The axisymmetric problem of thermoelasticity of a multilayer thermosensitive tube

We propose a method of solving stationary heat-conduction problems of contacting bodies with coefficient of thermal conductivity that are linear functions of the temperature and the corresponding problems of thermoelasticity based on the method of perturbations. We give a numerical analysis of the thermal stresses in a two-layer tube. Translated fromMatematichni Metodi ta Fiziko-mekhanichni Polya, Vol. 39, No. 1, 1996, pp. 97–103.     

Simple exact solutions applicable to microwave heating

Summary There is presently considerable interest in the utilization of microwave heating to novel industrial applications. Mathematically such problems involve Maxwell's equations coupled with the heat equation and for which all thermal, electrical and magnetic properties of the material are nonlinearly dependent upon temperature. Accordingly such problems are highly complex and very little theoretical work has been undertaken. The purpose of this paper is to obtain simple exact solutions applicable to microwave heating in the simplest situation, involving only one spatial dimension and assuming that all thermal, electrical and magnetic properties exhibit a power law dependence on temperature. Similarity solutions and other special solutions are examined. These generally result in highly nonlinear coupled systems of ordinary differential equations and although some new closed results are obtained in special cases, in general, such complex systems of ordinary differential equations need to be solved numerically. Roughly speaking, we show that stretching similarity solutions exist only if the power law electrical and thermal conductivities and magnetic permeability with indicesl, m andn respectively are such thatl+m+n=0. Similar constraints on the indices apply for the existence of other simple solutions.     

Solution of nonstationary heat-conduction problems for thermosensitive bodies under convective heat exchange

We propose a technique for solving nonstationary heat-conduction problems for thermosensitive bodies with simple nonlinearity (the coefficients of thermal conductivity and the heat capacity per unit volume depend on temperature, but the coefficient of thermal diffusivity is constant) heated by convective heat exchange from the surrounding medium. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 2, 1997, pp. 148–152.     

Sur un probleme d'optimisation de domaine

Let Ω be a bounded open subset of R ⁿ; with every measurable subset A of Ω is associated the real number En(A), related to capacity (for electrostatic problems) or to heat flux (for problems of thermal conduction). One investigates the minimization of En(A) when the measure of A is imposed. Penalty methods allows one to apply some techniques of convex analysis.     

The thermoelastic problem for steady-state vibrations of a plante in three-dimensional formulation

The perturbation method is used to reduce a coupled three-dimensional problem of stationary vibrations of thermoelasticity to a number of three-dimensional vibration problems of the theory of thermal stresses. We develop an algorthm for constructing the solution in the zeroth approximation on the basis of the method of homogeneous solutions. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 23, 1992, pp. 119–127.     

Error estimates of finite element methods for nonstationary thermal convection problems with temperature-dependent coefficients

Summary. General error estimates are proved for a class of finite element schemes for nonstationary thermal convection problems with temperature-dependent coefficients. These variable coefficients turn the diffusion and the buoyancy terms to be nonlinear, which increases the nonlinearity of the problems. An argument based on the energy method leads to optimal error estimates for the velocity and the temperature without any stability conditions. Error estimates are also provided for schemes modified by approximate coefficients, which are used conveniently in practical computations.Mathematics Subject Classification (2000): 65M12, 65M60, 76M10     

Determination of nonstationary thermal fields by the Ritz method with orthogonalization

The authors consider an application of the Ritz, method with orthogonalization to the determination of a nonstationary thermal field. In order to obtain a sequence of elliptic boundary-value problems they apply a subtle difference scheme to the time variable. An estimate is obtained for a linear minimization functional in an appropriate energy space.Translated from Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 29, pp. 55–58, 1989.     

Dynamics of a free boundary in a binary medium with variable thermal conductivity

We construct an asymptotic solution of the phase field system with variable thermal conductivity different in domains occupied by different phases. We show that, depending on relations between parameters characterizing the substance, the dynamics of the free interface between the phases is determined by solutions of the classical or modified Stefan problems. Translated fromMatematicheskie Zametki, Vol. 66, No. 2, pp. 231–241, August, 1999.     

Mathematical modeling of heat conduction processes for structural elements of nuclear power plants by the method of <Emphasis Type="Italic">R</Emphasis>-functions

Mathematical models of heat exchange in the motion of an incompressible viscous liquid through channels with a helical type of symmetry, including polyzonal finning of fuel elements, in curvilinear nonorthogonal coordinates are proposed. For a laminar flow in the domain of thermal stabilization, we reduce the three-dimensional problem to the two-dimensional problem and investigate the influence of a twist parameter on the distribution of a temperature field. Numerical solutions of model multiparameter problems of calculation of stationary temperature fields by the R-function method (RFM) are considered, which enables us to choose appropriate structural means of the RFM for the subsequent solution of real problems.     

Thermoelasticity problems in the mechanics of composite materials with large-scale deformation of the filler

A problem is posed and a method of solution developed for a class of thermoelasticity problems in the mechanics of composite materials with large-scale structural deformations, in which in the continuum approximation the composite material is modelled by a homogeneous orthotropic body with curvilinear orthotropy. As an application the authors establish the foundations of a theory of buckling for planar structural elements subjected to constant thermal fields.Translated from Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 29, pp. 16–25, 1989.     

Control-theoretic properties of structural acoustic models with thermal effects,I. Singular estimates

We consider a class of structural acoustics models with thermoelastic flexible wall. More precisely, the PDE system consists of a wave equation (within an acoustic chamber) which is coupled to a system of thermoelastic plate equations with rotational inertia; the coupling is strong as it is accomplished via boundary terms. Moreover, the system is subject to boundary thermal control. We show that—under three different sets of coupled (mechanical/thermal) boundary conditions—the overall coupled system inherits some specific regularity properties of its thermoelastic component, as it satisfies the same singular estimates recently established for the thermoelastic system alone. These regularity estimates are of central importance for (i) well-posedness of Differential and Algebraic Riccati equations arising in the associated optimal control problems, and (ii) existence of solutions to the semilinear initial/boundary value problem under nonlinear boundary conditions. The proof given uses as a critical ingredient a sharp trace theorem pertaining to second-order hyperbolic equations with Neumann boundary data.     

On the structure of the stresses inside an ellipsoidal irregularity in thermoelastic problems

We study the structure of the stresses inside an ellipsoidal irregularity. We obtain a result that generalizes the theorem on polynomial conservation for thermoelastic problems under homogeneous and linear external stress and temperature fields when there is perfect thermal and mechanical contact on the phase interface. We show by a specific example how significantly the stress structure inside the irregularity changes when the form of interaction on the boundary surface changes.Translated fromVychislitel'naya i Prikladnaya Matematika, Issue 71, 1990, pp. 70–77.     

Exact bounds for the solution branches of nonlinear eigenvalue problems

Summary A method is constructed which yields a strip containing the full solution sets of nonlinear eigenvalue problems of the formu=Tu.The strip can be narrowed iteratively, and the method applies for both stable and unstable branches. Its high degree of accuracy is demonstrated by numerical examples. In particular, a lower bound is given for the critical value at which criticality is lost in the thermal ignition problem for the unit ball.     

Thermal fracture of functionally gradient ceramics

An edge crack in a strip of functionally gradient ceramics (FGC) is studied under thermal loading conditions. Two FGC materials are considered, i.e., one with a spatial variation of shear modulus and the other with a spatial variation of thermal conductivity. Thermal stress intensity factors (TSIF) are numerically calculated based on singular integral equations derived for the dislocation density along the crack faces. It is shown that: (a) for the FGC with a graded shear modulus, the TSIF are reduced for crack lengths longer thanl _c b and remain approximately the same as those of a homogeneous material for shorter crack lengths, wherel _c is about 0.065 andb is the width of the strip; and (b) for the FGC with a thermal conductivity gradient, the TSIF are generally lower compared with those for the bonded two-layer material.     

Generalized p-Center problems: Complexity results and approximation algorithms

In an earlier paper, two alternative p-Center problems, where the centers serving costumers must be chosen so that exactly one node from each of p prespecified disjoint pairs of nodes is selected, were shown to be NP-complete. This paper considers a generalized version of these problems, in which the nodes from which the p servers are to be selected are partitioned into k sets and the number of servers selected from each set must be within a prespecified range. We refer to these problems as the ‘Set’ p-Center problems. We establish that the triangle inequality (Δ-inequality) versions of these problems, in which the edge weights are assumed to satisfy the triangle inequality, are also NP-complete. We also provide a polynomial time approximation algorithm for the two Δ-inequality Set p-Center problems that is optimal for one of the problems in the sense that no algorithm with polynomial running time can provide a better constant factor performance guarantee, unless P = NP. For the special case ‘alternative’ p-Center problems, which we refer to as the ‘Pair’ p-Center problems, we extend the previous results in several ways. For example, the results mentioned above for the Set p-Center problems also apply to the Pair p-Center problems. Furthermore, we establish and exploit a correspondence between satisfiability and the dominating set type of problems that naturally arise when considering the decision versions of the Pair p-Center problems.     

On <Emphasis Type="Italic">G</Emphasis>-invex multiobjective programming. Part II. Duality

This paper represents the second part of a study concerning the so-called G-multiobjective programming. A new approach to duality in differentiable vector optimization problems is presented. The techniques used are based on the results established in the paper: On G-invex multiobjective programming. Part I. Optimality by T.Antczak. In this work, we use a generalization of convexity, namely G-invexity, to prove new duality results for nonlinear differentiable multiobjective programming problems. For such vector optimization problems, a number of new vector duality problems is introduced. The so-called G-Mond–Weir, G-Wolfe and G-mixed dual vector problems to the primal one are defined. Furthermore, various so-called G-duality theorems are proved between the considered differentiable multiobjective programming problem and its nonconvex vector G-dual problems. Some previous duality results for differentiable multiobjective programming problems turn out to be special cases of the results described in the paper.     

Piecewise optimal distributed Controls for 2D Boussinesq equations

In this article, we study the dynamics of a piecewise (in time) distributed optimal control problem for the Boussinesq equations which model velocity tracking over time coupled to thermal dynamics. We also study the dynamics of semidiscrete approximation of this problem. We prove that the rates of velocity tracking coupled to thermal dynamics are exponential and that the difference between the solution of the semi‐discrete piecewise optimal control problem and the desired states in L² and H¹ norms decay to zero exponentially as n→∞. Copyright © 2000 John Wiley & Sons, Ltd.     

Solvability conditions and monotone iterative scheme for boundary-value problems related to nonlinear monotone potential operators

This article deals with boundary-value problems (BVPs) for the second-order nonlinear differential equations with monotone potential operators of type Au := ??(k(|?u|²)?u(x)) + q(u ²)u(x), x ∈ Ω ? R ⁿ . An analysis of nonlinear problems shows that the potential of the operator A as well as the potential of related BVP plays an important role not only for solvability of these problems and linearization of the nonlinear operator, but also for the strong convergence of solutions of corresponding linearized problems. A monotone iterative scheme for the considered BVP is proposed.     

The conditioning of the maximum entropy covariance matrix and its inverse

The maximum entropy covariance matrix is positive definite even when the number of variables p exceeds the sample size n. However, the inverse of this matrix can have stability problems when p is close to n, although these problems tend to disappear as p increases beyond n. We analyze such problems using the variance of the latent roots in a particular metric as a condition number.