Showing papers on "Spherical coordinate system published in 2016"

Tesseroids: Forward-modeling gravitational fields in spherical coordinates

Abstract: We have developed the open-source software Tesseroids, a set of command-line programs to perform forward modeling of gravitational fields in spherical coordinates. The software is implemented in the C programming language and uses tesseroids (spherical prisms) for the discretization of the subsurface mass distribution. The gravitational fields of tesseroids are calculated numerically using the Gauss-Legendre quadrature (GLQ). We have improved upon an adaptive discretization algorithm to guarantee the accuracy of the GLQ integration. Our implementation of adaptive discretization uses a “stack-based” algorithm instead of recursion to achieve more control over execution errors and corner cases. The algorithm is controlled by a scalar value called the distance-size ratio (D) that determines the accuracy of the integration as well as the computation time. We have determined optimal values of D for the gravitational potential, gravitational acceleration, and gravity gradient tensor by comparing the comp...

An Exact, Steady, Purely Azimuthal Equatorial Flow with a Free Surface

Abstract: The general problem of an ocean on a rotating sphere is considered. The governing equations for an inviscid, incompressible fluid, written in spherical coordinates that are fixed at a point on the rotating Earth, together with the free surface and rigid bottom boundary conditions, are introduced. An exact solution of this system is presented; this describes a steady flow that is moving only in the azimuthal direction, with no variation in this direction. However, this azimuthal velocity component has an arbitrary variation with depth (i.e., radius), and so, for example, an Equatorial Undercurrent (EUC) can be accommodated. The pressure boundary condition at the free surface relates this pressure to the shape of the surface via a Bernoulli relation; this provides the constraint on the existence of a solution, although the restrictions are somewhat involved in spherical coordinates. To examine this constraint in more detail, the corresponding problems in model cylindrical coordinates (with the equat...

Solution of the Dynamics of Liquids in the Large-Dimensional Limit.

Abstract: We obtain analytic expressions for the time correlation functions of a liquid of spherical particles, exact in the limit of high dimensions d The derivation is long but straightforward: a dynamic virial expansion for which only the first two terms survive, followed by a change to generalized spherical coordinates in the dynamic variables leading to saddle-point evaluation of integrals for large d The problem is, thus, mapped onto a one-dimensional diffusion in a perturbed harmonic potential with colored noise At high density, an ergodicity-breaking glass transition is found In this regime, our results agree with thermodynamics, consistently with the general random first order transition scenario The glass transition density is higher than the best known lower bound for hard sphere packings in large d Because our calculation is, if not rigorous, elementary, an improvement in the bound for sphere packings in large dimensions is at hand

A Closed-Form Solution for RSS/AoA Target Localization by Spherical Coordinates Conversion

Abstract: This letter addresses the problem of target localization in a 3-D space, utilizing combined measurements of received signal strength and angle of arrival (AoA). By using the spherical coordinate conversion and available AoA observations to establish new relationships between the measurements and the unknown target location, we derive a simple closed-form solution method. We then show that the proposed method has straightforward adaptation to the case where the target’s transmit power is also not known. Simulation results validate the outstanding performance of the proposed method.

Magneto-frictional modeling of coronal nonlinear force-free fields. i. testing with analytic solutions

Abstract: We report our implementation of the magneto-frictional method in the Message Passing Interface Adaptive Mesh Refinement Versatile Advection Code (MPI-AMRVAC). The method aims at applications where local adaptive mesh refinement (AMR) is essential to make follow-up dynamical modeling affordable. We quantify its performance in both domain-decomposed uniform grids and block-adaptive AMR computations, using all frequently employed force-free, divergence-free, and other vector comparison metrics. As test cases, we revisit the semi-analytic solution of Low and Lou in both Cartesian and spherical geometries, along with the topologically challenging Titov-Demoulin model. We compare different combinations of spatial and temporal discretizations, and find that the fourth-order central difference with a local Lax-Friedrichs dissipation term in a single-step marching scheme is an optimal combination. The initial condition is provided by the potential field, which is the potential field source surface model in spherical geometry. Various boundary conditions are adopted, ranging from fully prescribed cases where all boundaries are assigned with the semi-analytic models, to solar-like cases where only the magnetic field at the bottom is known. Our results demonstrate that all the metrics compare favorably to previous works in both Cartesian and spherical coordinates. Cases with several AMR levels perform in accordance with their effective resolutions. The magneto-frictional method in MPI-AMRVAC allows us to model a region of interest with high spatial resolution and large field of view simultaneously, as required by observation-constrained extrapolations using vector data provided with modern instruments. The applications of the magneto-frictional method to observations are shown in an accompanying paper.

Magneto-frictional modeling of coronal nonlinear force-free fields. ii. application to observations

Abstract: A magneto-frictional module has been implemented and tested in the Message Passing Interface Adaptive Mesh Refinement Versatile Advection Code (MPI-AMRVAC) in the first paper of this series. Here, we apply the magneto-frictional method to observations to demonstrate its applicability in both Cartesian and spherical coordinates, and in uniform and block-adaptive octree grids. We first reconstruct a nonlinear force-free field (NLFFF) on a uniform grid of 1803 cells in Cartesian coordinates, with boundary conditions provided by the vector magnetic field observed by the Helioseismic and Magnetic Imager (HMI) on board the Solar Dynamics Observatory (SDO) at 06:00 UT on 2010 November 11 in active region NOAA 11123. The reconstructed NLFFF successfully reproduces the sheared and twisted field lines and magnetic null points. Next, we adopt a three-level block-adaptive grid to model the same active region with a higher spatial resolution on the bottom boundary and a coarser treatment of regions higher up. The force-free and divergence-free metrics obtained are comparable to the run with a uniform grid, and the reconstructed field topology is also very similar. Finally, a group of active regions, including NOAA 11401, 11402, 11405, and 11407, observed at 03:00 UT on 2012 January 23 by SDO/HMI is modeled with a five-level block-adaptive grid in spherical coordinates, where we reach a local resolution of pixel−1 in an area of 790 Mm × 604 Mm. Local high spatial resolution and a large field of view in NLFFF modeling can be achieved simultaneously in parallel and block-adaptive magneto-frictional relaxations.

Novel fusion computing method for bio-medical image of WSN based on spherical coordinate

Abstract: In bio-medical field, embedded numerous sensing nodes can be used to monitor and interact with physical world based on signal analysis and processing Data from many different sources can be collected into massive data sets via localized sensor networks Understanding the environment requires collecting and analyzing data from thousands of sensors monitoring, this is big data environment The application of bio-medical image fusion for big-data computing has strong development momentum, big-data bio-medical image fusion is one of key problems, so the fusion method study is a hot topic in the field of signal analysis and processing The existing methods have many limitations, such as large delay, data redundancy, more energy cost, low quality, so novel fusion computing method based on spherical coordinate for big-data bio-medical image of WSN is proposed in this paper In this method, the three high-frequency coefficients in wavelet domain of bio-medical image are pre-processed This pre-processing strategy can reduce the redundant ratio of big-data bio-medical image Firstly, the high-frequency coefficients are transformed to the spherical coordinate domain to reduce the correlation in the same scale Then, a multi-scale model product (MSMP) is used to control the shrinkage function so as to make the small wavelet coefficients and some noise removed The high-frequency parts in spherical coordinate domain are coded by improved SPIHT algorithm Finally, based on multi-scale edge of bio-medical image, it can be fused and reconstructed Experimental results indicate the novel method is effective and very useful for transmission of big-data bio-medical image, which can solve the problem of data redundancy, more energy cost and low quality

Two-dimensional inflow-wind solution of black hole accretion with an evenly symmetric magnetic field

Abstract: We solve the two-dimensional magnetohydrodynamic (MHD) equations of black hole accretion with the presence of magnetic field. The field includes a turbulent component, whose role is represented by the viscosity, and a large-scale ordered component. The latter is further assumed to be evenly symmetric with the equatorial plane. The equations are solved in the $r-\theta$ plane of a spherical coordinate by assuming time-steady and radially self-similar. An inflow-wind solution is found. Around the equatorial plane, the gas is inflowing; while above and below the equatorial plane at a certain critical $\theta$ angle, $\theta\sim 47^{\circ}$, the inflow changes its direction of radial motion and becomes wind. The driving forces are analyzed and found to be the centrifugal force and the gradient of gas and magnetic pressure. The properties of wind are also calculated. The specific angular momentum of wind is found to be significantly larger than that of inflow, thus wind can transfer angular momentum outward. These analytical results are compared to those obtained by the trajectory analysis based on MHD numerical simulation data and good agreements are found.

Transversal Strapdown INS Based on Reference Ellipsoid for Vehicle in the Polar Region

Abstract: The transversal coordinate system has been proposed to solve the problem of navigation in the Polar Region for a strapdown inertial navigation system (INS). However, it is assumed that the Earth is a perfect sphere in the transversal coordinate system, which is not accurate for a high-precision INS. To solve this problem, we consider the Earth as a reference ellipsoid and then deduct its radii of curvature in our transversal coordinate system. In addition, with the aid of the Bessel geodetic projection method, transformations of navigation quantities from a local-level coordinate system to our transversal coordinate system are derived based on the reference ellipsoid. Numerical results indicate that our method based on the reference ellipsoid performs better than the method based on a reference sphere.

Spherical Coordinates Based Methods of Ground Extraction and Objects Segmentation Using 3-D LiDAR Sensor

Abstract: -This paper proposes new methods to extract ground points and segment non-ground objects based on the spherical coordinates of point clouds collected by the 3-D LiDAR sensor system. Ground points extraction and non-ground objects segmentation are two primary issues in environment perception of autonomous driving. Firstly, the new method to extract ground points is described in this paper. It is based on the breakpoints and turning points of the radial distance curve in the spherical coordinates without assuming the road is located in the lowest. Then the algorithm for non-ground objects segmentation is proposed, which uses azimuth angle and radial distance in the spherical coordinates as judgment criterion. This algorithm works efficiently and it can avoid over-segmentation and under-segmentation. Finally, the experimental results using the point clouds acquired by the sensor Velodyne HDL-32E are presented. Comparing to the existing methods, the results show the advantages of methods based on the spherical coordinates.

Robustness of oscillatory $\alpha^2$ dynamos in spherical wedges

Abstract: We study the connection between spherical wedge and full spherical shell geometries using simple mean-field $\alpha^2$ dynamos. We solve the equations for a one-dimensional time-dependent mean-field dynamo to examine the effects of varying the polar angle $\theta_0$ between the latitudinal boundaries and the poles in spherical coordinates. We investigate the effects of turbulent magnetic diffusivity and $\alpha$ effect profiles as well as different latitudinal boundary conditions to isolate parameter regimes where oscillatory solutions are found. Finally, we add shear along with a damping term mimicking radial gradients to study the resulting dynamo regimes. We find that the commonly used perfect conductor boundary condition leads to oscillatory $\alpha^2$ dynamo solutions only if the wedge boundary is at least one degree away from the poles. Other boundary conditions always produce stationary solutions. By varying the profile of the turbulent magnetic diffusivity alone, oscillatory solutions are achieved with models extending to the poles, but the magnetic field is strongly concentrated near the poles and the oscillation period is very long. By introducing radial shear and a damping term mimicking radial gradients, we again see oscillatory dynamos, and the direction of drift follows the Parker--Yoshimura rule. Oscillatory solutions in the weak shear regime are found only in the wedge case with $\theta_0 = 1^\circ$ and perfect conductor boundaries. A reduced $\alpha$ effect near the poles with a turbulent diffusivity concentrated toward the equator yields oscillatory dynamos with equatorward migration and reproduces best the solutions in spherical wedges.

Visualization of three-dimensional incompressible flows by quasi-two-dimensional divergence-free projections in arbitrary flow regions

Abstract: A visualization of three-dimensional incompressible flows by divergence-free quasi-two-dimensional projections of the velocity field onto three coordinate planes is revisited. An alternative and more general way to compute the projections is proposed. The approach is based on the Chorin projection combined with a SIMPLE-like iteration. Compared to the previous methodology based on divergence-free Galerkin–Chebyshev bases, this technique, formulated in general curvilinear coordinates, is applicable to any flow region and allows for faster computations. To illustrate this visualization method, examples in Cartesian and spherical coordinates, as well as post-processing of experimental 3D-PTV data, are presented.

Resonance energy transfer: The unified theory via vector spherical harmonics

Abstract: In this work, we derive the well-established expression for the quantum amplitude associated with the resonance energy transfer (RET) process between a pair of molecules that are beyond wavefunction overlap. The novelty of this work is that the field of the mediating photon is described in terms of a spherical wave rather than a plane wave. The angular components of the field are constructed in terms of vector spherical harmonics while Hankel functions are used to define the radial component. This approach alleviates the problem of having to select physically correct solution from non-physical solutions, which seems to be inherent in plane wave derivations. The spherical coordinate system allows one to easily decompose the photon’s fields into longitudinal and transverse components and offers a natural way to analyse near-, intermediate-, and far-zone RET within the context of the relative orientation of the transition dipole moments for the two molecules.

Diffusion Penetration Time for Transient Heat Conduction

Abstract: The time duration for processes involving transient thermal diffusion can be a critical piece of information related to thermal processes in engineering applications. Analytical solutions must be used to calculate these types of time durations because the boundary conditions in such cases can be effectively like semi-infinite conditions. This research involves an investigation into analytical solutions for six geometries, including one-dimensional cases for Cartesian, cylindrical, and spherical coordinates. The fifth case involves a heated surface on the inside of a hole bored through an infinite body, which is a one-dimensional problem in cylindrical coordinates. The sixth case involves two-dimensional conduction from a point heat source on the surface of a slab subjected to insulated boundary conditions elsewhere. The mathematical modeling for this case is done in cylindrical coordinates. For each geometric configuration, a relationship is developed to determine the time required for a temperature rise ...

Vibration control for a nonlinear three-dimensional flexible manipulator trajectory tracking

Abstract: This paper proposes an innovative approach to the trajectory tracking in three-dimensional space and vibration control problems in the presence of a nonlinear three-dimensional flexible manipulator based on the partial differential equation model. Unlike two-dimensional plane, we select spherical coordinates to describe the position of the end point in three-dimensional space. This novel approach makes it possible to realise the trajectory tracking by controlling the two angles in spherical coordinates, meanwhile, a vibration control scheme is proposed to restrain vibrations. In addition, the existence and uniqueness of solutions are demonstrated. Finally, the performance of the desired trajectory tracking, the proposed vibration control scheme and their convergence properties are demonstrated by numerical simulations.

Strongly magnetized rotating dipole in general relativity

Abstract: Context. Electromagnetic waves arise in many areas of physics. Solutions are difficult to find in the general case.Aims. We numerically integrate Maxwell equations in a 3D spherical polar coordinate system.Methods. Straightforward finite difference methods would lead to a coordinate singularity along the polar axis. Spectral methods are better suited for such artificial singularities that are related to the choice of a coordinate system. When the radiating object rotates like a star, for example, special classes of solutions to Maxwell equations are worthwhile to study, such as quasi-stationary regimes. Moreover, in high-energy astrophysics, strong gravitational and magnetic fields are present especially around rotating neutron stars.Results. To study such systems, we designed an algorithm to solve the time-dependent Maxwell equations in spherical polar coordinates including general relativity and quantum electrodynamical corrections to leading order. As a diagnostic, we computed the spin-down luminosity expected for these stars and compared it to the classical or non-relativistic and non-quantum mechanical results.Conclusions. Quantum electrodynamics leads to an irrelevant change in the spin-down luminosity even for a magnetic field of about the critical value of 4.4 × 109 T. Therefore the braking index remains close to its value for a point dipole in vacuum, namely n = 3. The same conclusion holds for a general-relativistic quantum electrodynamically corrected force-free magnetosphere.

A Gauss-Kronrod-Trapezoidal integration scheme for modeling biological tissues with continuous fiber distributions

Abstract: Fibrous biological tissues may be modeled using a continuous fiber distribution (CFD) to capture tension–compression nonlinearity, anisotropic fiber distributions, and load-induced anisotropy. The CFD framework requires spherical integration of weighted individual fiber responses, with fibers contributing to the stress response only when they are in tension. The common method for performing this integration employs the discretization of the unit sphere into a polyhedron with nearly uniform triangular faces (finite element integration or FEI scheme). Although FEI has proven to be more accurate and efficient than integration using spherical coordinates, it presents three major drawbacks: First, the number of elements on the unit sphere needed to achieve satisfactory accuracy becomes a significant computational cost in a finite element (FE) analysis. Second, fibers may not be in tension in some regions on the unit sphere, where the integration becomes a waste. Third, if tensed fiber bundles span a small regi...

2D Fourier series representation of gravitational functionals in spherical coordinates

Abstract: 2D Fourier series representation of a scalar field like gravitational potential is conventionally derived by making use of the Fourier series of the Legendre functions in the spherical harmonic representation. This representation has been employed so far only in the case of a scalar field or the functionals that are related to it through a radial derivative. This paper provides a unified scheme to represent any gravitational functional in terms of spherical coordinates using a 2D Fourier series representation. The 2D Fourier series representation for each individual point is derived by transforming the spherical harmonics from the geocentric Earth-fixed frame to a rotated frame so that its equator coincides with the local meridian plane of that point. In the obtained formulation, each functional is linked to the potential in the spectral domain using a spectral transfer. We provide the spectral transfers of the first-, second- and third-order gradients of the gravitational potential in the local north-oriented reference frame and also those of some functionals of frequent use in the physical geodesy. The obtained representation is verified numerically. Moreover, spherical harmonic analysis of anisotropic functionals and contribution analysis of the third-order gradient tensor are provided as two numerical examples to show the power of the formulation. In conclusion, the 2D Fourier series representation on the sphere is generalized to functionals of the potential. In addition, the set of the spectral transfers can be considered as a pocket guide that provides the spectral characteristics of the functionals. Therefore, it extends the so-called Meissl scheme.

Numerical modeling of a spherical buoy moored by a cable in three dimensions

Abstract: Floating facilities have been studied based on the static analysis of mooring cables over the past decades. To analyze the floating system of a spherical buoy moored by a cable with a higher accuracy than before, the dynamics of the cables are considered in the construction of the numerical modeling. The cable modeling is established based on a new element frame through which the hydrodynamic loads are expressed efficiently. The accuracy of the cable modeling is verified with an experiment that is conducted by a catenary chain moving in a water tank. In addition, the modeling of a spherical buoy is established with respect to a spherical coordinate in three dimensions, which can suffers the gravity, the variable buoyancy and Froude-Krylov loads. Finally, the numerical modeling for the system of a spherical buoy moored by a cable is established, and a virtual simulation is proceeded with the X- and Y-directional linear waves and the X-directional current. The comparison with the commercial simulation code ProteusDS indicates that the system is accurately analyzed by the numerical modeling. The tensions within the cable, the motions of the system, and the relationship between the motions and waves are illustrated according to the defined sea state. The dynamics of the cables should be considered in analyzing the floating system of a spherical buoy moored by a cable.

Spherical Vector Waves

Abstract: There is a special interest in finding solutions of the Maxwell equations in source-free, homogeneous, isotropic materials in the spherical coordinate system (r, 8, φ). The reason this is at least twofold: (1) we aim at developing efficient tools to solve scattering problems with spherical symmetries, and (2) outside the circumscribed sphere of the scatterer, the are naturally expanded in vector waves with spherical symmetries-we have already Chapter 4 that the scattered field in the far zone is a spherical wave. For these reasons, study solutions to the source-free Maxwell equations.

Efficient Fluid Simulation on the Surface of a Sphere

Abstract: For the purposes of computer graphics, we have developed a simulation tool to model fluid flow on the surface of a sphere with the inclusion of control parameters for the benefit of art directability. Difficulties associated with the use of spherical coordinates were surmounted by the use of locally modified consistent equations that result from an analysis of the singular equations in the neighborhood of the poles. The resulting system was solved efficiently for only a small additional cost when compared to a two-dimensional planar simulation.

Axisymmetric motion of a porous sphere through a spherical envelope subject to a stress jump condition

Abstract: The flow problem of an incompressible axisymmetrical quasisteady translation and steady rotation of a porous sphere in an eccentric spherical container is discussed using a combined analytical–numerical technique. A continuity of velocity components and normal stress together with the stress jump condition for the tangential stress are used at the interface between porous and clear-fluid regions. The fluid flow outside the particle is governed by the classical Stokes equations while the fluid flow inside the porous region is treated by Brinkman model. 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 on both the porous sphere and spherical envelope. Solutions for translational and rotational motion of porous eccentric spherical particle in a spherical envelope are obtained using the boundary collocation technique. The hydrodynamic drag force and couple exerted by the surrounding fluid on the porous particle which is proportional to the translational and angular velocities, respectively, are calculated with good convergence for various values of the ratio of porous-to-container radii, the relative distance between the centers of the porous and container, the stress jump coefficient, and a coefficient that is proportional to the permeability. In the limits of the motions of a porous sphere in a concentric container and near a container surface with a small curvature, the numerical values of the normalized drag force and the normalized coupling coefficient are in good agreement with the available values in the literature.

Highly effective stable evaluation of bandlimited functions on the sphere

Abstract: An algorithm for fast and accurate evaluation of band-limited functions at many scattered points on the unit 2-d sphere is developed. The algorithm is based on trigonometric representation of spherical harmonics in spherical coordinates and highly localized tensor-product trigonometric kernels (needlets). It is simple, fast, local, memory efficient, numerically stable and with guaranteed accuracy. Comparison of this algorithm with other existing algorithms in the literature is also presented.

Visualising three-dimensional volumetric data with an arbitrary coordinate system

Abstract: Astronomical data does not always use Cartesian coordinates. Both all-sky observational data and simulations of rotationally symmetric systems, such as accretion and protoplanetary discs, may use spherical polar or other coordinate systems. Standard displays rely on Cartesian coordinates, but converting non-Cartesian data into Cartesian format causes distortion of the data and loss of detail. I here demonstrate a method using standard techniques from computer graphics that avoids these problems with 3D data in arbitrary coordinate systems. The method adds minimum computational cost to the display process and is suitable for both realtime, interactive content and producing fixed rendered images and videos. Proof-of-concept code is provided which works for data in spherical polar coordinates.