonal, the cone becomes the elliptic cone, and the conical k-point shifts to a general position inside the Brillouin zone. The existence of the conical point crossing in the orthorhombic and the monoclinic lattices can be proved using some models, but not solely by symmetry considerations.

Dynamic analysis of thick and thin elliptic shell structures made of laminated composite materials

Elliptic geometries appear to be important components in engineering practices. The purpose of the present study is to examine the free vibration nature of laminated composite thick and moderately thick elliptic cones, cylinders and plates. A strong form approach, such as the generalized differential quadrature (GDQ) method is employed to carry out the numerical analyses. The geometric description of the structures under consideration is performed through the differential geometry which is a convenient and general mathematical tool to have parametric description of curved structures. Reference solutions are presented for laminated composite thick elliptic shells. 3D finite element models are used for proving the validity and the advantages of the present methodology. Since laminated composite structures are investigated, higher-order theories are considered to capture the nonlinear behavior of the material fibers through the shell thickness. In particular, a hierarchical expansion order of the kinematic displacements is adopted. The expansion order is a function of a free parameter. The numerical solution is found discretizing the dynamic equilibrium equations, written as functions of the displacement parameters, with GDQ method. This technique proved to have several advantages such as stability, accuracy and easy implementation as also demonstrated by the authors in the provided literature.

Traveling Crossflow Instability for the HIFiRE-5 Elliptic Cone

A 38.1%-scale model of the Hypersonic International Flight Research Experimentation Program’s Flight Five 2:1 elliptic cone flight vehicle was used to investigate the traveling crossflow instability in a Mach 6 quiet wind tunnel. Traveling crossflow waves were detected with pressure sensors mounted flush with the model surface. The crossflow instability phase speed and wave angle were calculated from the cross spectra of the three pressure sensors. Both quantities showed good agreement with linear stability theory. Duplicate runs at the same initial conditions showed excellent repeatability in traveling crossflow wave properties. Traveling crossflow waves in quiet flow showed very low levels of nonlinear interactions. No traveling crossflow waves were observed for any Reynolds number for elevated freestream noise levels, but transition occurred for a much lower Reynolds number than in quiet flow. Due to the lack of nonlinear growth in quiet flow and the absence of traveling crossflow waves in noisy flow, it appeared that the traveling crossflow instability was not the primary transition mechanism on the model for quiet or noisy flow in this wind tunnel.

HIFiRE-5 Flight Test Results

The Hypersonic International Flight Research Experimentation (HIFiRE-5) is a hypersonic flight test vehicle designed to investigate the aerothermodynamics of a three-dimensional geometry. It is a 7-deg minor-axis half-angle elliptic cone with a aspect ratio and 2.5 mm nose radius. The flight test occurred in April 2012, but the upper stage of the sounding rocket failed to ignite, resulting in a peak Mach number of about 3 instead of the target of 7. The instrumentation (almost 300 thermocouples and 50 pressure transducers) performed well and provided a wealth of supersonic aeroheating and boundary-layer transition data. The pressure transducers indicated the expected dependence upon angle of attack and yaw and offer a check for the inertial measurement unit. Heat flux was calculated from paired thermocouples and boundary-layer transition locations were identified from the heating rates. Two boundary-layer transition mechanisms were encountered during the supersonic descent. One mode leads to transition over the acreage of the vehicle and correlates well with ; the other mechanism causes the leading-edge boundary-layer transition to advance rapidly over a small range of freestream and is suspected to be roughness-induced. The good performance of the instrumentation increases confidence in the success of a repeat flight.

Fluid Dynamics of Flow Driven by Nodal Cilia Precessing Non-circular Cones

Nodal cilia play an important role in the left-right symmetry breaking at the early stage of the mammal embryos and show an apparent rotational motion. This study is about the ow induced by cilia sweeping out circular/elliptical cones above a no-slip plane in the low Reynolds number regime. Using the regularized Stokeslet method, we examine the properties of ows generated by a single cilium and multiple cilia, especially two cilia, which are assumed to be anchored to the embryonic heart wall. For a short-term Lagrangian uid tracer trajectory, epicycles are presented as a nodal cilium precesses in either a circular cone or a non-circular cone shape. For the longterm behavior, the trajectory is periodic around both the circular and elliptical cones. Compared to the circular cone cases, the vertical variances are enhanced in elliptical cone cases. Besides tracer trajectories, uid velocity elds are presented to demonstrate the ow structure. When

СРАВНЕНИЕ МЕТОДОВ НАЧЕРТАТЕЛЬНОЙ ГЕОМЕТРИИ И 3D КОМПЬЮТЕРНОГО ГЕОМЕТРИЧЕСКОГО МОДЕЛИРОВАНИЯ ПО ТОЧНОСТИ, СЛОЖНОСТИ И ЭФФЕКТИВНОСТИ

Urgent educational problem of inconsistency of content of the course of descriptive geometry (DG) with the trends in the development of CAD and geometric simulation is considered. While the development is aimed at expanding the scope of computer 3D geometric simulation the teaching of descriptive geometry projection methods continues despite the fact that they are in no demand in practical applications. To compare the estimations of versions of solutions in descriptive geometry and 3D five specific problems of the nomination in descriptive geometry of All-Russian Olympiad 2014 are considered. They are the construction of sphere, tangent to the paraboloid; determining the points of intersection of the ellipse defined by the axes, with a circle (without the ellipse); finding the reference points of the ellipse section and its projections; construction of a tetrahedron by one projection of its face; as well as the defining the points of intersection of the circle with an elliptical cone. The detailed versions of solutions of descriptive geometry and 3D are given for each problem. The comparison of versions of the solution according to the geometrical accuracy is done. It is offered to estimate not an abstract accuracy which corresponds to constructions made by aperfect caliper and a ruler but the real accuracy independently of methods of its achievement. It’s shown that with the computer realization of both variants their real accuracy is either equal or 3D is much better than DG. The estimation of the complexity of task solutions is given. The amount of information needed for the solution is taken as a criterion. The author makes the conclusion that generally 3D versions are easier than DG. The examples show that 3D solutions can be easily supplemented with researches of a solvable problem. Much higher visualization, accessibility and universality of 3D versions are highlighted. The author makes the conclusion about much higher general effectiveness of computer 3D variants and methods of solving the structural problems and the necessity of transfer to new training course of 3d theory. Keywords: descriptive geometry, computer graphics, 3D modeling, structural problems, geometrical accuracy, complexity and effectiveness of structural problems.