Contact problems of thermoelasticity for a half-space of a functionally gradient material

We propose a method of solution of problems of thermoelasticity for an inhomogeneous half-space. We assume that Poisson's ratio of a material in the half-space is constant, and Young's modulus and the coefficients of linear thermal expansion and thermal conduction vary exponentially with distance from the surface of the half-space. As an example, we consider the contact problem on sliding an inhomogeneous body along the surface of a rigid base with regard for frictional heating. I. Franko L'viv University, L'viv; Warsaw University, Warsaw. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 2, pp. 45–56, April–June, 1998.     

Boundary-value contact problem of thermoelastoplasticity for a two-layer eccentric cylindrical pipe

We formulate the plane two-dimensional static boundary-value contact problem of thermoelastoplasticity for a two-layer eccentric cylindrical pipe under the action of a temperature field and compressive normal stresses that are uniformly distributed on its lateral surfaces and present its approximate solution. We assume that the mechanical and thermophysical properties of the materials are temperature-independent, plastic strains arise on the interior lateral surface of the two-layer pipe and completely envelop it, and the material of the pipe is perfectly elastoplastic, incompressible in the domain of plasticity, and satisfies the Tresca-Saint-Venant plasticity condition. I. Franko L'viv State University, L'viv. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 2, pp. 57–66, April–June, 1998.     

On fundamental solutions of the Cauchy problem for a class of degenerate parabolic equations

We present the results of an investigation and some applications of fundamental solutions of the Cauchy problem for a new class of parabolic equations. In these equations: (i) there exist three groups of spatial variables, one basic and two auxiliary, (ii) different weights of spatial variables from the basic group with respect to the time variable are admitted, (iii) degeneracies in variables from the auxiliary groups are present, (iv) a degeneracy on the initial hyperplane is present. Pidstryhach Institute of Applied Problems in Mechanics and Mathematics, Ukrainian Academy of Sciences, L'viv; Ternopil' Academy of Economics, Ternopil'. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 2, pp. 13–19, April–June, 1998.     

Continuity equations for a deformable solid

For three-dimensional problems of the mechanics of a deformable solid in a Cartesian coordinate system, we show that there are only three continuity equations in terms of deformations and not six, as was considered earlier. We obtain continuity equations of the integro-differential type. They are reduced to the corresponding three of the six well-known Saint-Venant continuity differential equations only if the necessary conditions of consistency between displacements and deformations at the domain boundary are satisfied. As a results, the problem of closedness of the equations of elasticity and thermoelasticity in terms of stresses for three-dimensional problems as well as the equations of the mechanics of a deformable solid is finally solved as a whole. Pidstryhach Institute of Applied Problems in Mechanics and Mathematics, Ukrainian Academy of Sciences, L'viv. Translated from Matermatychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 2, pp. 117–123, April–June, 1998.     

Appearance and redistribution of internal stresses in coatings in the process of their application

We describe an original method for experimental determination of the formation of and changes in internal stresses that appear in electrometallized coatings in the process of their application. The regularities of changes in internal stresses depending on the thickness of coatings and composition of the blend of a powder wire are investigated. It is shown that residual tension stresses in coatings relax by means of plastic strain of low-strength structural components and/or due to the initiation and growth of cracks in brittle components. The characteristics of strength and crack resistance of coatings can be considerably improved by optimizing the composition of the blend of the powder wire. Karpenko Physicomechanical Institute, Ukrainian Academy of Sciences, L'viv. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 2, pp. 105-110. April–June, 1998     

Inverse problem of nondestructive optical testing of residual stresses in cyclindrical shells

We considered the inverse problem of determination of the stress-strain state of a circular cylindrical shell that consists of two heterogeneous parts joined by a circular seam. As the input data for this problem, the results of polarization-optical measurements are used. The problem is formulated within the framework of a model that considers residual stress as caused by the geometric incompatibility of elastic deformations. Pidstryhach Institute of Applied Problems in Mechanics and Mathematics, Ukrainian Academy of Sciences, L'viv. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 2, pp. 151–156, April–June, 1998.     

Branching of solutions of the problem of synthesis of a linear antenna with given directional amplitude diagram in a Hölder space

We investigate the branching of solutions of a nonlinear integral equation of the Hammerstein type which arises in the problem of synthesis of a linear antenna with given directional amplitude diagram. Systems of transcendental equations for determination of branching points of various types are deduced, and the number of branched solutions and their qualitative characteristics are analyzed. Pidtryhach Institute of Applied Problems in Mechanics and Mathematics, Ukrainian Academy of Sciences, L'viv. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 2, pp. 35–44, April–June, 1998.     

Generalized theorem on positive definiteness of energy in the general relativity theory

We introduce the notion of the energy of gravitational and material fields with respect to an arbitrary time-like vector field and a space-like hypersurface as an integral over a nonholonomic hypersurface. In the case of an asymptotically Minkowskian space, by developing the approach of Witten and Nester, we obtain an expression for the energy functional in terms of the spinors associated with a differential-geometric distribution. By applying the Sen-Witten generalized equation, we prove the nonnegativity of this functional. Pidstryhach Institute of Applied Problems in Mechanics and Mathematics, Ukrainian Academy of Sciences. L'viv. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 2, pp. 26–34, April–June, 1998.     

Mathematical model of an internal crack in an elastoplastic cylindrical shell

We propose a mathematical model that makes it possible to reduce the problem of the stressed state and limit equilibrium of a cylindrical anisotropic elastoplastic shell with internal crack to a system of nonlinear singular integral equations with discontinuous functions on the right-hand sides. We construct an algorithm for numerical solution of such systems together with the conditions of plasticity and boundedness of stresses near the crack. For a transversally isotropic shell, we carry out a numerical analysis of the dependence of the opening of the internal crack front on the load and geometric and mechanical parameters. Pidstryhach Institute of Applied Problems in Mechanics and Mathematics, Ukrainian Academy of Sciences L'viv. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 2, pp. 111–116, April–June, 1998     

Investigation of contact corrosion of anodic coatings depending on the type of polarization of materials of the base and coating

We propose a refined mathematical model for the fracture mechanism and prediction of durability of anodic coatings in seawater. Various types of polarization of materials of a coating and a base in electrolyte are considered. We investigate the influence of the thickness of electrolyte on fracture of anodic coatings. The results of the quantitative investigations performed enable us to propose certain recommendations concerning the optimal choice of the material of a coating depending on the geometric sizes of possible through damages in it. For noninteracting small strips-damages, we established the existence of a critical thickness of the electrolyte, for which the density of the corrosion current increased by 30–35%. Pidstryhach Institute of Applied Problems in Mechanics and Mathematics, Ukrainian Academy of Sciences, L'viv. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 2, pp. 90–95, April–June, 1998.     

Minimizing a quasi-concave function subject to a reverse convex constraint

Any constraintg(x)0 is called a reverse convex constraint ifg: R ⁿ R ¹ is a continuous convex function. This paper establishes a finite method for finding an optimal solution to a concave program with an additional reverse convex constraint. The method presented is a new approach to global optimization problems since it combines the idea of the branch and bound method with the idea of the cutting plane method.This paper is dedicated to Professor A. Pelczar     

Inference of a minimum size boolean function from examples by using a new efficient branch-and-bound approach

This paper deals with the problem of identifying a hidden Boolean function : 0, 1' 0, 1 from positive and negative examples. This problem is of paramount importance in many real life applications of artificial intelligence. The method proposed in this paper is based on a branch-and-bound approach. This approach is an expansion of some earlier work (Triantaphyllouet al., 1994). Computational results, comparing the new method with one based on Karmakar's interior point method, suggest that the new method is very efficient.     

Convergence rate of a penalty-function scheme

The penalty-function approach is an attractive method for solving constrained nonlinear programming problems, since it brings into play all of the well-developed unconstrained optimization techniques, If, however, the classical steepest-descent method is applied to the standard penalty-function objective, the rate of convergence approaches zero as the penalty coefficient is increased to yield a close approximation to the true solution.In this paper, it is shown that, ifm+1 steps of the conjugate-gradient method are successively repeated (wherem is the number of constraints), the convergence rate when applied to the penalty-function objective conveges at a rate predicted by the second derivative of the Lagrangian. This rate is independent of the penalty coefficient and, hence, the scheme yields reasonable convergence for a first-order method.This research was supported by National Science Foundation, Grant No. NSF-GK-1683.     

Nonlinear programming without a penalty function or a filter

A new method is introduced for solving equality constrained nonlinear optimization problems. This method does not use a penalty function, nor a filter, and yet can be proved to be globally convergent to first-order stationary points. It uses different trust-regions to cope with the nonlinearities of the objective function and the constraints, and allows inexact SQP steps that do not lie exactly in the nullspace of the local Jacobian. Preliminary numerical experiments on CUTEr problems indicate that the method performs well.      

Asymptotic error estimates of a linearized projection-difference method for a differential equation with a monotone operator

We study a projection-difference method of solving the Cauchy problem for an operatordifferential equation with a selfadjoint leading operator A(t) and a nonlinear monotone subordinate operator K(·) in a Hilbert space. This method leads to a solution of a system of linear algebraic equations at each time level. Error estimates are derived for approximate solutions as well as for fractional powers of the operator A(t). The method is applied to a model parabolic problem.     

Superconvergence of a stabilized finite element approximation for the Stokes equations using a local coarse mesh L2 projection

This article first recalls the results of a stabilized finite element method based on a local Gauss integration method for the stationary Stokes equations approximated by low equal‐order elements that do not satisfy the inf‐sup condition. Then, we derive general superconvergence results for this stabilized method by using a local coarse mesh L² projection. These supervergence results have three prominent features. First, they are based on a multiscale method defined for any quasi‐uniform mesh. Second, they are derived on the basis of a large sparse, symmetric positive‐definite system of linear equations for the solution of the stationary Stokes problem. Third, the finite elements used fail to satisfy the inf‐sup condition. This article combines the merits of the new stabilized method with that of the L² projection method. This projection method is of practical importance in scientific computation. Finally, a series of numerical experiments are presented to check the theoretical results obtained. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 28: 115‐126, 2012     

Point force in a stratified fluid in a cylinder under a magnetic field

The flow induced by a body moving in an inviscid incompressible density stratified fluid in an infinite circular cylinder under the influence of a uniform axial magnetic field is studied using the method of replacing the body by an isolated point force. This method was adopted by Childress and others in discussing the body effects in a viscous fluid. The solution is obtained using the Fourier transformation and the Lighthill’s radiation condition. The cases of weak and strong magnetic fields are discussed.     

Analysis of a fourth-order compact ADI method for a linear hyperbolic equation with three spatial variables

This paper is concerned with a three-level alternating direction implicit (ADI) method for the numerical solution of a 3D hyperbolic equation. Stability criterion of this ADI method is given by using von Neumann method. Meanwhile, it is shown by a discrete energy method that it can achieve fourth-order accuracy in both time and space with respect to H ¹- and L ²-norms only if stable condition is satisfied. It only needs solution of a tri-diagonal system at each time step, which can be solved by multiple applications of one-dimensional tri-diagonal algorithm. Numerical experiments confirming the high accuracy and efficiency of the new algorithm are provided.     

Nonsmooth equations approach to a constrained minimax problem

An equivalent model of nonsmooth equations for a constrained minimax problem is derived by using a KKT optimality condition. The Newton method is applied to solving this system of nonsmooth equations. To perform the Newton method, the computation of an element of the b-differential for the corresponding function is developed.This work has been supported by Shanghai Education Committee (04EA01).This revised version was published online in April 2005 with a corrected missing date string.     

Semilocal convergence of a continuation method in Banach spaces

This paper is concerned with the semilocal convergence of a continuation method between two third-order iterative methods, namely, the Halley’s and the convex acceleration of Newton’s method, also known as the Super-Halley’s method. This convergence analysis is discussed using the recurrence relations approach. This approach simplifies the analysis and leads to improved results. The convergence analysis is established under the assumption that the second Frëchet derivative satisfies Lipschitz continuity condition. An existence-uniqueness theorem is given. Also, a closed form of error bound is derived in terms of a real parameter α ∈ [0, 1]. Two numerical examples are worked out to demonstrate the efficacy of our approach. On comparing the existence and uniqueness region and error bounds for the solution obtained by our analysis with those obtained by using majorizing sequences [15], we observed that our analysis gives better results. Further, we have observed that for particular values of the α, our analysis reduces to those for the Halley’s method (α = 0) and the convex acceleration of Newton’s method (α = 1), respectively, with improved results.