Solution Of Three-dimensional Boundary Value Problem Research Articles

The peculiarities of the non-axisymmetric frequency spectra of finite elastic cylinders

A rigorous solution of three-dimensional boundary-value problem concerning the forced vibrations of a finite, elastic, isotropic cylinder is constructed analytically by means of the superposition method. With this solution, the resonances in non-propagating waves were investigated. Particularly, a survey of the frequency spectrum for an aluminum cylinder, vibrating with the circumferential order two, reveals the existence of a localized resonance, usually referred to as an end resonance, well below the cut-off frequency of the lowest real dispersion branch of an infinite cylinder. This phenomenon demonstrates the remarkable differences between the axisymmetric and non-axisymmetric end resonances of elastic cylinders. Comparison of the theoretical results with the experiments published elsewhere reveals an excellent agreement.

Solution of a three-dimensional boundary-value problem for a fractional differential heat conduction equation

We consider a three-dimensional axisymmetric problem for a differential heat conduction equation with fractional time derivatives. Using the method of homogeneous solutions and integral transformations, we obtain an asymptotic and a numerical solution of the problem. The results of calculations are presented. In recent years, the interest of researchers in the physicomechanical properties of materials with a fractal and multifractal structure has grown [5, 9]. Highly porous structured media, which are the result of many processes and phenomena of irreversible growth such as diffusion, aggregation, fracture, percolation, dynamic chaos, and dissolution, can serve as examples of fractals. In the literature, there exist several models for description of the mechanical, thermal, and electric properties of fractal media [5, 10]. They contain partial differential equation with fractional derivatives (with respect to space variables). The fractional differentiation index is related to the fractal dimensionality of the space. However, along with geometrical fractals, time fractals are also considered because there exist processes that have fractal nature in time. Among them are processes of charge transfer in amorphous media within the framework of the random-walk model with continuous time, impact wave processes in condensed media, anomalous diffusion, and heat conduction [9–14]. At present, in the literature, there are several approaches to the solution of differential equations with fractional derivatives. In [1], the Monte Carlo method in combination with the scheme of finite differences was used. The methods of integral transformations are efficient [10]. In [13], the problem of anomalous diffusion in an infinite medium with a cylindrical cavity was considered. Some general reasoning on the solution of fractional differential equations were presented in [11]. In the present work, we propose analytic and numerical procedures for the solution of a new threedimensional boundary-value problem of fractal heat conduction for a layer with a through cylindrical cavity. Numerical results that characterize the evolution of temperature with time for different orders of the derivative under the action of pulse load on the surface of the body are presented. Statement of the Problem In a curvilinear Cartesian system Ox 1 x 2 x 3 , we consider a layer �� < x 1 , x 2 <� , x 3 h, containing a

Nano- and micrometer-scale thin-film-interconnection failure theory and simulation and metallization lifetime prediction, Part 1: A general theory of vacancy transport, mechanical-stress generation, and void nucleation under electromigration in relation to multilevel-metallization degeneration and failure

The physical principles are considered of on-chip interconnection degeneration and failure in sub-1-µm technologies from the viewpoint of multilevel-metallization reliability. A general theory and simulation results are presented that deal with electromigration-induced failure of thin-film conducting tracks on the micro- and nanometer scales. They provide a detailed treatment of micro-, meso-, and nanoscopic mechanisms underlying the deformation and breaking of actual interconnection configurations. Parts 1 and 2 contain (i) the derivation and analysis of basic kinetic equations; (ii) the formulation and solution of three-dimensional boundary-value problems representing the transport of vacancies (both in the bulk and along grain boundaries) and the development of deformation and stress; (iii) models of void nucleation and development; (iv) an analysis of a multilevel-metallization lifetime; and (v) the identification of potential break locations in a multilevel metallization in relation to current density, metallization structure and layout, temperature, and the grain structure of the conducting material. Part 3 deals with electromigration in doped on-chip polycrystalline interconnections in terms of lengthening their lifetime. To predict electromigration resistance, models are constructed that represent the influence of texture and other grain-boundary properties on the effective charges of native and dopant ions.

Three-dimensional Coons macroelements in Laplace and acoustic problems

This paper introduces a new global functional set for the FEM solution of three-dimensional boundary-value problems. The main idea is to construct large isoparametric finite elements based on the interpolation formula, which was developed in 1960s by S.A. Coons for the numerical representation of arbitrary solid CAD regions bounded by six curvilinear surfaces. In this way, besides the geometry, Coons interpolation formula is used here for the global interpolation of the unknown potential within the whole solid region (problem area), a procedure that leads to large elements, called “macroelements”. For adequately smooth regions, the degrees of freedom appear only at the 12 boundary edges of the macroelement and can be used in the solution of both static (Laplace) and eigenvalue (acoustic) problems. The proposed approach is sustained by five numerical results where it is successfully compared with conventional finite elements and exact analytical solutions.

On the solution of three-dimensional boundary-value problems for layered bodies with nonplanar interfaces

The interfaces play an important role in various buildup bodies, and also in the composite materials and structural elements. Special monographs [7, 8] have been devoted to this question, presenting the results of scientific studies of the physical and chemical phenomena on the interfaces, the mechanical behavior, and the role of the interfaces in the damage processes, and also their influence on the basic mechanical properties of the composites. In many cases the interfaces deviate from the ideal geometric shapes: planar (in the layered composites), circular cylindrical (in the fibrous composites), and spherical (in the granular composites). Numerous theoretical and experimental studies confirm this. Thus, in the explosive welding of metals (and nonmetals) there form wavy surfaces, the sections of which may be close to sinusoids, for example in the welding of niobium and copper [9]. If the densities of the materials differ significantly, then the sinusoidal nature of the interface distorts as illustrated in [12] for the example of the welding of lead and steel. In addition, in view of the nature of the technological processes [10] the interfaces may become curved in the layered composite materials and deviate locally or periodically from the ideal coordinate planes. Theoretical and experimental studies have shown that the shape of the interface has a significant influence on the physical and mechanical processes and phenomena (bond strength, stress concentration, wave diffraction, thermal conduction, and so on). Numerous publications that are cited in the survey works [1, 3, 11] confirm this. A second variant of the boundary shape perturbation method was developed in [4, 5] for the solution of the three-dimensional boundary-value problems for nonorthogonal surfaces that are close to the coordinate planes. It was assumed that the equations of the interfaces are linear relative to the small parameter characterizing the degree of deviation from the coordinate planes. This narrowed significantly the class of the examined boundary-value problems and their practical importance. In the present work we examine the three-dimensional boundary-value problems of the mechanics of layered bodies with interfaces that are described by nonlinear equations relative to a small parameter. We construct in general form the recurrence relations and the differential operators of the boundary conditions, making it possible to solve the three-dimensional boundary-value problems with the accuracy that is required for applications. We examine particular cases and present one of the possible criteria for evaluating the accuracy of the approximate solutions that are obtained with the aid of the described variant of the boundary shape perturbation method.

Solution of three-dimensional boundary-value problems for laminated noncircular cylinders

UDC 539.3 A second variant of the boundary perturbation method was developed in the monograph [4] to solve three-dimensional boundary-value problems of the mechanics of deformable bodies bounded by nonorthogonal (in particular, noncircular cylindrical) surfaces (surfaces for which conditions of orthogonality are not satisfied at arbitrary points between the unit normal and the unit coordinate vectors). The method describes nonorthogonal surfaces by linear equations in a small parameter characterizing the amplitude of the deviation of the given surface from circular cylindrical shape. Numerous problems solved by the method and the corresponding numerical results were systematized and analyzed in [2, 4]. Although the assumptions made regarding the linearity of the equation of the interface relative to a small parameter simplified the mathematical calculations, it limited the generality of the approach, the range of problems that can be solved, and their practical value. Some of these deficiencies are corrected in the present study. Here, we generalize the second variant of the boundary-perturbation method to the case when the noncircular cylindrical interfaces of a laminated body are described by nonlinear parametric equations.

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.

Numerical determination of the fundamental eigenvalue for the Laplace operator on a spherical domain

Methods for obtaining approximate solutions for the fundamental eigenvalue of the Laplace-Beltrami operator (i.e., the membrane eigenvalue problem for the vibration equation) on the unit spherical surface are developed. Two types of spherical surface domains are considered: (1) the interior of a spherical triangle, and (2) the exterior of a great circle arc extending for less thanπ radians (a spherical surface with a slit). In both cases, zero boundary conditions are imposed. In order to solve the resulting second-order elliptic partial differential equations in two independent variables, a finite difference approximation is employed. The fundamental eigenvalue is approximated by iteration utilizing the power method and point successive overrelaxation. Some numerical results are given and compared, in certain special cases, with analytical solutions to the eigenvalue problem. The significance of the numerical eigenvalue results is discussed in terms of the singularities in the solution of three-dimensional boundary-value problems near a polyhedral corner of the domain.