Murman Kublashvili

The problem of a longitudinal shearing crack in an elastic body is reduced to calculation of some Cauchy-type integral.

For approximate calculation of this integral, the approximation scheme was developed and its accuracy was estimated. The results of the approximate integral calculation are presented.

There is proposed the development of a quadratic formula for singular integrals. It has a higher-order accuracy as compared with the classic quadratic formula for the method of discrete vortexes in the case of open contours.

High-order accuracy numerical solution algorithms for some flat problems of cracks using the method of singular integral equations are constructed.

The singular integral equation of the first kind in the case of open contours is considered. In the case of a non-zero index, high-order precision numerical schemes are built for it. The basis of the constructed algorithm is presented.

In this paper we investigate the question how the method of conformal mapping (MCM) can be applied for approximate solving of the generalized Dirichlet boundary problem for harmonic function. Under the generalized problem is meant the case when a boundary function has a finite number of first kind break points. The problem is considered for finite and infinite simply connected domains. It is shown that the method of fundamental solutions (MFS) is ineffective for solving of the considered problem from the point of view of the accuracy. We propose an efficient algorithm for approximate solving of the generalized problem, which is based on the MCM. Examples of application of the proposed algorithm and the results of numerical experiments are given.

An algorithm for the approximate solution of the Dirichlet generalized problem is proposed. The term “generalized” indicates that the boundary function has a finite number of first kind break points. The solution consists of the following stages:

1) the reduction of the Dirichlet generalized problem to an ordinary auxiliary problem for a harmonic function; 2) an approximate solution of the auxiliary problem by the modified version of the MFS (the method of fundamental solutions); 3) the construction of an approximate solution of the generalized problem from the solution of the auxiliary problem. An example is considered in which the break points are the cusp ones.

A Dirichlet generalized harmonic problem for finite right circular cylindrical domains is considered. The term “generalized” indicates that a boundary function has a finite number of first kind discontinuity curves. It is shown that if a finite domain is bounded by several surfaces and the curves are placed in arbitrary form, then the generalized problem has a unique solution depending continuously on the data. The problem is considered for the simple case when the curves of discontinuity are circles with centers situated on the axis of the cylinder. An algorithm of numerical solution by a probabilistic method is given, which in its turn is based on a computer simulation of the Wiener process. A numerical example is considered to illustrate the effectiveness and simplicity of the proposed method.

The Dirichlet ordinary and generalized harmonic problems for some 3D finite domains are considered. The term “generalized” indicates that a boundary function has a finite number of first kind discontinuity curves. An algorithm of numerical solution by the method of probabilistic solution (MPS) is given, which in its turn is based on a computer simulation of the Wiener process. Since, in the case of 3D generalized problems there are none exact test problems, therefore, for such problems, the way of testing of our method is suggested. For examining and to illustrate the effectiveness and simplicity of the proposed method five numerical examples are considered on finding the electric field. In the role of domains are taken ellipsoidal, spherical and cylindrical domains and both the potential and strength of the field are calculated. Numerical results are presented.

In the artcle is discussed the issue regarding the durability of the reinforced concrete elements, determined by using only the methods of the “reinforced concrete fracture mechanics”, where the element’s section with cracks and existing

deficiencies (hollows) has been reviewed, the length and the width of the crack have been calculated and the durability of the reinforced concrete element has been determined; The software for the calculation has been created.

In this paper, for determination of the electric and thermal stationary fields the Dirichlet ordinary and generalized harmonic problems are considered. The term “generalized” indicates that a boundary function has a finite number of first kind discontinuity curves. For numerical solution of boundary problems the method of probabilistic solution (MPS) is applied, which in its turn is based on a modeling of the Wiener process. The suggested algorithm does not require an approximation of a boundary function, which is main of its important properties. For examining and to illustrate the effectiveness and simplicity of the proposed method four numerical examples are considered on finding the electric and thermal fields. In the role of domains are taken: finite right circular cone and truncated cone; a rectangular parallelepiped. Numerical results are presented.

A number of quadrature processes connected with approximation of Cauchy type singular integrals are considered in relation with approximate solution of boundary problems of certain type. Namely, significant attention is paid to problems of unique solvability of approximating scheme, accuracy and similar questions related with boundary integral problems based on corresponding approximation.

The present paper is devoted to the analysis of an explicit analytic solution of the Dirichlet generalized harmonic problem for a finite right circular axisymmetric cylindrical ring. We intend to use it for testing. For construction of the mentioned solution, the following methods are applied: separation of variables, particular solutions and heuristic method. Since the heuristic method does not guarantee finding the best solution, because of this, properties of the noted solution were investigated. It is shown that the above-mentioned problem can be used in the role of a test with the help of the given analytic solution.