Basic Boundary Value Problems of Thermoelasticity for Anisotropic Bodies with Cuts. II

1995 ◽  
Vol 2 (3) ◽  
pp. 259-276
Author(s):  
R. Duduchava ◽  
D. Natroshvili ◽  
E. Shargorodsky
Keyword(s):  
Boundary Value Problems ◽  
Smooth Boundary ◽  
Boundary Value ◽  
Three Dimensional ◽  
Theory Of Elasticity ◽  
Arbitrary Configuration ◽  
Anisotropic Bodies ◽  
Present Part ◽  
Classical Potential Theory ◽  
Basic Boundary

Abstract In the first part [Duduchava, Natroshvili and Shargorodsky, Georgian Math. J. 2: 123–140, 1985] of the paper the basic boundary value problems of the mathematical theory of elasticity for three-dimensional anisotropic bodies with cuts were formulated. It is assumed that the two-dimensional surface of a cut is a smooth manifold of an arbitrary configuration with a smooth boundary. The existence and uniqueness theorems for boundary value problems were formulated in the Besov () and Bessel-potential () spaces. In the present part we give the proofs of the main results (Theorems 7 and 8) using the classical potential theory and the nonclassical theory of pseudodifferential equations on manifolds with a boundary.

Basic Boundary Value Problems of Thermoelasticity for Anisotropic Bodies with Cuts. I

1995 ◽  
Vol 2 (2) ◽  
pp. 123-140
Author(s):  
R. Duduchava ◽  
D. Natroshvili ◽  
E. Shargorodsky
Keyword(s):  
Boundary Value Problems ◽  
Displacement Vector ◽  
Boundary Value ◽  
Three Dimensional ◽  
Manifolds With Boundary ◽  
Hölder Continuous ◽  
Arbitrary Configuration ◽  
Anisotropic Bodies ◽  
Potential Methods ◽  
Basic Boundary

Abstract The three-dimensional problems of the mathematical theory of thermoelasticity are considered for homogeneous anisotropic bodies with cuts. It is assumed that the two-dimensional surface of a cut is a smooth manifold of an arbitrary configuration with a smooth boundary. The existence and uniqueness theorems for boundary value problems of statics and pseudo-oscillations are proved in the Besov and Bessel-potential spaces by means of the classical potential methods and the theory of pseudodifferential equations on manifolds with boundary. Using the embedding theorems, it is proved that the solutions of the considered problems are Hölder continuous. It is shown that the displacement vector and the temperature distribution function are Cα -regular with any exponent α < 1/2. This paper consists of two parts. In this part all the principal results are formulated. The forthcoming second part will deal with the auxiliary results and proofs.

scholarly journals Stability of mathematical models of the main problems of the anisotropic theory of elasticity

2020 ◽  
Vol 30 (1) ◽  
pp. 112-124
Author(s):  
A.V. Yudenkov ◽  
A.M. Volodchenkov
Keyword(s):  
Boundary Value Problems ◽  
Analytic Functions ◽  
Boundary Value ◽  
Research Result ◽  
Theory Of Elasticity ◽  
Boundary Problems ◽  
Main Research ◽  
Contour Shape ◽  
Variable Function ◽  
Anisotropic Bodies

The boundary problems of the complex-variable function theory are effectively used while investigating equilibrium of homogeneous elastic mediums. The most complicated systems of the boundary value problems correspond to the case when an elastic body exhibits anisotropic properties. Anisotropy of the medium results in the drift of boundary conditions of the function that in general disrupts analyticity of the functions of interest. The paper studies systems of the boundary value problems with drift for analytic vectors corresponding to the primal elastic problems (first, second and mixed problems). Systems of analytic vectors with drift are reduced to equivalent systems of Hilbert boundary value problems for analytic functions with weak singularity integrators. The obtained general solution of the primal boundary value problems for the anisotropic theory of elasticity allows us to check the above problems for stability with respect to perturbations of boundary value conditions and contour shape. The research is relevant as there is necessity to apply approximate numerical methods to the boundary value problems with drift. The main research result comes to be a proof of stability of the systems of the vector boundary value problems with drift for analytic functions on the Hölder space corresponding to the primal problems of the elastic theory for anisotropic bodies in the case of change in the boundary value conditions and contour shape.

Three-Dimensional Boundary Value Problems of Elastothermodiffusion with Mixed Boundary Conditions

1997 ◽  
Vol 4 (3) ◽  
pp. 243-258
Author(s):  
T. Burchuladze ◽  
Yu. Bezhuashvili
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Singular Integrals ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Three Dimensional ◽  
Boundary Surface ◽  
Mixed Boundary Conditions ◽  
Theory Of Elasticity ◽  
Classical Solutions

Abstract We investigate the basic boundary value problems of the connected theory of elastothermodiffusion for three-dimensional domains bounded by several closed surfaces when the same boundary conditions are fulfilled on every separate boundary surface, but these conditions differ on different groups of surfaces. Using the results of papers [Kupradze, Gegelia, Basheleishvili, and Burchuladze, Three-dimensional problems of the mathematical theory of elasticity and thermoelasticity, North-Holland Publishing Company, 1979, Russian original, 1976–Mikhlin, Multi-dimensional singular integrals and integral equations, 1962], we prove theorems on the existence and uniqueness of the classical solutions of these problems.

scholarly journals FEATURES OF THE NUMERICAL SOLUTION OF BOUNDARY VALUE PROBLEMS WITH THE USE OF PYTHON ALGORITHMIC PROGRAMMING LANGUAGE

2018 ◽  
Vol 14 (1) ◽  
pp. 126-136
Author(s):  
Vladimir L. Mondrus ◽  
Dmitry K. Sizov
Keyword(s):  
Numerical Solution ◽  
Boundary Value Problems ◽  
Programming Language ◽  
Difference Method ◽  
Boundary Value ◽  
Three Dimensional ◽  
Theory Of Elasticity ◽  
Object Oriented Programming ◽  
Python Language ◽  
Variational Difference Method

The solution of the majority of the practical tasks arising at calculation of constructions demands a numerical solution of boundary value problems. In article on examples of a solution of specific boundary value problems possibilities of a modern object-oriented programming language of Python are described. This lan-guage possesses convenient syntax and opportunities of flexible use of the existing libraries of the numerical methods allowing the user to prepare a program code and to start studying of a task in the shortest terms. Rich opportunities of graphic library Matplotlib are allowed to have graphic display of results of calculation. The Numpy library used in article is standard library for carrying out any matrix and vector calculations in the Python language, its application turns the Python language on Wednesday for programming of numerical calculations, similar to such widely known software products as Matlab and GNU Octave. In article the method of finite differences has solved a task for Laplace's equation, the describing field distribution of temperature on rectangular area; problem of search of a minimum of functionality of Dirikhle by a variational-difference method; a dy-namic problem of vibration of weight on a visco-elastic element at vibrations of the basis and a three-dimensional task of the theory of elasticity by a variational-difference method.

Basic boundary value problems of thermoelasticity for anisotropic bodies with cuts. II

1995 ◽  
Vol 2 (3) ◽  
pp. 259-276 ◽  
Author(s):  
R. Duduchava ◽  
D. Natroshvili ◽  
E. Shargorodsky
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Anisotropic Bodies ◽  
Basic Boundary
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