Arbitrary Coordinate System Research Articles

ОБТЕКАНИЕ ЦИЛИНДРА И АЭРОДИНАМИЧЕСКОГО ПРОФИЛЯ С УЧЕТОМ ЛАМИНАРНО-ТУРБУЛЕНТНОГО ПЕРЕХОДА

A simulation of the flow around a cylinder and a NACA 4412 airfoil was carried out using the γγ-Reθ laminar-turbulent transition model and without it. Numerical simulation was performed on the basis of the Reynolds-averaged Navier-Stokes equations (URANS), using the Spalart-Allmaras differential one-parameter turbulence model. The system of initial equations was written with respect to an arbitrary curvilinear coordinate system. The coordination of the pressure and velocity fields was carried out using the method of artificial compressibility modified to calculate non-stationary problems. The integration of the system of initial equations was carried out numerically using the control volume method. For convective flows, a countercurrent Rogers-Kwak approximation was used based on the third-order Roe scheme. In turbulence models, the TVD scheme with a third-order ISNAS flow limiter was used to approximate convective terms. Comparison of the calculation results for a flow past a cylinder is carried out using the laminar-turbulent transition model and without it. It is shown that at low Reynolds numbers, when the flow around a cylinder is laminar, and the track isturbulent use of the Spalart-Allmaras turbulence model leads to the development of a turbulent boundary layer on the cylinder and, consequently, to a change in the position of the separation point. The incorrect position of the separation point affects the pressure distribution in the bottom of the cylinder and, as a result, the integral aerodynamic characteristics. The use of the transition model allows one to adequately reconstruct the laminar separation near the leading edge of the airfoil with its subsequent reattachment. The use of the Spalart-Allmaras model alone leads to excessive generation of turbulent viscosity. It is shown that the use of the γ-Reθ laminar-turbulent transition model the qualitatively and quantitatively improves the results of numerical simulation. The results of numerical simulation of the flow around a circular cylinder and the NACA 4412 aerodynamic airfoil agree well with experimental data in a wide range of Reynolds numbers.

Study of the elastic properties of hexagonal metal single crystals

Hexagonal metals (e.g., Be, Zr, Ti) are widely used in the nuclear industry, space and aircraft engineering (in manufacturing of the structural elements operating under extreme conditions). A promising way to improve the quality of products made of them is to improve the physical properties of materials using the natural anisotropy of metal single crystals. The results of studying anisotropy and a comparative analysis of the technical characteristics of the elastic properties of single crystals of hexagonal metals are presented. The equations of the elastic compliance matrix components are derived in the explicit form for arbitrary crystallographic direction proceeding from transformations of the elastic compliance tensor in the principal axes to a new arbitrary coordinate system with a subsequent use of Euler angles. Analytical expressions are presented for the technical characteristics of the elastic properties (shear and Young's moduli, Poisson's ratio) of the single crystals of 10 hep metals for an arbitrary crystallographic direction. The axial symmetry of the characteristics about the hexagonal axis is revealed. The sums of the elastic compliance coefficients which determine the shear moduli and the Poisson's ratios in two mutually perpendicular directions are constant in any crystallographic plane of the single crystal. A comparative analysis of the anisotropy of the elastic properties of single crystals of the studied group of metals revealed auxetic properties of Zn and Be single crystals and the region of crystallographic directions of uniaxia tension, leading to an auxetic effect The auxetic effect in Zn was observed under tension in the directions of the plane perpendicular to the hexagonal axis of the single crystal. The planes of the auxetic effect manifestation in Be single crystals are perpendicular to the directions making an angle of 45° with the hexagonal axis.

Aerodynamics of the turbulent flow around a multi­element airfoil in cruse configuration and in takeoff and landing configuration

Numerical modeling of multi-element airfoil's aerodynamics employs the Reynolds averaged Navier-Stokes equations of incompressible environment, which are related via a single-parametric differential turbulence model by Spalart-Allmaras. The system of initial equations has been recorded with respect to an arbitrary curvilinear coordinate system. The pressure and velocity fields have been aligned by using an artificial compressibility method modified for calculating the nonstationary problems. The system of initial equations has been integrated numerically by applying a control volume method. The counter-flow Rogers-Kwak approximation has been used for convection flows, based on the Roe scheme of third-order accuracy. The turbulence models, in order to approximate the convection components, employed a TVD scheme with the third-order flow limiter ISNAS. The paper reports results from calculating a turbulent flow around a multi-element airfoil in a wide range of the angles of attack. The result of the current research is the performed analysis of the flow field around a multi-element airfoil, pressure coefficients, the lifting force, as well as the drag force. Physical features in a flow structure at flowing around the multi-element airfoil 30P30N have been identified. In the investigated range of the angles of attack, flowing around a airfoil in the takeoff and landing configuration is stationary in nature except for the regions where the flow is detached from sharp edges, such as the slat's inside part and a region in the tail part of the main profile. There are recirculation currents within these regions. With an increase in the angle of attack, dimensions of the detachable zone at the slat's inner surface decrease while remaining almost unchanged in the tail part of the main profile. At the top surface of the main profile there forms a jet of air, due to the acceleration of the flow between the slat and the leading edge of the main profile. The existence of a gap between the main profile and the flap leads to the interference of jet currents at the upper surface of the slat. It has been shown that the takeoff and landing configuration demonstrates the higher values of the lifting force coefficient than the cruise configuration, especially at large angles of attack. The calculation results agree well with the data by other authors.

Second strain gradient theory in orthogonal curvilinear coordinates: Prediction of the relaxation of a solid nanosphere and embedded spherical nanocavity

In this paper, Mindlin’s second strain gradient theory is formulated and presented in an arbitrary orthogonal curvilinear coordinate system. Equilibrium equations, generalized stress-strain constitutive relations, components of the strain tensor and their first and second gradients, and the expressions for three different types of traction boundary conditions are derived in any orthogonal curvilinear coordinate system. Subsequently, for demonstration, Mindlin’s second strain gradient theory is represented in the spherical coordinate system as a highly-practical coordinate system in nanomechanics. Second strain gradient elasticity have been developed mainly for its ability to capture the surface effects in the presence of micro-/nano- structures. As a numeric illustration of the theory, the surface relaxation of spherical domains in Mindlin’s second strain gradient theory is considered and compared with that in the framework of Gurtin–Murdoch surface elasticity. It is observed that Mindlin’s second strain gradient theory predicts much larger value for the radial displacement just near the surface in comparison to Gurtin–Murdoch surface elasticity.

Lattice Boltzmann models based on the vielbein formalism for the simulation of flows in curvilinear geometries

In this paper, we consider the Boltzmann equation with respect to orthonormal vielbein fields in conservative form. This formalism allows the use of arbitrary coordinate systems to describe the space geometry, as well as of an adapted coordinate system in the momentum space, which is linked to the physical space through the use of vielbeins. Taking advantage of the conservative form, we derive the macroscopic equations in a covariant tensor notation, and show that the hydrodynamic limit can be obtained via the Chapman-Enskog expansion in the Bhatnaghar-Gross-Krook approximation for the collision term. We highlight that in this formalism, the component of the momentum which is perpendicular to some curved boundary can be isolated as a separate momentum coordinate, for which the half-range Gauss-Hermite quadrature can be applied. We illustrate the capabilities of this formalism by considering two applications. The first one is the circular Couette flow between rotating coaxial cylinders, for which benchmarking data are available for all degrees of rarefaction, from the hydrodynamic to the ballistic regime. The second application concerns the flow in a gradually expanding channel. We employ finite-difference lattice Boltzmann models based on half-range Gauss-Hermite quadratures for the implementation of diffuse reflection, together with the fifth-order weighted essentially nonoscillatory and third-order total variation diminishing Runge-Kutta numerical methods for the advection and time stepping, respectively.

Marker-free coregistration of UAV and backpack LiDAR point clouds in forested areas

Unmanned aerial vehicle Laser Scanning (ULS) and Backpack Laser Scanning (BLS) are two emerging mobile mapping technologies applicable for monitoring forested environments in unprecedented detail from complementary perspectives. Although ground-based backpack techniques provide detailed information about the forest understory and terrain, the measured point clouds based on SLAM techniques are stitched together gradually and normally expressed in a less-accurate arbitrary coordinate system. Conversely, ULS point clouds are acquired from above and usually georeferenced, yet the point density and penetrability near the ground may still suffer from dense overstory despite the low attitude operation. Coregistering the ground and aerial point clouds in the ULS coordinate system therefore provides a method for fusing understory and overstory information at single tree level without the time consuming procedure of applying ground control points. Since the ULS and BLS acquisition viewpoints differ greatly, standard coregistration methods requiring 3D point-level correspondences are likely to fail. This paper presents an object-level coregistration approach which instead operates on two sets of tree positions, with the goal of finding the optimal 3D transformation (consisting of rotation, translation and scaling) between the respective coordinate systems. The entire task is decomposed into separate problems of computing the common Z axis, estimating the scale, and 2D coregistration. In contrast to existing methods, our approach does not require additional information such as tree diameters or heights. We evaluated our method on real test plots involving diverse stem densities and tree species situated in forest farm of the eastern coastal region of Jiangsu, China. The tree positions for ground and aerial data were obtained respectively by cylinder fitting and tree segmentation. On 3 coniferous (dawn redwood) plots, 46–81% trees were matched with a distance below 50 cm, and mean position deviation of 27–36 cm. For 4 broadleaf (poplar) plots, no more than 50% trees were matched below a 1 m threshold and mean error of 54–67 cm, which can be attributed to the broadleaf trees’ more irregular shape and lack of a well defined tree top. Moreover, we show that the introduction of scaling into the transform can increase the matched tree count by up to 20 percentage points and decrease the mean matched distance by up to 13% compared to a strictly rigid transform.

Modified couple stress theory in orthogonal curvilinear coordinates

The formulations for the modified couple stress theory (MCST) are consistently derived in general orthogonal curvilinear coordinate systems. In particular, the expressions for the rotation vector, higher-order strain, and stress tensors, i.e., the rotation gradient tensor and the deviatoric part of the symmetric couple stress tensor, and the classical strain and stress tensors are derived for an arbitrary orthogonal curvilinear coordinate system. Additionally, using the theory of surfaces, the formulations for the MCST are derived for general doubly curved coordinates, which are more convenient to use for shells of arbitrary curvature. The expressions for special cases, i.e., cylindrical and spherical shells, are obtained. The MCST expressions derived in this study are comprehensive and generally and can be used for consistent utilisation of the MCST in any orthogonal curvilinear coordinate system.

Influence of drift gyroscope on the accuracy of complementary filter

The paper deals with the analysis and synthesis of a complementary filter. The focus is on studying the effect of gyro drift on the accuracy of the filter. It is shown that the presence of errors can be associated with an additional rotation matrix, which is the sum of the unit and small skew-symmetric matrix. The structure of the moment of the complementary filter correction is substantiated. The feature of filter functioning in an arbitrary reference coordinate system is presented. In an arbitrary coordinate system there will be the effect of drifts with respect to all axes of the reference coordinate system. The efficiency of a magnetometer introduction into the complementary filter is investigated. It is shown that due to the proper choosing оf the transfer function in the accelerometer correction channel, it is possible to eliminate the systematic error not only from the gyroscope signal, but also from the magnetometer signal.

A least-squares method for the inverse reflector problem in arbitrary orthogonal coordinates

In this article we solve the inverse reflector problem for a light source emitting a parallel light bundle and a target in the far-field of the reflector by use of a least-squares method. We derive the Monge–Ampère equation, expressing conservation of energy, while assuming an arbitrary coordinate system. We generalize a Cartesian coordinate least-squares method presented earlier by Prins et al. [13] to arbitrary orthogonal coordinate systems. This generalized least-squares method provides us the freedom to choose a coordinate system suitable for the shape of the light source. This results in significantly increased numerical accuracy. Decrease of errors by factors up to 104 is reported. We present the generalized least-squares method and compare its numerical results with the Cartesian version for a disk-shaped light source.

Three-dimensional transformation optics for arbitrary coordinate systems: transforming conductive materials and boundaries.

A three-dimensional transformation optics method, leading to homogeneous materials, applicable to any non-Cartesian coordinate systems or waveguides/objects of arbitrary cross-sections is presented. Both the conductive boundary and internal material of the desired device is determined by the proposed formulation. The method is applicable to a wide range of waveguide, radiation, and cloaking problems, and is demonstrated for circular waveguide couplers and an external cloak. An advantage of the present method is that the material properties are simplified by appropriately selecting the conductive boundaries. For instance, a right-angle circular waveguide bend is presented which uses only one homogenous material. Also, transformation of conductive materials and boundaries are studied. The conditions in which the transformed boundaries remain conductive are discussed. In addition, it is demonstrated that negative infinite conductivity can be replaced with positive conductivity, without affecting the field outside the conductive boundary. It is also observed that a negative finite conductivity can be replaced with a positive one, by accepting some small errors. The general mathematical procedure and formulation for calculating the parametric surface equations of the conductive peripheries are presented.

A note on Navier-Stokes equations with nonorthogonal coordinates

There are many fluid flow problems involving geometries for which a nonorthogonal curvilinear coordinate system may be the most suitable. To the authors’ knowledge, the Navier–Stokes equations for an incompressible fluid formulated in terms of an arbitrary nonorthogonal curvilinear coordinate system have not been given explicitly in the literature in the simplified form obtained herein. The specific novelty in the equations derived here is the use of the general Laplacian in arbitrary nonorthogonal curvilinear coordinates and the simplification arising from a Ricci identity for Christoel symbols of the second kind for flat space. Evidently, however, the derived equations must be consistent with the various general forms given previously by others. The general equations derived here admit the well-known formulae for cylindrical and spherical polars, and for the purposes of illustration, the procedure is presented for spherical polar coordinates. Further, the procedure is illustrated for a nonorthogonal helical coordinate system. For a slow flow for which the inertial terms may be neglected, we give the harmonic equation for the pressure function, and the corresponding equation if the inertial effects are included. We also note the general stress boundary conditions for a free surface with surface tension. For completeness, the equations for a compressible flow are derived in an appendix. doi:10.1017/S144618111700058X

Thermoelastic-plastic flow equations in general coordinates

The equations governing the thermoelastic-plastic flow of isotropic solids in the Prandtl-Reuss and small anisotropy approximations in Cartesian coordinates are generalized to arbitrary coordinate systems. In applications the choice of coordinates is dictated by the symmetry of the solid flow. The generally invariant equations are evaluated in spherical, cylindrical (including uniaxial), and both prolate and oblate spheroidal coordinates.

Strategies for Improving and Evaluating Robot Registration Performance.

In a dynamic robotic workcell environment where robots may be frequently relocated or may need to collaborate with other robots, it is simpler and more robust to program robots in an external or unifying reference frame. The process of robot registration involves finding the location of a robot with respect to another reference frame. For instance, if parts are in known locations on a table and a robot can locate itself with respect to the table (an external reference frame), then the robot will also know the location of the parts. Furthermore, if two or more robots can locate themselves with respect to the table, then each robot will not only know the location of the parts, but also the location of every other robot. This knowledge facilitates the coordination of robot motions and robot collaboration and eases the integration of additional robots into the workcell. Since robot registration is critically necessary and occurs frequently, its process needs to be inexpensive, fast, and accurate. This paper details the requirements for a relatively inexpensive and fast robot registration experience, along with detailing strategies that incur significant improvements to registered robot positioning accuracy with minimal overhead. A quantitative verification process is presented to evaluate the performance impacts of these strategies. Peg-in-hole experiments are conducted to validate the notion that more accurate robot registration translates to more reliable task-level performance.

A NOTE ON NAVIER–STOKES EQUATIONS WITH NONORTHOGONAL COORDINATES

There are many fluid flow problems involving geometries for which a nonorthogonal curvilinear coordinate system may be the most suitable. To the authors’ knowledge, the Navier–Stokes equations for an incompressible fluid formulated in terms of an arbitrary nonorthogonal curvilinear coordinate system have not been given explicitly in the literature in the simplified form obtained herein. The specific novelty in the equations derived here is the use of the general Laplacian in arbitrary nonorthogonal curvilinear coordinates and the simplification arising from a Ricci identity for Christoffel symbols of the second kind for flat space. Evidently, however, the derived equations must be consistent with the various general forms given previously by others. The general equations derived here admit the well-known formulae for cylindrical and spherical polars, and for the purposes of illustration, the procedure is presented for spherical polar coordinates. Further, the procedure is illustrated for a nonorthogonal helical coordinate system. For a slow flow for which the inertial terms may be neglected, we give the harmonic equation for the pressure function, and the corresponding equation if the inertial effects are included. We also note the general stress boundary conditions for a free surface with surface tension. For completeness, the equations for a compressible flow are derived in an appendix.

Critical comments to the Oliver–Pharr measurement technique of hardness and elastic modulus by instrumented indentations and refinement of its basic relations

A critical analysis was made of the Oliver and Pharr method for determination of the hardness and elastic moduli of materials by instrumented indentations with the continuous recording of the P–hdiagrams (P is the force acting on the indenter, h is the approach of the indenter and a sample). Mistakes and insufficient justification were revealed in the basic theoretical relations of this method. In particular, this refers to an incorrect definition of the depth of the elastic contact hc, which is the base of these relalations. New refined basic relations and formulas for the determination of hardness and elastic modulus are given, in which the above defects are eliminated and which are based only on the assumption of elastic unloading of the indenter according to the classic theories of the elastic contact. In addition to the foregoing, using the data of the P–h diagram measured in the arbitrary laboratory coordinate system, an improved method of the stable determination of the contact stiffness S = dP/dh at the P–h diagram position have been proposed in the commonly accepted theoretical coordinate system in which its basic classic model relations are recorded. These refinements have been derived without additional assumptions to hypotheses to the Oliver and Pharr method and additional experimental measurements.

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.

A new profile roughness measurement approach for involute helical gears

With increasing quality requirements of involute gear contact surfaces resulting from grinding, superfinishing, polishing and coatings of various types, surface characterization at the roughness scale is becoming more important. Typical gear roughness measurements are made in an arbitrary coordinate system using roughness measuring instruments which are difficult to relate to gear meshing action and therefore gear functionality. A method has been developed which allows well defined, repeatable measurements to be taken using conventional roughness measuring instruments and then transform the results into the functional gear meshing coordinate system. As part of the method a mathematical model describing the gear geometry and alignment method was developed. The method has been validated by standard gear profile measurement.

The elastic theory of shells using geometric algebra.

We present a novel derivation of the elastic theory of shells. We use the language of geometric algebra, which allows us to express the fundamental laws in component-free form, thus aiding physical interpretation. It also provides the tools to express equations in an arbitrary coordinate system, which enhances their usefulness. The role of moments and angular velocity, and the apparent use by previous authors of an unphysical angular velocity, has been clarified through the use of a bivector representation. In the linearized theory, clarification of previous coordinate conventions which have been the cause of confusion is provided, and the introduction of prior strain into the linearized theory of shells is made possible.

Visualizing Three-dimensional Volumetric Data with an Arbitrary Coordinate System

Astronomical data does not always use Cartesian coordinates. Both all-sky observational data and simulations of rotationally symmetric systems, such as accretion and protoplanetary disks, may use spherical polar or other coordinate systems. Standard displays rely on Cartesian coordinates, but converting non-Cartesian data into Cartesian format causes distortion of the data and loss of detail. Here, I demonstrate a method using standard techniques from computer graphics that avoids these problems with three-dimensional data in arbitrary coordinate systems. The method adds minimum computational cost to the display process and is suitable for both realtime, interactive content, and producing fixed rendered images and videos. Proof-of-concept code is provided which works for data in spherical polar coordinates.

INVARIANTS IN ELASTIC PROPERTIES OF METAL SINGLE CRYSTALS OF CUBIC SYMMETRY

The equations of the elastic compliance matrix components are derived for arbitrary crystallographic directions determined by the Euler angles for cubic single crystals using the approach [4] developed by the authors which consists in transformation of the elastic compliance tensor in the principal axes into an arbitrary coordinate system with subsequent use of the Euler angles. Euler’s angles are applied in the following format: rotation around the hexagonal axis z (azimuth angle α), inclination of the hexagonal axis to an arbitrary position z ’ (polar angle β), rotation around the new position of z ’ axis (shear direction angle γ). Analysis of the equations derived for the components of elastic compliance matrix for cubic single crystals revealed invariant combinations of the components which, in turn, revealed relations between technical characteristics of the elastic properties of cubic single crystals: Young’s modulus along an arbitrary crystallographic direction, shear modulus and Poisson’s coefficients in a plane perpendicular to the selected crystallographic direction, shear modulus under torsion around this direction, and others. The validity of the obtained relations is verified in the calculations done for arbitrary crystallographic directions and various cubic single crystals. Calculations for single crystals of copper and nickel are presented.