Parabolic Coordinate System Research Articles

Neural Network Prediction for Ice Shapes on Airfoils Using iceFoam Simulations

In this article the procedure and method for the ice accretion prediction for different airfoils using artificial neural networks (ANNs) are discussed. A dataset for the neural network is based on the numerical experiment results—obtained through iceFoam solver—with four airfoils (NACA0012, General Aviation, Business Jet, and Commercial Transport). Input data for neural networks include airfoil and ice geometries, transformed into a set of parameters using a parabolic coordinate system and Fourier series expansion. Besides input features include physical parameters of flow (velocity, temperature, droplets diameter, liquid water content, time of ice accretion) and angle of attack. The novelty of this work is in that the neural network dataset includes various airfoils and the data augmentation technique being a combination of all time slices. Several artificial neural networks (ANNs), fully connected networks (FCNNs), and convolutional networks (CNNs) were trained to predict airfoil ice shapes. Two different loss functions were considered. In order to improve performance of models, batch normalization and dropout layers were used. The most accurate results of ice shape prediction were obtained using CNN and FCNN that applied batch normalization and dropout layers to output neurons of each layer.

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.

Accidental degeneracy of the hydrogen atom and its non-accidental solution in parabolic coordinates

The degeneracy associated with dynamical symmetry of a potential can be identified in quantum mechanics, by solving the Schrödinger equation analytically, using the method of separation of variables in at least two different coordinate systems, and in classical mechanics by solving the Hamilton–Jacobi equation. In the present pedagogical review, the notion of separability and superintegrability of a potential, with profound implications, is discussed. In an earlier tutorial paper, we addressed the n2-fold degeneracy of the hydrogen atom using the Casimir operators corresponding to the SO(4) symmetry of the 1/r potential. The present paper is a sequel to that work, in which we solve the Schrödinger equation for the hydrogen atom using separation of variables in the parabolic coordinate systems. In doing so, we take the opportunity to revisit some excellent works on symmetry and degeneracy in classical and quantum physics, if only to draw attention to these insightful studies, which unfortunately miss even a mention in most undergraduate and even graduate level courses in quantum mechanics and atomic physics.

Generation and Tunable Focal Shift of the Hybridly Polarized Vector Optical Fields with Parabolic SymmetrySupported by the National Natural Science Foundation of China (Grant Nos. 11534006, 11674184, 11774183, 11804187 and 11904199), the Natural Science Foundation of Shandong Province (Grant No. ZR2019BF006), and the Collaborative Innovation Center of Extreme Optics.

Based on a parabolic coordinate system, we theoretically design and experimentally generate hybridly polarized vector optical fields with parabolic symmetry of the first and second kinds, which can further enrich the family of vector optical fields. The wavefront of this new-kind vector optical field contains circular, elliptic and linear polarizations, and the polarizations can keep the same or change along the parabolic curves. Then we present the realization of tunable focal shift with the hybridly polarized vector optical field, and show a specific law of the focal shift of the focused hybridly polarized vector optical field with the parabolic symmetry. We hope these results can provide a new way to flexibly modulate focal fields, which can be applied in realms such as optical machining, optical trapping and information transmission.

Spectra of a Rydberg Atom in Crossed Electric and Magnetic Fields

Contemporary spectroscopic studies of astrophysical and laboratory plasmas frequently deal with extremely large values of principle quantum numbers of atomic systems. These atomic states are very sensitive to electric and magnetic fields of the surrounding medium. While interpreting the spectra of such excited atomic systems, one faces the problem of a huge array of radiative transitions between highly excited atomic levels. Moreover, external electric and magnetic fields significantly complicate the problem because of the absence of standard selection rules typical for the spherical quantization. The analytical expression in the parabolic representation for dipole matrix elements obtained by Gordon contains hyper-geometric series and it has a very complex structure. The matrix elements that involve the presence of electric and magnetic fields are calculated while using a representation closely related to the parabolic quantization on two different axes. This matrix element depends in a complex way on the transition probabilities in the parabolic coordinate system (Gordon’s formulas) and the Wigner d-functions. This circumstance leads to even greater computational difficulties. A method of simplification of these complicated expressions for transition probabilities is demonstrated. The semiclassical approximation for coordinate matrix elements (Gulayev) and recurrence properties of the Wigner d-functions are used. The Hnβ line is under consideration. Specific calculations for the transition 10–8 in the case of parallel and perpendicular fields are presented.

Experimental observation of three-dimensional non-paraxial accelerating beams.

We experimentally realize three-dimensional non-paraxial accelerating beams associated with different coordinate systems. They are obtained by Fourier transforming a phase-modulated wave front in an aberration-compensated system. The phase pattern is encoded to include the phase and amplitude modulation for the accelerating beams with additional correction phase for the aberration compensation. These beams propagate along a circular trajectory, but they exhibit rather complex intensity patterns corresponding to the shape-invariant solutions in parabolic, prolate spheroidal and oblate spheroidal coordinate systems.

AN ANALYTICAL APPROXIMATION FOR THE LOCAL FLOW BETWEEN COUNTER-ROTATING CYLINDER PAIR

Flow around a pair of cylinders is of interest for various applications. Especially with the development of self-propelled, autonomous vehicles, this topic has gained further importance. There have been experimental, numerical and, to a much less extent, theoretical studies of this flow and its implications on the cylinders. However, the complexity of the problem does not allow a general solution, and case-by-case solutions have been produced. In this study, a theoretical approach was embraced, in which differential geometry is employed to model the local flow between a pair of counter-rotating cylinders. In order to obtain practical formulae for a steady, laminar, incompressible flow, Navier-Stokes equations were simplified and expressed in a new, parabolic coordinate system. Then, further simplifications due to the symmetrical nature of the problem were applied. Finally, boundary conditions were imposed while performing the integrals, and the desired equations for velocity and pressure were reached. Unlike the previous studies that are limited to very slow flows, the equations obtained in this study are applicable in the entire incompressible, laminar regime, albeit only between the cylinders. Using these equations, the effects of the rotation speed, cylinder spacing and cylinder radius were studied and presented graphically.

On the use of the axially symmetric paraboloidal coordinate system in deriving some properties of Stark states of hydrogenic atoms and ions

While the parabolic coordinate system has been widely applied in providing the quantum mechanical basis for the description of the Stark states, following the pioneering studies of Schrödinger and Epstein of the Stark effect, use of the alternative, closely related axially symmetric paraboloidal coordinate system in atomic physics appears to have been practically confined to the so-called ‘Old Quantum Theory’ of Sommerfeld. The aim of the present study is to show that the axially symmetric paraboloidal coordinate system, which has found practical application in electromagnetic theory, can also be used quite readily for deriving the properties of the Stark states. In this paper, we consider the application of this coordinate system both from the standpoint of wave mechanics and the operator calculus of the O(4) symmetry group applied to the Coulomb problem, obtaining the analogue of the Kramers–Pasternack recursion relations, which may readily be used in the determination of coordinate matrix elements, besides examining some properties of the Runge–Lenz–Pauli vector operator, in particular in relation to the Pauli identity. The generalised momenta corresponding to the curvilinear coordinates are defined, and the canonically conjugate (Hermitian) momentum operators derived. The discussion is illustrated in conclusion by a straight-forward numerical example, which relates the present treatment to corresponding results obtained in spherical polar coordinates, for both the radial matrix elements and the radial Heisenberg uncertainty product. The use here of ladder techniques in both the coordinate matrix elements and in the projection of the state functions on different coordinate systems would again provide a ready means of efficient calculation for highly excited states in dilute astrophysical plasmas, complementing the discussion in earlier papers (Hey 2007 J. Phys. B: At. Mol. Opt. Phys. 40 4077–96; Hey 2015 J. Phys. B: At. Mol. Opt. Phys. 48 185701), when one wishes to avoid numerical problems associated with very high principal quantum numbers in the analysis of radio recombination spectra from H II regions (Hey 2006 J. Phys. B: At. Mol. Opt. Phys. 39 2641–64).

Exact eigenstates of a nanometric paraboloidal emitter and field emission quantities.

The progress in field emission theory from its initial Fowler-Nordheim form is centred on the transmission coefficient. For the supply (of electrons) function one still uses the constant value due to a supply of plane-waves states. However, for emitting tips of apex radius of 1-5 nm this is highly questionable. To address this issue, we have solved the Schrödinger equation in a sharp paraboloidally shaped quantum box. The Schrödinger equation is separable in the rotationally parabolic coordinate system and we hence obtain the exact eigenstates of the system. Significant differences from the usual Cartesian geometry are obtained. (1) Both the normally incident and parallel electron fluxes are functions of the angle to the emitter axis and affect the emission angle. (2) The WKB approximation fails for this system. (3) The eigenfunctions of the nanoemitter form a continuum only in one dimension while complete discretization occurs in the other two directions. (4) The parallel electron velocity vanishes at the apex which may explain the recent spot-size measurements in near-field scanning electron microscopy. (5) Competing effects are found as the tip radius decreases to 1 nm: The electric field increases but the total supply function decreases so that possibly an optimum radius exists.

Simple method for numerical solving of Schroedinger equation for hydrogen atom in electric field

A propagation numerical method for determining energy eigenvalues and eigen wave functions for hydrogen atom in constant and uniform electric field is described in this paper. Solution is presented for 3-D Schroedinger equation in natural parabolic co-ordinate system. Criteria for accepting eigenvalues are introduced, and results are compared with previous papers.

A mass-independent conformal quantum cloak

An obstacle to constructing a quantum cloak with the transformation design method for steering a matter wave is the device involving the anisotropic modulation of the particle mass. It is the purpose of this paper to show that a narrow-acceptance-angle quantum cloak generated by a kind of conformal mapping could be mass-independent. This greatly simplifies the construction of the device since it can now be finished only by applying a potential field to a region of the shell. As demonstrations of the conclusion, an elliptical and a parabolic conformal quantum cloak are designed from the elliptical and parabolic coordinate systems. Their performance is inspected by probability currents. It is shown that the conformal devices can serve as novel splitters for a 2D matter wave. Finally, we discuss the difficulty of constructing a steering device from the hyperbolic coordinate system.

Approximated Complementary Cloak With Diagonally Homogeneous Material Parameters Using Shifted Parabolic Coordinate System

In this communication, a parabolic shaped complementary cloak is proposed which makes the object invisible by reducing its scattering under different incidence angles of electromagnetic wave. The proposed design is based on transforming a line segment into an enclosed space that is surrounded by the core region. The enclosed space is symmetrically divided into two regions, namely, cloaked region and complementary region. The shifted parabolic coordinate system introduced in the present work enhances the area of cloaked region and provides diagonally anisotropic but optimized homogeneous material properties in the core region. The cloaked region is mapped onto the complementary region using a simple one directional folding transformation, which yields homogeneous and isotropic material properties in the complementary region and also eases the creation of antiobject. Numerical results confirm the cloaking effect of the proposed approximate complementary cloak. The metamaterial unit cell designs are also presented for realizing the proposed material properties of core and complementary regions.

Analytical study on deep penetration induced by focused moving high-energy beam

AbstractThis paper investigates the focal location effects on the penetration depth of molten region surrounding a paraboloid of revolution-shaped cavity (i.e. keyhole of this model) irradiated by a moving focused energy beam, which profile of intensity is assumed to be Gaussian distribution. Considering the momentum balance at the base of the keyhole, a quasi-steady-state thermal model relative to a constant-speed moving high-energy beam and paraboloid of revolution-shaped cavity is developed in a parabolic coordinate system. The analytical solution is obtained for this model with the adiabatic condition directly set on the workpiece surface for semi-infinite domain instead of the image method for infinite domain using the separation-of-variables method. The analytical solution of this model gives a reasonable prediction for the cavity temperatures. The predicted relation of the penetration depth to the focal location agrees with the available measured data. The effects of focal convergence angle and spot size on the penetration depth are also discussed.

Parabolic scaling beams.

We generalize the concept of diffraction free beams to parabolic scaling beams (PSBs), whose normalized intensity scales parabolically during propagation. These beams are nondiffracting in the circular parabolic coordinate systems, and all the diffraction free beams of Durnin's type have counterparts as PSBs. Parabolic scaling Bessel beams with Gaussian apodization are investigated in detail, their nonparaxial extrapolations are derived, and experimental results agree well with theoretical predictions.

Analytical Modeling of Gaseous Slip Flow in Parabolic Microchannels

The paper presents an analytical solution of velocity, mass flow rate, and pressure distribution for fully developed gaseous slip flow in nonsymmetric and symmetric parabolic microchannels. The flow is considered to be steady, laminar, and incompressible with constant fluid properties. Fully developed gaseous slip flow in microchannels of parabolic cross section is solved analytically for various aspect ratios using a parabolic cylindrical coordinate system on applying the method of separation of variables. Prior to apply separation of variables, Arfken transform [Arfken, 1970, Mathematical Methods for Physicists, Academic Press, Orlando, FL, Ch. 2] was used on momentum equations and first-order slip boundary conditions at each channel wall were imposed. A simple model is proposed to predict the friction factor and Reynolds number product fRe for slip flow in parabolic microchannels. Through the selection of a characteristic length scale, the square root of cross-sectional area and the effect of duct shape have been minimized. The results of a normalized Poiseuille number for symmetric parabolic microchannels (ɛ=1) shows good agreement with the previous results [Morini et al., 2004, “The Rarefaction Effect on the Friction Factor of Gas Flow in Micro/Nano-Channels,” Superlattices Microstruct., 35(3–6), pp. 587–599; Khan and Yovanovich, 2008, “Analytical Modeling of Fluid Flow and Heat Transfer in Microchannel/Nanochannel Heat Sinks,” J. Thermophys. Heat Transf., 22(3), pp. 352–359] for rectangular microchannels. The developed model can be used to predict mass flow rate and pressure distribution of slip flow in parabolic microchannels.

Three-dimensional nonparaxial beams in parabolic rotational coordinates

We introduce a class of three-dimensional nonparaxial optical beams found in a parabolic rotational coordinate system. These beams, representing exact solutions of the nonparaxial Helmholtz equation, have inherent parabolic symmetries. Assisted with a computer-generated holography, we experimentally demonstrate the generation of different modes of these beams. The observed transverse beam patterns along the propagation direction agree well with those from our theoretical predication.

Superintegrable Stäckel Systems on the Plane: Elliptic and Parabolic Coordinates

Recently we proposed a generic construction of the additional integrals of motion for the St\"ackel systems applying addition theorems to the angle variables. In this note we show some trivial examples associated with angle variables for elliptic and parabolic coordinate systems on the plane.

Large amplitude free vibration of a shear deformable laminated composite parabolic plate with parabolically orthotropic plies

The large amplitude free vibration of a laminated composite parabolic plate with parabolically orthotropic plies is investigated for the first time. The effects of out-of-plane shear deformations, rotary inertia, and geometrical nonlinearity are taken into account. The geometry of the plate is described, and the analysis performed in the parabolic coordinate system. The problem is solved numerically using a new parabolic hierarchical finite element. The nonlinear equations of free motion are mapped from the time domain into the frequency domain using the harmonic balance method. The resultant nonlinear equations are solved iteratively using the linearized updated mode method. Results for the fundamental linear and nonlinear frequencies are obtained for symmetric and antisymmetric laminates with clamped and simply supported edges. Comparisons are made with the finite element method for clamped and free isotropic parabolic plates and show excellent agreement. The aspect ratio, thickness ratio, moduli ratio, number of plies, layup sequence, and boundary conditions are shown to affect the hardening behavior.

Elastodynamic Scalar Potential Functions in Cylindrical Coordinate Systems

The aim of this paper is to present scalar potential functions for three-dimensional elastodynamic problems in transversely isotropic media defined by different cylindrical coordinate systems. To do so, the standard cylindrical coordinate system, elliptical cylindrical coordinate system and parabolic cylindrical coordinate system are considered. By virtue of Poisson’s representation, Lame solution and Chadwick-Trowbridge, the completeness of such representations is addressed. The representations given here are useful in the study of wave scattering from the circular, elliptical and parabolic edges.

Using molecular-dynamics simulations to understand and improve the treatment of anharmonic vibrations. II. Developing and assessing new Debye–Waller factors

Two new anharmonic forms for the Debye-Waller factor, aimed at modelling curvilinear and asymmetric motion, have been introduced. These forms permit the refinement of structures with these types of anharmonic motion using a small number of additional parameters. Molecular-dynamics-derived numerical probability density functions (PDFs) have been used to assess the merit of these new functions in real space. The comparison is favourable particularly for the curvilinear PDF based on a parabolic coordinate system change of a trivariate Gaussian distribution. The initial results also suggest that high-order even terms from the Gram-Charlier series may be important for modelling methyl-group libration. The molecular-dynamics data sets provide useful insights into the nature of anharmonic thermal motion. Addressing the problem in real space allows intuitive PDFs to be developed but numerical methods may be necessary for these methods to be implemented in refinement programs as an analytical Debye-Waller factor cannot always be obtained.