Orthogonal Coordinate System Research Articles

Unsupervised detection and mapping of sparks in the Electrochemical Discharge Machining (ECDM) process

Material removal in electrochemical discharge machining is caused by sparks generated in a tool immersed in an electrolytic solution. Being the primary machining agent in this non-contact machining process, mapping the locations of microscopic sparks is of great interest. The distribution of sparks around the tool surface could give insights into the machined hole properties like the size, surface finish, and depth as compared to the machining parameters such as applied voltage, tool size, rotation speed, and feed rate. This paper is focused on detecting sparks in photographs of the ECDM process captured using a high-speed camera. A novel approach of using a tri-planar reflective surface for capturing the location of sparks in 3D space using a 2D camera output is attempted. Traditional spark detection methods use neural network classifiers that need labeled data for training. This labeled data often comes from human intervention and contains inherent biases that could lead to misclassification. In this paper, an unsupervised spark detection methodology is demonstrated, which eliminates the need for human intervention and relies on the number of neighboring pixels detected in regions of interest (ROIs). The feasibility of using adaptive background modeling to classify thousands of images and identify the ones with sparks is demonstrated in this work. The masking technique combining effects of erosion followed by dilation is used to determine the exact boundaries of the spark contours in every image. Centroids for each of these contours are then transformed from the skewed coordinate system as observed in camera images, to a three-dimensional orthogonal coordinates system centered around the tool. The same procedure is repeated for various voltages to benchmark the distribution of sparks around a tool tip in an ECDM process.

Physical interpretation of invariance of vector differential operators in three orthogonal coordinate systems and a case study of wall and globe of death

In the present study, the effect of operators on fundamental physical quantities in three orthogonal coordinate systems was analyzed using differential equations. The mathematical expression for the position vector and its magnitude, velocity, &acceleration in three orthogonal coordinate systems were obtained from the geometry of the figure and differentiation. The derivations of operation of vector differential operators (curl and diversions) on position vectors were derived, and their physical significance was illustrated. Similarly, the operation of the Laplacian operator on the magnitude of the position vector was also studied. The study highlighted the invariance in the results of all operations in three orthogonal coordinate systems. An investigation has been performed to describe the wall of death and the globe of death problem in appropriate orthogonal coordinate systems. The interpretations in terms of velocity, acceleration, and the net force acting on the vehicle are explained in detail.

Rheological analysis of pressure-driven Jeffrey fluid flow between corrugated porous curved walls with slip constraints

Pressure-driven movement is a fundamental concept with numerous applications in various industries, scientific disciplines, and fields of engineering. Its proper execution is vital for promoting revolutionary innovations and providing solutions in numerous sectors. Therefore, this article scrutinizes the pressure-driven flow of a magnetized Jeffrey fluid between two curved corrugated walls. The geometry of the channel is represented mathematically in an orthogonal curvilinear coordinate system. The corrugation grooves are described by sinusoidal functions with phase differences between the corrugated curved walls. The boundary perturbation method is used to find the analytical solution for the velocity and temperature taking the corrugation amplitude as the perturbation parameter. Furthermore, the volumetric flow rate, skin friction coefficient, and local Nusselt numbers are precisely calculated numerically for a variety of parameters, with the results presented comprehensively in tabular form. The impact of dissimilar parameters, such as the curvature parameter, wave number, magnetic parameter, Darcy number, thermal radiation, heat source/sink parameter, Jeffery fluid parameter, and amplitude parameter, on the flow fields is analyzed through graphical and tabular forms and discussed in detail. The results show that the velocity profile increases due to the curvature parameter and the Jeffrey fluid parameter. However, it decreases due to the magnetic parameter. The temperature distribution rises with the thermal slip and heat source/sink parameters. Meanwhile, it declined for the radiation parameter and the curvature parameter. The model can be used to simulate blood flow in arteries with varying geometries and magnetic fields, aiding in the study of cardiovascular diseases and the design of medical devices such as stents.

Rheological study of water-based Cu nanofluid between two corrugated curved walls under constant pressure gradient

Nanofluids have captured the attention of scientists due to their superior thermophysical properties compared to ordinary liquids. Consequently, nanofluids serve as suitable cooling agents applicable to systems requiring swift response to thermal changes, such as vehicle engines. Therefore, the current article scrutinizes the characteristics of heat transfer on the pressure-driven flow of magnetized nanofluid between two curved corrugated surfaces in the presence of heat generation/absorption. Copper oxide nanoparticles (Cu) have been combined with pure water to form a nanofluid called Cu/H2O. The geometry of the channel is represented mathematically in an orthogonal curvilinear coordinate system. The corrugation grooves are described by sinusoidal functions with phase differences between the corrugated curved walls. The boundary perturbation method is used to find the analytical solution for the velocity field, temperature field, and volumetric flow rate taking the corrugation amplitude as the perturbation parameter. The impact of dissimilar parameters such as the curvature parameter (0.5≤k≤10), wave number (1≤α≤3), magnetic parameter (1≤M≤5), and wave amplitude (0.1≤ε≤0.8) on the flow fields are analyzed through graphs and discussed in detail. The results show that the peak of the velocity increases with the radius of curvature and the width of the channel for a constant pressure gradient. If we increase the magnetic parameter from 1 to 4, the velocity profile at the specified point decreases by 30 %. If we increase the heat source/sink parameter from 2 to 5, the temperature profile at the specified point increases by approximately 17 %. The flow rate is increased by the corrugations for any phase difference between the corrugated curved walls depending on the corrugation wavenumber and the channel radius of curvature.

Einthoven's triangle adapted for horses: Proposal for the Delta configuration.

Reliable ECGs are crucial for diagnosing arrhythmias, yet a lack of standardization impedes arrhythmia diagnosis and treatment in horses. To objectively determine an optimal position of Einthoven's triangle for ECG recordings in horses at rest, which can form the basis for standardized ECG recording and improve diagnosis and treatment of arrhythmias. The study involved 72 healthy, warmblood horses aged between 3 and 20 years. In view of future 12-lead studies and vectorcardiography, requiring an orthogonal system, Einthoven's triangle was positioned around the heart, in the transverse plane. Therefore, 11 electrodes were placed encircling the thorax behind the olecranon, to construct triangles with a horizontal base. Electrocardiogram recordings from different triangles were analyzed. Signal processing involved filtering, R peak detection, and median complex generation. Principal component analysis (PCA) and Euclidean distance measures were employed for data analysis. The left mid-thoracic and ventral regions had high PCA scores, indicating high information content. Base-down triangles exhibited higher summed Euclidean distances, contributing to enhanced diagnostic capabilities. A base-down triangle, called "Delta (Δ) configuration" emerged as most informative, while meeting all criteria. The base-down "Delta configuration" is the optimal Einthoven's triangle adapted for horses, providing large amplitudes and potential to provide basic insights into the mechanisms and origins of cardiac arrhythmias. Because the Delta configuration is positioned in the transverse plane, it forms the ideal basis for 12-lead ECG recordings that provide vectorcardiograms in an orthogonal coordinate system. Standardizing electrode positioning could improve ECG data comparability in equine cardiology.

A study on the pressure‐driven flow of magnetized non‐Newtonian Casson fluid between two corrugated curved walls of an arbitrary phase difference

AbstractPressure‐driven movement is a fundamental concept with numerous applications in various industries, scientific disciplines, and fields of engineering. Its proper execution is vital for promoting revolutionary innovations and providing solutions in numerous sectors. Therefore, this article scrutinizes the pressure‐driven flow of magnetized Casson fluid between two curved corrugated walls. The geometry of the channel is represented mathematically in an orthogonal curvilinear coordinate system. The corrugation grooves are described by sinusoidal functions with phase differences between the corrugated curved walls. The boundary perturbation method is used to find the analytical solution for the velocity field and volumetric flow rate, taking the corrugation amplitude as the perturbation parameter. The results show that the peak of the velocity increases with the radius of curvature and the width of the channel for a constant pressure gradient. The velocity exhibited a declining trend due to an increase in the Casson fluid parameter. For a sufficiently large corrugation wavenumber, the flow rate decreases, and the phase difference becomes irrelevant. However, the reduction in flow can be minimized by decreasing the channel radius of curvature. In general, a smooth curved channel will give the maximum flow rate for a large corrugation wavenumber. The model can be used to simulate blood flow in arteries with varying geometries and magnetic fields, aiding in the study of cardiovascular diseases and the design of medical devices like stents.

More considerations about the symmetry of the stress tensor of fluids.

Regarding a recent dispute about the symmetry of the stress tensor of fluids, more considerations are presented. The usual proofs of this symmetry are reviewed, and contradictions between this symmetry and the mechanism of gas viscosity are analyzed for simple gas flows. It is emphasized that these proofs depend on the theorem of angular momentum that assumes that internal forces between any two fluid particles are always along the line connecting them. Without this assumption, from Newton's laws of motion one can only obtain the theorem of angular momentum with an additional term. It is proved within classical continuum mechanics that this additional term represents the total moment of internal forces, and its volume density is similar to the body couple introduced in generalized continuum mechanics. When discrete structure of matter is considered, this term corresponds to the total moment of the forces exerted on the nuclei by the internal electrons. This moment of internal forces may lead to nonsymmetry of the stress tensor and is, in general, nonzero as long as shear stress exists. A nonsymmetrical stress tensor suggested in the literature is discussed in terms of its effects in eliminating the contradictions and simplifying the Navier-Stokes equation. The derivation of this stress tensor for ideal gases based on the kinetic theory of gas molecules is presented, and its form in a general orthogonal curvilinear coordinate system is given. Finally, a possible experimental verification of this nonsymmetrical stress tensor is discussed.

Classification of Dupin cyclidic cubes by their singularities

Triple orthogonal coordinate systems having coordinate lines as circles or straight lines are considered. Technically, they are represented by trilinear rational quaternionic maps and are called Dupin cyclidic cubes, naturally generalizing the bilinear rational quaternionic parametrizations of principal patches of Dupin cyclides. Dupin cyclidic cubes and their singularities are studied and classified up to Möbius equivalency in Euclidean space.

Modeling a triclinic lattice elastic body based on the linear couple stress theory

Triclinic lattice structures with generalized parallelepiped unit cells are crucial targets in understanding the influence of lattice structure on the mechanical behaviors of lattice elastic bodies. This study models a three-dimensional elastic body with a triclinic lattice microstructure comprising three uniform elastic members rigidly joined at a single point. Based on the linear couple stress theory, a specialized case of the Cosserat theory, this study derives elasticity tensors in the constitutive equations of the triclinic lattice elastic body that depend on the crossing angles between the members. To derive the angular-dependent elasticity tensors, coordinate transformations between oblique and orthogonal coordinate systems are effectively employed in formulating the substitution of deformation and force fields in the continuum for the fields in the lattice structure. Evaluating the elasticity tensors reveals the influences of the crossing angles on the anisotropic tensile and torsional properties, the existence of the angle-covariant lattice number manipulation that cancels out the size effect, and several types of geometrical shape dependence. This study could contribute to the theoretical evaluation of elastic behaviors of both artificially engineered and naturally formed materials with lattice microstructures.

3D nonlocal solution with layerwise approach and DQM for multilayered functionally graded shallow nanoshells

A nonlocal layerwise 3D bending solution for multilayered functionally graded shallow nanoshells is presented. Eringen’s nonlocal elasticity theory is used for approaching the small length scale effect. Curvilinear orthogonal coordinate system is employed for writing the constitutive and equilibrium equations. The middle-plane of displacements are assumed as a Navier solution which are only valid for simply supported spherical, cylindrical panels and rectangular plates. The thickness domain is discretized by the well-known Chebyshev–Gauss–Lobatto, and its derivatives are calculated numerically by the Differential Quadrature method (DQM). Lagrange interpolation polynomials are utilized as the basis functions for DQM. The traction conditions at the top and the bottom of the shell panel and continuity conditions of the interlaminar adjacent layers for transverse stresses are considered, so this method presents layerwise capabilities. The results are compared with higher order theories proposed in the literature, and due to the 3D nature of the presented formulation, benchmark solutions are provided.

Autoencoders for discovering manifold dimension and coordinates in data from complex dynamical systems

While many phenomena in physics and engineering are formally high-dimensional, their long-time dynamics often live on a lower-dimensional manifold. The present work introduces an autoencoder framework that combines implicit regularization with internal linear layers and L 2 regularization (weight decay) to automatically estimate the underlying dimensionality of a data set, produce an orthogonal manifold coordinate system, and provide the mapping functions between the ambient space and manifold space, allowing for out-of-sample projections. We validate our framework’s ability to estimate the manifold dimension for a series of datasets from dynamical systems of varying complexities and compare to other state-of-the-art estimators. We analyze the training dynamics of the network to glean insight into the mechanism of low-rank learning and find that collectively each of the implicit regularizing layers compound the low-rank representation and even self-correct during training. Analysis of gradient descent dynamics for this architecture in the linear case reveals the role of the internal linear layers in leading to faster decay of a ‘collective weight variable’ incorporating all layers, and the role of weight decay in breaking degeneracies and thus driving convergence along directions in which no decay would occur in its absence. We show that this framework can be naturally extended for applications of state-space modeling and forecasting by generating a data-driven dynamic model of a spatiotemporally chaotic partial differential equation using only the manifold coordinates. Finally, we demonstrate that our framework is robust to hyperparameter choices.

Geometric Characteristics of Surfaces with Curved Trapezoidal Plan

A method of forming a curved orthogonal coordinate system on a plane and a technique of constructing new surface shapes with curved trapezoidal plans are presented. Multiple examples of curved trapezoidal plans based on different directrix curves and surfaces with the given plans, including combinations of surfaces with different conjugate directrix curves, are illustrated. The proposed technique of surface forming may be used in architecture and construction for development of thin-walled space structures in both urban and industrial buildings. But for the analysis of thin shells, geometric characteristics of the middle surface of the shell are usually used. Vector equation of surfaces with curved trapezoidal plan was used to obtain the formulas for the fundamental form coefficients and surface curvatures. Examples of calculation of the fundamental form coefficients and curvatures of surfaces with particular directrix curves and vertical coordinate functions are presented.

Data-driven state-space and Koopman operator models of coherent state dynamics on invariant manifolds

The accurate simulation of complex dynamics in fluid flows demands a substantial number of degrees of freedom, i.e. a high-dimensional state space. Nevertheless, the swift attenuation of small-scale perturbations due to viscous diffusion permits in principle the representation of these flows using a significantly reduced dimensionality. Over time, the dynamics of such flows evolves towards a finite-dimensional invariant manifold. Using only data from direct numerical simulations, in the present work we identify the manifold and determine evolution equations for the dynamics on it. We use an advanced autoencoder framework to automatically estimate the intrinsic dimension of the manifold and provide an orthogonal coordinate system. Then, we learn the dynamics by determining an equation on the manifold by using both a function-space approach (approximating the Koopman operator) and a state-space approach (approximating the vector field on the manifold). We apply this method to exact coherent states for Kolmogorov flow and minimal flow unit pipe flow. Fully resolved simulations for these cases require $O(10^3)$ and $O(10^5)$ degrees of freedom, respectively, and we build models with two or three degrees of freedom that faithfully capture the dynamics of these flows. For these examples, both the state-space and function-space time evaluations provide highly accurate predictions of the long-time dynamics in manifold coordinates.

A correction method of magnetic gradient tensor system to improve magnet localization accuracy

The magnetic gradient tensor systems using vector magnetic sensors are easily affected by single sensor error and sensor array misalignment, significantly affecting target localization accuracy. To address this problem, based on the differential measurement theory of magnetic gradient tensor, a vector sensor error model was established, and the single sensor errors were corrected using the total least square (TLS) ellipsoid fitting method. Besides, a misalignment error correction model for the sensor is obtained by constructing a rotation matrix for the magnetic sensor's three-axis orthogonal coordinate system transformation, and the tensor system can be aligned by performing the TLS estimation of the rotation angle. After calibration, the experimental results show that the outputs of each sensor have high coincidence and coaxiality. The root mean square error (RMSE) of the total magnetic intensity (TMI) is reduced to within 100 nT, and the RMSE of the tensor component is reduced to within 85 nT/m. Within a range of 5 m, the magnetic target positioning error is reduced to less than 25.1 cm. This technology can improve the measurement accuracy of the magnetic gradient tensor system with high stability and reliability.

МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ НЕИЗОТЕРМИЧЕСКОГО ТЕЧЕНИЯ ДВУХФАЗНЫХ СРЕД В ОБЛАСТЯХ СЛОЖНОЙ ГЕОМЕТРИЧЕСКОЙ ФОРМЫ

Mathematical modeling of non-isothermal flow of two-phase media in areas of complex geometric shape, which are often working units of various types of technological equipment, is considered. To construct a mathematical model describing the flow of a two-phase medium in channels and pipes of complex shape, the equations of conservation of mass, momentum and energy of the mechanics of multiphase media were used with the corresponding closing relations for the force of interphase interaction, the physical parameters of the two-phase medium and the rheological law of the state of the mixture in the form of the Ostwald-de Ville. The conservation equations, taking into account the flow features close to one-dimensional, are written in an orthogonal coordinate system associated with the region, which allows them to be significantly simplified. An algorithm for conducting a computational experiment has been constructed. As a numerical method for conducting a computational experiment, a modified method of equal flow surfaces was used. In this case, changes in the physical parameters of a two-phase medium depending on temperature are taken into account, which significantly complicates the calculations. Numerical calculations were carried out for areas of parabolic and conical shapes, which are determined in the general conservation equations written in a curvilinear orthogonal coordinate system by specifying the corresponding Lame coefficients. In this case, the location and number of equal flow surfaces varied over a wide range, and the input values of the longitudinal velocity and temperature on the streamlines were specified. A computational experiment was implemented for various values of the parameters of the flow region, the rheological law of the state of the medium, taking into account changes in the effective viscosity of the medium from temperature, heat transfer modes and the flow of a two-phase medium. As a result of the computational experiment, various flow regimes and the influence of various geometric characteristics of the flow region, mass forces, and parameters of the rheological law of the state of the medium on the hydrodynamic situation were studied. The various longitudinal velocity and temperature profiles specified at the input developed to a certain steady state, which do not depend on the choice of the initial profile. The temperature values of the medium on the streamlines monotonically approach their asymptotes. The locations of the asymptotic values are determined by the temperature values on the channel walls and the initial temperature of the medium at the inlet. At the same wall temperatures, a symmetrical pattern of heating of the medium is observed, but when the wall temperatures are different, no symmetry is observed. The results of mathematical modelling and computational experiment can be used in the calculation of various technological processes occurring in non-isothermal flows of two-phase media in areas of complex geometric shape.

Global well-posedness of the 3D incompressible Navier-Stokes and Euler equations in orthogonal curvilinear coordinate systems

Global well-posedness of the 3D incompressible Navier-Stokes and Euler equations in orthogonal curvilinear coordinate systems

Волны Рэлея на гладкой кривой поверхности

A model has been developed that describes the propagation of surface waves on smooth curved surfaces. This model is developed using the hyperbolic wave equation and is applicable to studying the propagation of surface waves on any curved surface described by a related orthogonal curvilinear coordinate system and for any values of the product of the wave vector and the radius of curvature. The behavior of Rayleigh waves on canonical curved surfaces (cylinder, sphere) is simulated, and the results are used to quantify the effect of curvature on the speed of Rayleigh waves. The condition for the existence of a Rayleigh wave is the validity of the values of the wave vectors K obtained from the dispersion equations. The actual values of the wave vectors K determine the waves propagating along the surface without carrying the energy deep into the body. The speed of the surface wave increases with increasing curvature of the concave surface, and decreases on the concave surface. The Rayleigh wave has limited growth as the positive radius of curvature decreases. At positive values of the radius of curvature (outer surface), as the radius decreases, a maximum is observed, and then the speed of waves decreases and breaks of curves are observed at negative curvature (inner surface), which is associated with resonance along the circumference of the envelope of a cylinder or great circle of a sphere.

Improvement of methods of interception of enemy UAVs

The main result of this work is the creation of a methodology for determining the coordinates, velocities and trajectories of spatial movements of moving objects by means of kinematic design and the development of principle schemes for the optimal placement of radar search equipment for the successful detection of enemy UAVs. Its purpose is to increase the defense capability of military units, populated areas of Ukraine and their infrastructure by resisting attacks by attack and sabotage reconnaissance unmanned aerial vehicles (UAVs). The protection scheme is based on the determination of coordinates, movement speeds and trajectories of spatial movements of enemy aircraft by means of kinematic design. The established coordinates of enemy UAVs are reported to command posts equipped with liquidator drones (kamikaze) and small arms or to control points of anti-aircraft missile systems (SAMS), which eliminate these aircraft. The proposed improvement of the widespread method of radar search for moving objects is based on the introduction of additional search radar equipment and mathematical apparatus for precise calculations of coordinates and parameters of spatial movements of moving objects by means of kinematic design. Two paired radar search systems, equipped with modern high-speed computing equipment with appropriate software, are able to "screen out" false return electromagnetic signals. This makes it possible to clearly single out the wanted moving object and subsequently track its movement in space. Corresponding mathematical dependencies are proposed for calculating the distance of a moving object to radar stations, its coordinates in the introduced orthogonal coordinate system, speed and acceleration of movement, that is, instantaneous coordinates and parameters of spatial movements of moving objects. This makes it possible to increase the effectiveness of detecting enemy attack and sabotage-reconnaissance unmanned aerial vehicles, including Iranian-made drones of the Sahed-136 model.

Orthogonal Curvilinear Coordinate Systems and Torsion-Free Sheaves over Reducible Spectral Curves

Orthogonal Curvilinear Coordinate Systems and Torsion-Free Sheaves over Reducible Spectral Curves

Orthogonal signed-distance coordinates and vector calculus near evolving curves and surfaces

We provide an elementary derivation of an orthogonal coordinate system for boundary layers around evolving smooth surfaces and curves based on the signed-distance function. We go beyond previous works on the signed-distance function and collate useful vector calculus identities for these coordinates. These results and provided code enable consistent accounting of geometric effects to arbitrary order for boundary layer asymptotics in a wide range of physical systems.