Solution Of Axisymmetric Problem Research Articles

Solutions for Axisymmetric Problems of Layered Generalized Gibson Subgrade

This article presents the solutions for displacements and stress of layered generalized Gibson subgrade subjected to an axisymmetric surface load. We assume that the material has constant Poisson’s ratios (μ=0.5), and its shear moduli varies linearly with depth. During the solutions, the Hankel transform in a cylindrical co-ordinate system is employed. The relationship between the solution and half-space solution is also examined and discussed. At last, the analysis flow is presented .

Numerical Solution of Axisymmetric Problem for an Elastic Footing

Based on formulas of elastic mechanics, the axisymmetric problem applied to an elastic footing is analyzed. Using Hankel transform and finite difference method, the arbitrary points’ displacement and stress solutions of the axisymmetric elastic footing problem is obtained. Compared with recursive substitution method and transfer matrix method, this method is simple because it avoids complex formulas and exponential function calculation. The half space model is more flexible to simulate an actual footing. The numerical solutions are compared with Boussinesq’s Equation and results show that finite difference method is applicable to solve axisymmetric elastic footing problem. But in the application of finite difference method, there are a few factors that could cause calculation errors. Effective measures are needed to obtain exact solutions.

Local heating of a moving multilayer cylinder

A mathematical model of thermal state kinetics of a long multilayer cylindrical body moving in the axial direction at the high-temperature heating of a local section on its external surface is constructed. The tentative problems are solved and their results are discussed on the basis of this model and the previously developed methods for the numerical solution of axisymmetric problems of nonstationary heat conduction.

Axisymmetric Body-Force Fields in Elastodynamics of Transversely Isotropic Media

In this paper, a complete elastodynamic solution for axisymmetric problems under axial body-forces in terms of two retarded potential functions in transversely isotropic media is extended to the case of general torsionless axisymmetry. Allowing for both axial and radial distributed internal loads through the use of an extra potential, the new solution retains its completeness via the theory of repeated wave equations. By virtue of its analytical design, the formulation can be reduced to the corresponding elastostatic case by simply suppressing the time-dependence of its potentials as well as the case of isotropy. In the limiting case of the latter material condition, the proposed representation for elastostatic problem degenerates to a recent extension of Love’s potential function.

Exterior axisymmetric Neumann problem for the Laplace equation: Numerical algorithms without saturation

The development of conceptually new—unsaturated—numerical methods is closely related to Babenko’s work [1]. A distinctive feature of these methods is that they lack the principal error term. As a result, they are able to adjust automatically to any natural correctness classes of problems [2]. Babenko’s fundamental ideas are used to develop an unsaturated method for the numerical solution of an external axisymmetric Neumann problem for the Laplace equation. In the case of C ∞ -smooth solutions, the method has an absolutely sharp exponential error estimate up to a slowly growing factor. This result is fundamental, since the sharpness of the error estimate is determined by the asymptotics of the Aleksandrov width of the compact set of C ∞ -smooth functions containing the desired solution. This asymptotics also has the form a decaying exponential function [3]. The method is validated by computing the nonseparated axisymmetric potential flow of an inviscid incompressible fluid past an ellipsoid with an aspect ratio of 1000 (see [4]). Note that any saturated methods (i.e., with a principal error term), such as finite differences, finite elements, the quadrature method, etc., fail to produce a satisfactory solution to the problem even for an aspect ratio of 25. Success in the solution of this axisymmetric problem is reached solely due to achievements in the theory of constructive function approximation on the interval. 1. Let x ∈ � 3 , x = ( x , y , z ) , and ω ⊂ � 3 be a domain with an axis of symmetry z bounded by a smooth closed surface of revolution ∂ω . The meridional section of ∂ω is the curve γ : [0, π ] → { r ( s ), z ( s )}, γ ( s ) ∈ C ∞ [0, π ] . Then γ (0) and γ ( π ) are the poles of ∂ω . The values r = and z are invariant under the group of rotax 2 y 2 + tions of ω about the z axis. The normal N on ∂ω is assumed to be directed inward � 3 \ ω . The locations of the points x = ( r , z ) and ξ = ( ρ , ζ ) on γ are defined by coordinates s and σ , respectively: r = r ( s ), z = z ( s ); ρ = r ( σ ), ζ = z ( σ ) . If x ∈ ∂ω , then N ≡ N ( x ) = N ( s ) . The direct value of F ( x ) on ∂ω (if any) is denoted by ( x ) = F ( x ) | x ∈ ∂ω . The limiting value of the derivative = ∇ x F 〈 N 〉 from the outside of ∂ω and its direct value on ∂ω (if they exist) are designated as ( N – )( x ) and ( N )( x ), respectively. The exterior Neumann problem is to find a function Φ ( x ) that obeys the conditions

The method of approximate fundamental solutions for axisymmetric problems with Laplace operator

The paper presents a new numerical technique for solving axisymmetric problems with Laplace operator. It is similar to the method of fundamental solutions but it is based on the use of special basis functions which satisfy the majority of the boundary conditions of the problem considered. This reduces the number of unknowns and the size of the collocation matrix considerably. As it is shown in the paper, this technique can also be applied successfully in the cases when the solution domain has infinite boundaries in z or r directions. Numerical examples justifying the method are presented.

RADIATIVE HEAT TRANSFER IN AXISYMMETRIC GEOMETRIES USING AN UNSTRUCTURED FINITE-VOLUME METHOD

The finite-volume method is extended to compute radiation in axisymmetric geometries using unstructured polyhedral meshes. Control angle overhangs, resulting from the nonalignment of the angular discretization with cell faces, are accounted for in the differencing. Conservative discretization of the angular redistribution term is proposed, and allows the solution of axisymmetric problems on the same two-dimensional mesh as the fluid flow. The method is shown to be equivalent to the mapping procedure of Chui and Railhby. The scheme is applied to a number of benchmark problems and is shown to perform satisfactorily.

The solution of axisymmetric problems near singular points and computation of stress intensity factors

The computation of the stress intensity factors for axially symmetric domains, in the vicinity of singular points in the framework of linear elastostatics, is presented. It is shown that the decomposition of the solution in the neighborhood of any singular point, in an axisymmetric domain, is identical to the decomposition corresponding to plane strain condition, and an extraction vector, to be used in conjunction with the contour integral method and the p-version of the finite element method, is developed for crack singularities. Although the contour integral is shown to be path dependent for axisymmetric 2-D domains, in the limiting case it is valid for the computation of the crack stress intensity factors. Numerical examples support our analysis, extending the application of the superconvergent contour integral in conjunction with the p-version of the finite element method to the axisymmetric crack case.

Monte Carlo solution of axisymmetric potential problems

An extension of the Monte Carlo procedures to the solution of axisymmetric problems is presented. The stochastic techniques are illustrated with specific practical applications to the solution of Laplace's and Poisson's equations. The method requires no input data and provides results which are more accurate than those obtained using the finite element method. >

A general solution of axisymmetric problem of arbitrary thick spherical shell and solid sphere

In this paper, the axisymmetric problems of arbitrary thick spherical shell and solid sphere are studied directly from equilibrium equations of three-dimensional problem, and the general solutions in forms of Legendre series for thick spherical shell and solid sphere are given by using the method of weighted residuals.

A simplified approach to estimating the stress state of flexible cylindrical shells under nonaxisymmetric load

An approach is proposed to approximate numerical analysis of the stress state of a geometrically nonlinear cylindrical shell under an unsymmetric load. An approximate technique is also proposed to reduce the solution of a two-dimensional nonlinear problem of a cylindrical shell with unsymmetric parameters to the solution of axisymmetric problems. A numerical example is considered.

An approximate method of reducing the solutions of axisymmetric problems of flow in porous media to plane problems

An approximate method of reducing the solutions of axisymmetric problems of flow in porous media to plane problems

Use of triangular three-node nonconforming elements for solution of axisymmetric problems of the theory of elasticity

Use of triangular three-node nonconforming elements for solution of axisymmetric problems of the theory of elasticity

Analytic solutions of axisymmetric problems in elastic cylinders

A general scheme encompassing expansion methods for the sleeve is presented: one homogeneous condition for each contour portion establishes real eigenvalues ensuring orthogonality to Fourier-Bessel expansions. Solutions by integral equations may be regarded as limiting cases, with summations replaced by integrations. Special attention is devoted to the alternative use of complex eigensolutions, subjected to two homogeneous conditions on the cylindrical boundary.

Scattering of a stationary torsion wave by a cavity in the form of a body of revolution in an infinite elastic space

Axisymmetric problems of the diffraction of torsion waves reduce in the case of a harmonic wave to the solution of the scalar Helmholtz equation with certain boundary conditions. The impossibility of obtaining exact solutions (with the exception of a sphere and spheroids) requires the inclusion of approximate methods of solution, among which are the method of perturbing the boundary shape [i, 2], methods based on reduction of the original boundary-value problem to integral equations [3, 7], and others. In the case of spheroids the wave functions which arise in the separation of variables are obtained by rather complicated methods, and approximate methods of solution of these problems turn out to be more effective. A method of solution of axisymmetric problems of the scattering of stationary torsion waves by cavities of various shapes whose surfaces are free of stresses is proposed in this article. The method is based on the use of solutions with a singularity of the center-of-rotation type which are distributed over a segment of the axis of symmetry which lies completely within the cavity.

On the obtaining of axisymmetric flows from plane-parallel flows

The obtaining of axisymmetric flows of incompressible and compressible fluids from plane-parallel flows by means of integral transformations relating harmonic and p-harmonic functions [1] is considered. A transformation is found that carries plane-parallel flows from elementary singularities into axisymmetric flows. It is shown that this transformation makes it possible to obtain the general form of the solution of axisymmetric problems of flow past bodies from the solution of plane-parallel problems.

Numerical solution of axisymmetric problems of the dynamics of thin orthotropic shells of revolution

Numerical solution of axisymmetric problems of the dynamics of thin orthotropic shells of revolution

Solution of the axisymmetric problem of the theory of elasticity for transversely isotropic bodies with the aid of generalized analytic functions

Abstract The solution of the axisymmetric problem for transversely isotropic bodies, expressed in terms of generalized analytic functions, is constructed. We obtain formulas for the displacements and stresses, similar to the corresponding formulas of the plane problem. The representation of the generalized analytic functions by analytic ones, are indicated, and the analogue of the Cauchy-type integral which gives the possibility of reducing the boundary value problems to integral equations, is presented. As an example, we consider the action of forces which are distributed along a circumference in the interior of a transversely isotropic space. The plane problems of the theory of elasticity for transversely isotropic bodies are solved effectively with the aid of analytic functions of a complex variable [1], In [2, 3] the solution of axisymmetric and nonaxisymmetric problems for bodies of revolution with the aid of analytic functions and contour integrals,was considered. In the case of an isotropic elastic medium, the solution of axisym-metric problems with the aid of a class of generalized analytic functions [4] was proposed.

Solution of axisymmetric problems of the theory of elasticity. I

On the basis of these results we can draw the following conclusion. The use of a layer of less strong material, for the joint of tubular testpieces of different materials, allows us to formulate the problem of the choice of the geometrical and physical parameters of the insert which guarantees the strength of the joint material, the calculated failure stresses being determined according to the momentless theory for the less strong testpiece.

The fundamental mixed problem of the axisymmetric theory of elasticity

Analytic and p-analytic functions of a complex variable have been used in the solution of axisymmetric problems in the theory of elasticity (see, for example, [1 to 5]). In [6 and 7], the same results were accomplished by using generalized analytic functions which do not differ essentially from the functions introduced in [8]. In this manner, Fredholm integral equations were obtained for the first and second fundamental problems for simply as well as multiply connected bodies of revolution. The method given below deals with the fundamental mixed problem in which the applied forces are specified on one part of the boundary while the displacements are given on the other part. The singular integral equation which is obtained is analogous to the corresponding equation in the plane theory of elasticity [9 and 10]. This equation is then investigated, and the existence of a solution is proved.