Elliptic Integrals Research Articles

Analytical Method for Groundwater Seepage through and Beneath a Fully Penetrating Cut-off Wall Considering Effects of Wall Permeability and Thickness

A fully penetrating cut-off wall is a vertical seepage barrier that fully penetrates an aquifer and is embedded in an underlying aquitard to a certain depth. Groundwater seepage with this type of wall occurs through three paths: leakage through the body of the wall in the aquifer, leakage through the body of the wall embedded in the aquitard, and seepage under the wall. Seepage through the first path can be simply treated as one-dimensional flow. However, due to the mutual influence of seepage through the latter two paths, the seepage problem is complicated and still needs to be studied. An analytical method is proposed to solve this problem in this study. Mathematic expressions for flow rate and head value are obtained by superposition of drawdowns of two exact models, namely, the model with only leakage through the wall body and the model with only seepage under the wall, respectively. Exact solutions are quoted or derived for the exact models, but they involve Legendre’s elliptic integrals of the first and third kinds. To facilitate an engineering application, approximate models of the exact models are introduced and their solutions are applied to the analytical formulas. The accuracy and applicability of the proposed method are verified compared with the numerical method. The proposed method provides a simple but effective method for quickly estimating the quantity of seepage in the aquitard (including leakage through the wall body and seepage under the wall) when simultaneously considering the effects of wall permeability and thickness.

Magnetic stiffness calculation for the corresponding force between two current-carrying circular filaments arbitrarily oriented in the space

In this article, sets of analytical formulas for calculation of nine components of magnetic stiffness of corresponding force arising between two current-carrying circular filaments arbitrarily oriented in the space are derived by using Babic’s method and the method of mutual inductance (Kalantarov–Zeitlin’s method). Formulas are presented through integral expressions, whose kernel function is expressed in terms of the elliptic integrals of the first and second kinds. Also, we obtained an additional set of expressions for calculation of components of magnetic stiffness by means of differentiation of Grover’s formula of the mutual inductance between two circular filaments with respect to appropriate coordinates. The derived sets of formulas were mutually validated and results of calculation of components of magnetic stiffness agree well to each other.

A novel local ellipse method for ellipse-based tangential contact model applied to wheel-rail contact

ABSTRACT In order to simplify wheel-rail rolling contact for engineering computation, it usually applies an ellipse-based tangential contact model to non-elliptical contact patch. This paper proposes a novel local ellipse method to reach this transition due to contact stiffness difference in elliptic and non-elliptical contact. The method is similar to STRIPES that regards the contact patch as some strips along the rolling direction and locating at the center of their respective virtual ellipses. However, it provides an alternative way for the determination of the local virtual ellipse using elliptic integral, normal contact pressure and longitudinal contact boundary. One novel aspect is the lateral curvature at each strip is solely determined and needs no additional modification. This method is combined with FaStrip and evaluated by CONTACT. It thus provides an efficient pre-process of simplified tangential contact model to be applied to non-elliptical contact in vehicle-track dynamics modelling and wheel-rail damage prediction.

Geodesic motion in Euclidean Schwarzschild geometry

This paper performs a systematic investigation of geodesic motion in Euclidean Schwarzschild geometry, which is studied in the equatorial plane. The explicit form of geodesic motion is obtained in terms of incomplete elliptic integrals of first, second and third kind. No elliptic-like orbits exist in Euclidean Schwarzschild geometry, unlike the corresponding Lorentzian pattern. Among unbounded orbits, only unbounded first-kind orbits are allowed, unlike general relativity where unbounded second-kind orbits are always allowed.

Convexity and Monotonicity Involving the Complete Elliptic Integral of the First Kind

Convexity and Monotonicity Involving the Complete Elliptic Integral of the First Kind

New analysis and numerical values for the classical dam problem

The classical, two dimensional, dam problem was solved analytically and independently by Davison in 1932 and Hamel in 1934 and involved double integrals. In 1938 Polubarinova-Kochina produced solutions involving single integrals and in 1940 showed that the Davison-Hamel solutions could be reduced to ones obtained by her method. Missing from the conversion was a linkage between scale constants which is given in this paper. Initial calculations by these authors were difficult because of evaluations of elliptic integrals and numerical quadratures. As a free boundary problem it is now a test case for a multitude of numerical methods. Surveying analytical and numerical approaches, it is found that both are limited in accuracies and are overwhelmingly concerned with the location of the free boundary and its seepage point. Analytical expressions are here given for the gradients near the three critical points of free boundary entry and exit and the intersection of seepage face and tailwater. New numerical results to six decimal places are given for tables and contours on and within the dam boundary for potential and stream functions and orthogonal potential gradients.

Computed multivalues of AGM reveal periodicities of inverse functions

The article shows how two choices are possible whenever computing the geometric mean, and the repetition of this process can in general yield 2-to-the power N different values when the choices are compounded in the first N steps of evaluation of the arithmeticgeometric mean. This happens not only in the simple AGM involved in the computation of the complete elliptic integral of the first kind, but also in analogous methods for the computation of the complete and incomplete elliptic integrals of the first and second kind.

Off-axis electric field due to cylindrical geometries of charge distribution

Off-axis electric field due to cylindrical distribution of charge is studied in various geometries, including solid cylinder, cylindrical shell, disk, and ring. The electric fields of the charged solid cylinder and cylindrical shell can be found by numerical calculations, but closed forms expressed in terms of elliptic integrals are derived for the charged disk and ring. By investigating the radial and axial components of the electric fields systematically we reveal many interesting and new features that are implicit or unseen in the ordinary approach based on the cylindrical symmetry.

Matrix Green's function solution of closed-form singularity for functionally graded and transversely isotropic materials under circular ring force vector

The paper develops the Green's function solution in matrix form for functionally graded and transversely isotropic material (FGTIM) due to the loading of a circular ring force vector. The force vector is concentrated along a horizontal circular ring and acts at any location in the interior of the FGTIM material occupying either a fullspace or a halfspace. Without loss of generality, the variations of the FGTIM's five elastic parameters are expressed as five arbitrary step functions with the depth as the independent variable. The two-dimensional Fourier integral transforms in the cylindrical coordinate system are used for the mathematical formulation and derivation in matrix form. The Green's function solution of the displacement and stress fields is explicitly expressed in the matrix form in terms of classical improper Hankel transform integrals of the orders of 0 to 3. The kernel functions of the Hankel transform integrals are explicitly expressed in the forms of backward transfer matrix. Their mathematical properties are analyzed analytically and numerically. The singular terms associated with the improper Hankel transform integrals are analytically isolated and expressed in the exact closed-form in terms of the complete elliptic integrals of the first, second and third kind. Such closed-form singular terms can be expressed as the matrix Green's function solution for the corresponding bi-material due to the same circular ring force vector. Numerical results show that the computation of the matrix Green's function solution can be achieved with high accuracy and efficiency and the heterogeneity and anisotropy can have significant effects on the elastic fields in the FGTIM induced by the circular ring force vector.

Probability of Occurrence of Some Planar Random Quasi-homogeneous Vector Fields

The objective of this work is the study of the probability of occurrence of phase portraits in a family of planar quasi-homogeneous vector fields of quasi degree q, that is a natural extension of planar linear vector fields, which correspond to q=1. We obtain the exact values of the corresponding probabilities in terms of a simple one-variable definite integral that only depends on q. This integral is explicitly computable in the linear case, recovering known results, and it can be expressed in terms of either complete elliptic integrals or of generalized hypergeometric functions in the non-linear one. Moreover, it appears a remarkable phenomenon when q is even: the probability to have a center is positive, in contrast with what happens in the linear case, or also when q is odd, where this probability is zero.

A generic inequality for basic special functions

We introduce a generic inequality of the form and show that many special functions such as gamma and polygamma functions, Riemann zeta function, beta and incomplete beta functions, Gauss and confluent hypergeometric functions, elliptic integrals and error function satisfy such type of the inequality. We also obtain new inequalities of this type for inverse trigonometric functions and .

Analytical study of gravitational lensing in Kerr-Newman black-bounce spacetime

We investigate the equatorial deflection angle of light rays propagating in Kerr-Newman black-bounce spacetime. Furthermore, we analyze the light ray trajectories and derive a closed-form formula for deflection angle in terms of elliptic integrals. The deflection angle increases with the decrease of charge and regularisation parameter for a particular impact parameter. We also study the strong field limit of the deflection angle. Using this strong deflection angle formula and lens equation, we find the radius of the first Einstein ring and study its dependence on the charge and the regularisation parameter. We demonstrate that the charge has a robust effect on the size of the Einstein rings, but the effect of the regularization parameter on the ring size is negligible. We also investigate the non-equatorial lensing and the caustic structures for small polar inclination, and the same observations appear to hold. These results directly affect the observational appearance of the Kerr-Newman black-bounce.

Null and timelike geodesics in the Kerr-Newman black hole exterior

We study the null and time-like geodesics of the light and the neutral particles respectively in the exterior of Kerr-Newman black holes. The geodesic equations are known to be written as a set of first-order differential equations in Mino time from which the angular and radial potentials can be defined. We classify the roots for both potentials, and mainly focus on those of the radial potential with an emphasis on the effect from the charge of the black holes. We then obtain the solutions of the trajectories in terms of the elliptical integrals and the Jacobian elliptic functions for both null and time-like geodesics, which are manifestly real functions of the Mino time that the initial conditions can be explicitly specified. We also describe the details of how to reduce those solutions into the cases of the spherical orbits. The effect of the black hole's charge decreases the radii of the spherical motion of the light and the particle for both direct and retrograde motions. In particular, we focus on the light/particle boomerang of the spherical orbits due to the frame dragging from the back hole's spin with the effect from the charge of the black hole. To sustain the change of the azimuthal angle of the light rays, say for example $\Delta \phi=\pi$ during the whole trip, the presence of the black hole's charge decreases the radius of the orbit and consequently reduces the needed values of the black hole's spin. As for the particle boomerang, the particle's inertia renders smaller change of the angle $\Delta \phi$ as compared with the light boomerang. Moreover, the black hole's charge also results in the smaller angle change $\Delta \phi$ of the particle than that in the Kerr case. The implications of the obtained results to observations are discussed.

Analytic solution for perturbed Keplerian motion under small acceleration using averaging theory

A novel approach is developed for analytic orbit propagation based on asymptotic expansion with respect to a small perturbative acceleration. The method improves upon existing first order asymptotic expansions by leveraging on linear systems and averaging theories. The solution starts with the linearization of Gauss planetary equations with respect to both the small perturbation and the six orbital elements. Then, an approximate solution is obtained in terms of secular and short period components. The method is tested on a low-thrust maneuver scenario consisting of a Keplerian orbit perturbed by a constant tangential acceleration, for which a solution can be obtained in terms of elliptic integrals. Results show that the positional propagation error is about one order of magnitude smaller with respect to state-of-the-art methods. The position accuracy for a LEO orbit, apart from pathological cases, is typically in the range of tens of meters for a tangential acceleration of 10−7 km/s2 after 5 orbital periods propagation.

Applications of Caputo operators in the evaluation of Clebsch–Gordan-type multiple elliptic integrals

We apply a method of semi-integration by parts (SIBP) that we had previously derived and formulated using the semi-derivative and semi-primitive operators. We obtain many new results on integrals involving the complete elliptic integrals as integrand factors, building on the work of Glasser, Cantarini, Wan, and Zhou. Our main results have not appeared in past literature concerning integrals involving products of the complete elliptic integral(s) of the first and/or second kinds. Furthermore, many previously known identities for such integrals are special cases of our SIBP identity for analytic functions.

λ‐spheres as a new reference model for the geoid

We propose here a new reference model approximating the actual shape of the Earth. It is based on ‐spheres as an alternative to the rotational ellipsoids. Relying on quite convenient parametrization of the ‐spheres, we have obtained the concise expression for its ‐coordinate as a combination of the incomplete elliptic integrals of all three kinds. Next, we have derived the geodesic equations on the ‐spheres and solved them in the analytical form using the mechanical method developed in our previous papers. As a result, three kinds of geodesics have been obtained for different values of the integration constant , that is, the meridians, the Equator, and the slant geodesics. We have also found the conditions that should be fulfilled in order to obtain a closed slant geodesics, whereas the meridians and the Equator are closed by construction. Finally, we have presented the comparison of the newly proposed ‐sphere's model of the geoid with the most popular ellipsoidal reference models.

To calculation of full elliptic integrals third sortsin applied researches

Introduction. When carrying out theoretical and applied researches on modeling of hydrogeophysical filtrational processes and other potential streams quite often there is a need of analytical determination of values of “impossible” elliptic integrals that is connected with labor-consuming mathematical calculations, cross nonlinear interpolating of these special tables and schedules, etc. Use at the solution of applied tasks for their definition of computer programs yields only discrete results in “figures” (but not in “symbols”) without a possibility of their subsequent transformations for receiving the common analytical decision in elementary functions.
 
 Materials and methods. Therefore, despite existence of various computing programs, analytical decisions are demanded so far as they have great opportunities for identification of internal physical relationships of cause and effect in solvable tasks and the subsequent their use in mathematical transformations.
 
 Results. In work estimated dependences in elementary functions by determination of values of full elliptic integral 3rd sorts with a sufficient accuracy (~ << 2–4 %) for carrying out applied (and theoretical) researches are offered. 
 
 Conclusions. L. Milne-Thomson is brought comparison of the received results with exact data of the Mathcad program for a wide interval of changes of parameter of integral and theoretical decisions.

Approximations related to the complete p-elliptic integrals

Abstract In this paper, the authors present some monotonicity properties for certain functions involving the complete p-elliptic integrals of the first and second kinds, by showing the monotonicity and concavity-convexity properties of certain combinations defined in terms of K p {{\mathscr{K}}}_{p} , E p {{\mathscr{E}}}_{p} and the inverse hyperbolic tangent arth p {{\rm{arth}}}_{p} , which is of importance in the computation of the generalized pi and in the elementary proof of Ramanujan’s cubic transformation. By these results, several well-known results for the classical complete elliptic integrals including its bounds and logarithmic inequalities are remarkably improved.

About the structure of the discrete and continuous Eringen’s nonlocal elastica

The Eringen’s nonlocal elastica equation does not possess a Lagrangian formulation. In this article, we find a variational integrating factor which enables us to provide a Lagrangian and Hamiltonian structure associated to this equation. Explicit expressions of the solutions in terms of elliptic integrals of the first kind are then deduced. We then derive discrete version of the Eringen’s nonlocal elastica preserving the Lagrangian and Hamiltonian structure and compare it with Challamel’s and co-worker definition of a discrete Eringen’s nonlocal elastica.

Curvature Correction of a Notched Continuum Robot Based on a Static Model Considering Large Deformation and Friction Effect

Continuum robots are often used as wrist joints in medical robots because of their high dexterity and flexibility. Especially, the notched continuum robot (NCR) is used in the miniaturized wristed surgical robot. The Piecewise Constant Curvature (PCC) assumption is often used in the design of NCR. However, due to the friction effect, ideal PCC is difficult to achieve. Static analysis is a necessary means to correct the curvature of NCR. The static modeling of NCR is often based on the theory of small deformation. However, this cannot obtain accurate solutions at large bending angles. In this paper, a static model of a triangular-notched continuum robot is proposed. It presents a curvature correction method of NCR, considering large deformation. In addition, the friction effect is considered in the correction of PCC. The static model is derived from the end notch. Based on the Coulomb friction model, the recurrence relationship of the force on the cable is obtained. Then the elliptic integral solution corresponding to the large deformation assumption is calculated. The deformation parameters of the NCR are obtained by numerical iteration. Finally, the capability and validity of the static model proposed in this paper are verified in the experiment. This paper is of great significance for establishing an accurate static model for curvature correction and design of the notched continuum robot.