Method In Spherical Coordinates Research Articles

3D gravity inversion based on mixed-norm regularization in spherical coordinates with application to the lunar Moscoviense Basin

Three-dimensional gravity inversion has been widely used to infer density structures and tectonic movements of the earth and Moon. However, two problems (the nonuniqueness and low depth resolution) of current gravity inversion methods still exist and are not completely resolved, which affect the reliability of the inversion results and the corresponding geologic interpretation. To improve the depth resolution of the 3D gravity inversion, we propose an efficient inversion method in spherical coordinates based on a mixed smooth and focused regularization with depth-weighting functions. In the inversion, we also use the kernel matrix equivalence strategy and the fast kernel-vector multiplication method based on a fast Fourier transform in each iteration to increase computational efficiency. Two synthetic inversion examples indicate that the proposed method can recover more complex density structures compared with the often used smooth-constraint inversion approach. Finally, the proposed inversion approach is applied to the latest lunar gravitational field model GL1500E to study the 3D density structures of the Moscoviense Basin. The inverted results indicate that the density structure with high-positive anomalies is principally distributed at depths ranging from 5 to 70 km, forming a giant asymmetrical high-density structure inclining from the southwest to the northeast. We may conclude that the Moscoviense Basin was formed by the double impact, where the second collision occurred in the southwest of the basin center of the first collision.

BiFold: A Python code for the calculation of double-folded (bifold) potentials with density-in/dependent nucleon-nucleon interactions

BiFold calculates the density-dependent (DDM3Yn, BDM3Yn, CDM3Yn) or independent double-folded potentials between two colliding spherical nuclei. It is written in a Python package form to give the ability to use the potentials directly in a nuclear reaction/structure code. In addition to using Woods-Saxon/Fermi or Gaussian functions, the code also allows the definition of nuclear matter densities using pre-calculated densities in a data file. The manuscript provides an overview of the double folding model and the use of the code. Program summaryProgram Title: BiFoldCPC Library link to program files:https://doi.org/10.17632/rmdx86p9dg.1Developer's repository link:https://github.com/mkarakoc/BiFoldCode Ocean capsule:https://codeocean.com/capsule/5286231Licensing provisions: GPLv3Programming language: Python 3.xNature of problem: BiFold calculates the real part of the nuclear potential between two colliding spherical nuclei by integrating a density-independent/dependent nucleon-nucleon (NN) interaction [1,2,3] over the nuclear matter densities of the two nuclei. The code is based on M3Y Reid/Paris NN interactions [1,2,3] by default, but it is possible to define custom NN interactions when necessary.Solution method: The code uses the Fourier transform method in spherical coordinates to calculate the potential. The method simplifies the sixfold integration [1] and makes the calculation significantly faster. The integration is done by default using Simpson's integration method, but Filon's integration method is also available.

Gravity Observations and Apparent Density Changes before the 2017 Jiuzhaigou Ms7.0 Earthquake and Their Precursory Significance.

An Ms7.0 earthquake struck Jiuzhaigou (China) on 8 August 2017. The epicenter was in the eastern margin of the Tibetan Plateau, an area covered by a dense time-varying gravity observation network. Data from seven repeated high-precision hybrid gravity surveys (2014–2017) allowed the microGal-level time-varying gravity signal to be obtained at a resolution better than 75 km using the modified Bayesian gravity adjustment method. The “equivalent source” model inversion method in spherical coordinates was adopted to obtain the near-crust apparent density variations before the earthquake. A major gravity change occurred from the southwest to the northeast of the eastern Tibetan Plateau approximately 2 years before the earthquake, and a substantial gravity gradient zone was consistent with the tectonic trend that gradually appeared within the focal area of the Jiuzhaigou earthquake during 2015–2016. Factors that might cause such regional gravitational changes (e.g., vertical crustal deformation and variations in near-surface water distributions) were studied. The results suggest that gravity effects contributed by these known factors were insufficient to produce gravity changes as big as those observed, which might be related to the process of fluid material redistribution in the crust. Regional change of the gravity field has precursory significance for high-risk earthquake areas and it could be used as a candidate precursor for annual medium-term earthquake prediction.

High-frequency leaky whispering-gallery modes in embedded elastic spheres

The goal of this paper is to investigate the characteristics of high-frequency whispering-gallery modes in embedded elastic spheres, that is, surrounded by an infinite elastic matrix. Due to several modeling difficulties, the high-frequency regime of embedded spheres has remained unexplored in elasticity. Our approach consists of formulating a specific finite-element method in spherical coordinates. The basic idea is to discretize only the radial coordinate while describing analytically the angular distribution of elastodynamic fields. Then, we also introduce a radial perfectly matched layer to cope with the unbounded nature of the external medium. Our approach yields a linear matrix eigensystem, simple and costless to solve. In order to identify general trends, both stiff and soft configurations are considered, corresponding to a sphere stiffer and softer than the external medium, respectively. Including material loss, our results highlight the behavior of leaky elastic whispering-gallery modes in the high-frequency regime. This work is motivated by the well-known behavior of whispering-gallery modes in optical resonators, reaching high $Q$ factors as the frequency increases. Identifying high-$Q$-factor whispering-gallery modes in elastic spheres could find promising applications for sensing the mechanical properties of external media.

A New 3D Shaping Method for Low-Thrust Trajectories between Non-Intersect Orbits

This paper proposes a new shape-based method in spherical coordinates to solve three-dimensional rendezvous problems. Compared with the existing shape-based methods, the proposed method does not need parameter optimization. Moreover, it improves the flexibility of orbit fitting, greatly reduces the velocity increment and maximum thrust acceleration, and ensures the orbit safety to a certain extent. The shaping function can provide the initial estimate for numerical trajectory optimization and improve the convergence rate in a certain range when combined with the normalization method. The superiority of the proposed method over the existing methods is demonstrated by two numerical examples. Its effectiveness at initial estimation generation in the indirect optimization of a low-thrust trajectory is demonstrated by the third example.

Relativistic ionization probabilities and photoelectron distributions of hydrogenlike ions in superstrong electromagnetic fields

The ionization probabilities and photoelectron distributions of the hydrogen atom and hydrogenlike ions ${\mathrm{He}}^{+}$ and ${\mathrm{Ne}}^{9+}$ in superstrong electromagnetic fields are obtained by solving the time-dependent Dirac equation with the help of the generalized pseudospectral method in spherical coordinates. A simple transformation of the original radial Dirac Hamiltonian is suggested that removes the spurious states. The ionization probabilities are calculated both within and beyond the dipole approximation for two pulse durations and various peak field strengths scaled with respect to the nuclear charge for each target. Performance of two analytic parameters previously suggested to estimate the nondipole effects is assessed against the numerical data obtained in this study. The photoelectron energy and angular distributions exhibit the above-threshold ionization peaks as well as the low-energy structure. In the superstrong field regime, the low-energy structure builds up with increasing field strength while the above-threshold ionization peaks decrease. The nondipole effects in the photoelectron distributions are also revealed and analyzed.

Interaction of circularly polarized light with an absorptive electromagnetic conductor sphere – Radiation force and spin torque

The concept of a perfect electromagnetic conductor (PEMC) material, which is a class of a magneto-electric metamaterial, has been recently examined in the context of the optical radiation force theory [J. Quant. Spectr. Rad. Transfer, 233 (2019) 21–28; ibid 256 (2020) 107280], with particular emphasis on the rotary polarization effect and the contribution of the cross-polarized waves to the transfer of linear momentum, assuming an illumination of linearly-polarized plane progressive waves. The present investigation shows that additional cross-interference factors, which cannot be ignored, contribute to the time-averaged optical radiation force exerted on an absorptive electromagnetic conductor (AEMC) sphere when the incident field becomes circularly-polarized. Moreover, the analysis is extended to investigate the co-polarized, cross-polarized and cross-interference components of the time-averaged optical spin radiation torque. The multipole series expansion method in spherical coordinates is utilized to derive exact mathematical partial-wave series for the co-polarized, cross-polarized and cross-interference components of the radiation force and torque based on the integration of Maxwell’s radiation stress tensor and its moment over a spherical surface of large radius enclosing the sphere suspended in a lossless medium of wave propagation. Numerical results for the radiation force and spin torque efficiencies and their corresponding co-polarized, cross-polarized and cross-interference components illustrate the analysis with emphasis on the size parameter of the sphere ka, its electromagnetic admittance M as well as its imaginary part amounting to absorption. The computations are of importance from the standpoint of the fundamentals of the linear and angular momenta of circularly-polarized electromagnetic waves transferred to an AEMC and related applications in optical tweezers and particle manipulation of objects exhibiting circular dichroism.

Joint tomographic inversion of crustal structure beneath the eastern Tibetan Plateau with ambient noise and gravity data

SUMMARY We have developed a joint tomographic inversion method with seismic surface wave dispersion and gravity data for obtaining more reliable crustal 3-D shear wave structures. We take the eikonal-based direct surface wave tomographic method and adaptive gravity modelling method in spherical coordinates in the inverse problem. Based on the empirical relations between seismic velocity and density parameters, our method combines surface wave dispersion curves (i.e. surface wave traveltimes at different periods) and Bouguer gravity anomaly data together to invert for 3-D shear wave velocity structures. In our method, off-great-circle propagation of the surface wave and the earth's curvature is considered in the forward modelling. Synthetic tests suggest that the joint tomographic method could improve the reliability and obtain more convincing results than individual seismic surface wave tomography. The gravity data can provide more constraints into the model resolution and help restore the crustal anomalies better. The inversion results in the eastern Tibetan Plateau and Sichuan basin indicate complex distributions of low-velocity zone in the mid-crust of the eastern Tibetan Plateau and a craton-like basement of the Sichuan basin, which supports the crust channel flow model. Although both the 3-D shear wave velocity model from joint inversion and the individual seismic surface wave inversion can fit the surface wave data almost equally well, the joint inversion model can better match the gravity data We also found that the 3-D model from joint inversion in this study shows similar structural characteristics with the surface wave tomographic model, which indicates the icing on the cake effects of gravity data.

Relativistic ionization dynamics of hydrogenlike ions in strong electromagnetic fields: Generalized pseudospectral method for the time-dependent Dirac equation

We perform a theoretical and computational study of relativistic hydrogenlike ions subject to linearly polarized strong electromagnetic fields. The time-dependent Dirac equation is solved with the help of the generalized pseudospectral method in spherical coordinates. When solving the Dirac equation numerically with basis-set methods, spurious eigenstates usually show up. We suggest a simple transformation of the original discretized Dirac Hamiltonian that removes such states and keeps highly accurate true eigenstates. We calculate the ionization probabilities of the hydrogen atom and hydrogenlike ions ${\mathrm{Ne}}^{9+}$ and ${\mathrm{Ar}}^{17+}$ for various peak field strengths and pulse durations scaled with respect to the nuclear charge of the target. Calculations are performed both within and beyond the dipole approximation. We analyze the nondipole effects and find the region of the Lorentz deflection parameter where the dipole approximation is still applicable.

Optical radiation force on a perfect electromagnetic conductor (PEMC) sphere

This work extends the scope of the previous study on the electromagnetic radiation force on a perfect electromagnetic conductor (PEMC) cylinder [F.G. Mitri, J. Quant. Spectr. Rad. Transfer, 233 (2019) 21–28] to the case of a sphere of arbitrary size, illuminated by linearly-polarized plane progressive waves. In contrast with the PEMC cylinder case, the analysis shows that the effect of the cross-polarized waves compensates that of the co-polarized waves such that the optical radiation force is unaffected by the change of the electromagnetic admittance parameter M of the sphere. The multipole series expansion method in spherical coordinates is used to derive exact mathematical series for the co-polarized and cross-polarized components of the longitudinal radiation force. Numerical results for the radiation force efficiency and its co-polarized and cross-polarized components demonstrate the individual contributions. The results show that the cross-polarized component of the radiation force efficiency can be positive or negative as the dimensionless size ka varies. Moreover, it vanishes for ka ≈ 1.78 regardless of the admittance parameter of the PEMC sphere. Notice that the total force (i.e. the sum of the co-polarized and cross-polarized components) is always repulsive (i.e., positive). It is also verified that the results are in complete agreement with the law of energy conservation applied to scattering, where the extinction and scattering energy efficiencies are equal, and independent of M. Furthermore, an equivalent expression for the longitudinal radiation force efficiency is derived stemming from an analysis of the asymmetry parameter and the law of energy conservation applied to scattering. Additional computations for the co-polarized and cross-polarized components of the scattering asymmetry parameter are presented and discussed. The results are of some importance from the standpoint of the basics of the electromagnetic radiation force theory and related applications in particle manipulation.

Dynamic acoustic tractor beams II: Oscillatory powers, cross sections, and energy efficiencies of a spherical non-viscous liquid droplet

A previous work on dynamic acoustic tractor beams [Mitri, J. Appl. Phys. 117, 094903 (2015)] demonstrated the feasibility of pulling a liquid sphere toward a finite circular piezo-disk transducer and vibrating it around an equilibrium position at the beat frequency of two interfering waves driven at slightly different frequencies. The analysis is extended here to investigate the oscillatory phenomenon from the standpoint of energy conservation applied to scattering. Expressions for dynamic scattering, extinction and absorption powers, and their related cross sections and energy efficiencies are formally defined and derived based on the partial-wave series expansion method in spherical coordinates and the short-term time averaging (STTA) procedure. A circular flat piston transducer insonifying a spherical liquid hexane droplet (with arbitrary radius) centered on the axis of wave propagation is considered. Numerical predictions for dynamic extinction and scattering energy efficiencies illustrate the theory. A percentage error criterion related to extinction and scattering energy efficiencies is defined. Computations are also performed to quantify the numerical inaccuracy introduced by the difference-frequency component of the STTA. This criterion must be always utilized as a benchmark tool to validate dynamic radiation force computations. The results show that the percent error (induced by the difference-frequency component of the STTA process) increases as the dimensionless difference-frequency Δka increases. The present analysis is of some importance to validate dynamic radiation force numerical computations from the standpoint of energy conservation, used in the design and optimization of dynamic/oscillatory acoustic tractor beams and single-beam tweezers.

The Wave Concept Iterative Process Method in Spherical Coordinates

In this paper, we present a global development of the iterative method WCIP in spherical coordinates, also some special parameters that we will interpreted.

Role of excited states in molecular alignment-dependent ionization.

We introduce an ab initio approach and the modified strong-field approximation to investigate the alignment-dependent ionization of H2+(1πu) exposed to different few-cycle laser fields. The ab initio calculations are performed by the B-splines one-center method and the Crank-Nicolson method in spherical coordinates. It is shown that the peak ionization probabilities appear around alignment angles 50° and 40° at the laser intensities 3×1013 W/cm2 and 5×1013 W/cm2, respectively, and the above distinct features come from the resonant excitation of the molecular ion, which is confirmed by calculation including and excluding the state 2σg in the basis expansion. Furthermore, the results obtained by including the state 2σg in the ab initio simulations can be qualitatively reproduced by the modified molecular length gauge strong-field approximation (SFA) taking account of the 1πu and 2σg states simultaneously. Analysis indicates that a part of electron is directly emitted from the 1πu orbital and another portion of electron is released from 2σg orbital and other excited state after the single-photon resonant transition between 1πu and 2σg orbitals.

Lagrange-Mesh Method for Deformed Nuclei With Relativistic Energy Density Functionals

The application of relativistic energy density functionals to the description of nuclei leads to the problem of solving self-consistently a coupled set of equations of motion to determine the nucleon wave functions and meson fields. In this work the Lagrange-mesh method in spherical coordinates is proposed for the numerical calculation. The essential field equations are derived from the relativistic energy density functional and the basic principles of the Lagrange-mesh method are delineated for this particular application. The numerical accuracy is studied for the case of a deformed relativistic harmonic oscillator potential with axial symmetry. Then the method is applied to determine the point matter distributions and deformation parameters of self-conjugate even-even nuclei from 4He to 40Ca.

General optimal shape of plasmonic nano particles based absorption coefficient using a linearization approach

In this paper, the problem of determining the shape of plasmonic nano particle for maximum absorption coefficient in the visible part of light spectrum is considered. In order to determine a general optimal shape, a new linearization solution method in spherical coordinates is presented. Indeed, by using this approach which is based on an embedding process, the solution is determined via an appropriate finite linear programming problem by an iterative method based on particle swarm optimization (PSO) algorithm. This approach in comparison to the other methods has some advantages: the optimization is global and does not need any initial shape and mesh designing. This approach can be useful also in optical applications such as plasmonic nano antenna, plasmonic solar cell and plasmonic sensor. The numerical simulation results also show the efficiency of the proposed method especially in comparison.

Effect of velocity slip on settling of assemblages of spherical particles in power-law liquids at low to moderate Reynolds numbers

Combined effects of the velocity slip, volume fraction of spherical particles and power-law behavior index on the flow and drag phenomena of assemblages of spherical particles in power-law liquids with velocity slip at the interface are numerically investigated. The effects of varying degrees of the velocity slip at interfaces of the fluid–solid particles are incorporated in the solver by making use of a linear slip velocity model. Further, the assemblages of particles are mathematically simplified as clusters of mono-size spherical particles equidistant from each other and the effect of neighboring particles on the target particle nullified at midpoint between the centers of two neighboring spherical particles. Within the combination of these two models, the dimensionless governing conservation equations of mass and momentum for power-law fluids are solved using a simplified marker and cell method in spherical coordinates on a staggered grid arrangement. The convective and non-Newtonian terms of the momentum equation are discretized using quadratic upstream interpolation for convective kinematics scheme and a second-order central differencing scheme, respectively. The presented numerical solver is extensively tested for its reliability via routes of the grid independence and validation with the existing literature counterparts. Furthermore, extensive new results were obtained in the range of dimensionless parameters such as the holdup of spheres, \(\Phi = 0.1{-}0.5\); dimensionless slip number, \(\lambda = 0.01{-}100\); Reynolds number, \(Re = 0.1{-}200\); and power-law behavior index, \(n = 0.6{-}1.6\). Briefly, some results indicate that the drag coefficients of assemblages of spheres decrease with the decreasing slip number and/or decreasing power-law index and/or with the decreasing holdup of the spheres.

Axial acoustic radiation force on rigid oblate and prolate spheroids in Bessel vortex beams of progressive, standing and quasi-standing waves

The analysis using the partial-wave series expansion (PWSE) method in spherical coordinates is extended to evaluate the acoustic radiation force experienced by rigid oblate and prolate spheroids centered on the axis of wave propagation of high-order Bessel vortex beams composed of progressive, standing and quasi-standing waves, respectively. A coupled system of linear equations is derived after applying the Neumann boundary condition for an immovable surface in a non-viscous fluid, and solved numerically by matrix inversion after performing a single numerical integration procedure. The system of linear equations depends on the partial-wave index n and the order of the Bessel vortex beam m using truncated but converging PWSEs in the least-squares sense. Numerical results for the radiation force function, which is the radiation force per unit energy density and unit cross-sectional surface, are computed with particular emphasis on the amplitude ratio describing the transition from the progressive to the pure standing waves cases, the aspect ratio (i.e., the ratio of the major axis over the minor axis of the spheroid), the half-cone angle and order of the Bessel vortex beam, as well as the dimensionless size parameter. A generalized expression for the radiation force function is derived for cases encompassing the progressive, standing and quasi-standing waves of Bessel vortex beams. This expression can be reduced to other types of beams/waves such as the zeroth-order Bessel non-vortex beam or the infinite plane wave case by appropriate selection of the beam parameters. The results for progressive waves reveal a tractor beam behavior, characterized by the emergence of an attractive pulling force acting in opposite direction of wave propagation. Moreover, the transition to the quasi-standing and pure standing wave cases shows the acoustical tweezers behavior in dual-beam Bessel vortex beams. Applications in acoustic levitation, particle manipulation and acousto-fluidics would benefit from the results of the present investigation.

Axisymmetric scattering of an acoustical Bessel beam by a rigid fixed spheroid.

Based on the partial-wave series expansion (PWSE) method in spherical coordinates, a formal analytical solution for the acoustic scattering of a zeroth-order Bessel acoustic beam centered on a rigid fixed (oblate or prolate) spheroid is provided. The unknown scattering coefficients of the spheroid are determined by solving a system of linear equations derived for the Neumann boundary condition. Numerical results for the modulus of the backscattered pressure (θ = π) in the near field and the backscattering form function in the far field for both prolate and oblate spheroids are presented and discussed, with particular emphasis on the aspect ratio (i.e., the ratio of the major axis over the minor axis of the spheroid), the half-cone angle of the Bessel beam, and the dimensionless frequency. The plots display periodic oscillations (versus the dimensionless frequency) because of the interference of specularly reflected waves in the backscattering direction with circumferential Franz' waves circumnavigating the surface of the spheroid in the surrounding fluid. Moreover, the 3-D directivity patterns illustrate the near- and far-field axisymmetric scattering. Investigations in underwater acoustics, particle levitation, scattering, and the detection of submerged elongated objects and other related applications utilizing Bessel waves would benefit from the results of the present study.

Acoustic radiation force on oblate and prolate spheroids in Bessel beams

A formal theoretical analysis is developed using the partial-wave series expansion (PWSE) method in spherical coordinates, which allows accurate evaluation of the acoustic radiation force (ARF) of a Bessel beam incident upon a rigid oblate or prolate spheroid, centered on its axis of wave propagation. The scattering coefficients for either the oblate or the prolate spheroid are determined based on Neumann’s boundary condition for a rigid immovable surface, and used to compute the ARF function, which is the radiation force per unit characteristic energy density and surface cross-section of the spheroid. Numerical results are performed with particular emphasis on the waves’ amplitude ratio describing the evolution from progressive (traveling), quasi-standing and pure Bessel standing waves, the half-cone angle β of the beam, and the aspect ratio (i.e. the distance from the center to pole along the symmetry axis a divided by the equatorial radius b) of the spheroid. Unlike the results obtained in the Rayleigh limit (i.e., ka≪1, where k is the wavenumber of the incident illuminating waves), calculations for the ARF functions for progressive, quasi-standing and standing Bessel waves for ka>1, generally reveal larger amplitudes for an oblate rather than a prolate spheroid having the same surface cross-section. Exceptions are also noted for Bessel beams with a large half-cone angle. Potential applications are in acoustic levitation of dense spheroids in air, particle dynamics, and other related research.

Implicit dual-time stepping method for a solar wind model in spherical coordinates

In this paper, an implicit dual-time stepping scheme based on the finite volume method in spherical coordinates with a six-component grid system is developed to model the steady state solar wind. By adding a pseudo-time derivative to the magnetohydrodynamics equations for the solar wind plasma, the governing equations are solved implicitly at each physical time step by advancing in pseudo time. As a validation, ambient solar wind for Carrington rotation 2048 has been studied. Numerical tests with different Courant factors show its capability of producing structured solar wind and that the physical time step can be enlarged to be one hundred times that of the original one. Our numerical results have demonstrated overall good agreements with the observations.