Orthogonal Coordinate System Research Articles

The method for celestial bodies’ center of mass position relative to their figures determination on the basis of harmonic analysis of the expansion in spherical functions in order to refine the physical libration parameters

In this paper the problem of the lunar center of mass relative to the center of its figure determination on the basis of space observations is considered, since the Moon is the most studied celestial object and there is a complete database on it. The future prospects for lunar laser ranging and radio interferometry require development of adequate theoretical support for modern technologies. The aim of these studies is the distances’ measurement between the Moon and the Earth with an accuracy of 1 mm. Thus, determination of the lunar center of mass position, represented in this paper, and development of the selenocentric system will allow to solve the above mentioned problem more accurately and reliably. The new values of the lunar center of mass relative to its center of figure in orthogonal selenographic coordinate system Δξ, Δη, Δζ have been determined; they are: -1.75, -0.75, 0.11 km respectively.

A case-based decision theory based process model to aid product conceptual design

In new product development, the rapid proposal of innovative solutions represents an important phase. This in turn relies on creative ideas, their evaluation, refinement and embodiment of worthwhile directions. This study aims to describe a CBDT based process model for product conceptual design that concentrates on rapidly generating innovations with the support of decision-making rationale. Case-based decision theory (CBDT), derived from case-based reasoning, is applied in this paper as a core method to aid design engineers to make an informed decision quickly, thus accelerating the design process. In the process of utilizing CBDT to support a decision, as for the similarity function, the proper value assignment methods to the selected attribute set for calculation are discussed. In order to assist with innovative solution, aspects of the theory of inventive problem solving (TRIZ) are integrated into the case-based reasoning process. Accordingly, a CBDT-TRIZ model is developed. Quality-function deployment is used to translate customer wants into relevant engineering design requirements and thus formulating the design specification. Image-Scale is used to offer an orthogonal coordinates system to aid evaluation. Finally, a case study is used to demonstrate the validity of the proposed process model based on the design of a cordless hand-tool for garden and lawn applications.

Numerical analysis of different fluted fins for condensation on a vertical tube

A numerical analysis of laminar film condensation of pure vapor on an isothermal vertical fluted tube surface is presented using a surface-fitted orthogonal curvilinear coordinate system. Based on the assumptions employed in Nusselt's analysis on a flat plate, a liquid film thickness equation for film-wise condensation on fluted surface is obtained by simplifying the Navier-Stokes equations. By defining the fluted fin surface curvatures, different fin shapes can be obtained. A relationship between curvature parameters of the sinuous fluted tube such as fin height and fin arc length is obtained. A general condensation performance comparison method for different fin geometries which is set up using the Fourier's series approximation is presented by defining the fin surface curvature parameters. By solving the liquid film thickness equation, the curvature variation effects on a condensing surface and the stable film profiles are investigated. The calculations show that the standard sinuous fin's crest region area is crucial for total condensation heat transfer. It is demonstrated that the effect of surface tension because of curvature change on fluted sinuous fins is important in heat transfer enhancement as compared to other rectangular, triangular and other sinuous fin surfaces. For a constant base diameter tube, it was determined that heat transfer enhancement of the sinuous fins can be achieved by increasing the number of fins and the fin length. However, it is not a valid method to lengthen the fin height for rectangular or triangular fin when the liquid flow does not reach flooding. A more useful outside tube fin arrangement, which takes full advantage of fin tip effect and flooding flow characteristic, is presented.

Investigation of coupled instability for shear flexible FG sandwich I-beams subjected to variable axial force

The general thin-walled beam model is presented to study the flexural-torsional coupled buckling behavior of shear flexible sandwich I-beams made of functionally graded materials based on an orthogonal Cartesian coordinate system. The derived beam model includes the transverse shear, the restrained warping induced shear deformations, the structural coupling coming from the material anisotropy, and the variable axial force effect. Material properties of the beam such as Young’s and shear moduli are assumed to be graded across the wall thickness. The strain energy and the potential energy due to the variable axial force are introduced. The seven coupled instability equations are derived from the energy principle. To solve the instability problem, three types of finite beam elements, namely, linear, quadratic, and cubic elements are employed with the scope to discretize the governing equations. In order to verify the accuracy and efficiency of the beam model presented by this study, numerical results are presented and compared with results from other researchers. By numerical examples, the bisymmetric and mono-symmetric I-beams with two types of material distributions are considered to investigate the effects of shear deformation, variable axial force, gradient index, thickness ratio of ceramic, boundary conditions, material ratio, and span-to-height ratio on the buckling behavior of FG sandwich I-beams. Particularly, the crossover phenomenon in buckling modes with changes in gradient index and thickness ratio of ceramic in flanges is investigated.

Numerical Modeling of Marine Circulation, Pollution Assessment and Optimal Ship Routes

Methods and technology have been developed to solve a wide range of problems in the dynamics of sea currents and to assess their “impact” on objects in the marine environment. Technology can be used for monitoring and forecasting sea currents, for solving the problems of minimizing risks and analyzing marine disasters associated with the choice of the optimal course of the ship, and assessing the pollution of coastal zones, etc. The technology includes a numerical model of marine circulation with improved resolution of coastal zones, a method for solving the inverse problem of contamination of the sea with a passive impurity, and a variational algorithm for constructing the optimal trajectory of the vessel. The methods and technology are illustrated by solving problems of Baltic Sea dynamics. The model of sea dynamics is governed by primitive equations that are solved on a grid with an improved resolution of the selected coastal zone—in this case, the Gulf of Finland. The equations of the model are formulated in a bipolar orthogonal coordinate system with an arbitrary arrangement of poles and the sigma coordinate in the vertical direction. An increase in the horizontal resolution of the allocated zone is achieved due to the displacement of the north pole in the vicinity of the city of St. Petersburg. A class of dangerous technogenic situations and natural phenomena (sea accidents, which can be investigated with the help of the proposed methodology), includes tanker accidents in the case of a possible collision with a stationary object (with “dynamic danger”) or a moving object (including another ship), accidents on oil-producing platforms and oil pipelines, and coastal pollution.

Knee Kinematics Following Anatomic Single vs. Double Bundle ACL Reconstruction: A Randomized Clinical Trial

Objectives:A randomized clinical trial was conducted to compare knee kinematics during gait and running 24 months after ACL reconstruction, using either single-bundle (SB) or double-bundle (DB) quadriceps tendon grafts. We hypothesized that DB reconstruction would better restore kinematics than SB reconstruction, in comparison to the uninjured, contralateral knee.Methods:Subjects were between 14 and 50 years of age, participated in at least 100 hours of Level 1 or 2 sports activities and presented within 12 months of injury to both bundles of the ACL (with or without meniscal injury). Exclusion criteria included prior injury or surgery of the ipsilateral or contralateral knee or greater than a grade 1 concomitant knee ligament injury. Subjects were randomized to undergo SB or DB ACL reconstruction with a 10 mm quadriceps tendon autograft harvested with a patellar bone block. A single, anatomically placed femoral tunnel was used for all cases. For DB ACL reconstruction, the soft tissue portion of the graft was split and passed through two anatomically placed tibial tunnels. Biplane radiographic images were acquired 24 months after surgery while subjects performed downhill running on a treadmill (3.0 m/s, 10 degree slope, 150 images/s) and level gait (1.3 m/s, 100 images/s). Subject specific bone models were generated from computed tomography images and matched to the biplane radiographs using a previously validated model-based tracking process to determine tibiofemoral kinematics. Rotations of the tibia relative to the femur were calculated using the rotational component of the Joint Coordinate System originally described by Grood and Suntay. Displacements of the tibia relative to the femur were expressed in an orthogonal anatomical coordinate system fixed to the tibia. Primary outcome variables were based on previous findings of abnormalities in knee kinematics after ACL injury/reconstruction, and included peak knee external rotation, adduction and anterior translation during heelstrike to mid-stance. Three trials were collected for each limb and each task, and averaged for statistical analysis. Differences between SB and DB kinematics were determined using Wilcoxon Signed Rank tests, with significance level p < 0.05.Results:No significant differences were found in any of the primary kinematic variables between single and double bundle anatomic ACL reconstruction (Table). Differences between reconstructed and contralateral (uninjured) limbs were small, averaging less than 2 degrees in rotation and 2 mm in translation. Rotational differences were smaller during gait than during downhill running.Conclusion:Conclusions: Contrary to the study hypothesis, DB reconstruction was not found to be superior to SB reconstruction. While some abnormalities remained (particularly during the more stressful downhill running evaluation), both anatomical reconstructions were equally effective at restoring normal knee kinematics.kinematic differences between limbs (affected-contralateral) during gait and downhill runningSingle BundleSingle BundleSingle BundleDouble BundleDouble BundleDouble BundleNMeanStd DevNMeanStd DevWilcoxon pRunningPeak Knee Adduction25-0.231.1221-0.211.160.71RunningPeak External Rotation25-1.564.0421-0.173.120.20RunningPeak Anterior Translation250.572.62212.022.750.13GaitPeak Knee Adduction26-0.061.1921-0.051.110.83GaitPeak External Rotation26-0.444.3421-0.113.550.77GaitPeak Anterior Translation260.832.97212.002.560.20

Viscous laminar flow in smooth coil tubes

A mathematical model of incompressible viscous laminar flow in smooth coil tubes is proposed and the results of its numerical realization in a nonorthogonal helical coordinate system are presented. This coordinate system is free of singularities in the domain of definition of the unknown functions, that is, the pressure and the velocity components, which makes it possible to refine the existing distributions of the axial component and the secondary crossflows obtained using the well-known orthogonal coordinate system having a singularity at the center of the coil channel. The momentum transport equation is written in the projections on the axes of the natural basis of the coordinate system, which makes it possible to subdivide the system of equations into two alternately solved subsystems. The distributions of the axial and two transverse components show that at the center of coil tubes the transverse components are comparable with the axial velocity (the transverse components can be as high as half the mean-flow-rate velocity and one third of it at the center of the channel).

An integrated solution for reducing ill-conditioning and testing the results in non-linear 3D similarity transformations

To find 3D similarity transformation parameters (a scale, three rotational angles, and three translation elements) between two orthogonal coordinate systems in 3D is an ill-posed non-linear inverse problem by means of common points (their Cartesian components are known in the both systems). The problem can be solved via Linearized Least Squares (LLS) or Direct (non-iterative) Least Squares (DLS). Since the parameters in LLS take different quantities (and units) from each other, the condition problems can arise during the solution of normal equations. In this paper, we propose a combined solution to reducing ill-conditioning and to perform precision analysis and global outlier test in LLS accordingly. The way is based on column norming and uses the normalized unknowns instead of the original ones at the solution stage of the normal equations. While the global outlier test is fulfilled on the normalized unknowns, the original unknowns and their precisions obtained using the normalized matrix with linear transformation to the normalized unknowns and by the error propagation law to their variance-covariance matrix. A direct method (Direct Least Squares, DLS) is used to compute the initial values of LLS for reducing iteration number (to 1–3 cycles) with a 1.E-7 threshold in the paper. And, when the solution of 3D-ST having large rotational angles and uncertainties, it is also shown that DLS goes to a worse solution than LLS by means of a simulated transformation problem.

Unified Construction of Green’s functions for Poisson’s equation in inhomogeneous media with diffuse interfaces

Green’s functions for Poisson’s equation in inhomogeneous media with material interfaces have many practical applications. In the present work, we focus on Green’s functions for Poisson’s equation in inhomogeneous media with diffuse material interfaces where a gradual and continuous transition in the material constant is assumed in a small region around the interfaces between different materials. We present a unified general framework for calculating Green’s functions for Poisson’s equation in such inhomogeneous media and the framework can apply to all eleven orthogonal coordinate systems in which the three-dimensional Laplace equation is separable. Within this framework, the idea on how to design the so-called quasi-harmonic diffuse interface is discussed, formulations for building Green’s function for Poisson’s equation in an inhomogeneous medium with such a diffuse interface is elaborated, and a robust numerical method for calculating Green’s functions for Poisson’s equation in inhomogeneous media with general diffuse interfaces is developed. Several practically relevant separable coordinate systems are briefly surveyed at the level of definition and basic facts relevant for implementing the unified framework in these coordinate systems.

Saturated-unsaturated groundwater modeling using 3D Richards equation with a coordinate transform of nonorthogonal grids

• Coordinate transform of nonorthogonal grids for complex computational domain. • Derivation of Richards equation in arbitrary coordinates and boundary conditions. • Saturated-unsaturated groundwater modeling using 3D Richards equation • Four test bench simulations: hillslope, coupled flow, tilted V-Catchment and slab. Saturated-unsaturated flow under a complex terrain is usually solved using the Richards equation. Finite difference or finite volume methods are commonly employed for discretization of Richards equation because of simplicity of coding. Complex subsurface boundary geometries lead to nonorthogonal grids in curvilinear grid systems, which leads to difficulty in discretization and mesh generation. This paper develops a vertical coordinate transform, enabling a computational domain regular in the vertical direction. As a result, the grid of curvilinear surfaces can be successfully transformed to a computational grid that allows solution of the Richards equation with efficient computation and simpler coding. The anisotropic Richards equation in the Cartesian coordinate system is transformed to the equation in the arbitrary coordinate system and further expressed as a form appropriate to the orthogonal coordinate system. The generalized third boundary condition is transformed to a form suited to the orthogonal coordinate system. The finite volume method is used to solve the Richards equation in the orthogonal coordinate system. Four cases are used to validate the present orthogonal coordinate system. The computational results from the orthogonal coordinate system are in good agreement with the results from Ansys Fluent solved in a Cartesian coordinate system for the subsurface flow case. For the coupled case of hill slopes, a good agreement between the computational results and the experimental data is obtained. The present results for V-titled catchment and slab case accord well with the results obtained from HydroGeoSphere and PAWS. The present algorithm can improve grid generation for solution of Richards equation in a hydrological model for a complex domain.

Mass transfer enhancement in non‐dispersive solvent extraction with helical hollow fiber enabling Dean vortices

In this work mass transfer enhancement of non‐dispersive solvent extraction by use of helical hollow fiber membranes (HHFM) was investigated by means of experiment and model simulation. Purified terephthalic acid wastewater treatment by extraction with p‐xylene as solvent was chosen as the application case. Experiments showed that extraction efficiency of the HHFM was doubly enhanced compared with that of the straight hollow fiber. A comprehensive mathematical model of the HHFM extraction was developed in an orthogonal helical coordinate system with an analytical solution of the 3D velocities. Model simulation revealed that Dean vortices circulate the peripheral fluid to the center, which enhances the mass transfer in the lumen side where radial diffusion is the rate determining step of the extraction. Relations of effluent impurity concentration and enhancement factor with the Graetz number and dimensionless curvature, were obtained by model simulation. Optimal parameters were selected for HHFM extraction design. © 2017 American Institute of Chemical Engineers AIChE J, 63: 3479–3490, 2017

A Potential Enstrophy and Energy Conserving Scheme for the Shallow-Water Equations Extended to Generalized Curvilinear Coordinates

An energy and potential enstrophy conserving finite-difference scheme for the shallow-water equations is derived in generalized curvilinear coordinates. This is an extension of a scheme formulated by Arakawa and Lamb for orthogonal coordinate systems. The starting point for the present scheme is the shallow-water equations cast in generalized curvilinear coordinates, and tensor analysis is used to derive the invariant conservation properties. Preliminary tests on a flat plane with doubly periodic boundary conditions are presented. The scheme is shown to possess similar order-of-convergence error characteristics using a nonorthogonal coordinate compared to Cartesian coordinates for a nonlinear test of flow over an isolated mountain. A linear normal mode analysis shows that the discrete form of the Coriolis term provides stationary geostrophically balanced modes for the nonorthogonal coordinate and no unphysical computational modes are introduced. The scheme uses centered differences and averages, which are formally second-order accurate. An empirical test with a steady geostrophically balanced flow shows that the convergence rate of the truncation errors of the discrete operators is second order. The next step will be to adapt the scheme for use on the cubed sphere, which will involve modification at the lateral boundaries of the cube faces.

О создании единых глобальных прямоугольных координат для всей России

This article deals with the problem of creating unitary global space orthogonal geocentric coordinatesystem throughout Russia on the base of a geoid. This problem arose on Resolution of The Russian Federation Government. The point is that geodesic coordinates B, L and H are acceptable for moving and navigation of transport, but orthogonal coordinates X and Y are in the equatorial plane and have nothing to do with designing and construction. Adaptive systems unite global orthogonal coordinates. That is why the authors set the goal of using polyhedral projections. All maps compiled and used in Russian cartographic Gauss and Crueger projection could be attributed to that. For many border squares errors will be very little. The authors used very little borders 20 by 20 km and the known topo-centric methods. As a result they create an adaptive system of unitary global orthogonal coordinates.

ОРТОГОНАЛЬНАЯ КРИВОЛИНЕЙНАЯ СИСТЕМА КООРДИНАТИ ПОСТРЕНИЕ ПОВЕРХНОСТЕЙ НА ТРАПЕЦИЕВИДНО-КРИВОЛИНЕЙНЫХ ПЛАНАХ

Thin space constructions on curved plans there are used at building of public structures, trade centers. Sport constructions. Working-out the methods of forming of the surfaces on the curved plans that is one of the modern task of the architecture and town building. The orthogonal coordinate system that is formed at the plane with some plane directrix curve and the system of the right lines orthogonal to the directrix curve there is regarded at the stat. The coordinate system forms some trapezium-curved segment. Taking some function of the vertical coordinate there is possible to receive different surfaces at the curved plans. Conjugating different directrix curves, it’s possible to receive the combined surfaces. The article presents a system of orthogonal coordinates of curvilinear-trapezoidal planes and methods for forming surfaces on these planes. Surfaces with a function of the vertical coordinate of a general view are considered, and surfaces on combined plans of segments of the same type are shown.

Implementation of an Approximate Conformal UPML in 2-D DGTD

Using the numerical discrete technique with unstructured grids, conformal perfectly matched layer (PML) absorbing boundary in the discontinuous Galerkin time-domain (DGTD) can be set flexibly so as to save lots of computing resources. Based on the DGTD equations in an orthogonal curvilinear coordinate system, the processes of parameter transformation for 2-D UPML between the coordinate systems of elliptical and Cartesian are given; and the expressions of transition matrix are derived. The calculation scheme of conductivity distribution in elliptic cylinder absorbing layer is given, and the calculation coefficient of DGTD in elliptic UPML is calculated. Furthermore, the 2-D iterative formulas of DGTD and that of auxiliary equation in the elliptical cylinder UPML are derived; the conformal UPML calculation in DGTD is realized. Numerical results show that very good accuracy and computational efficiency are achieved by using the method in this paper. Compared to the rectangular computational region, both the memory and computation time of conformal UPML absorbing boundary are reduced by more than 20%.

EXTENDED CLARKE TRANSFORMATION FOR n-PHASE SYSTEMS

The article describes how to convert space vectors written in a stationary multiphase system, consisting of a number of phases where n &gt; 3, to the stationary alfa beta orthogonal coordinate system. The transformation of vectors from a stationary n-phase system to the stationary alfa beta orthogonal coordinate system is defined The inverse transformation of a vector written in the orthogonal coordinate system to a stationary n-phase system is also defined. The application of the extended Clarke transformation allows control calculations to be performed in both stationary alfa beta or rotating dq orthogonal coordinate systems. This gives the possibility of performing different control strategies. It has a practical application for drive systems with five-phase, six-phase or dual three-phase motors.

Ferromagnetic Flow of Viscous Fluid in a Slot between Fixed Surfaces of Revolution

AbstractIn this paper the steady laminar flow of viscous incompressible ferromagnetic fluid is considered in a slot between fixed surfaces of revolution having a common axis of symmetry. The boundary layer ferromagnetic equations for axial symmetry are expressed in terms of the intrinsic curvilinear orthogonal coordinate system x, θ ,y.The method of perturbation is used to solve the boundary layer equations. As a result, the formulae defining such parameters of the flow as the velocity components vx, vy, and the pressure , were obtained.

Transient Solution for the Energy Balance in Porous Media Considering Viscous Dissipation and Expansion/Compression Effects Using Integral Transforms

Monitoring of downhole flowing temperatures in oil wells is gaining attention in the recent years due to the possibility of exploring these data for reservoir characterization and determination of inflow profiles along the well completions, leading to an increased interest in the development of solutions for the equations governing the thermal behavior of a reservoir. In this work, it is proposed to use the generalized integral transform technique (GITT) to provide solutions for the energy balance equation, considering the thermal effects related to fluid flow. A formal and general solution for the energy balance in the porous media is presented and validated. It is presented the application of the proposed solution to one-dimensional and two-dimensional problems in the Cartesian coordinate system. The two-dimensional problem, which considers heat transfer to the surrounding impermeable formations, is tackled by a single domain formulation. The mathematical approach taken in these solutions is rigorous, valid for all flow regimes (transient, late-transient and pseudo-steady-state/steady-state) and for any orthogonal coordinate system, presenting the possibility of achieving differentiable and stable solutions with controlled accuracy. The solution comprises an important contribution to support the application of temperature data to reservoir engineering problems.

Change in Anterior and Posterior Curvature After Cataract Surgery.

To analyze the change in anterior and posterior corneal curvature after cataract surgery using a Placido-dual rotating Scheimpflug device. In a prospective cross-sectional study, corneal curvature was measured using the Galilei G4 device (Ziemer Ophthalmic Systems, Port, Switzerland) preoperatively and 1 week and 1, 3, and 6 months after cataract surgery with a temporal limbal self-sealing 2.2-mm incision. The surgically induced astigmatism (SIA) was determined on the anterior and posterior surfaces. Fifty-nine patients (68 eyes) were assessed. Based on the anterior corneal surface, 16 (23.5%) eyes had a vertically steep meridian (with-the-rule [WTR] astigmatism), 32 (47.1%) had a horizontally steep meridian (against-the-rule [ATR] astigmatism), and 20 (29.4%) had oblique astigmatism. Based on the posterior corneal surface, 2 (2.9%) eyes had a horizontally steep meridian (ATR astigmatism), 62 (91.2%) had a vertically steep meridian (WTR astigmatism), and 4 (5.9%) had oblique astigmatism. SIA of the anterior and posterior corneal surfaces was 0.61 ± 0.33 and 0.20 ± 0.17 diopters (D), respectively. However, there was no significant difference between the preoperative and the 6-month postoperative data in the Jackson coefficient orthogonal coordinate system for the anterior and posterior corneal surfaces. SIA of WTR astigmatism of the posterior cornea was 0.19 ± 0.16 D at 6 months. Sixty-one of 62 eyes with WTR astigmatism in the posterior corneal surface still showed WTR astigmatism after cataract surgery. The tendency of SIA of the posterior cornea may not be uniform, but type of posterior corneal astigmatism did not change in most cases after the 2.2-mm temporal limbal incision cataract surgery. [J Refract Surg. 2016;32(11):754-759.].

The Size-Dependent Electromechanical Coupling Response in Circular Micro-Plate due to Flexoelectricity

AbstractFlexoelectricity, the coupling of strain gradient to polarization, enhances the properties of piezoelectric response desirable for advanced MEMS dramatically even in centrosymmetric dielectrics. In this paper, the general formulations of the flexoelectric couple stress theory presented by Hadjesfandiari in orthogonal curvilinear coordinate system are derived, and are then specified for the case of cylindrical coordinates. A size-dependent flexoelectric model of circular plate is established based on the current formulations in cylindrical coordinates. The governing equations, boundary conditions and initial conditions are derived by applying Hamilton's principle. The static bending and free vibration problems of a simply supported axisymmetric circular plate are carried out to illustrate the applicability of the present model. Numerical results reveal that a homogeneous electric field between the up and down surfaces of the circular plate is induced indeed. The generated deflection, induced voltage and natural frequency show obvious size effect, but the size effect is almost diminishing as the thickness of the plate is far greater than the material length scale parameter. As the increase of the flexoelectric coefficient, the induced voltage increases evidently and the generated deflection and the natural frequency increase weakly.