Orthogonal Coordinate System Research Articles

Third-order superintegrable systems with potentials satisfying only nonlinear equations

The conditions for superintegrable systems in two-dimensional Euclidean space admitting separation of variables in an orthogonal coordinate system and a functionally independent third-order integral are studied. It is shown that only systems that separate in subgroup type coordinates, Cartesian or polar, admit potentials that can be described in terms of nonlinear special functions. Systems separating in parabolic or elliptic coordinates are shown to have potentials with only non-movable singularities.

A new three-dimensional terrain-following tidal model of free-surface flows

A three-dimensional hydrodynamic model is presented which combines a terrain-following vertical coordinate with a horizontally orthogonal curvilinear coordinate system to fit the complex bottom topography and coastlines near estuaries, continental shelves, and harbors. To solve the governing equations more efficiently, we improve the alternating direction implicit method, which is extensively used in the numerical modeling of horizontal two-dimensional shallow water equations, and extend it to a three-dimensional tidal model with relatively little computational effort. Through several test cases and realistic applications, as presented in the paper, it can be demonstrated that the model is capable of simulating the periodic to-and-fro currents, wind-driven flow, Ekman spirals, and tidal currents in the near-shore region.

Experimental and numerical investigation of nanofluid heat transfer in helically coiled tubes at constant wall heat flux

The current study focuses on numerical and experimental investigation of the laminar, steady state flow in helical tubes with constant wall heat flux boundary condition. Pressure drop and convection heat transfer behavior of water–silver nanofluid in helical coils were evaluated and compared to the pure water. In the numerical part of the study, incompressible Navier–Stokes equations, derived from an orthogonal helical coordinate system were solved by finite difference method with projection algorithm using FORTRAN programming language. Homogeneous model with constant effective properties was used for nanofluid. The Local Nusselt number in the entrance region of the helical tubes and the effects of curvature and torsion ratios and Reynolds number have been discussed. In the experimental part, six helical heat exchangers with different curvature and torsion ratios were designed, capable of providing a constant wall heat flux. Pressure drop measurement and fully developed heat transfer coefficient and Nusselt number calculations were carried out and compared with the numerical results.The results show that for laminar flow with constant wall heat-flux boundary condition, the helical tubes with larger curvature ratio lead to higher heat transfer enhancement and higher pressure drop increase, when employing the nanofluid instead of the base fluid. Moreover, compared to using metal Ag nanofluid in straight tubes, utilization of base fluid in helical coils with greater curvature increases heat transfer more effectively. Finally, we conclude that since using Ag nanofluid increases the heat transfer of helical heat exchangers about 3.5–3.8%, utilizing this nanofluid could enhance the thermal performance of heat exchangers, which can benefit the industry.

Heat and mass transfer simulation for thin-film two-phase emulsion flow over heated surfaces

The thermohydrodynamic situation in a thin-film two-phase emulsion flow over heated, arbitrarily shaped surfaces has been investigated with account taken of heat and mass transfer to the environment. The process has been described using conservation equations of multiphase flow mechanics. These equations have been written and simplified in an arbitrary orthogonal coordinate system tied to the flow surface, with the specific features of the flow taken into consideration. The problem has been solved semi-analytically, approximately, using the Slezkin method. Numerical calculations have been carried out using the mathematical model constructed, and significant results for particular applications of the process have been obtained.

Measuring module of the Cherenkov water detector NEVOD

Quasispherical Module (QSM) of Cherenkov water detector NEVOD represents six low-noise FEU-200 photomultipliers with flat photocathodes (15 cm in diameter), oriented along the axes of orthogonal coordinate system. Such configuration allows to register Cherenkov radiation arriving from any direction with almost equal efficiency. The results of measurements of QSM characteristics in the sensitive volume of the NEVOD detector during the registration of Cherenkov radiation of single muons at different distances and angles are discussed.

Spectral collocation for multiparameter eigenvalue problems arising from separable boundary value problems

In numerous science and engineering applications a partial differential equation has to be solved on some fairly regular domain that allows the use of the method of separation of variables. In several orthogonal coordinate systems separation of variables applied to the Helmholtz, Laplace, or Schrödinger equation leads to a multiparameter eigenvalue problem (MEP); important cases include Mathieu's system, Lamé's system, and a system of spheroidal wave functions. Although multiparameter approaches are exploited occasionally to solve such equations numerically, MEPs remain less well known, and the variety of available numerical methods is not wide. The classical approach of discretizing the equations using standard finite differences leads to algebraic MEPs with large matrices, which are difficult to solve efficiently.The aim of this paper is to change this perspective. We show that by combining spectral collocation methods and new efficient numerical methods for algebraic MEPs it is possible to solve such problems both very efficiently and accurately. We improve on several previous results available in the literature, and also present a MATLAB toolbox for solving a wide range of problems.

Application of 2-D sediment model to fluctuating backwater area of Yangtze River

Based on the characteristics of backflow, a two-dimensional mathematical model of sediment movement was established. The complexity of the watercourse boundary at the confluence of the main stream and the tributary was dealt with using a boundary-fitting orthogonal coordinate system. The basic equation of the two-dimensional total sediment load model, the numerical calculation format, and key problems associated with using the orthogonal curvilinear coordinate system were discussed. Water and sediment flow in the Chongqing reach of the Yangtze River were simulated. The calculated water level, flow velocity distribution, amount of silting and scouring, and alluvial distribution are found to be in agreement with the measured data, which indicates that the numerical model and calculation method are reasonable. The model can be used for calculation of flow in a relatively complicated river network.

A High-Order Numerical Method to Study Three-Dimensional Hydrodynamics in a Natural River

AbstractA high-order numerical method for three-dimensional hydrodynamics is presented. The present method applies high-order compact schemes in space and a Runge-Kutta scheme in time to solve the Reynolds-averaged Navier-Stokes equations with the k-ε turbulence model in an orthogonal curvilinear coordinate system. In addition, a two-dimensional equation is derived from the depth-averaged momentum equations to predict the water level. The proposed method is first validated by its application to simulate flow in a 180° curved laboratory flume. It is found that the simulated results agree with measurements and are better than those from SIMPLEC algorithm. Then the method is applied to study three-dimensional hydrodynamics in a natural river, and the simulated results are in accordance with measurements.

Euler angles and crystal symmetry

For the description of (single) crystal orientations, e.g. as measured by electron backscatter diffraction (EBSD) & X‐ray diffraction (XRD), Euler angles are still generally used to import and export data. However, because of the lack of standard definitions for the unit cell reference settings and specimen axes, several transformation descriptions exist which produce different sets of Euler angles for the same orientation. There is also no recommended region within the minimal Euler orientation space into which orientations should be placed. This is the reason why different sets of Euler angles for the same orientation are generated by the available software packages for indexing EBSD patterns. These issues are reviewed and addressed.The influence of crystal symmetry in form of chiral (enantiomorphic) groups is discussed, as well as how multiple, but symmetry‐equivalent sets of Euler angles can be reduced in order to deliver a unique orientation description. The Euler coloring algorithms applied to EBSD map data is critically discussed. The specific case of cubic symmetry, especially the effect of the three‐fold rotation on the Euler space is investigated in more detail for the highest‐symmetric chiral group 432. Recommendations for standard settings of the unit cell to orthogonal coordinate system transformation are given which exploit inherent symmetry.

Asymptotic analysis of narrow escape problems in nonspherical three-dimensional domains.

Narrow escape problems consider the calculation of the mean first passage time (MFPT) for a particle undergoing Brownian motion in a domain with a boundary that is everywhere reflecting except for at finitely many small holes. Asymptotic methods for solving these problems involve finding approximations for the MFPT and average MFPT that increase in accuracy with decreasing hole sizes. While relatively much is known for the two-dimensional case, the results available for general three-dimensional domains are rather limited. This paper addresses the problem of finding the average MFPT for a class of three-dimensional domains bounded by the level surface of an orthogonal coordinate system. In particular, this class includes spheroids and other solids of revolution. The primary result presented is a two-term asymptotic expansion for the average MFPT of such domains containing an arbitrary number of holes. Steps are taken towards finding higher-order asymptotic expansions for both the average MFPT and the MFPT in these domains. The results for the average MFPT are compared to full numerical calculations performed with the comsol Multiphysics finite element solver for three distinct domains: prolate and oblate spheroids and biconcave disks. This comparison shows good agreement with the proposed two-term expansion of the average MFPT in the three domains.

An eigenvalue relation in spheroidal wave functions related to potential theory and its applications to contact problems

Eigenvalue and related integral relations for an integral operator with a symmetrical kernel, defined in a circular region, in the form of the ratio of an exponential function of the distance between two points to its distance, are established by methods of generalized potential theory, related to the Helmholtz equation, in an orthogonal system of coordinates of an oblate spheroid, where one of the coordinate surfaces is degenerate in a plane doubly covered circular disc. These relations extend corresponding relations for the integral operator with a symmetric kernel in the form of the Weber–Sonin integral and contain spheroidal wave functions. Using the integral relations obtained, an exact solution of the integral equation of the contact problem of the indentation of a punch, circular in plan, into a linearly deformed base of an elastic half-space with a kernel identical with a kernel, which varies exponentially with distance, is constructed. Other applications of the eigenvalue and related integral relations obtained are also indicated.

Stability Analysis and Design of a Nonlinear Controller for Hot Rolling Coiler

For the new style hot rolling coiler which adopt AC asynchronous motor as the driving force and with using the algorithm based on differential geometry design nonlinear controller, precise coiling tension control in the rolling process of strip steel is achieved. In this paper, under the rotating orthogonal coordinate system, the fifth-order nonlinear motor model is selected as the controlled plant. By multi-input multioutput (MIMO) exact feedback linearization (EFL) algorithm, the nonlinear model is transformed to a linear one. In terms of small-gain theorem, it is the first to prove that the nonlinear coiler engine that contains the controller has characteristics of input-to-state stability. Experimental results show that the algorithm can be used for high order tracking control system with time-varying parameters. Even without the traditional flux orientation calculation, the output signals are decoupled. With this controller, the tension deviation is restricted to less than 3% and average rotational speed bias was decreased from 0.5% to 0.1% that ensure high-quality plate cut and surface of strip products.

The Numerical Simulation of Hydrodynamics in the Sancha River Mouth

The sancha river mouth is located at the intersection of qinhuai river and Yangtze river. The flow movement of sancha River mouth are affected by application of sancha river gate and Yangtze River. The flow characteristics of the river mouth is very complex. The numerical simulation is used to study the flow movement. The 2-D depth-averaged mathematical model has been established. The govering equations and numerical simulation of flow movement are given in the boundary-fitting orthogonal coordinate systems. The model verification has done by the field data. The flow movement are computed for different application mode of sancha river gate and Yangtze river level. The mainstream line variation and local inverse flow are analyzed for the sancha river mouth.

High-Performance Indirect Current Control Scheme for Railway Traction Four-Quadrant Converters

In the recent development of railway traction drives, four-quadrant converters are widely adopted at the grid side as front-end converters. Since the switching frequency of such megawatt-rated converters is limited to several hundreds of hertz, the bandwidth of the current controller is usually restricted. In addition to feedback control of instantaneous current, indirect current control (ICC) is also an alternative in high-power low-switching-frequency applications. Suffering from the inherent cross coupling between active and reactive power as well as sluggish response of ac current, traditional ICC is not a perfect solution for power conversion. To realize the independent control of active and reactive power, a phasor diagram arranged in an orthogonal coordinate system is introduced. In order to improve the dynamic performance of ICC, a transient-free current control method is proposed after analyzing the characteristic of ac current response in detail. In addition, a large-signal model of dc voltage loop, which is based on the conception of average power control, is set up. Finally, robustness to circuit parameters and adaptability to weak grids are certified for the transient-free controller. Both simulation and experimental results are presented to prove the validity of the proposed control scheme.

On numerical analysis of axisymmetric thick circular cylindrical shells based on higher order shell theories by segmentation method

Numerical analysis is carried out in this paper for thick circular cylindrical shells using higher order shell theory. In orthogonal curvilinear coordinate system, all equations are derived with the inclusion of the additional quadratic and cubic terms in the Taylor’s series expansions of the both in-plane as well as the transverse displacement components for the improvement of bending behaviour of the shell. Assuming ( h/ R)2 ≪ 1, a rigorous formulation involving the reduction of a three-dimensional elasticity problem to a two-dimensional one, based partly upon the Reissner’s variational principle, is presented. These equations are algebraically manipulated to be in the form of a coupled system of first-order differential equations in terms of the intrinsic dependent variables. These are then solved by a segmentation method – numerical integration technique for various combinations of material and geometric parameters. The theory is shown to result in a partial differential equation system of sixteenth order.

Regional bone density variations in osteoarthritic glenoids: a comparison of symmetric to asymmetric (type B2) erosion patterns

Accurate characterization of regional variations in bone density in symmetric and asymmetric (B2) glenoid erosion patterns can assist with surgical planning, reaming, and component implantation. The purpose of this study was to characterize regional bone density and porosity in symmetric and asymmetric (B2) osteoarthritic glenoids. Symmetric (n = 25) and asymmetric (B2) (n = 25) erosion patterns were compared by computed tomography-based imaging software. An orthogonal coordinate system separated each glenoid into quadrants. In addition, a linear best-fit line defined the line-of-erosion between the paleoglenoid and neoglenoid in the asymmetric cohort. All glenoids were further divided into volumes at depths of 0 to 2.5 mm and 2.5 to 5 mm. Average bone density was measured in Hounsfield units. Bone voids or cysts were included to quantify regional porosity as the fraction of void volume to total glenoid volume. For the symmetric cohort, there were no significant differences in bone density between quadrants at either depth (P ≥ .089). For the asymmetric cohort, bone density was significantly higher in the posterior quadrants compared with the anterior quadrants (P < .001), especially posteroinferiorly (P ≤ .007) at both depths. In addition, the neoglenoid had significantly higher density and lower void fraction compared with the paleoglenoid (P < .001). There were also significant differences in void fraction between quadrants for both cohorts, at both depths (P ≤ .004). This study demonstrates that osteoarthritic glenoids with symmetric erosion have uniform subarticular bone density. However, asymmetric (B2) erosion patterns have potentially important regional variations in bone density and porosity, with the densest bone with the least porosity found posteroinferiorly or in the neoglenoid region.

Integrated calibration of magnetic gradient tensor system

Measurement precision of a magnetic gradient tensor system is not only connected with the imperfect performance of magnetometers such as bias, scale factor, non-orthogonality and misalignment errors, but also connected with the external soft-iron and hard-iron magnetic distortion fields when the system is used as a strapdown device. So an integrated scalar calibration method is proposed in this paper. In the first step, a mathematical model for scalar calibration of a single three-axis magnetometer is established, and a least squares ellipsoid fitting algorithm is proposed to estimate the detailed error parameters. For the misalignment errors existing at different magnetometers caused by the installation process and misalignment errors aroused by ellipsoid fitting estimation, a calibration method for combined misalignment errors is proposed in the second step to switch outputs of different magnetometers into the ideal reference orthogonal coordinate system. In order to verify effectiveness of the proposed method, simulation and experiment with a cross-magnetic gradient tensor system are performed, and the results show that the proposed method estimates error parameters and improves the measurement accuracy of magnetic gradient tensor greatly.

Skeleton-based viewpoint invariant transformation for motion analysis

Viewpoint variation has been a major challenge in dealing with comparison-based image processing. Reduction or total removal of viewpoint variation has been the common pursuit of many processing applications such as human motion analysis and gesture analysis. By exploiting three-dimensional (3-D) skeletal joints infor- mation provided by RGB-D cameras such as Kinect, this study proposes a skeleton-based viewpoint invariant transformation (SVIT) technique that transforms the 3-D skeleton data into an orthogonal coordinate system constructed by the three most stable joints of detected person's upper torso: the left shoulder, the right shoulder, and the spine. The proposed 3-D transformation can eliminate observation variations caused not only by view- point differences but also individual differences in body swing movement. With reference to the human activity database MSRDailyAct3D and our recording data, experiments on human motion analysis were designed and conducted to contrast the effectiveness of the proposed SVIT as a preprocessing step before comparing the test data and the sample data, with very different viewpoints. This results show that, with SVIT as a pre-processing for activity analysis, the performance of correct activity identification has increased from 45% to 60%. SVIT out- performs other state-of-the-art view invariant human motion analysis based on human body centric approaches. © 2014 SPIE and IS&T (DOI: 10.1117/1.JEI.23.4.043021)

RESEARCH ON THE DISTRIBUTION OF PRESSURE FIELD ON THE BASILAR MEMBRANE IN THE PASSIVE SPIRAL COCHLEA

The cochlea is the important auditory organ of the inner ear. It is responsible for transforming the acoustic signals into neural impulses that travel along the auditory nerve to the brain. The role of, perhaps, the most characteristic feature of the cochlea, its three-dimensional (3D) helical structure, has remained elusive. To address this problem, the present paper develops a 3D spiral cochlea mathematical model using orthogonal coordinate system. Based on the method of separation of variables and conformal transformation, equations of three cases for the velocity potential are derived to solve the steady flow problem of lymph in the cochlea. Then, the distribution of pressure field on the basilar membrane (BM) is obtained. By comparing the analytical results with FE analyses results, the derived formulas are demonstrated to be accurate and reliable. The conclusion can be drawn that the spiral shape and physical dimension of the cochlea have a significant influence on the distribution of pressure field. Interestingly, near the helicotrema, the velocity potential of the first case plays a leading role in pressure distribution on the BM. Therefore, it may enhance the vibration of BM and improve hearing ability in the low-frequency parts of human ears. The proposed model could provide an approach for further investigation of fluid-structure interaction problem in the cochlea.

Linear calibration method of magnetic gradient tensor system

Based on the detailed analysis of systematic errors, mathematical model of error parameters is constructed and linear calibration method is proposed for magnetic gradient tensor system. Firstly, nonlinear mathematical model of error parameters for single vector magnetometer is constructed based on scalar calibration, and least square solution is deduced by two nonlinear conversions without any mathematical simplification. Then outputs of four tri-axial magnetometers are calibrated to sensor’s orthogonal coordinate respectively. Secondly, a least square estimation is proposed for the misalignment errors between different magnetometers according to the rotation matrix comprising conversion of different orthogonal coordinate system. After calibration, outputs of tri-axial magnetometers are acquired along the ideally platform frame-orthogonal coordinate system and these enable calibration of magnetic gradient tensor system. Simulations and experiments show that the proposed linear calibration method can accurately solve the detailed error parameters and decrease measurement errors of magnetic gradient tensor system remarkably.