Solution In Cartesian Coordinates Research Articles

Explicit solutions in Cartesian coordinates for an elliptic hole in an infinite elastic plate

Explicit solutions in Cartesian coordinates for an elliptic hole in an infinite elastic plate

Two-dimensional Harmonic Modelling for Electro-magnetic Solution in Cartesian Coordinates

This paper first presents a general two-dimension (2D) harmonic analytical solution for the magnetic field of electric machines in the Cartesian coordinates. In this solution, the relative permeance is directly considered in Laplace and Poisson’s equations, and the particular solutions in Cartesian coordinates are solved. By applying the complex Fourier separation method, with the boundary and interface conditions, the magnetic field in the inhomogeneous region is solved from system equations. Numerical examples validate the presented method and the obtained results have a satisfactory agreement with the finite element analysis. The proposed model in this paper has a significant value for modelling electric machines, such as linear permanent magnet (PM) machines and axial flux PM machines.

Analytical solution of flows in porous media for transpiration cooling applications

Abstract

Application of Geodesic Grids for Modeling the Hydrodynamic Processes in Spherical Objects

We propose a new numerical method that bases on the mathematical apparatus of geodesic grids. This approach allows us to simulate spherical objects without any singularities that occur in using spherical or cylindrical coordinates. Solution of hyperbolic equations is described in detail. The method is expanded to solve the equations of hydrodynamics and tested on the Sedov point explosion problem. The numerical method and the approach to grid construction make it possible to compute a rotation invariant solution in Cartesian coordinates. This in turn allows us to use this approach effectively for simulating various spherical astrophysical objects.

The direct geodesic problem and an approximate analytical solution in Cartesian coordinates on a triaxial ellipsoid

Abstract In this work, the direct geodesic problem in Cartesian coordinates on a triaxial ellipsoid is solved by an approximate analytical method. The parametric coordinates are used and the parametric to Cartesian coordinates conversion and vice versa are presented. The geodesic equations on a triaxial ellipsoid in Cartesian coordinates are solved using a Taylor series expansion. The solution provides the Cartesian coordinates and the angle between the line of constant v and the geodesic at the end point. An extensive data set of geodesics, previously studied with a numerical method, is used in order to validate the presented analytical method in terms of stability, accuracy and execution time. We conclude that the presented method is suitable for a triaxial ellipsoid with small eccentricities and an accurate solution is obtained. At a similar accuracy level, this method is about thirty times faster than the corresponding numerical method. Finally, the presented method can also be applied in the degenerate case of an oblate spheroid, which is extensively used in geodesy.

Geodesic equations and their numerical solution in Cartesian coordinates on a triaxial ellipsoid

Abstract In this work, the geodesic equations and their numerical solution in Cartesian coordinates on an oblate spheroid, presented by Panou and Korakitis (2017), are generalized on a triaxial ellipsoid. A new exact analytical method and a new numerical method of converting Cartesian to ellipsoidal coordinates of a point on a triaxial ellipsoid are presented. An extensive test set for the coordinate conversion is used, in order to evaluate the performance of the two methods. The direct geodesic problem on a triaxial ellipsoid is described as an initial value problem and is solved numerically in Cartesian coordinates. The solution provides the Cartesian coordinates and the angle between the line of constant λ and the geodesic, at any point along the geodesic. Also, the Liouville constant is computed at any point along the geodesic, allowing to check the precision of the method. An extensive data set of geodesics is used, in order to demonstrate the validity of the numerical method for the geodesic problem. We conclude that a complete, stable and precise solution of the problem is accomplished.

Using pseudo-spectral method on curved grids for SH-wave modeling of irregular free-surface

This paper uses curved grids in the context of a pseudo-spectral method to model 2-D SH-wave propagation in a medium with an irregular free-surface. A technique originally developed for computational fluid dynamics (CFD) is first used to generate curved grids following the irregular free-surface. Then the SH-wave equations in curved coordinates are derived, together with numerical boundary conditions at a free-surface are presented. Finally, a Fourier-Chebyshev pseudo-spectral method is used to simulate SH-wave propagation in curved coordinates and solutions in Cartesian coordinates are obtained. This numerical scheme is tested against the analytical solution for Lamb's problem in a uniform elastic half space with a sloping surface. An excellent agreement is found, which confirms the accuracy of the method present here. In addition, the scheme is applied to the model of topography with discontinuous derivative, and the results are compared with those obtained from the modeling on conventional curved grids by algebraic transform. It shows that the new scheme is more accurate than that on algebraic grids.

Spatial Correlation Functions in Three-Dimensional APS With $\cos^n\theta$

This letter describes a generalized formula for spatial correlation of three-dimensional incoming waves with an angular power spectrum of cosn θ. Based on the proposed formula, spatial correlation performances at antennas or an antenna array can be approximated by simple computation with closed-form solutions in Cartesian coordinates. The validity of the proposed method is verified numerically.

Poromechanics Solutions to Plane Strain and Axisymmetric Mandel-Type Problems in Dual-Porosity and Dual-Permeability Medium

The two-dimensional Mandel-type problem geometry is well-known to bio-geomechanicians for testing rocks, cartilages, and bones with solutions in Cartesian coordinates for rectangular specimens or polar coordinates for cylindrical and disk samples. To date, all existing solutions are only applicable to single-porosity and single-permeability models, which could fall short when the porous material exhibits multiporosity and/or multipermeability characteristics, such as secondary porosity or fracture. This paper extends the plane strain and axisymmetric Mandel-type solutions from single-to dual-porosity and dual-permeability poromechanics. The solutions are presented in explicit analytical forms and account for arbitrary time-dependent external loading conditions, e.g., cyclic and ramping. The derived analytical solutions and results exhibit general behaviors characterized by two time scales. Stresses, pore pressures, and displacements are plotted for various time scale ratios to illustrate the interplaying effects of permeability and stiffness contrast of both porous regions, in addition to the interporosity exchange, on the overall responses of the system. Also, examples with realistic loading conditions for laboratory testing or field simulation such as cyclic and ramping are provided to demonstrate the engineering applications of the presented dual-poroelastic formulation and solutions.

General expressions for internal deformation due to a moment tensor in an elastic/viscoelastic multilayered half-space

SUMMARY In the framework of elasticity theory any indigenous source can be represented by a moment tensor. We have succeeded in obtaining general expressions for internal deformation due to a moment tensor in an elastic/viscoelastic multilayered half-space under gravity. First, starting from Stokes’ classical solution, we obtained the expressions for static displacement fields due to a moment tensor in an infinite elastic medium. Then, performing the Hankel transformation of the static solution in Cartesian coordinates, we derived static displacement potentials for a moment tensor in cylindrical coordinates. Second, representing internal deformation fields by the superposition of a particular solution calculated from the displacement potentials and the general solution for an elastic multilayered half-space without sources, and using the generalized propagator matrix method, we obtained exact expressions for internal elastic deformation fields due to a moment tensor. Finally, applying the correspondence principle of linear viscoelasticity to the elastic solution, we obtained general expressions for quasi-static internal deformation fields due to a moment tensor in an elastic/viscoelastic multilayered half-space. The moment tensor can be generally decomposed into the three independent force systems corresponding to isotropic expansion, crack opening and shear faulting, and so the general expressions include internal deformation fields for these force systems as special cases. As numerical examples we computed the quasi-static internal displacement fields associated with dyke intrusion, episodic segmental ridge opening and steady plate divergence in an elastic‐viscoelastic two-layered half-space. We also demonstrated the usefulness of the source representation with moment tensor through the numerical simulation of deformation cycles associated with the periodic occurrence of interplate earthquakes in a ridge-transform fault system.

Three-dimensional analytical calculation of induction machines with multilayer rotor structure in cylindrical coordinates

An analytical three-dimensional solution of the field problem of induction machines with multilayer rotor structure is presented in cylindrical coordinates. The curvature of the machine is no longer neglected as against the known solution in cartesian coordinates. This results in much higher accuracy for machines with a large air-gap width or small bore radius. In addition, the analysis is performed for an arbitrary number of rotor layers and for arbitrary speed of each rotor layer.

NUMERICAL SOLUTION OF THREE-DIMENSIONAL STABILITY PROBLEMS FOR ELASTIC BODIES

A solution in Cartesian coordinates to plane and spatial stability problems for composites is obtained within the framework of the second variant of the three-dimensional linearized theory of stability of deformable bodies. Two mechanical models are used: a homogeneous anisotropic medium with averaged mechanical characteristics and a piecewise-homogeneous medium with orthotropic linearly elastic components. To solve the problems, a mesh approach is applied. Discrete models are constructed using the concept of a base scheme. The calculated results are analyzed

Harmonic Green’s functions of a semi-infinite plate with clamped or free edges

Harmonic Green’s functions for a thin semi-infinite plate with clamped or free edges are developed starting from either simply supported or roller supported solutions and applying corrections to account for boundary excitation. This is achieved by connecting the solutions in terms of polar coordinates with the solutions in Cartesian coordinates. The formal solutions in the form of improper wave-number integrals are numerically evaluated using adaptive Clenshaw–Curtis integration. Alternate solutions obtained for each boundary condition compare well. Only harmonic point loads are considered in this article but the methodology may be extended to moment excitation and distributed loads. The methodology developed here will form the basis for advancing the ray tracing technique for vibration analysis of finite plates.

The celestial bodies motion theories: on the required accuracy of the motion theories of the perturbing bodies

The equation for calculation of the required accuracy of the perturbing bodies motion theories is obtained. The equation relates the accuracy required to take into account perturbing acceleration, acting on the perturbed body, with the accuracy of the motion theory of the perturbing body. The solutions for estimation of the required accuracy both for the inner and the external cases in the spherical coordinates are coincided. The solution for the calculation of the required accuracy for the general case (combining the inner and the external cases) in Cartesian coordinates is obtained. The special cases for the solution in Cartesian coordinates are studied. As an example, the estimations of the required accuracy of the motion theories of the solar system planets for some perturbed bodies (the near-Earth asteroid 4179 Toutatis, the main belt asteroid 208 Larcimosa, the trojan asteroid 588 Achilles, the centaur asteroid 5145 Pholus, the Kuiper belt asteroid 1995 QZ9, the comet Halley) are obtained. The conditions of the use of the obtained results are discussed.

Stresses and displacements in an isotropic or transverse isotropic semi-infinite body submitted to flexible or rigid rectangular loads at its surface: a general solution in cartesian coordinates (In French) : Van Cauwelaert, F Rev Fr GeotechN28, 1984, P7–25

Stresses and displacements in an isotropic or transverse isotropic semi-infinite body submitted to flexible or rigid rectangular loads at its surface: a general solution in cartesian coordinates (In French) : Van Cauwelaert, F Rev Fr GeotechN28, 1984, P7–25

The settling of a sphere along the axis of a long square duct at low Reynolds' number

A theoretical treatment is presented for the determination of the drag upon a sphere settling along the axis of a long square duct under the condition that the creeping motion equations are applicable. In order to obtain the second reflection velocity field, it was necessary to develop a new general solution in cartesian coordinates to the creeping motion equation, applicable within the domain of a long square duct. Using the second reflection velocity field solution an Faxen's law, a third reflection (first correction to the Stokes' value) drag correction is obtained. The results show that the drag correction for a square container is quite close to (but smaller than) the drag correction produced by a cylinder whose diameter is the same as the duct width.

Optimal low-thrust rendezvous control.

A study is made of the coplanar rendezvous maneuver between a thrusting vehicle and a passive vehicle in an elliptic orbit. In the terminal phase of rendezvous, where the equations of relative motion can be linearized, an analytic solution for the fuel optimal thrust program is found for power-limited vehicles when the time to maneuver is fixed. The calculations are carried out in polar coordinates where the solution is found to be closer to the exact nonlinear motion than the solution in Cartesian coordinates used in previous analyses. The optimal power-limited thrust program for a practical example is compared to the exact numerical solution obtained using a Newton-Rhapson algorithm. I. Introduction T HE problem of rendezvous between a maneuvering space vehicle and a nonmaneuvering target has received considerable attention in recent years. The first investigations were of the free motion using linearized equations1'2 and perturbation theory to take into account nonlinear effects.35 Fuel optimal maneuvers have been studied to find both the structure of these maneuvers and the absolute bound on fuel consumption for comparisons with suboptimal schemes used in practice because of severe mission restrictions. As in general orbital transfer theory, the optimal control problem has been studied for both low-thrust (power-limited) and high-thrust, constant-exhaust velocity propulsion systems. Of particular importance has been the terminal rendezvous maneuver when the vehicles are relatively close together. Tschauner and Hempel6 and Tschauner7 have presented some analytical results for high-thrust rendezvous for both circular and elliptic target orbits and Paiwonsky and Woodrow8 and Goldstein, Greene, and Johnson9 have presented numerical results for the time optimal problem. For the low-thrust case, Edelbaum10 has found solutions for small changes in all orbital parameters when the thrust is of the same magnitude as the small orbital changes. Billik11 has derived the terminal low-thrust optimal control for circular target orbits and Gobetz12 has done the same for elliptic target orbits using linearized equations in Lagrange's planetary variables. Numerical studies of optimal low-thrust planetary transfer trajectories have been carrried out by Melbourne and Sauer.13 In this study, we shall extend the work of Billik to elliptic target orbits and complement the work of Gobetz, where the linearizing assumptions are different. Also, another new set of coordinates is used such that the solution to the linearized equations better approximates the nonlinear motion. The solution to the homogeneous linear equations of motion2 for elliptic target orbits permits the reduction of the low-thrust rendezvous optimal control problem to quadrature. When the eccentricity is small, a power series expansion can be made in this parameter to evaluate the resulting definite integrals.