Homogenization of the signorini boundary-value problem in a thick plane junction

We consider a mixed boundary-value problem for the Poisson equation in a plane thick junction Ω_ε that is the union of a domain Ω₀ and a large number of ε-periodically located thin rods. The nonuniform Signorini conditions are given on the vertical sides of the thin rods. The asymptotic analysis of this problem is made as ε → 0, i.e., in the case where the number of thin rods infinitely increases and their thickness tends to zero. With the help of the integral identity method, we prove the convergence theorem and show that the nonuniform Signorini conditions are transformed (as ε → 0) into the limiting variational inequalities in the domain that is filled up with thin rods when passing to the limit. The existence and uniqueness of a solution to this nonstandard limit problem are established. The convergence of the energy integrals is proved as well. Published in Neliniini Kolyvannya, Vol. 12, No. 1, pp. 44–58, January–March, 2009.     

Solution of the inverse boundary-value problem of aerohydrodynamics with allowance for a boundary layer

The problem of constructing the contour of a wing profile in a viscous (incompressible or compressible) flow from the velocity distribution, given in terms of the arc abscissa, is solved in the approximation of boundary layer theory. The solvability conditions are obtained. Numerical calculations are carried out. Wing profile contours are constructed from velocity distributions that ensure the nonseparation of the flow. The effect of viscosity and compressibility on the solution of the problem is considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 28–32, July–August, 1989.The authors are grateful to G. Yu. Stepanov for useful discussions.     

Homogenization of the Robin problem in a thick multilevel junction

We consider a mixed boundary-value problem for the Poisson equation in a plane two-level junction that is the union of a domain ₀ and a large number 2N of thin rods with variable thickness of order = (N ^–1). The thin rods are divided into two levels, depending on their length. In addition, the thin rods from each level are -periodically alternated. We investigate the asymptotic behavior of the solution as 0 under the Robin conditions on the boundaries of the thin rods. By using some special extension operators, a convergence theorem is proved.Published in Neliniini Kolyvannya, Vol. 7, No. 3, pp. 336–355, July–September, 2004.     

Asymptotic expansion of solutions of a singularly perturbed boundary-value problem

For linear singularly perturbed systems of ordinary differential equations, we construct an asymptotic expansion of a solution by using the method of boundary functions. Using pseudoinverse matrices and projections, we find all terms of the asymptotic expansion in the noncritical case.Translated from Neliniini Kolyvannya, Vol. 7, No. 2, pp. 155–168, April–June, 2004.     

Asymptotic analysis of boundary-value problems in thin perforated domains with rapidly varying thickness

For a second-order symmetric uniformly elliptic differential operator with rapidly oscillating coefficients, we study the asymptotic behavior of solutions of a mixed inhomogeneous boundary-value problem and a spectral Neumann problem in a thin perforated domain with rapidly varying thickness. We obtain asymptotic estimates for the differences between solutions of the original problems and the corresponding homogenized problems. These results were announced in Dopovidi Akademii Nauk Ukrainy, No. 10, 15–19 (1991). The new results obtained in the present paper are related to the construction of an asymptotic expansion of a solution of a mixed homogeneous boundary-value problem under additional assumptions of symmetry for the coefficients of the operator and for the thin perforated domain.     

Influence of boundary conditions on the solution of a hyperbolic thermoelasticity problem

We consider a series of problems with a short laser impact on a thin metal layer accounting various boundary conditions of the first and second kind. The behavior of the material is modeled by the hyperbolic thermoelasticity of Lord–Shulman type. We obtain analytical solutions of the problems in the semi-coupled formulation and numerical solutions in the coupled formulation. Numerical solutions are compared with the analytical ones. The analytical solutions of the semi-coupled problems and numerical solutions of the coupled problems show qualitative match. The solutions of hyperbolic thermoelasticity problems are compared with those obtained in the frame of the classical thermoelasticity. It was determined that the most prominent difference between the classical and hyperbolic solutions arises in the problem with fixed boundaries and constant temperature on them. The smallest differences were observed in the problem with unconstrained, thermally insulated edges. It was shown that a cooling zone is observed if the boundary conditions of the first kind are given for the temperature. Analytical expressions for the velocities of the quasiacoustic and quasithermal fronts as well as the critical value for the attenuation coefficient of the excitation impulse are verified numerically.     

Finite element approximation of a two-point boundary-value problem in nonlinear elasticity

In this paper finite element approximations of a one-dimensional nonlinear boundary-value problem in finite elasticity are considered. Under reasonable conditions on the form of the constitutive equations, it is shown that optimal rates of convergence can be obtained inW ₂ ¹. A criterion for the monotonicity of the stress operator is established. In addition,L _p -estimates are also derived. A numerical example is included in which results are obtained that confirm some of the theoretical estimates.

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Résumé On considère ici l'approximation par éléments finis pour le problème à limite, non linéare, à une dimension, d'élasticité finie. Sous les conditions raisonnables sur la forme des équations constitutives, c'est montré que l'on peut obtenir les taux optimaux de convergence dansW ₂ ¹. On établit un critère pour la monotonicité de l'opérateur constraint. D'ailleurs, on conduit aussiL _p -estimations. Un exemple numérique qui confirme quelques des estimations théoriques est inclus comme résultat.

     

Asymptotic experssion of the solution of general boundary value problem for higher order elliptic equation with perturbation both in boundary and in operator

In this paper basing on (1) and (2), we study the singular perturbation of general boundary value problem for higher order elliptic equation with perturbation both in the boundary and in the operator, so as to establish the asymptotic expression involving two parameters. Thus, the iterative process of finding the asymptotic solution is derived and the estimation of the remainder term is given out, we extend and improve the previously published papers.     

Asymptotic form of the solution to the problem of multicomponent mixture flow through a porous medium with a boundary layer

In problems of two-phase mixture flow through a porous medium in a subterranean stratum a boundary layer phenomenon arises caused by the fact that relative phase motion exists in the system, and so having no analogy with the single-phase case. The physical nature of boundary layer phenomena is explained, and an asymptotic solution is constructed for the self-similar problem with an arbitrary number of components in the system, by using the method of matched asymptotic forms. The conditions are established for the motions of a multicomponent and a binary mixture to be equivalent, and a study is made of the role of convective factors in the formation of averaged working indices for the stratum.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 94–100, July–August, 1985.     

Singular perturbation solution of boundary-value problem for a secord-order differential-difference equation

In this paper, the method of two-variables expansion is used to construct boundary layer terms of asymptotic solution of the boundary-value problem for a second-order DDE. The n-order formal asymptotic solution is obtained and the error is estimated. Thus the existence of uniformly valid asymptotic solution is proved.     

Analytical solution for bending problem of moderately thick composite annular sector plates with general boundary conditions and loadings using multi-term extended Kantorovich method

An analytical solution for bending of composite sector plates is presented using multi-term extended Kantorovich method (MTEKM). The governing equations are derived using the displacement field of the first-order shear deformation theory and converted into two sets of coupled ordinary differential equations (ODEs) utilizing MTEKM. Next, an analytical iterative procedure is presented for solving the derived sets of ODEs based on state-space method. Numerous examples are studied by the present method, and as special cases, solid sector and rectangular plates are also investigated. Next, the results obtained by the present method are compared to those of finite element method and other results available in the literature. It is found that the present method has a high convergence rate as well as good accuracy in all cases.     

Conditions for the existence of a solution of a Noether boundary-value problem for a second-order system

We consider a weakly nonlinear boundary-value problem for a system of second-order ordinary differential equations. We find a sufficient condition for the existence of at least one solution of this problem and propose a convergent iterative algorithm for the determination of its solution. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 3, pp. 368–375, July–September, 2006.     

Asymptotic solution of the equations for a three-dimensional multicomponent laminar boundary layer with large injection

The asymptotic method of outer and inner expansions is used to analyze the flow of a multicomponent gas in a three-dimensional boundary layer on a smooth blunt body with large injection. Asymptotic expressions are derived for the friction coefficients, the heat and diffusion fluxes of the components on the surface of the body, and the velocity, temperature, and concentration profiles of the components across the layer of injected gases. It is shown that with large injection the limiting (bottom) streamlines on the surface of the body coincide in the first approximation with the vectorial lines of the pressure gradient.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 47–56, March–April, 1975.The author is indebted to G. A. Tirskii for a discussion of the work.     

Solving the elastic bending problem for a plate with mixed boundary conditions

The bending problem for an arbitrarily outlined thin plane with mixed boundary conditions is solved. A technique based on the methods of potentials and balancing loads is proposed for constructing Green’s function for the Germain-Lagrange equation. This technique ensures high accuracy of approximate solutions, which is checked against Levi’s solution for rectangular plates __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 5, pp. 104–112, May 2006.