Some fundamental problems of the theory of thermoviscoelasticity

Methods of solving quasistatic problems of the linear theory of thermoviscoelasticity are discussed. Special attention is given to the method of approximations. Modern methods of solving contact problems with a variable boundary and problems with a time-dependent boundary are briefly reviewed. Methods of solving problems with a nonuniform temperature field are outlined. The basic relations of the nonlinear theory of thermoviscoelasticity are given. Its various modifications and simplifications associated with the introduction of fairly general assumptions are examined. Methods of solving problems in certain nonlinear theories are noted. Nonisothermal (coupled) problems of thermoviscoelasticity and questions relating to the general theory with physical and geometric nonlinearity are discussed.Lomonosov Moscow State University. Translated from Mekhanika Polimerov, No. 1, pp. 41–58, January–February, 1971.     

An approximate method of solving dynamical problems of thermoviscoelasticity

A generalization of Whitham's asymptotic method is proposed for solving systems of integro-differential equations describing the dynamic behavior of bodies composed of nonlinear viscoelastic materials. The possibility of using this approach to solve coupled problems of thermoviscoelasticity is indicated. A number of examples are considered including the propagation of a harmonic disturbance in a rod of nonlinear viscoelastic material under isothermal conditions and the coupled problem of the high-frequency vibration of a rod of linear viscoelastic material, whose properties depend on temperature in accordance with the time-temperature superposition principle.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Mekhanika Polimerov, No. 1, pp. 58–65, January–February, 1973.     

The Optimal Homotopy Analysis Method for solving linear optimal control problems

In this paper, an Optimal Homotopy Analysis Method (Optimal HAM) is applied to solve the linear optimal control problems (OCPs), which have a quadratic performance index. This approach contains at most two convergence-control parameters which depend on the control system and is computationally rather efficient. A squared residual error for the system is defined, which can be used to find the unknown optimal convergence-control parameters by using Mathematica package BVPh (version 2.0). The results of comparisons among the proposed method, the homotopy perturbation method (HPM), the Adomian decomposition method (ADM), the differential transform method (DTM) and the homotopy analysis method (HAM) provide verification for the validity of the proposed approach. Moreover, numerical results are presented by several examples involving scalar and 2nd-order systems to clarify the efficiency and high accuracy of the proposed approach.     

Method of elastic solutions of problems of thermoviscoelasticity

The quasi-static mixed boundary value problem is considered for a viscoelastic body with time-dependent properties and a singular nondegenerate relaxation kernel. The existence of a time-temperature (or similar) analogy is not assumed. A method of reducing the viscoelastic problem to the solution of a series of uniform elastic problems is proposed and its convergence proved. The method is illustrated with reference to examples of solutions of specific viscoelastic problems.Lomonosov Moscow State University. Translated from Mekhanika Polimerov, No. 4, pp. 603–614, July–August, 1970.     

On a method of solving two-dimensional problems of the linear theory of viscoelasticity

We describe the basic propositions of the linear theory of viscoelasticity. We give transformation formulas for the resolvent integral operators of viscoelasticity with an arbitrary analytic kernel of difference type. The method of computing the irrational operator functions is illustrated by determining the real parameters of the two-dimensional stressed state of an orthotropic plate. Three figures. Bibliography: 7 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 27, 1997, pp. 86–96.     

Applying mod-k-cuts for solving linear ordering problems

The linear ordering problem consists of finding an ordering of the nodes of the weighted complete digraph on n nodes such that the sum of the weights of the arcs compatible with the ordering is maximized. In this paper, we report about the usefulness of mod-k cuts in a branch-and-cut algorithm for solving linear ordering problems to optimality.      

Coupled problems of thermoviscoelasticity

The author derives a heat flux equation with allowance for the dissipation of internal forces on the basis of the fundamental equations of continuum mechanics and the thermodynamics of irreversible processes.Moscow Lomonosov State University. Translated from Mekhanika Polimerov, No. 3, pp. 415–421, May–June, 1969.     

New multiplier method for solving linear complementarity problems

A new multiplier method for solving the linear complementarity problem LCP(q, M) is proposed. By introducing a Lagrangian of LCP(q, M), a new smooth merit function ϑ(x, λ) for LCP(q, M) is constructed. Based on it, a simple damped Newton-type algorithm with multiplier self-adjusting step is presented. When M is a P-matrix, the sequence {ϑ(x ^k, λ ^k)} (where {(x ^k, λ ^k)} is generated by the algorithm) is globally linearly convergent to zero and convergent in a finite number of iterations if the solution is degenerate. Numerical results suggest that the method is highly efficient and promising. Selected from Numerical Mathematics (A Journal of Chinese Universities), 2004, 26(2): 162–171     

Sinc-Galerkin method for solving linear sixth-order boundary-value problems

There are few techniques available to numerically solve sixth-order boundary-value problems with two-point boundary conditions. In this paper we show that the Sinc-Galerkin method is a very effective tool in numerically solving such problems. The method is then tested on examples with homogeneous and nonhomogeneous boundary conditions and a comparison with the modified decomposition method is made. It is shown that the Sinc-Galerkin method yields better results.

     

Numerical methods for solving linear least squares problems

A common problem in a Computer Laboratory is that of finding linear least squares solutions. These problems arise in a variety of areas and in a variety of contexts. Linear least squares problems are particularly difficult to solve because they frequently involve large quantities of data, and they are ill-conditioned by their very nature. In this paper, we shall consider stable numerical methods for handling these problems. Our basic tool is a matrix decomposition based on orthogonal Householder transformations.Reproduction in Whole or in Part is permitted for any Purpose of the United States government. This report was supported in part by Office of Naval Research Contract Nonr-225(37) (NR 044-11) at Stanford University.     

A linearized theory of thermoviscoelasticity

A generalized linearized theory of thermoviscoelasticity, including the effect of heat formation, is presented. The linearized equations of motion, of state, and for the energy are given together with the linearized boundary conditions for large initial deformations. Attention is drawn to the fact that the equations which have been derived can be used for the solution of problems concerning the stability of viscoelastic bodies, the propagation of waves in viscoelastic materials which are subjected to deformation, and problems concerning the stress-deformed state of viscoelastic elements. The problem of the propagation of plane waves in viscoelastic materials which are subjected to deformation is considered as an example.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Mekhanika Polimerov, No. 2, pp. 214–221, March–April, 1972.     

Studies in the theory of thermoviscoelasticity

Recent work on thermoviscoelasticity in the Department of Theory of Elasticity of the Mechanics and Mathematics Faculty of M. V. Lomonosov Moscow State University is reviewed.     

Potential method in the steady vibrations problems of the theory of thermoviscoelasticity for Kelvin-Voigt materials with voids

In this paper the linear theory of thermoviscoelasticity for Kelvin-Voigt materials with voids is considered. The basic internal and external boundary value problems (BVPs) of steady vibrations are formulated. The uniqueness and existence theorems for classical solutions of the above mentioned BVPs are proved by using the potential method (boundary integral equation method) and the theory of singular integral equations. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Use of the results of auxiliary experiments in solving problems in the linear theory of viscoelasticity

The approximate method of solving problems of the theory of linear viscoelasticity with arbitrary creep and relaxation kernels, proposed in [2], is substantiated and generalized. The essence of this method consists in the approximation of the functions depending on the Laplace — Carson transforms of the mechanical characteristics of a viscoelastic body by means of certain combinations of the transforms of the creep and relaxation kernels. The expressions obtained as a result of the approximation enable the inverse transforms of the unknown functions to be found without difficulty.Moscow Lomonosov State University. Translated from Mekhanika Polimerov, Vol. 4, No. 6, pp. 963–969, November–December, 1968.     

Quasilinearization method for coupled dynamic problems of thermoviscoelasticity

Use of the quasilinearization method is proposed for the solution of coupled dynamic (particularly, quasistatic) problems of thermoviscoelasticity under cyclic loading. The coupled problems under consideration here include vibrations of a one-dimensional body (beam, plate, shell) and shear vibrations of a hollow cylinder made of a material with temperature-dependent properties, with the principle of temperature-time analogy applicable in the latter case. The quasilinearization method is shown to be a fast converging one when applied to the solution of these problems.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Mekhanika Polimerov, No. 2, pp. 310–316, March–April, 1976.     

A genetic algorithm for solving linear fractional bilevel problems

Bilevel programming has been proposed for dealing with decision processes involving two decision makers with a hierarchical structure. They are characterized by the existence of two optimization problems in which the constraint region of the upper level problem is implicitly determined by the lower level optimization problem. In this paper a genetic algorithm is proposed for the class of bilevel problems in which both level objective functions are linear fractional and the common constraint region is a bounded polyhedron. The algorithm associates chromosomes with extreme points of the polyhedron and searches for a feasible solution close to the optimal solution by proposing efficient crossover and mutation procedures. The computational study shows a good performance of the algorithm, both in terms of solution quality and computational time.     

Collocation methods for solving linear differential-algebraic boundary value problems

Summary. We consider boundary value problems for linear differential-algebraic equations with variable coefficients with no restriction on the index. A well-known regularisation procedure yields an equivalent index one problem with d differential and a=n-d algebraic equations. Collocation methods based on the regularised BVP approximate the solution x by a continuous piecewise polynomial of degree k and deliver, in particular, consistent approximations at mesh points by using the Radau schemes. Under weak assumptions, the collocation problems are uniquely and stably solvable and, if the unique solution x is sufficiently smooth, convergence of order min {k+1,2k-1} and superconvergence at mesh points of order 2k-1 is shown. Finally, some numerical experiments illustrating these results are presented. Received October 1, 1999 / Revised version received April 25, 2000 / Published online December 19, 2000     

Nonlinear relaxation methods for solving symmetric linear complementarity problems

A family of iterative algorithms is presented for the solution of the symmetric linear complementarity problem,

Computational methods for solving two-stage stochastic linear programming problems

Approximating a given continuous probability distribution of the data of a linear program by a discrete one yields solution methods for the stochastic linear programming problem with complete fixed recourse. For a procedure along the lines of [8], the reduction of the computational amount of work compared to the usual revised simplex method is figured out. Furthermore, an alternative method is proposed, where by refining particular discrete distributions the optimal value is approximated.

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Zusammenfassung Für das zweistufige Modell der stochastischen linearen Programmierung mit vollständiger Kompensation werden Verfahren untersucht, die sich aus der Annäherung einer gegebenen stetigen Wahrscheinlichkeitsverteilung der Daten durch endlich diskrete Verteilungen ergeben. Beim Vorgehen nach [8] wird die Reduktion des Rechenaufwandes im Vergleich zur üblichen revidierten Simplexmethode ermittelt. Als Alternative wird ein Verfahren vorgeschlagen, in dem durch sukzessive Verfeinerung speziell gewählter diskreter Verteilungen der Optimalwert monoton angenähert wird.

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Herrn Prof. Dr. Eduard Stiefel in kollegialer Verbundenheit