Three-dimensional Boundary Value Problems Research Articles

The method of axisymmetric generalized analytical functions in the analysis of dynamic processes

A method for analysing various dynamic processes based on axisymmetric generalized analytical functions. Such functions satisfy a system of equations describing axisymmetric processes, which is canonical for two-dimensional dynamic processes in curvilinear layers with certain non-homogeneous behaviour of the characteristics and also for some kinds of anisotropy. The method, applied to some three-dimensional boundary-value seepage problems in a piecewise-homogeneous medium, produced solutions of these problems in a finite form.

Maximum depth of keyhole melting of metals by a laser beam

A calculation is reported of the maximum depth and diameter of a narrow crater formed in a stationary metal target exposed to high-power cw CO2 laser radiation. The energy needed for erosion of a unit volume is assumed to be constant and the energy losses experienced by the beam in the vapor–gas channel are ignored. The heat losses in the metal are allowed for by an analytic solution of the three-dimensional boundary-value heat-conduction problem of the temperature field in the vicinity of a thin but long crater with a constant temperature on its surface. An approximate solution of this problem by a method proposed earlier by one of the present authors was tested on a computer. The dimensions of the thin crater were found to be very different from those obtained earlier subject to a less rigorous allowance for the heat losses.

Testing convergence schemes on the solution of two-dimensional boundary value problems in elasticity

This paper is the first of a three-part series in which solutions are given to two-dimensional and three-dimensional boundary value problems in elasticity. Using a new convergence technique developed here, the stress in the near zone for the case of a two-dimensional semi-infinite slab subject to static loading will be evaluated. In the next article this same technique will be applied in two ways to evaluate the dynamic response of a three-dimensional solid cylinder subject to time-dependent end loading. In addition, the third article will discuss a resonance phenomenon associated with the reflection of an infinite wave train off the free end of a semi-infinite cylindrical bar.

Rigorous formulation of scattering and guidance by dielectric grating waveguides: general case of oblique incidence

First a rigorous formulation is presented for the scattering of a uniform plane wave by an infinite dielectric grating waveguide, under the most general condition of oblique incidence. The results are then applied to the analysis of the guidance of waves by the dielectric grating waveguide. By using a simple coordinate rotation, the TE and TM Floquet mode functions determined previously for the special case of principal-plane incidence are combined to treat the general case of oblique incidence. In terms of the coupling between the known Floquet mode functions of both polarizations, the general case of oblique incidence is formulated in an exact fashion, as a three-dimensional boundary-value problem, so that the hybrid nature of waves supported by the structure can be investigated rigorously.

Two-level hierarchically preconditioned conjugate gradient methods for solving linear elasticity finite element equations

In this paper we study and compare some preconditioned conjugate gradient methods for solving large-scale higher-order finite element schemes approximating two- and three-dimensional linear elasticity boundary value problems. The preconditioners discussed in this paper are derived from hierarchical splitting of the finite element space first proposed by O. Axelsson and I. Gustafsson. We especially focus our attention to the implicit construction of preconditioning operators by means of some fixpoint iteration process including multigrid techniques. Many numerical experiments confirm the efficiency of these preconditioners in comparison with classical direct methods most frequently used in practice up to now.

Equivalent circuit of a waveguide-coaxial joint of the turnstile type

We shall present a scheme for solving the three-dimensional vector boundary value problem of a waveguide-coaxial joint of the turnstile type; we construct its equivalent circuit and investigate the transformation properties of this unit on the basis of numerical calculations.

Generalization of the method of perturbation of the form of the boundary in the mechanics of deformable bodies

Introduction. Under the title "A Method of Pertubation of the Form of the Boundary" there,appeared in the scientific literature an approximate analytical method of solution of boundary value problems of continuum mechanics for noncanonical domains (domains not allowing a solution by the method of direct separation of variables). Its conceptual basis was contained in a paper by Guz' [4], published in 1962, in which, for the first time, an effective, approximate analytical method was proposed for studying stress concentration around curvilinear holes in shells. This idea proved to be sufficiently fruitful, and the developed method so general, that with no principal difficulties it was extended not only to threedimensional boundary value problems of elasticity theory for bodies of revolution [6, 13] and noncircular cylinders [7, 15], but also to broad classes of three-dimensional boundary value problems of the mechanics of deformable b6dies [12] and related problems of continuum mechanics [5, 9] for noncanonical domains. The recursion relationships and differential operators constructed in an arbitrary approximation [16] make it possible to solve stated problems in principle with specified accuracy. A broadening of the possibilities of applying the method proposed in [4] (subsequently referred to in [8] as a first variant of the method of perturbation of the form of the boundary) was enhanced through its application, along with other analytical methods, to the solution of boundary value problems for media with complicated elastic properties. Thus, along with the methods of successive approximations and the perturbation of elastic properties, it was applied for the first time to the solution of planar [I0] and spatial [14] nonlinear physical problems of elasticity theory and also to the solution of boundary value problems for curvilinear-orthotropic bodies of noncanonical form [18]. The indicated first version of the method of perturbation of the form of the boundary, along with an integral Laplace transform with respect to the time, was also applied to the solution of spatial [27] and planar [23] related problems of the mechanics of saturated porous media with noncanonical processing.

Numerical solution of three-dimensional boundary-value problems by generalized approximate inverse matrix techniques

Generalized Approximate Inverse Matrix (GAIM) techniques based on the concept of LU-type sparse factorization procedures are introduced for calculating explicitly approximate inverses of large sparse unsymmetric matrices of regular structure without inverting the factors L and U. Explicit first and second-order iterative methods in conjunction with modified forms of the GAIM techniques are presented for solving numerically three-dimensional initial/boundary-value problems on multiprocessor systems. Applications of the new methods on a 3D boundary-value problem is discussed and numerical results are given.

Analytical solutions for the three-dimensional side-fringing field of magnetic recording heads

Using conical coordinates, and in the limit of small gap length, the three-dimensional boundary value problem for the side-fringing field of beveled recording heads is reduced to two dimensions. This makes it possible to use analytic functions of a complex variable, and find simple expressions, in terms of elementary functions, for the scalar potential and three-dimensional magnetic fringing field. For a given bevel angle, the spectral response for side reading from in-plane and perpendicular magnetization depends on a single dimensionless parameter, and a quasi-exponential falloff of this response is calculated as a function of the bevel angle.

On the Numerical Solution of Three-Dimensional Boundary Value Problems by Separation of Variables

The classical method of separation of variables is subject to certain deficiencies as a numerical method for the solution of the Dirichlet problem for the Laplace equation. The method is only appli...

Three-dimensional axisymmetric elastic stresses in hollow pressurized torus with varied thickness.

The three-dimensional axisymmetric elastic boundary value problem of the torus of a closed hollow circular cross section with varied thickness under uniform internal pressure is analyzed by the indirect fictitious boundary integral method. The configuration parameters are the radius of the torus, the mean meridional radius of the hollow circular cross section, the thickness, and the eccentricity between the outer and the inner circles of the cross section. Moreover, the leveling of the stresses in the hollow torus is investigated.

Generalized coupled azimuthal potentials for electromagnetic fields in inhomogeneous media

The original coupled azimuthal potential (CAP) formulation utilized two continuous potentials to represent the Fourier azimuthal modes of the vector field components in inhomogeneous axisymmetric material media. This formulation has since been used as the basis for finite-element solutions of radiation, scattering, and penetration problems. A generalization of the CAP formulation is presented that removes the restriction to axisymmetry of the inhomogeneous media while retaining the advantage of vector field generation in terms of only two scalar potentials. The discrete form of this generating relation is cast into a matrix transport equation, which is solved by the use of natural basis function. Variational and Galerkin approaches are considered for the solution of the given potentials. Computational examples, using finite elements, are given for the case of a simple three-dimensional boundary-value problem. >

A general boundary integral approach to elliptical boundary value problems

Based upon basic principles of continuum theory a unified direct boundary integral representation of three-dimensional elliptical boundary value problems is presented. Within this framework many interesting engineering problems, e.g. elastostatics, manetostatic and heat conduction may be considered. The special analysis of the problem under consideration appears in the fundamental solutions. A general procedure for derivation of fundamental solutions with regard to an effecient numerical realisation of the boundary element method is discussed. In the appendix a FORTRAN IV code concerning the calculation of fundamental solutions in anisotropic elastostatics is presented.

A semi-iterative multigrid method and its implementation as a program for solving spatial boundary value problems

A version of the multigrid method for solving three-dimensional boundary value problems is described. An operator formulation of the method is given. The main characteristics of a universal program package which has been developed on the basis of this method are presented together with examples of real problems in the theory of elasticity and heat conduction which have been solved.

Three-dimensional axisymmetric elastic stresses in hollow pressurized torus (Comparison of shell theory solution)

The three-dimensional axisymmetric boundary value problem of the torus of a closed hollow circular cross section under uniform internal pressure is analysed by the indirect fictitious-boundary integral method. The configuration parameters are the mean radius of the torus, the mean meridional radius of the hollow circular cross section, and the thickness. Moreover, using the results, the applicability of the shell theory solution by Novozhilov is investigated.

Three-Dimensional axisymmetric elastic stresses in pressure vessels with torispherical drumheads (comparison of elasticity, photoelasticity, and shell theory solutions)

The three-dimensional axisymmetric boundary value problem of pressure vessels with torispherical drumheads is analysed easily and accurately by the use of the ‘indirect fictitious-boundary integral method’. The analysed configurations are similar to each other but have different thicknesses. The comparison of the resent results with photoelasticity and shell theory solutions makes it clear that the shell theory based on Love's first approximation is very reasonable. Additionally, a vessel with hemispherical drumheads with varying thickness from cylinder to head, which is noteworthy for reducing the maximum stresses in the vessel, is studied.

Solution of three-dimensional boundary value problems for magnetostatics by a constrained Fredholm integral equation

In this paper, a constrained boundary integral equation is presented for the analysis of three-dimensional magnetostatic problems. This formulation uses a simple layer of charge on the interface surface to represent the permeable object. A constraining condition that sets the total integrated surface charge to zero is used to make this formulation unique. In addition, a method of generating exact analytical solutions for use in validating the boundary element code is presented. A test comparison using this method will be shown.

The solution of a thermoelasticity problem for inhomogeneous bodies in terms of stresses by the perturbation method

Using the general properties of positive-definiteness of the elasticity theory operator, the convergence in an energy metric is proved for the solution of a thermoelasticity problem for a three-dimensional continuously inhomogeneous isotropic body by the perturbation method. The convergence of a solution by the perturbation method was proved earlier /1/ for the plane problem of the theory of elasticity of inhomogeneous bodies on the basis of properties for representing harmonic functions in terms of an integral with a weak singularity and the properties of multidimensional singular integrals. A comparison is given in /2/ between the exact solution and a solution constructed by the perturbation method for a special case of the plane problem, and their agreement in the domain of convergence is noted. Another approach that relies on the theory of integral equations is described in /3/, where an analysis is given of three-dimensional boundary value problems of the linear theory of elasticity and thermoelasticity for homogeneous and piecewise-homogeneous media, including existence and uniqueness theorems for the solutions.

Complete model of the process of propagation of long waves and their interaction with a vertical wall

Asymptotic analysis of the nonlinear three-dimensional boundary-value problem of potential theory is carried out and a complete system of equations describing the process of propagation of long surface waves is obtained. Approximate solutions of the problems for both traveling and standing waves are constructed to the third approximation.

Enhancement of Remotely Sensed Temperature Fields by Wind Observations from a VHF Radar Network

Abstract Ground-based remote sensing devices have recently been developed which provide high-resolution tropospheric wind measurements and coarse-resolution radiometric temperature measurements under most weather conditions. A variational analysis scheme for inferring missing details in the three-dimensional temperature field from concurrent wind observations is proposed. The scheme is based on the solution of a three-dimensional boundary value problem and thus requires input from a network of profilers, rather than one individual instrument. Since observing networks of this kind do not presently exist, the scheme is tested on objectively analyzed, thermally degraded radiosonde data. The ultimate purpose of the thermal enhancement procedure is to improve the dynamic balance between mass and wind fields observed by future ground- or space-based profiler networks and to lessen the initialization shock if these data are used in numerical prediction models.