Showing papers on "Spherical coordinate system published in 2019"

Exact, Free-Surface Equatorial Flows with General Stratification in Spherical Coordinates

Abstract: This paper is concerned with the construction of a new exact solution to the geophysical fluid dynamics governing equations for inviscid and incompressible fluid in the equatorial region. This solution represents a steady purely-azimuthal flow with a free-surface. The novel aspect of the solution we derive is that the flow it prescribes accommodates a general fluid stratification: the density may vary both with depth, and with latitude. The solution is presented in the terms of spherical coordinates, hence at no stage do we invoke approximations by way of simplifying the geometry in the governing equations. Following the construction of our explicit solution, we employ functional analytic considerations to prove that the pressure at the free-surface defines implicitly the shape of the free-surface distortion in a unique way, exhibiting also the expected monotonicity properties. Finally, using a short-wavelength stability analysis we prove that certain flows defined by our exact solution are stable for a specific choice of the density distribution.

Accurate real-time interpolation of 5-axis tool-paths with local corner smoothing

Abstract: 5-axis machining tool-paths, when programmed in workpiece coordinates, translational and rotational tool motion in terms of Cartesian tool center points (TCPs), and unit tool orientation vectors (ORI). This paper presents a computationally efficient real-time trajectory generation algorithm for 5-axis machine tools to interpolate translational and rotational tool motion synchronously for accurate 5-axis machining. Finite Impulse Response (FIR) filters are used to generate jerk limited motion trajectories in real-time. Linear translational motion of the tool center point (TCP) is interpolated by FIR filtering of Cartesian velocity pulses in G01 blocks. In order to generate constant speed tool axis rotation, spherical linear interpolation is used, and unit tool orientation vectors (ORI) are filtered directly in the spherical coordinates. Precise tool motion synchronization is realized by matching time-constants of FIR filters utilized for translational and rotational interpolation. Non-stop path interpolation is achieved by locally blending consecutive linear G01 commands. Instead of fitting geometric blending curves and solving feed scheduling problem, smoothing functionality of FIR filtering is used, and a direct 1-step path smoothing algorithm is proposed for real-time implementation. The algorithm considers path blending errors in Cartesian (Euclidian) as well as in spherical (orientation) coordinates due to transient response of the FIR filter. As a result, both tool-tip and the tool-orientation errors are controlled accurately. Effectiveness of the developed algorithms are validated in simulations and also experimentally on an open-NC controlled 5-axis machine tool.

On the Nonlinear, Three-Dimensional Structure of Equatorial Oceanic Flows

Abstract: A systematic development, based on the construction of an asymptotic solution of the Euler equation, written in rotating, spherical coordinates (φ,θ,r), is used to investigate the flows of ...

Regression Versus Classification for Neural Network Based Audio Source Localization

Abstract: We compare the performance of regression and classification neural networks for single-source direction-of-arrival estimation. Since the output space is continuous and structured, regression seems more appropriate. However, classification on a discrete spherical grid is widely believed to perform better and is predominantly used in the literature. For regression, we propose two ways to account for the spherical geometry of the output space based either on the angular distance between spherical coordinates or on the mean squared error between Cartesian coordinates. For classification, we propose two alternatives to the classical one-hot encoding framework: we derive a Gibbs distribution from the squared angular distance between grid points and use the corresponding probabilities either as soft targets or as cross-entropy weights that retain a clear probabilistic interpretation. We show that regression on Cartesian coordinates is generally more accurate, except when localized interference is present, in which case classification appears to be more robust.

Tensor calculus in spherical coordinates using Jacobi polynomials. Part-I: Mathematical analysis and derivations

Abstract: This paper presents a method for accurate and efficient computations on scalar, vector and tensor fields in three-dimensional spherical polar coordinates. The method uses spin-weighted spherical harmonics in the angular directions and rescaled Jacobi polynomials in the radial direction. For the 2-sphere, spin-weighted harmonics allow for automating calculations in a fashion as similar to Fourier series as possible. Derivative operators act as wavenumber multiplication on a set of spectral coefficients. After transforming the angular directions, a set of orthogonal tensor rotations put the radially dependent spectral coefficients into individual spaces each obeying a particular regularity condition at the origin. These regularity spaces have remarkably simple properties under standard vector-calculus operations, such as gradient and divergence. We use a hierarchy of rescaled Jacobi polynomials for a basis on these regularity spaces. It is possible to select the Jacobi-polynomial parameters such that all relevant operators act in a minimally banded way. Altogether, the geometric structure allows for the accurate and efficient solution of general partial differential equations in the unit ball.

On the computation of gravitational effects for tesseroids with constant and linearly varying density

Abstract: The accurate computation of gravitational effects from topographic and atmospheric masses is one of the core issues in gravity field modeling. Using gravity forward modeling based on Newton’s integral, mass distributions are generally decomposed into regular mass bodies, which can be represented by rectangular prisms or polyhedral bodies in a rectangular coordinate system, or tesseroids in a spherical coordinate system. In this study, we prefer the latter representation because it can directly take the Earth’s curvature into account, which is particularly beneficial for regional and global applications. Since the volume integral cannot be solved analytically in the case of tesseroids, approximation solutions are applied. However, one well-recognized issue of these solutions is that the accuracy decreases as the computation point approaches the tesseroid. To overcome this problem, we develop a method that can precisely compute the gravitational potential $$\left( V\right) $$ and vector $$\left( V_x, V_y, V_z\right) $$ on the tesseroid surface. In addition to considering a constant density for the tesseroid, we further derive formulas for a linearly varying density. In the near zone (up to a spherical distance of 15 times the horizontal tesseroid dimension from the computation point), the gravitational effects of the tesseroids are computed by Gauss–Legendre quadrature using a two-dimensional adaptive subdivision technique to ensure high accuracy. The tesseroids outside this region are evaluated by means of expanding the integral kernel in a Taylor series up to the second order. The method is validated by synthetic tests of spherical shells with constant and linearly varying density, and the resulting approximation error is less than $$10^{-4}\,\hbox {m}^2\,\hbox {s}^{-2}$$ for V, $$10^{-5}\,\hbox {mGal}$$ for $$V_x$$ , $$10^{-7}\,\hbox {mGal}$$ for $$V_y$$ , and $$10^{-4}\,\hbox {mGal}$$ for $$V_z$$ . Its practical applicability is then demonstrated through the computation of topographic reductions in the White Sands test area and of global atmospheric effects on the Earth’s surface using the US Standard Atmosphere 1976.

Efficient 3-D Large-Scale Forward Modeling and Inversion of Gravitational Fields in Spherical Coordinates With Application to Lunar Mascons

Abstract: An efficient forward modeling algorithm for calculation of gravitational fields in spherical coordinates is developed for 3D large‐scale gravity inversion problems. 3D Gauss‐Legendre quadrature (GLQ) is used to calculate the gravitational fields of mass distributions discretized into tesseroids. Equivalence relations in the kernel matrix of the forward‐modeling are exploited to decrease storage and computation time. The numerical tests demonstrate that the computation time of the proposed algorithm is reduced by approximately two orders of magnitude, and the memory requirement is reduced by Nλ times compared with the traditional GLQ method, where Nλ is the number of the model elements in the longitudinal direction. These significant improvements in computational efficiency and storage make it possible to calculate and store the dense Jacobian matrix in 3D large‐scale gravity inversions. The equivalence relations can be applied to the Taylor series method or combined with the adaptive discretization to ensure high accuracy. To further illustrate the capability of the algorithm, we present a regional synthetic example. The inverted results show density distributions consistent with the actual model. The computation took about 6.3 hours and 0.88 GB of memory compared with about a dozen days and 245.86 GB for the traditional 3D GLQ method. Finally, the proposed algorithm is applied to the gravity field derived from the latest lunar gravity model GL1500E. 3D density distributions of the Imbrium and Serenitatis basins are obtained, and high‐density bodies are found at the depths 10‐60 km, likely indicating a significant uplift of the high‐density mantle beneath the two mascon basins.

Explicit and exact solutions concerning the Antarctic Circumpolar Current with variable density in spherical coordinates

Abstract: We use spherical coordinates to devise a new exact solution to the governing equations of geophysical fluid dynamics for an inviscid and incompressible fluid with a general density distribution and subjected to forcing terms. The latter are of paramount importance for the modeling of realistic flows, that is, flows that are observed in some averaged sense in the ocean. Owing to the employment of spherical coordinates we do not need to resort to approximations (e.g., of f- and β-plane type) that simplify the geometry in the governing equations. Our explicit solution represents a steady purely azimuthal stratified flow with a free surface that—thanks to the inclusion of forcing terms and the consideration of the Earth’s geometry via spherical coordinates—makes it suitable for describing the Antarctic Circumpolar Current and enables an in-depth analysis of the structure of this flow. In line with the latter aspect, we employ functional analytical techniques to prove that the free surface distortion is defined in a unique and implicit way by means of the pressure applied at the free surface. We conclude our discussion by setting out relations between the monotonicity of the surface pressure and the monotonicity of the surface distortion that concur with the physical expectations.

Modeling of Short-Range Ultraviolet Communication Channel Based on Spherical Coordinate System

Abstract: The study on short-range ultraviolet communication (UVC) channel modeling is often based on prolate-spheroidal coordinate system in the previous studies. In this letter, we propose a single scattering coplanar model based on spherical coordinate system, where the transmitter beam and receiver field of view (FOV) can point arbitrarily under the limitation that the sum of transceiver elevation angles and half of their own FOV is less than 180°, and the transmitter beam and receiver FOV are above ground. We utilize the number of intersections between a special ray on the transmitter cone and the receiver conical surface to distinguish the UVC scenarios, which makes the analysis of UVC channel modeling comprehensible. Meanwhile, we validate the proposed model in comparison with the single scattering model and the Monte Carlo simulation model through numerical calculations. The results show that the proposed model agrees well with both models. This letter presents a new method to estimate the path loss of short-range UVC and provides guidelines for the experimental system design.

On a two-point boundary-value problem in geophysics

Abstract: We present some results on the existence and uniqueness of solutions to a two-point boundary-value problem that models gyre flows in rotating spherical coordinates.

Three-phase model of particulate composites in second gradient elasticity

Abstract: Generalized self-consistent method for particulate composites is realized in the framework of Mindlin's second gradient elasticity theory (SGET). Effective properties of the macroscopically isotropic medium containing spherical inclusions are determined based on the three-phase sphere model, in which the spherical inclusion surrounded by matrix shell is embedded in the effective medium. Analytical solutions for the hydrostatic pressure and pure shear problems for the considered model are found based on the generalized Papkovich-Neuber potentials in the spherical coordinate system . Eshelby integral formula generalized for SGET is used as a closing energetic equation to estimate the effective properties of the composite material. Positive size effects for the effective elastic moduli of composites with small size inclusions are predicted based on obtained analytical solutions. Strain concentration in the matrix phase is investigated and found to be a non-monotonic function of the inclusion size. Obtained results are verified by the finite element simulations realized for the special case of SGET with simplified constitutive equations.

Large-scale oceanic currents as shallow-water asymptotic solutions of the Navier-Stokes equation in rotating spherical coordinates

Abstract: We show that a consistent shallow-water approximation of the incompressible Navier-Stokes equation written in a spherical, rotating coordinate system produces, at leading order in a suitable limiting process, a general linear theory for wind-induced ocean currents which goes beyond the limitations of the classical Ekman spiral. In particular, we obtain Ekman-type solutions which extend over large regions in both latitude and longitude; we present examples for constant and for variable eddy viscosities. We also show how an additional restriction on our solution recovers the classical Ekman solution (which is valid only locally).

Axisymmetric creeping motion caused by a spherical particle in a micropolar fluid within a nonconcentric spherical cavity

Abstract: The problems of the quasisteady translation and steady rotation of a solid spherical particle located at a non-concentric position of a spherical cavity filled with an incompressible micropolar fluid are investigated semi-analytically in the limit of low Reynolds numbers. General solutions are constructed from the superposition of the basic solutions in the two spherical coordinate systems based at the centers of the particle and cavity. The boundary conditions on the particle surface and cavity wall are satisfied by a collocation numerical method. The hydrodynamic drag force and torque exerted by the fluid on the particle which are proportional to the translational and angular velocities respectively are obtained numerically with good convergence for a range of values of the ratio of particle-to-cavity radii, the relative distance between the centers of the particle and cavity and micropolarity parameter. In the limit of the motion of a spherical particle in a concentric position in the cavity and in the lubrication limit, the hydrodynamic drag force and torque are in good agreement with the available results in the literature. As expected, the boundary-corrected drag force and torque exerted on the particle is a monotonic increasing function of the micropolarity parameter.

Spherical Coordinates Transform-Based Motion Model for Panoramic Video Coding

Abstract: Panoramic video is produced and displayed as if it is supported on sphere, but it is usually mapped to two-dimensional (2D) plane for compression purpose. The mapping from sphere to 2D plane inevitably causes deformation of video content, which shall be taken into account in panoramic video coding. In this paper, we propose a new motion model based on spherical coordinates transform to explicitly address the deformation problem. Different from the traditional 2D translational motion model, our model assumes a 3D translational motion, which is characterized by a 2D motion vector and a relative depth parameter. Based on spherical coordinates transform, we can deduce the motion compensation and motion estimation algorithms for our proposed motion model. We then integrate the algorithms into the joint exploration model (JEM) scheme for panoramic video coding, and perform experiments on some typical equirectangular projection format video sequences. Experimental results show that our proposed method leads to up to 10.7% and on average 2.6% BD-rate reduction compared to the JEM anchor on our test sequences, with no increase of decoding time.

Study of an elliptic partial differential equation modelling the Antarctic Circumpolar Current

Abstract: We consider the ocean flow of the Antarctic Circumpolar Current. Using a recently-derived model for gyres in rotating spherical coordinates, and mapping the problem on the sphere onto the plane using the Mercator projection, we obtain a boundary-value problem for a semi-linear elliptic partial differential equation. For constant and linear oceanic vorticities, we investigate existence, regularity and uniqueness of solutions to this elliptic problem. We also provide some explicit solutions. Moreover, we examine the physical relevance of these results.

Electrostatically tunable small-amplitude free vibrations of pressurized electro-active spherical balloons

Abstract: Designing tunable resonators is of practical importance in active/adaptive sound generation, noise control, vibration isolation and damping. In this paper, we propose to exploit the biasing fields (induced by internal pressure and radial electric voltage) to tune the three-dimensional and small-amplitude free vibration of a thick-walled soft electro-active (SEA) spherical balloon. The incompressible isotropic SEA balloon is characterized by both neo-Hookean and Gent ideal dielectric models. The equations governing small-amplitude vibrations under inhomogeneous biasing fields can be linearized and solved in spherical coordinates using the state–space formalism, which establishes two separate transfer relations correlating the state vectors at the inner surface with those at the outer surface of the SEA balloon. By imposing the mechanical and electric boundary conditions, two separate analytical frequency equations are derived, which characterize two independent classes of vibration for torsional and spheroidal modes, respectively. Numerical examples are finally conducted to validate the theoretical derivation as well as to investigate the effects of both radial electric voltage and internal pressure on the resonant frequency of the SEA balloon. The reported analytical solution is truly and fully three-dimensional, covering from the purely radial breathing mode to torsional mode to any general spheroidal mode, and hence provides a more accurate prediction of the vibration characteristics of tunable resonant devices that incorporate the SEA spherical balloon as the tuning element.

NADA-FLD: a general relativistic, multidimensional neutrino-hydrodynamics code employing flux-limited diffusion

Abstract: We present the new code NADA-FLD to solve multi-dimensional neutrino-hydrodynamics in full general relativity (GR) in spherical polar coordinates. The energy-dependent neutrino transport assumes the flux-limited diffusion (FLD) approximation and evolves the neutrino energy densities measured in the frame comoving with the fluid. Operator splitting is used to avoid multi-dimensional coupling of grid cells in implicit integration steps involving matrix inversions. Terms describing lateral diffusion and advection are integrated explicitly using the Allen-Cheng or the Runge-Kutta-Legendre method, which remain stable even in the optically thin regime. We discuss several toy-model problems in one and two dimensions to test the basic functionality and individual components of the transport scheme. We also perform fully dynamic core-collapse supernova (CCSN) simulations in spherical symmetry. For a Newtonian model we find good agreement with the M1 code ALCAR, and for a GR model we reproduce the main effects of GR in CCSNe already found by previous works.

The Antarctic Circumpolar Current as a shallow-water asymptotic solution of Euler's equation in spherical coordinates

Abstract: We develop a consistent shallow-water approximation of the incompressible Euler equation in a rotating frame in spherical coordinates, coupled with the appropriate boundary conditions, as a model for the flow of the Antarctic Circumpolar Current. By means of the stereographic projection we come to the boundary value problem for a semilinear elliptic partial differential equation, coupled with the Dirichet or Neumann conditions on the boundary of the considered domain. For recent progress in direction of study of the above mentioned problems we refer to Marynets (2018a, 2018b, 2018c) .

Modeling infrasonic propagation through a spherical atmospheric layer-Analysis of the stratospheric pair.

Abstract: Methods are developed to calculate acoustic propagation paths through an atmospheric layer surrounding a spherical globe in order to more accurately model the propagation of infrasonic signals, which are often observed after propagating hundreds or thousands of kilometers. A generalized curvilinear coordinate system is used to define the ray tracing equations from the eikonal equation for a moving, inhomogeneous atmosphere, and the specific case of spherical coordinates is applied to obtain a system of coupled equations describing geometric propagation paths in an atmospheric layer surrounding a globe. Comparison with propagation predictions using a Cartesian geometry shows that even for relatively short infrasonic propagation distances of a few hundred kilometers, differences introduced by the change in geometry are significant. Characteristics of the stratospheric pair are considered, and it is found that differences in the upward and downward legs of the propagation paths corresponding to the fast and slow stratospheric arrivals are changed such that spherical coordinate geometry predicts a decrease in the relative arrival time between the two. Analysis of regional infrasonic signals produced by a series of surface explosions shows that predictions obtained using spherical geometry are more accurate in cases where the tropospheric and stratospheric waveguides are accurately characterized.

Rectilinear oscillations of two spherical particles embedded in an unbounded viscous fluid

Abstract: The problem of the rectilinear oscillations of two spherical particles along the line through their centers in an axi-symmetric, viscous, incompressible flow at low Reynolds number is considered. The particles oscillate with the same frequency and with different amplitudes. In addition, the particles may differ in their sizes. In order to solve the Stokes equations for the flow field, a general solution is constructed from the superposition of the basic solutions in the two spherical coordinate systems based at the centers of the particles. A collocation technique is used to satisfy the boundary conditions on the surfaces of the particles. The solution is valid for all values of the frequency parameter subject to the conditions that justify the use of the unsteady Stokes equations. Numerical results displaying the in phase and the out of phase force amplitudes acting on each particle are obtained with good convergence for various values of the physical parameters of the problem. The results are tabulated and represented graphically. Our results agree well with the existing solutions of the steady motion of two spherical particles and with the oscillations of single particle.

A Particle Module for the PLUTO Code. III. Dust

Abstract: The implementation of a new particle module describing the physics of dust grains coupled to the gas via drag forces is the subject of this work. The proposed particle-gas hybrid scheme has been designed to work in Cartesian as well as in cylindrical and spherical geometries. The numerical method relies on a Godunov-type second-order scheme for the fluid and an exponential midpoint rule for dust particles which overcomes the stiffness introduced by the linear coupling term. Besides being time-reversible and globally second-order accurate in time, the exponential integrator provides energy errors which are always bounded and it remains stable in the limit of arbitrarily small particle stopping times yielding the correct asymptotic solution. Such properties make this method preferable to the more widely used semi-implicit or fully implicit schemes at a very modest increase in computational cost. Coupling between particles and grid quantities is achieved through particle deposition and field-weighting techniques borrowed from Particle-In-Cell simulation methods. In this respect, we derive new weight factors in curvilinear coordinates that are more accurate than traditional volume- or area-weighting. A comprehensive suite of numerical benchmarks is presented to assess the accuracy and robustness of the algorithm in Cartesian, cylindrical and spherical coordinates. Particular attention is devoted to the streaming instability which is analyzed in both local and global disk models. The module is part of the PLUTO code for astrophysical gas-dynamics and it is mainly intended for the numerical modeling of protoplanetary disks in which solid and gas interact via aerodynamic drag.

Gravitational field calculation in spherical coordinates using variable densities in depth

Abstract: We present a new methodology to compute the gravitational fields generated by tesseroids (spherical prisms) whose density varies with depth according to an arbitrary continuous function. It approximates the gravitational fields through the Gauss-Legendre Quadrature along with two discretization algorithms that automatically control its accuracy by adaptively dividing the tesseroid into smaller ones. The first one is a preexisting two dimensional adaptive discretization algorithm that reduces the errors due to the distance between the tesseroid and the computation point. The second is a new density-based discretization algorithm that decreases the errors introduced by the variation of the density function with depth. The amount of divisions made by each algorithm is indirectly controlled by two parameters: the distance-size ratio and the delta ratio. We have obtained analytical solutions for a spherical shell with radially variable density and compared them to the results of the numerical model for linear, exponential, and sinusoidal density functions. The heavily oscillating density functions are intended only to test the algorithm to its limits and not to emulate a real world case. These comparisons allowed us to obtain optimal values for the distance-size and delta ratios that yield an accuracy of 0.1% of the analytical solutions. The resulting optimal values of distance-size ratio for the gravitational potential and its gradient are 1 and 2.5, respectively. The density-based discretization algorithm produces no discretizations in the linear density case, but a delta ratio of 0.1 is needed for the exponential and most sinusoidal density functions. These values can be extrapolated to cover most common use cases, which are simpler than oscillating density profiles. However, the distance-size and delta ratios can be configured by the user to increase the accuracy of the results at the expense of computational speed. Lastly, we apply this new methodology to model the Neuquen Basin, a foreland basin in Argentina with a maximum depth of over 5000 m, using an exponential density function.

Tensor calculus in spherical coordinates using Jacobi polynomials. Part-II: Implementation and examples

Abstract: We present a simulation code which can solve a broad range of partial differential equations in a full sphere. The code expands tensorial variables in a spectral series of spin-weighted spherical harmonics in the angular directions and a scaled Jacobi polynomial basis in the radial direction, as described in Vasil et al. (2018; hereafter, Part-I). Nonlinear terms are calculated by transforming from the coefficients of the spectral series to the value of each quantity on the physical grid, where it is easy to calculate products and perform other local operations. The expansion makes it straightforward to solve equations in tensor form (i.e., without decomposition into scalars). We propose and study several unit tests which demonstrate the code can accurately solve linear problems, implement boundary conditions, and transform between spectral and physical space. We then run a series of benchmark problems proposed in Marti et al. (2014), implementing the hydrodynamic and magnetohydrodynamic equations. We are able to calculate more accurate solutions than reported in Marti et al. (2014) by running at higher spatial resolution and using a higher-order timestepping scheme. We find the rotating convection and convective dynamo benchmark problems depend sensitively on details of timestepping and data analysis. We also demonstrate that in low resolution simulations of the dynamo problem, small changes in a numerical scheme can lead to large changes in the solution. To aid future comparison to these benchmarks, we include the source code used to generate the data, as well as the data and analysis scripts used to generate the figures.