Ellipsoidal Coordinate System Research Articles

A semi-analytical approach for the computation of ellipsoidal harmonics

It is known that the separation of variables of the Laplace’s equation in the ellipsoidal coordinate system leads to the Lamé equation, with solutions the Lamé functions of the first and second kind, which cannot be produced analytically for a degree higher than the seventh. These compose the corresponding ellipsoidal harmonic eigenfunctions, which are necessary for the complete description and solution of physical problems in three-dimensional domains with ellipsoidal boundaries. In this work, an efficient algorithm for calculating the Lamé functions of any degree and order is proposed, adopting compact and dimensionless expressions. The developed technique has been successfully applied to important applications in electrostatics and biophysics with excellent results. In addition, important complementary issues are analyzed, regarding the derivatives of the Lamé functions, the calculation of surface integrals, the Lamé functions of the second kind and at the same time we provide a closed-form expression for the normalization constants in terms of the polynomial coefficients of the Lamé functions.

Mathematical Model of Satellite-Inertial Mobile Computational Gravimetry

The article proposes a mathematical model of a hybrid system installed on board of a moving object and represented by an inertial sensor of the vector of specific forces — a three-component newtonometer with orthogonal sensitivity axes and a network of receivers of a navigation satellite system (HSS). The purpose of this hybrid system is the temporal estimations of the nearEarth gravitational field on the trajectory of the object. Within the Newtonian mechanics the possibility of choosing an inertial reference system with a beginning at the center of mass of the Earth is assumed; сomplementary to the PZ-90 (Russia) and WGS-84 (USA) standards, the following are introduced: 1) an ellipsoidal (geodesic) coordinate system, rigidly connected with the solid Earth; 2) two movable accompanying right-angled trihedrals with a common origin as a point whose movement forms the trajectory of the object; the instrument one, is rigidly connected with the object as a solid body, and is thus freely oriented, other one are geographic, and permanently oriented to the cardinal points (East, North, Zenith). In the presentation of kinematics, attention is drawn to the fact that the variability of the absolute linear velocity in inertial space explained by the motion of an object relative to solid Earth and its own rotation is characterized by rotational vectors of relative and portable velocities which identify the vectors of relative and portable angular rotational velocities of the undeformable geographic trihedron and are represented projections on its axis. The causality of the trajectory is determined by Newton’s second law; in projections on the axis of the moving geographic trihedron, component-wise recording of the equations of dynamics is performed, which are resolved with respect to the components of the strength of the gravitational field. It is noted that the previous article by the authors is devoted to the problem of a high-precision estimate of the kinematic parameters of motion equations. In such a context it is indicated that direct calculation of the gravitational field strength requires the convertting of Newtonometer measurements from the instrument trihedron to the geographic one. The required linear transformation for this is constructed considering the possibilities of multiposition of the NSS object. A computational experiment is described and some of its results are presented.

Perturbative Approach for the Analysis of Charge Distribution on Arbitrarily Shaped Conductors

In this paper, we analyze the electrostatic charge distribution on arbitrarily shaped conductor surfaces. Following a perturbative approach, we derive an approximate analytical formulation of the problem. We start from the known case of a conducting ellipsoid, we adopt a deformed ellipsoidal coordinate system, and we search for the zero-order approximated solution of the problem. We also focus on arbitrary-shaped thin foils, showing that the charge density is divergent on their borders. We then define the applicability range of the proposed approach expressing the contour equation as the Fourier series. Finally, we present a detailed error analysis for several polygonal contours, comparing the analytical results with those obtained via a numerical analysis based on the Finite Element Methods (FEM).

Low‐frequency dipolar electromagnetic scattering by a solid ellipsoid in lossless environment

AbstractElectromagnetic wave scattering phenomena for target identification are important in many applications related to fundamental science and engineering. Here, we present an analytical formulation for the calculation of the magnetic and electric fields that scatter off a highly conductive ellipsoidal body, located within an otherwise homogeneous and isotropic lossless medium. The primary excitation source assumes a time‐harmonic magnetic dipole, precisely fixed and arbitrarily orientated that operates at low frequencies and produces the incident fields. The scattering problem itself is modeled with respect to rigorous expansions of the electromagnetic fields at the low‐frequency regime in terms of positive integral powers of the real wave number of the ambient. Obviously, the Rayleigh static term and a few dynamic terms are sufficient for the purpose of the present work, as the additional terms are neglected due to their minor contribution. Therein, the classical Maxwell's theory is suitably modified, leading to intertwined either Laplace's or Poisson's equations, accompanied by the impenetrable boundary conditions for the total fields and the limiting behavior at infinity. On the other hand, the complete spatial anisotropy of the three‐dimensional space is secured via the introduction of the genuine ellipsoidal coordinate system, being appropriate for tackling incrementally such scattering boundary value problems. The nonaxisymmetric fields are obtained via infinite series expansions in terms of ellipsoidal harmonic eigenfunctions, providing handy closed‐form solutions in a compact fashion, whose validity is verified by a straightforward reduction to simpler geometries of the metal object. The main idea is to demonstrate an efficient methodology, according to which the constructed analytical formulae can offer the appropriate environment for a fast numerical estimation of the scattered electromagnetic fields that could be useful for real data inversion.

Steady flow in a rapidly rotating spheroid with weak precession: I

The celebrated Poincaré (1910) steady solution, the uniform vorticity flow in a precessing spheroid, which is realized in the inviscid interior region outside the boundary layer in the strong spin and weak precession limit, is derived, as a unique solution analytically without any prior assumption on the spatial structure, and its singular behavior for a sphere is resolved. Assuming that the spin and precession axes are respectively parallel and perpendicular to the symmetry axis of the spheroid, we denote the spin and precession angular velocities by and respectively, and define the Reynolds number Re = and the Poincaré number , where a is the equatorial radius and ν is the kinematic viscosity of fluid. An ellipsoidal coordinate system is introduced, which makes it possible to express the solution explicitly in a separable form. It is shown that only the uniform vorticity flow can develop as a steady flow in the inviscid limit. The vorticity, relative to the spinning spheroid, is in magnitude and points to or against the z-axis for an oblate (c < 1) or a prolate (c > 1) spheroid. This expression of vorticity magnitude diverges to infinity for a sphere (c = 1), where the inviscid solution is degenerate. This apparent singular behavior of the vorticity is resolved by introducing viscosity and analyzing the flow by invoking the torque balance equation for a spheroid close to a sphere in the limit , where . It is found that the vorticity vector changes the orientation smoothly by 180° over and that the parallel and the perpendicular (to the spin axis) components of vorticity is expressed, up to the nontrivial leading order, as the twice of −0.7876Po2/[(2.620δ)2 + (0.2585δ + 1 − c)2] and , respectively. The latter is included in Busse ()'s formula, whereas the former is new. An analysis of the same flow as the present paper from a different approach was performed recently by Zhang et al (). Unfortunately, however, an inconsistency is found in their results.

Model of Hybrid Satellite-Inertial Navigation System with Partial Structure

A mathematical model of a hybrid navigation system (GNS) consisting of a three-component block of linear newtonometers (accelerometers) physically simulating a vector-based measurer of non-gravitational nature forces and on-board GLONASS receivers that positioning a moving object in an ellipsoidal coordinate system is presented and investigated. The absence of a gyroscopic angular velocity sensors unit, traditional for the classical schemes of the inertial navigation method, and the presence of no more than two onboard satellite positioning devices (receivers) make it possible to characterize the considered GNS as a partial structure system. As a basic element of a mathematical model for estimating linear and angular parameters of an object’s motion, the developed procedure of multiple numerical differentiation of temporal data acquired from on-board satellite receivers, which functions stably irrespective of the magnitude of the discretization step of the problem in time, was used. The developed GNS makes it possible to qualitatively evaluate both the trajectory parameters (location, velocity, acceleration and forces causing the trajectory) as well as the parameters of the spatial orientation of the object (Euler-Krylov angles and its derivatives) with a two-positioning technique. The results of the computational experiment are given. The field of application of the research results is numerical-analytical planning of trajectories, determination of motion parameters and control of moving objects for various purposes and basing.

The effective ellipsoid: a method for calculating the permittivity of composites with multilayer ellipsoids

Abstract In order to calculate the effective permittivity of a mixture with multilayer ellipsoids, this paper presents a self-consistent approximation (SCA) on the basis of the Bruggeman’s analytical model. The effective permittivity of a mixture with aligned multilayer ellipsoids is derived directly from the linear system of equations, which are built using the boundary condition of the electric field on the confocal ellipsoidal interface in the ellipsoidal coordinate system. Furthermore, for a mixture with multilayer ellipsoids oriented randomly, an effective ellipsoid is introduced to substitute the original multilayer ellipsoid, and the permittivity of the effective ellipsoid is derived by jointly solving the two linear systems of equations for the situation of the original multilayer ellipsoid and that of the effective ellipsoid, then the effective permittivity of the mixture can be calculated by the existing Maxwell-Garnett formula. After comparisons, it is revealed that there is a good agreement between this SCA method and existing theories.

Theoretical development of elliptic cross-sectional hyperboloidal harmonics and their application to electrostatics

The analytic computation of electric and magnetic fields near corners and edges is important in many applications related to science and engineering. However, such complicated situations are hard to deal with, since they accumulate charges and consequently they mathematically represent singularities. In order to model this singular behavior, we introduce a novel method, which is related to the geometry and the analysis of the ellipsoidal coordinate system. Indeed, adopting the benefits of the corresponding coordinate surfaces, we use a general non-circular double cone, being the asymptote of a two-sided hyperboloid of two sheets with elliptic cross section, which matches almost perfectly the particular physics and captures the corresponding essential features in a fully three-dimensional fashion. To this end, our analytical technique employs the ellipsoidal geometry and adapts the ellipsoidal functions (solutions of the well-known Lamé equation) so as to construct a new set of the so-called elliptic cross-sectional hyperboloidal harmonics, supplemented by the appropriate orthogonality rules on every constant coordinate surface. By first recollecting the key results of the coordinate system and the related potential functions, including the indispensable orthogonality results, we demonstrate our method to the solution of two boundary value problems in electrostatics. Both refer to a non-penetrable two-hyperboloid of elliptic cross section and its double-cone limit, the first one being charged and the second one scattering off a plane wave. Closed form expressions are derived for the related fields, while the already known formulae from the literature are readily recovered, all cases being followed by the appropriate numerical implementation.

Точность определения координат мобильного сканирующего комплекса в зависимости от формы используемой системы координат

Inertial coordinate system and geodetic (terrestrial) coordinate system are used in processing of results of topographic survey, carried out with a mobile scanning complex. Mobile scanning complex geodetic coordinates, in turn, are presented in geodetic three-dimensional rectangular coordinate system form, in geodetic ellipsoidal coordinate system form and in the form of coordinates on a geodetic projection plane. The results of research, carried out earlier [4–7], suggest that the coordinate transformation on large areas distorts geodetic points coordinates. The article presents the results of similar investigations, but applied to a local area, limited by a mobile scanning complex surveying area. The accuracy of the mobile scanning complex coordinates is characterized by the mobile scanning complex coordinates errors cofactor matrix. It turned out that the local site sequential coordinate transformation procedure from one coordinate system to another coordinate system does not introduce any distortion into the mobile scanning complex coordinates.

Revisiting the Low-Frequency Dipolar Perturbation by an Impenetrable Ellipsoid in a Conductive Surrounding

This contribution deals with the scattering by a metallic ellipsoidal target, embedded in a homogeneous conductive medium, which is stimulated when a 3D time-harmonic magnetic dipole is operating at the low-frequency realm. The incident, the scattered, and the total three-dimensional electromagnetic fields, which satisfy Maxwell’s equations, yield low-frequency expansions in terms of positive integral powers of the complex-valued wave number of the exterior medium. We preserve the static Rayleigh approximation and the first three dynamic terms, while the additional terms of minor contribution are neglected. The Maxwell-type problem is transformed into intertwined potential-type boundary value problems with impenetrable boundary conditions, whereas the environment of a genuine ellipsoidal coordinate system provides the necessary setting for tackling such problems in anisotropic space. The fields are represented via nonaxisymmetric infinite series expansions in terms of harmonic eigenfunctions, affiliated with the ellipsoidal system, obtaining analytical closed-form solutions in a compact fashion. Until nowadays, such problems were attacked by using the very few ellipsoidal harmonics exhibiting an analytical form. In the present article, we address this issue by incorporating the full series expansion of the potentials and utilizing the entire subspace of ellipsoidal harmonic eigenfunctions.

Nonsingular expansions of the gravity potential and its derivatives at satellite altitudes in the ellipsoidal coordinate system

The series in ellipsoidal harmonics for derivatives of the Earth’s gravity potential are used only on the reference ellipsoid enveloping the Earth due to their very complex mathematical structure. In the current study, the series in ellipsoidal harmonics are constructed for first- and second-order derivatives of the potential at satellite altitudes; their structure is similar to the series on the reference ellipsoid. The point P is chosen at a random satellite altitude; then, the ellipsoid of revolution is described, which passes through this point and is confocal to the reference ellipsoid. An object-centered coordinate system with the origin at the point P is considered. Using a sequence of transformations, the nonsingular series in ellipsoidal harmonics is constructed for first and second derivatives of the potential in the object-centered coordinate system. These series can be applied to develop a model of the Earth’s potential, based on combined use of surface gravitational force measurements, data on the satellite orbital position, its acceleration, or measurements of the gravitational force gradients of the first and second order. The technique is applicable to any other planet of the Solar System.

A GPS Positioning Method for Underwater Targets Using Dual Acoustic Devices

Since the existing underwater GPS positioning technology can only position underwater targets equipped with acoustic transponders, this study proposed a method of positioning underwater combining targets based on an ultra short baseline forwardlooking sonar acoustic device. The mathematical model of the obtaining longitude and latitude coordinates of the underwater target in the WGS84 ellipsoidal coordinate system was established. A group of data was given based on the measuring state, and Monte Carlo simulation was carried out to study the effects of the random errors in the indirect measurements in the algorithm on the positioning accuracy and the combined effect of the random errors in the indirect measurements in the algorithm on the positioning accuracy at different geological locations. The preset target was measured in the offshore waters of Dalian, confirming the validity of the proposed algorithm.

The aging problem of twins in the ellipsoidal coordinates

In this paper, we have tried to explore the twin paradox in the ellipsoidal coordinate system. We have investigated that the traveling twin travels away from the gravitational source in the first case while in the other it moves towards the source. We then tried to compare it with the special relativistic case where the static twin ages faster as compared to his traveling counterpart.

Comments on Charron et al.'s three recent articles on deriving dynamically consistent equation sets

Charron and co‐authors propose the preservation of tensor covariance as the criterion for dynamically consistent approximation of the governing equations for global atmospheric and oceanographic flow. Earlier criteria, based on vector differential analysis, involve the preservation of individual conservation properties. Here, the two approaches are compared for the interested, but possibly perplexed, reader. Differences regarding terminology and some consistency issues relating to certain ‘quasi‐shallow’ models are discussed. However, the overall similarity of the results of the two approaches and of the variational approach using Hamilton's principle of least action is evident. Relevant aspects of an approximate global model that uses an analytically tractable ellipsoidal geopotential coordinate system are also discussed.

Regularization of the expression for a gradient of Earth’s gravity potential in the local ellipsoidal coordinate system

The results of further development of the theory of modeling the Earth’s gravitational field by series in ellipsoidal harmonics, which better approximate the Earth’s gravity potential as compared with traditional series of spherical harmonics, are presented. In the preceding study of the authors, new expansions with respect to ellipsoidal harmonics were constructed for the gravity potential and its derivatives. They depend on the hypergeometric Gauss series having the higher rate of convergence in comparison to the previously used series of other authors. In this work, the further improvement of convergence of the hypergeometric series is performed, and based on the results, the expression for the Earth’s gravity potential on the surface of the reference ellipsoid is constructed in the local north-oriented ellipsoidal coordinate system. This expression has no analytical singularities in contrast to previously known expansions that cannot be applied in polar regions on the Earth’s surface and in outer space. The new expression for the potential gradient may be used for constructing global models of the Earth’s gravitational field which take into account the data gathered in the polar regions.

Complete solution to elastic field around an elliptic punch

AbstractThe problem of a rigid flat indentor resting on a transversely isotropic medium under external forces is reexamined. The contacting surface is an elliptic region. Both normal and tangential indentation are considered. Unlike the usual approach, we do not use ellipsoidal co‐ordinate system. The integral equations for prescribed normal and tangential indentation are first solved in cartesian co‐ordinate system for the unknown stress on the contact plane. A series of transformation and known results of integrals involving Bessel functions are then used to derive exact and complete solutions to the displacement and stress inside the half space. The result is exact in the sense that it involves a finite integration which must be evaluated numerically. Such a complete solution to the elliptic punch problem in cartesian co‐ordinate has not been obtained before. We also obtain the asymptotic form of the field near the border of the contact region. The corresponding result for an isotropic medium is obtained by a limiting process.

Low-frequency dipolar excitation of a perfect ellipsoidal conductor

This paper deals with the scattering by a perfectly conductive ellipsoid under magnetic dipolar excitation at low frequency. The source and the ellipsoid are embedded in an infinite homogeneous conducting ground. The main idea is to obtain an analytical solution of this scattering problem in order to have a fast numerical estimation of the scattered field that can be useful for real data inversion. Maxwell equations and boundary conditions, describing the problem, are firstly expanded using low-frequency expansion of the fields up to order three. It will be shown that fields have to be found incrementally. The static one (term of order zero) satisfies the Laplace equation. The next non-zero term (term of order two) is more complicated and satisfies the Poisson equation. The order-three term is independent of the previous ones and is described by the Laplace equation. They constitute three different scattering problems that are solved using the separated variables method in the ellipsoidal coordinate system. Solutions are written as expansions on the few analytically known scalar ellipsoidal harmonics. Details are given to explain how those solutions are achieved with an example of numerical results.

Two-center overlap integrals, three dimensional adaptive integration, and prolate ellipsoidal coordinates

Numerical, adaptive algorithm evaluating the overlap integrals between the Numerical Type Orbitals (NTO) is presented. The described algorithm exploits the properties of the prolate ellipsoidal coordinates, which are the natural choice for two-center overlap integrals. The algorithm is designed for numerical atomic orbitals with the finite support. Since the cusp singularity of the atomic orbitals vanish in the prolate ellipsoidal coordinate system, the adaptive integration algorithm in \({\mathbb{R}^3}\) generates small number of subdivisions. The efficiency and reliability of the algorithm is demonstrated for the overlap integrals evaluated for the selected pairs of Slater Type Orbitals (STO).

Electromagnetic Induction From Highly Permeable and Conductive Ellipsoids Under Arbitrary Excitation: Application to the Detection of Unexploded Ordnances

The secondary field produced by 3-D highly permeable and conductive objects is computed in the electromagnetic induction regime, with the purpose of modeling unexploded ordnances (UXOs) and surrounding clutter. The analytical formulation is based on the ellipsoidal coordinate system that is able to model real 3-D geometries as opposed to bodies of revolutions like within a spheroidal approach. At the frequencies of interest (tens of hertz to hundreds of kilohertz), conduction currents in the soil are negligible, and the fields are computed in the magnetoquasistatic regime based on the Laplace equation. Inside the objects, where the wave equation governs the field distribution, the currents are assumed to have a small penetration depth, allowing for the analytical simplification of the field components, which become decoupled at the surface. This approximation, which is valid across the entire frequency spectrum because of the high permeability and conductivity, avoids the necessity of using ellipsoidal wave functions and results in a considerable saving of computational time. Numerical results favorably compare with numerical and experimental data, which proves the usefulness of our method to model UXOs in clutter-contaminated soils. Finally, the optimization approach used to match our numerical predictions with experimental data demonstrates the possibility of remotely inferring the material properties of objects.

The Internal and External Metrics of a Rotating Ellipsoid Under Post-Newtonianian Approximation

Under the post-Newtonian approximation, the internal and external metrics of a rigidly rotating oblate spheroid filled by a uniform and incompressible perfect fluid are obtained. And the analytic solutions of post-Newtonian metric components are derived by using the series expansion in an ellipsoidal coordinate system. For this specific problem, there are only finite terms remaining in the series expansion, so the obtained results can be used to study particle motion under these metrics.