Orthogonal Coordinate System Research Articles

Spatial Constitutive Modeling of AA7050-T7451 with Anisotropic Stress Transformation.

The mechanical properties of anisotropic materials are generally characterized based on the orthotropy or transverse isotropy. However, the two-dimensional plane stress problems cannot comprehensively characterize the anisotropy of materials. In this study, based on the theory of elasticity and the transformation of the three-dimensional space coordinate system, combined with the projection relationship of the Cauchy stress tensor of an arbitrary section, the transformation relationship of the elastic modulus, shear modulus, and stress–strain between the orthogonal and load coordinate systems are obtained. The orthotropic Johnson-Cook (JC) constitutive model of AA7050-T7451 aluminum alloy is modified by fitting, and the constitutive relationship at any spatial angle is theoretically calculated by combining the obtained spatial coordinate transformation matrix. The generated spatial constitutive model is verified and modified through experiments. The results demonstrate that the theoretical mechanical properties and the modified spatial constitutive model can accurately reflect the effect of the spatial angle on the material stress distribution.

Quantitative evaluation of the functional activity in multilayered brain structures using histological sections

Subject of study. We develop an algorithm and software to estimate optical density as a correlate of neural activity in multilayered brain structures. Aim of study. We develop a method and program algorithm for quantitative evaluation of the optical density in the layers of neural structures from images of histological brain preparations that correlates with the functional activity of the cells. Method. Optical density was measured locally in successive areas of a layer or several layers of the neural structure, which are curvilinear-shaped in the brain slices. An expert researcher selects the trajectory of optical density measurement in the digitized image of the slice in an interactive mode. The results of density measurements were plotted in an orthogonal coordinate system. Capabilities enabling automated curve smoothing, extremum seeking, and calculation of distances between extrema were developed. Results. An algorithm for implementing the developed program is presented. The successive stages of operation are illustrated through the evaluation of the optical density in the visual cortex layer from an image of a brain slice as an example. In addition, the evaluation results of the optical density in layers of a lateral geniculate body are presented. Practical significance. The developed method and program algorithm are required for reconstructing the functional activity of the curved layers of different brain structures to investigate and simulate the organization patterns of the neural networks of the human brain.

Mathematical modeling of rheostat-reactor start of wound-rotor induction motors

Introduction. Wound-rotor induction motors are less common compared squirrel-cage induction motors. However, they occupy a significant share among electric drives with difficult starting conditions. Their advantage is obtaining a high starting electromagnetic torque at lower values of starting currents. Problem. Due to the possibility of including different devices in the rotor circuit, it is possible to shape the starting characteristics according to the needs of the technological process. Due to a narrower range of applications of electric drives based on wound-rotor induction motors, they are less investigated. Selection of parameters of starting and regulating devices, included in the rotor circuit, is carried out by simplified methods, which do not satisfy modern requirements to regulated electric drives. Goal. The paper aims to develop mathematical models and methods for calculating the dynamic modes and static characteristics of the wound-rotor induction motor with a reactor in the rotor circuit. Methodology. In the developed algorithms, the mathematical model of the motor is presented by the differential equations made for electric circuits in a system of orthogonal coordinates that allows excluding angular coordinate from equations of electric equilibrium. The elements of the Jacobi matrix of equilibrium equations of motor circuits are eigenvalues, and mutual is the differential inductances of electrical circuits, which are determined based on the magnetization characteristics of the main magnetic flux and leakage fluxes of the rotor and stator circuits. Results. Mathematical models for the study of starting modes of wound rotor induction motor allow to calculate transients and static characteristics and, on their basis, to carry out design synthesis of starting reactors, which provide the law of change of electromagnetic torque during start-up operating conditions. Originality. The mathematical basis of the developed algorithms is the method of solving nonlinear systems of equations by Newton method in combination with the method of continuation by parameter. The developed mathematical models and software made on their basis have high speed that allows to carry out high-reliability calculation of starting modes taking into account saturation of a magnetic circuit of the motor. Practical value. The developed algorithms do not require significant computing resources, have high speed, and can be used both for the design synthesis of start-control devices and control of the electric drive in real time and to predict its course.

Discrete cyclic systems and circle congruences

We discuss integrable discretizations of 3-dimensional cyclic systems, that is, orthogonal coordinate systems with one family of circular coordinate lines. In particular, the underlying circle congruences are investigated in detail and characterized by the existence of a certain flat connection. Within the developed framework, discrete cyclic systems with a family of discrete flat fronts in hyperbolic space and discrete cyclic systems, where all coordinate surfaces are discrete Dupin cyclides, are investigated.

Graded infill design within free-form surfaces by conformal mapping

The infill design for free-form surfaces is challenging due to the mismatch of both the geometric feature and physical response between the curved surface on the macroscale and the regular-shaped infill unit cells on the meso/micro scale. Thus, this study proposes a novel optimization method to provide insight into the optimal design for the free-form surfaces consisting of micro-structured infills, where both the microstructural geometry and macroscopic distribution of the spatial-varying infills are optimized concurrently. In this method, a local level sets approach is proposed to generate a series of graded and connectable infill microstructures that are related to the volume fraction of each unit cell. The cubic polynomials interpolation is utilized to predict the effective properties for those graded infills rapidly. A computational conformal mapping technology is used to map the geometries between a 3D free-form surface and a regular 2D parameter space, such that the graded infills can be properly filled into the parameter domain, and then conformally mapped back onto the free-form surface to allow the infill unit cells to fit the curved surface without losing any geometric feature. Conformal shell finite element analysis defined in an orthogonal curvilinear coordinate system is derived in the lower-dimensional parameter domain to calculate the physical responses for the infill optimization during the conformal mapping. The infill optimization formulation for the free-form surface is established, which is recast as a 2D optimization problem defined in the parameter domain, aiming at minimizing the structural compliance subject to a global volume constraint. Thus, the dimension of the 3D free-form surface design problem is reduced. Several examples of the free-form surface demonstrate the features of the proposed method.

Solving Maxwell eigenvalue problems for three dimensional isotropic photonic crystals with fourteen Bravais lattices

In this paper, we present a unified finite difference framework to efficiently compute band structures of three dimensional linear non-dispersive isotropic photonic crystals with any of 14 Bravais lattice structures to a reasonable accuracy. Specifically, we redefine a suitable orthogonal coordinate system, and meticulously reformulate the Bloch condition for oblique Bravais lattices, and clearly identify the hierarchical companion matrix structure of the resulting discretized partial derivative operators. As a result, eigen-decompositions of discretized partial derivative operators and notably the discretized double-curl operator of any size, become trivial, and more importantly, the nullspace free method for the Maxwell’s equations holds naturally in all 14 Bravais lattices. Thus, the great difficulty arising from high multiplicity of zero eigenvalues has been completely overcome. On the basis of these results, we perform calculations of band structures of several typical photonic crystals to demonstrate the efficiency and accuracy of our algorithm.

Dynamics Analysis of Penetration against Pebble-Concrete Target.

The classical continuum mechanics theory cannot sufficiently describe the effect of pebbles on projectile, which leads to a large calculation error. In this paper, an orthogonal curvilinear coordinate system is constructed, which effectively describes and perfects the normal cavity expansion theory. A couple stress theory based on the normal cavity expansion is proposed in which not only the tangential movements but also the rotations of the concrete medium are considered. According to the high-speed impact of pebble concrete, combined with dynamic equations and the FE simulation, the theoretical and simulation results of pebble particles scale on warhead resistance are compared. It is shown that, the larger the scale of pebble particles, the stronger the effect of rotation on the resistant force applied on the warhead.

Observations of Strongly Modulated Surface Wave and Wave Breaking Statistics at a Submesoscale Front

Abstract Ocean submesoscale currents, with spatial scales on the order of 0.1–10 km, are horizontally divergent flows, leading to vertical motions that are crucial for modulating the fluxes of mass, momentum, and energy between the ocean and the atmosphere, with important implications for biological and chemical processes. Recently, there has been considerable interest in the role of surface waves in modifying frontal dynamics. However, there is a crucial lack of observations of these processes, which are needed to constrain and guide theoretical and numerical models. To this end, we present novel high-resolution airborne remote sensing and in situ observations of wave–current interaction at a submesoscale front near the island of O’ahu, Hawaii. We find strong modulation of the surface wave field across the frontal boundary, including enhanced wave breaking, that leads to significant spatial inhomogeneities in the wave and wave breaking statistics. The nonbreaking (i.e., Stokes) and breaking induced drifts are shown to be increased at the boundary by approximately 50% and an order of magnitude, respectively. The momentum flux from the wave field to the water column due to wave breaking is enhanced by an order of magnitude at the front. Using an orthogonal coordinate system that is tangent and normal to the front, we show that these sharp modulations occur over a distance of several meters in the direction normal to the front. Finally, we discuss these observations in the context of improved coupled models of air–sea interaction at a submesoscale front.

Derivative of Spatial Variables in Orthogonal Curvilinear System with Respect to the Ones in Cartesian Coordinate System

Based on the transformation of the spatial variables between the orthogonal curvilinear coordinate system and the Cartesian coordinate system, the derivation of spatial variables in orthogonal curvilinear system to the ones in Cartesian coordinate system is derived. The derivatives of the spatial variables in cylindrical, spherical, and elliptic cylindrical coordinate system are derived respectively.

Asymptotic Modeling of a Viscous Laminar Flow Around Thin Airfoils: Resolution and Experimental Treatment in Case of Supersonic Flow

In this paper, starting from the conservation equations in a 2D orthogonal curvilinear coordinate system, an approached asymptotic model for perturbed potential flow equation is established. The study considers a high-speed steady laminar irrotational flow of a compressible and viscous fluid, over thin airfoils at low incidence in the standard atmosphere. This study is aimed to contribute to developing a fully two-dimensional general model that is efficiently applicable to industrial scale and scientific research, in a laminar flow, defined on thin geometries as it takes into consideration both compressibility and viscous effects. The singular perturbation model is solved in case of supersonic viscous flow, based on asymptotic and singular perturbation methods. Asymptotic matching of different layers is done through the triple deck theory. Experimental measurements have been conducted, with the AF300 supersonic wind tunnel, on a NACA 43013 airfoil model fitted with two pressure taps. The comparison of the local Mach numbers between the asymptotic model and the experimentation shows the accuracy of the model in case of laminar flow but diverge relatively as the flow becomes turbulent towards the trailing edge.

Thin liquid films on a rotary bell atomizer in surface-following coordinates

This article analyzes a published formulation of the Navier–Stokes equations cast into surface-following coordinates and provides some additional mathematical background to follow the article. Ubiquitous in the paint shops of automotive plants around the world, a high-speed rotary bell is succinctly described as a rapidly spinning concave axisymmetric surface with liquid paint supplied from a port coinciding with the center of rotation. The spinning surface transfers momentum to the paint film causing it to flow outward. Upon reaching the bell periphery, it is flung off, subsequently forming an atomized spray transferred to an automotive body through advection and electrostatics. Common analytical frameworks of rotating films were spherical or cylindrical coordinate systems where the wetted surface profile of the bell was constrained to follow a coordinate axis. This led to solutions for films modeled with conical, disk-like, or partial hemispherical profiles. An alternative was a more general case using a surface-following orthogonal curvilinear coordinate system along with its derived vector operators. In the unique case of a thin film, these results validated a simpler pattern found in common coordinate systems.

3D hygro-elastic shell model for the analysis of composite and sandwich structures

The present work proposes the study of the hygrometric loading effects in the static analysis of multilayered composite and sandwich plates and shells. The employed model is based on a general exact 3D shell theory valid for one-layered and multilayered sandwich and composite plates, cylinders and cylindrical/spherical shell panels. The employed 3D equilibrium equations are developed in an orthogonal mixed curvilinear coordinate system for spherical shells in order to obtain the other simpler geometries as particular cases. A layer-wise approach is employed and equilibrium equations are solved in the thickness direction via the exponential matrix method. The partial derivatives in the in-plane directions are exactly calculated by assuming simply-supported edges and harmonic forms for the variables. The presented 3D shell model is extended to hygro-elastic cases by considering moisture content amplitudes imposed at the external surfaces of the structures in steady-state conditions. The moisture content profile through the thickness of the structures can be included in the 3D shell model using three different methodologies: it can be calculated by solving the steady-state 3D or 1D version of the Fick diffusion law, or it can be “a priori” assumed as linear through the thickness. Therefore, the developed system includes three non-homogeneous second order differential equilibrium equations that are solved after the introduction of opportune mathematical layers to consider the curvature effects through the thickness. The reduction to a first order differential equation system is obtained by redoubling the number of variables. The exponential matrix method is employed to accurately define both general and particular solutions. The implemented 3D hygro-elastic shell model is firstly validated and then it is employed with confidence for new benchmarks where the effects of thickness ratio, geometry, lamination scheme, material, thickness layer and imposed moisture content values at the external surfaces are investigated for composite and sandwich plates and shells. Showed results demonstrate the importance of an accurate developing of the elastic part of the 3D shell model combined with an appropriate evaluation of the moisture content profile through the thickness. Assumed linear moisture content profiles are correct only for thin one-layered isotropic structures, the use of the 1D Fick diffusion law allows only the correct definition of the material layer effect, the use of its 3D version allows the correct definition of both material and thickness layer effects.

3D Stress Analysis of Multilayered Functionally Graded Plates and Shells under Moisture Conditions

This paper presents the steady-state stress analysis of single-layered and multilayered plates and shells embedding Functionally Graded Material (FGM) layers under moisture conditions. This solution relies on an exact layer-wise approach; the formulation is unique despite the geometry. It studies spherical and cylindrical shells, cylinders, and plates in an orthogonal mixed curvilinear coordinate system (α, β, z). The moisture conditions are defined at the external surfaces and evaluated in the thickness direction under steady-state conditions following three procedures. This solution handles the 3D Fick diffusion equation, the 1D Fick diffusion equation, and the a priori assumed linear profile. The paper discusses their assumptions and the different results they deliver. Once defined, the moisture content acts as an external load; this leads to a system of three non-homogeneous second-order differential equilibrium equations. The 3D problem is reduced to a system of partial differential equations in the thickness coordinate, solved via the exponential matrix method. It returns the displacements and their z-derivatives as a direct result. The paper validates the model by comparing the results with 3D analytical models proposed in the literature and numerical models. Then, new results are presented for one-layered and multilayered FGM plates, cylinders, and cylindrical and spherical shells, considering different moisture contents, thickness ratios, and material laws.

SPHEROIDAL BASIS OF THE GENERALIZED MIK-KEPLER PROBLEM

Super integrated systems have an extremely important property: they allow the separation of variables in the Hamilton-Jacobi and Schrödinger equations in several orthogonal coordinate systems. The choice of a specific coordinate system is dictated by considerations of convenience, for example, the spectroscopic problem of hydrogen-like systems uses a spherical coordinate system, when considering the Stark effect - a parabolic coordinate system, and in the two-center problem - elongated spheroid coordinates. This abundance of separation of variables in the Schrödinger equation for super integrated systems leads to the problem of interphasic decompositions, i.e. there is a need to move from one wave function to another. The generalized MIC-Kepler problem in spherical coordinates is considered as an explicit form of the additional motion integral and the generalized MIC-Kepler problem in spheroid coordinates is given Λ ̂=M ̂+(R√(μ_0 ))/ℏ Ω ̂^((s) ) main function of which is the spheroid basis and three-membered recurrent relations are derived to which the decomposition coefficients of the spheroid basis according to spherical and parabolic bases as well.

Numerical evaluation of virtual mass force coefficient of single solid particles in acceleration

• Single solid sphere in acceleration is numerically simulated based on the vorticity-stream function formulation. • Particle acceleration changes significantly the surrounding flow field and the stress tensor on the particle surface. • Virtual mass force coefficient is evaluated from the numerical simulation. • A correlation of total drag coefficient with the Reynolds number and acceleration number is proposed for tentative use in particulate flow simulation. Virtual mass force is an indispensable component in the momentum balance involved with dispersed particles in a multiphase system. In this work the accelerating motion of a single solid particle is mathematically formulated and solved using the vorticity-stream function formulation in an orthogonal curvilinear coordinate system. The total drag coefficient was evaluated from the numerical simulation in a range of the Reynolds number ( Re ) from 10 to 200 and the dimensionless acceleration ( A ) between −2.0 to 2.0. The simulation demonstrates that the total drag is heavily correlated with A , and large deceleration even drops the drag force to a negative value. It is found that the value of virtual mass force coefficient ( C V ) of a spherical particle is a variable in a wide range and difficult to be correlated with A and Re. However, the total drag coefficient ( C DV ) is successfully correlated as a function of Re and A , and it increases as A is increased. The proposed correlation of total drag coefficient may be used for simulation of solid–liquid flow with better accuracy.

Calculus: High-Dimensional Numerical and Symbolic Calculus in R

The R package calculus implements C++ optimized functions for numerical and symbolic calculus, such as the Einstein summing convention, fast computation of the Levi-Civita symbol and generalized Kronecker delta, Taylor series expansion, multivariate Hermite polynomials, high-order derivatives, ordinary differential equations, differential operators and numerical integration in arbitrary orthogonal coordinate systems. The library applies numerical methods when working with R functions or symbolic programming when working with characters or expressions. The package handles multivariate numerical calculus in arbitrary dimensions and coordinates and implements the symbolic counterpart of the numerical methods whenever possible, without depending on external computer algebra systems. Except for Rcpp, the package has no strict dependencies in order to provide a stable self-contained toolbox that invites re-use.

Derivations of Vector Area and Volume Elements in Curved Coordinate Systems for Flux Vector Fields Helping Eye Disease

Our aim in this paper is to interest retinal eye specialists in preventing dry macula degeneration by a special flurry vector field through open or closed curved surfaces. The flux of vector fields through surfaces is based on vector element area and volume element. Therefore, we explain a few geometrical derivations of area and volume elements in curved orthogonal coordinate systems. We hope that by derivation of a spatial vector field flurry against drusen through open or closed surfaces due to the Gauss theorem might select drusen under eye retina cells without destroying the cells and prevent macula degeneration. A changed flurry of a magnetic or electric vector field through a closed line causes an electric or magnetic vector field on the surface closed by the line. We also hope that derivation by Stokes’ and Greens’ theorems, with the help of iron, might help eye cells to get in life.

Coupled flow and heat transfer of power-law nanofluids on non-isothermal rough rotary disk subjected to magnetic field

We study the coupled flow and heat transfer of power-law nanofluids on a non-isothermal rough rotating disk subjected to a magnetic field. The problem is formulated in terms of specified curvilinear orthogonal coordinate system. An improved BVP4C algorithm is proposed, and numerical solutions are obtained. The influence of volume fraction, types and shapes of nanoparticles, magnetic field and power-law index on the flow, and heat transfer behavior are discussed. The obtained results show that the power-law exponents (PLE), nanoparticle volume fraction (NVF), and magnetic field inclination angle (MFIA) have almost no effects on velocities in the wave surface direction, but have small or significant effects on the azimuth direction. The NVF has remarkable influences on local Nusselt number (LNN) and friction coefficients (FC) in the radial direction and the azimuth direction (AD). The LNN increases with NVF increasing while FC in AD decreases. The types of nanoparticles, magnetic field strength, and inclination have small effects on LNN, but they have remarkable influences on the friction coefficients with positively correlated heat transfer rate, while the inclination is negatively correlated with heat transfer rate. The size of the nanoparticle shape factor is positively correlated with LNN.

Explicit Two-Piece Quadratic Current–Voltage Characteristic Model for Solar Cells

A novel explicit model for the current–voltage ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I–V</i> ) characteristic of solar cells is proposed, consisting of two quadratic equation segments associated with inverse orthogonal coordinate systems and joined at the maximum powerpoint. Salient features of the proposed model are its high accuracy and simple parameter computation solely from the cell notable points. Comparison carried out against the most relevant explicit <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I–V</i> models suggests the proposed method has greater flexibility in fitting a wide range of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I–V</i> characteristic fill factors with high accuracy. It is also shown that the model enables explicit and precise computation of the solar cell operating point in terms of load and ambient conditions, which is a major advantage when carrying out simulations.

On steady motions of an ideal fibre-reinforced fluid in a curved stratum. Geometry and integrability

It is shown that the kinematic equations governing steady motions of an ideal fibre-reinforced fluid in a curved stratum may be expressed entirely in terms of the intrinsic Gauss equation, which assumes the form of a partial differential equation of third order, for the surface representing the stratum. In particular, the approach adopted here leads to natural non-classical orthogonal coordinate systems on surfaces of constant Gaussian curvature with one family of coordinate lines representing the fibres. Integrable cases are isolated by requiring that the Gauss equation be compatible with another third-order hyperbolic differential equation. In particular, a variant of the integrable Tzitzéica equation is derived which encodes orthogonal coordinate systems on pseudospherical surfaces. This third-order equation is related to the Tzitzéica equation by an analogue of the Miura transformation for the (modified) Korteweg–de Vries equation. Finally, the formalism developed in this paper is illustrated by focussing on the simplest ‘fluid sheets’ of constant Gaussian curvature, namely the plane, sphere and pseudosphere.