Dirichlet problem for axisymmetric potential fields in a disk of the meridian plane. II

We develop new methods for the solution of boundary-value problems in the meridian plane of an axisymmetric potential solenoidal field with regard for the nature and specific features of axisymmetric problems. We determine the solutions of the Dirichlet problems for an axisymmetric potential and the Stokes flow function in a disk in an explicit form. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 52, No. 6, pp. 748–757, June, 2000.     

On Integral Representations of an Axisymmetric Potential and the Stokes Flow Function in Domains of the Meridian Plane. II

We obtain new integral representations for an axisymmetric potential and the Stokes flow function in an arbitrary simply-connected domain of the meridian plane. The boundary properties of these integral representations are studied for domains with closed rectifiable Jordan boundary.     

On Integral Representations of an Axisymmetric Potential and the Stokes Flow Function in Domains of the Meridian Plane. I

We obtain new integral representations for an axisymmetric potential and the Stokes flow function in an arbitrary simply-connected domain of the meridian plane. The boundary properties of these integral representations are studied for domains with closed rectifiable Jordan boundary.     

The convergence of V-cycle multigrid algorithms for axisymmetric Laplace and Maxwell equations

We investigate some simple finite element discretizations for the axisymmetric Laplace equation and the azimuthal component of the axisymmetric Maxwell equations as well as multigrid algorithms for these discretizations. Our analysis is targeted at simple model problems and our main result is that the standard V-cycle with point smoothing converges at a rate independent of the number of unknowns. This is contrary to suggestions in the existing literature that line relaxations and semicoarsening are needed in multigrid algorithms to overcome difficulties caused by the singularities in the axisymmetric Maxwell problems. Our multigrid analysis proceeds by applying the well known regularity based multigrid theory. In order to apply this theory, we prove regularity results for the axisymmetric Laplace and Maxwell equations in certain weighted Sobolev spaces. These, together with some new finite element error estimates in certain weighted Sobolev norms, are the main ingredients of our analysis.

     

Quadrature formulas for periodic functions and their application to the boundary element method

Two-dimensional and axisymmetric boundary value problems for the Laplace equation in a domain bounded by a closed smooth contour are considered. The problems are reduced to integral equations with a periodic singular kernel, where the period is equal to the length of the contour. Taking into account the periodicity property, high-order accurate quadrature formulas are applied to the integral operator. As a result, the integral equations are reduced to a system of linear algebraic equations. This substantially simplifies the numerical schemes for solving boundary value problems and considerably improves the accuracy of approximation of the integral operator. The boundaries are specified by analytic functions, and the remainder of the quadrature formulas decreases faster than any power of the integration step size. The examples include the two-dimensional potential inviscid circulation flow past a single blade or a grid of blades; the axisymmetric flow past a torus; and free-surface flow problems, such as wave breakdown, standing waves, and the development of Rayleigh-Taylor instability.     

Dirichlet Problem for an Axisymmetric Potential in a Simply Connected Domain of the Meridian Plane

We develop a method for the reduction of the Dirichlet problem for an axisymmetric potential in a simply connected domain of the meridian plane to a Cauchy singular integral equation. In the case where the boundary of the domain is smooth and satisfies certain additional conditions, we regularize the indicated singular integral equation.     

A numerical method for analyzing the quasistationary field above a stratified medium with an axisymmetric elevation

We consider the quasistationary electromagnetic field above a plane-parallel stratified medium with an axisymmetric elevation. The three-dimensional vector problem is reduced to scalar problems in the azimuthal halfplane for the first two field harmonics. A system of one-dimensional integral equations of the first kind is obtained.Translated from Vychislitel'naya Matematika i Matematicheskoe Obespechenie EVM, pp. 176–181, 1985.     

Symmetry-breaking at positive solutions

We consider symmetry-breaking bifurcations at positive solutions of semilinear elliptic boundary value problems in a ball, with Dirichlet boundary conditions. We prove that symmetry-breaking can only occur by bifurcation of axisymmetric solutions. We also obtain sufficient conditions for such symmetry-breaking.Dedicated to Professor H. Knobloch at the occasion of his 60th birthday     

Mixed problems of axisymmetric elasticity theory for some bodies of revolution

We consider the problem of axisymmetric elasticity theory for a space with an elongated ellipsoidal cavity with mixed boundary conditions of smooth contact on the cavity surface and the main mixed problem of axisymmetric elasticity theory for a hyperboloidal layer formed by the two surfaces of a two-cavity hyperboloid of revolution symmetrical about the plane z = O. The problems are solved by the method of p-analytical functions. The solution of the first problem is reduced to solving a Fredholm integral equation of the second kind. We investigate the behavior of the normal stress near the boundary lines. The solution of the second problem is reduced to solving a system of two Fredholm integral equations of the second kind. Existence and uniqueness of the solution is proved for this system.Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 67, pp. 88–101, 1989.     

理想塑性轴对称问题的一般方程

本文引用速度势函数,将理想塑性轴对称问题化为二个非线性的偏微分方程,根据导得的方程讨论了Haar-Kármán假设对于Mises屈服准则及与其相关联的流动法则的协调性问题。     

The study of three-dimensional gas flow in a nozzle using the completely implicit method

We study three-dimensional potential gas flow in a nozzle. The matrix of the system of linear equations obtained in approximating the equation for the velocity potential by symmetric differences has a strongly sparse form. This property is used for an approximate expansion of it as a product of triangular matrices. To guarantee stability in the supersonic region of flow additional terms of artificial viscosity type are introduced into the equation for the velocity potential. We study gas flows in axisymmetric and three-dimensional nozzles: elliptic, superelliptic, and nozzles with nonsymmetric subsonic part. Comparison of the results with the data of other authors has shown the high effectiveness of such an approach. Translated fromMetody Matematicheskogo Modelirovaniya, 1998, pp. 76–86.     

关于椭圆型轴(球)对称问题及有限元解法的注记

椭圆型轴(球)对称问题的有限元方法,已有不少工作,见[1—4]。但这些工作都是分别进行的,没有进行统一的处理,而且往往是直接进行误差估计。本文给出一个框架,使得这种问题的研究变得十分简单,而且对n维的轴(球)对称问题进行了统一的处理。     

分析土壤半无限域的一种有限层方法(Ⅱ)——几种特殊情形及算例*

本文就几种特殊情形用作者所研究的一种用于横观各向同性体动力学的有限层法作了简化分析,分别讨论了二维问题、轴对称问题以及静力问题,并推广到介质具有粘性性质的情形.对于轴对称情形,本文还给出两个算例,表明作者所研究的有限层法用于分析半无限域层状土壤介质是可行的,因而为研究土壤与结构相互作用问题提供了一条新途径.     

Flux intensity functions for the Laplacian at axisymmetric edges

We derive explicit representation formulas for the computation of flux intensity functions for mixed boundary value problems for the Poisson equation in axisymmetric domains with edges. We rely on the decomposition of the boundary value problems in three dimensions by means of partial Fourier analysis with respect to the rotational angle into boundary value problems in the two‐dimensional meridian domain of . Utilizing smooth cutoff functions, the solutions of the reduced problems are analyzed semi‐analytically near corners of the plane meridian domain, and the edge flux intensity functions are constructed via Fourier synthesis and convergence analysis. The formulas are also applicable in the case of crack fronts. The constructive nature of the formulas provides in a straightforward way an efficient strategy for the accurate computation of edge flux intensity functions in axisymmetric domains. A demonstration example that illustrates the application of the formulas is presented. Copyright © 2012 John Wiley & Sons, Ltd.     

The IIM in polar coordinates and its application to electro capacitance tomography problems

In a numerical study of the electric field inside an Electro Capacitance Tomography (ECT) device and other applications, often a Poisson equation with a discontinuous coefficient needs to be solved in polar coordinates. This paper is devoted to the Immersed Interface Method (IIM) in polar coordinates and the application to the solution of the electric potential inside an ECT device. The numerical algorithm is based on a finite difference discretization on a uniform polar coordinates grid. The finite difference scheme is modified at grid points near and on the interface across which the coefficient is discontinuous so that the natural jump conditions are satisfied. The algorithm and analysis here is one step forward in applying the IIM for 3D problems in axisymmetric situations or in the spherical coordinates. Numerical examples against exact solutions and the application to ECT problems are also presented.     

横观各向同性轴对称问题的通解

本文按应力求解轴对称问题,以统一的格式导出了一系列有实用价值的通解,其中有的是已有的著名的通解,有的尚未见文献报导.同时证明了各种通解的完备性.     

The method of successive approximation of a free boundary in problems of cavitation flow

We discuss a numerical method of “recursion on the shape of the cavity” for solving problems of cavitation flow past planar and axisymmetric objects according to the Ryabushinskii scheme. We give a comparison of a closed-form solution with the numerical results. Translated fromDinamicheskie Sistemy, Vol. 11, 1992.     

Diffraction of sound on a spherical shell with a ring-shaped rib

We obtain a closed-form solution of the axisymmetric problem of the echo-signal from a thin spherical shell containing an elastic ring. We analyze the influence of the reinforcing rib on the frequency dependence of the scattering amplitude in the axial direction and also on the directional characteristics in regimes of potential and resonance scattering.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 35, 1992, pp. 143–146.     

正交各向异性轴对称位势问题的Trefftz有限元分析

Trefftz有限元法（Trefftz finite element method, TFEM）因其独特的优良品质而备受关注.针对正交各向异性轴对称位势问题,提出了一种4节点四边形环状单元．在该单元模型中,首先假设两套独立的位势插值模式:即单元域内场和网线场,然后代入修正变分泛函并利用Gauss散度定理消除区域积分,最后根据驻值原理导得只含边界积分的单元刚度方程．数值算例表明了该单元的准确性、稳定性以及对网格畸变的不敏感性.