# Vol 20, No 3 (2016)

**Year:**2016**Articles:**11**URL:**https://journals.eco-vector.com/1991-8615/issue/view/1221

### The evaluation of the order of approximation of the matrix method for numerical integration of the boundary value problems for systems of linear non-homogeneous ordinary differential equations of the second order with variable coefficients. Message 1. Boundary value problems with boundary conditions of the first kind

#### Abstract

We present the first message of the cycle from two articles where the rearrangement of the order of approximation of the matrix method of numerical integration depending on the degree in the Taylor’s polynomial expansion of solutions of boundary value problems for systems of ordinary differential equations of the second order with variable coefficients with boundary conditions of the first kind were investigated. The Taylor polynomial of the second degree use at the approximation of derivatives by finite differences leads to the second order of approximation of the traditional method of nets. In the study of boundary value problems for systems of ordinary differential equations of the second order we offer the previously proposed method of numerical integration with the use of matrix calculus where the approximation of derivatives by finite differences was not performed. According to this method a certain degree of Taylor polynomial can be selected for the construction of the difference equations system. The disparity is calculated and the order of the method of approximation is assessed depending on the chosen degree of Taylor polynomial. It is theoretically shown that for the boundary value problem with boundary conditions of the first kind the order of approximation method increases with the degree of the Taylor polynomial and is equal to this degree only for its even values. For odd values of the degree the order of approximation is less by one. The theoretical conclusions are confirmed by a numerical experiment for boundary value problems with boundary conditions of the first kind.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2016;20(3):389-409

389-409

### On decisions of Schwartz' problem for J-analytic functions with the same Jordan basis of real and imaginary parts of J-matrix

#### Abstract

Boundary Schwartz' problem for J-analytic functions was studied within this scientific work. These functions are solutions of linear complex system of partial differential equations of the first order. It was considered, that the real and imaginary parts of J-matrix are put into triangular form by means of one and the same complex transformation. The main theorem proved a criterion for eigenvalues of J-matrix. Shall this criterion be fulfilled within the complex plane within the boundaries defined by Lyapunov line, there is a decision on Schwartz' problem and it is the only one. The equal form of this criterion was found, which in many cases is more convenient for check. While proving the theorem, known facts about boundary properties of λ-holomorphic functions are applied. The proof itself is based on the method of direct and reverse reduction of Schwarz' problem to Dirichlet’s problem for real valued elliptic systems of partial differential equations of the second order. Examples of matrices are given, whereby the specified criterion is fulfilled.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2016;20(3):410-422

410-422

### Nonlocal transformation of the internal quantum particle structure

#### Abstract

The analysis of the integral wave equation, having path integral kernel, has resulted, that collapse phenomenon is based on the nonlocal transformation of the internal structure of a quantum particle, considering in the form of the matter fields collection. This nonlocality allows to escape the contradiction between the reduction quantum mechanics postulate and special relativity. It is shown, that the wave function transformation, corresponding to von Neumann’s reduction, has the deterministic nature and the quantum mechanics stochasticity is a consequence of a macroscopic measurer presence in the measuring process. Besides it is demonstrated, that the decogerence phenomenon has the same mechanism of the wave function transformation. EPR-type experiment is described in detail and the possibility of the faster-then light communication is proved, as well the possible rules of thumb of this communication are proposed.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2016;20(3):423-456

423-456

### The stability of monolithic lining of the vertical mine workings with the initial porosity of the material and non-elastic compression work skeleton

#### Abstract

A mathematical model has been made that describes the basic stress-strain state of the monolithic lining vertical excavation for materials with a porous structure, the skeleton of which has compressed the hardening elastoplastic properties. The deformation of the porous medium under the action of given radial compressive loads is divided into two interconnected parts: the elastic deformation of the porous medium and the inelastic deformation of the compressed matrix. The problem of determining the fields of stresses and displacements lining vertical production at each stage of deformation is solved within the framework of the plane strain. It does not take into account the effects due to the fact that the excavation has a finite depth. The equations define the field of stresses and displacements in the first and second stages of the deformation. The conditions of compatibility are the continuity conditions of the selected components of stresses and displacements in the elastic-plastic boundary and plastic strains are equal to zero on it. Within the framework of the exact three-dimensional equations of stability, the stability of the ground state of the monolithic lining vertical excavation in rock mass with tight pores has been studied. The estimation of influence on the value of the boundary between the elastic and plastic deformation of the initial porosity of the media and the yield strength of the material has been explained. The main component of the stress state of the coordinate values for different values of the initial solution pores and other physical and mechanical and geometric parameters of the material and the design have been studied and verified.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2016;20(3):457-474

457-474

### Variant of multimodal vibration damping of electroviscoelastic structures by appropriate choice of external electric circuit parameters

#### Abstract

In technical applications it takes place the problem of vibration damping in certain regions of the structure, at the location of optical sensors for instance, at any external dynamic excitations with no mass increase and no changes in spectral portrait. In order to solve these problems it is widespread the use of special damping devices: piezoelectric elements connected to external electric circuits and attached to the structure. It became possible due to piezoelectric effect, which provides transformation of part of energy of vibrations into electric one, which is dissipated in external electric circuit. So that by using appropriate electric circuits one may dissipate internal energy and therefore reduce structural vibrations in definite frequency range. As a rule, external circuit of single branch, which shunts single piezoelectric element, allows vibration damping on one certain frequency. Due to the fact, that practical applications usually include requirements of damping of several modes by one and the same technical devices, the problem of multimodal vibration damping in smart-structures is rather acute. The objective of this paper is the study of possibility of vibration damping on several modes by using single external series RL-circuit, connected to electrodes of single piezoelectric element on the basis of solution of problems on natural and forced steady-state vibrations of electroelastic systems with external electric circuits.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2016;20(3):475-495

475-495

### Dual plane problems for creeping flow of power-law incompressible medium

#### Abstract

In this paper, we consider the class of solutions for a creeping plane flow of incompressible medium with power-law rheology, which are written in the form of the product of arbitrary power of the radial coordinate by arbitrary function of the angular coordinate of the polar coordinate system covering the plane. This class of solutions represents the asymptotics of fields in the vicinity of singular points in the domain occupied by the examined medium. We have ascertained the duality of two problems for a plane with wedge-shaped notch, at which boundaries in one of the problems the vector components of the surface force vanish, while in the other-the vanishing components are the vector components of velocity, We have investigated the asymptotics and eigensolutions of the dual nonlinear eigenvalue problems in relation to the rheological exponent and opening angle of the notch for the branch associated with the eigenvalue of the Hutchinson-Rice-Rosengren problem learned from the problem of stress distribution over a notched plane for a power law medium. In the context of the dual problem we have determined the velocity distribution in the flow of power-law medium at the vertex of a rigid wedge, We have also found another two eigenvalues, one of which was determined by V. V. Sokolovsky for the problem of power-law fluid flow in a convergent channel.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2016;20(3):496-507

496-507

### Limit analysis and optimal support of reinforced three-layer circular plates of different resistant materials under non-uniform loading

#### Abstract

Within the model of an ideal rigid-plastic body the limit behavior of the hybrid composite circular plates is considered. The exact solution of the problem of bending is built for three-layer reinforced circular plates having different angular structure reinforcement at the top and bottom layer. The material of the middle layer and the binder in the upper and lower layers has a yield stress in compression much greater than in tension. In this case the condition of plasticity for the main moments that are based on the structural model of the reinforced layer with one-dimensional states of stress in the fibers has the form of a rectangle of type Johansen condition. The plates are hinge supported along the internal annular contour and have the rigid circular insert in the central part. The plates are under load non-uniformly distributed over the surface of the plate. It is shown that there are a few schemes of limit deformation of the plate, depending on the location of the internal support and on distribution of load. The conditions of implementation are defined for all schemes. The main moments and the velocities of the deflections of the plate are defined at different locations of the internal support. The simple analytic expressions are obtained for the limit load. The optimal location of support is determined. The optimal support is such support, at which the plate has a maximum limit load. It is shown that the optimal position of the support corresponds to the formation of plastic hinge on it. It is obtained that with increase in the applied distributed load in several times, the limit loads will be reduced in the same times and the optimal location of the support will not change. Numerical examples are given. The solution can be useful in engineering practice to evaluate the bearing capacity of three-layer reinforced concrete plates.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2016;20(3):508-523

508-523

### Long-term strength curves generated by the nonlinear Maxwell-type model for viscoelastoplastic materials and the linear damage rule under step loading

#### Abstract

The nonlinear Maxwell-type constitutive relation with two arbitrary material functions is formulated for viscoelastoplastic materials and studied analytically in uni-axial case to reveal capabilities of the model and its applicability scope. Its coupling with a number of fracture criteria is analyzed in order to simulate creep rupture under constant and piecewise-constant loading and to compare creep life estimates arising as a result. The limit strain criterion, the critical dissipation criterion and two proposed new families of failure criteria taking into account a strain history (i.e. a whole creep curve) are considered. Long-term strength curves equations generated by each one of the four chosen failure criteria are derived. Their general qualitative properties are analyzed and compared to each other under minimal restrictions on material functions of the constitutive relation. It is proved that qualitative properties of all theoretic long-term strength curves coincide with basic properties of typical test long-term strength curves of viscoelastoplastic materials. For every failure criteria considered herein, rapture time under step-wise loading is evaluated for arbitrary material functions and compared to the lifetime yielding from the linear damage accumulation rule (i.e. “Miner’s rule”). General formulas for cumulative damage (“Miner’s sum”) deviations from unity are obtained for all failure criteria coupled with the nonlinear Maxwell-type constitutive relation. Their dependences on material functions and loading program parameters are examined. In particular, it is proved that the linear damage rule is exactly valid for the critical dissipation criterion whatever material functions, number of loading steps and stress levels are chosen. On the contrary, for the limit strain criterion, the linear damage rule is never valid for two-step loading and cumulative damage at rapture instant is greater or less than unity depending on the sign of stress jump.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2016;20(3):524-543

524-543

### Block regularization Kaczmarz method

#### Abstract

This article focuses on the modification of the iterative version of Kaczmarz block algorithm for solving the problem of regularization, which is a fairly effective method for large-scale problems. An important characteristic of iterative methods is the speed of convergence, which depends on the condition number of the original problem. The main drawback of many iterative methods is the large condition number, while methods based on normal equations have the condition number of the system equal to the square of the condition number of the original problem. At the present time to increase the speed of convergence of iterative methods different types of preconditioners are used reducing the condition number of the system. The disadvantages of this approach is manifested in high computational complexity and the lack of universal preconditioner, which could be applied to any iterative method. One of the most effective approaches for improving the convergence rate of the method is to use a block variant of the method used. In this regard, in this paper we propose a modification of the original block Kaczmarz method for the regularization of the problem, which will reduce the computational complexity, and thus increase the rate of convergence of the algorithm. The article provides a detailed derivation of the proposed modification of the method and the proof of the convergence of the proposed variant of the block Kaczmarz method.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2016;20(3):544-551

544-551

### Aeroelastic stability of plate interacting with a flowing fluid

#### Abstract

The paper presents the results of a numerical study of the dynamic behavior of the deformable plate interacting both with the external supersonic gas flow and the internal fluid flow. The constitutive relations describing the behavior of ideal compressible fluid in the case of small perturbations are written in terms of the perturbation velocity potential and transformed using the Bubnov-Galerkin method. The aero- and dynamic pressures are calculated based on the quasi-static aerodynamic theory. The strains in the plate evaluated following the Timoshenko hypotheses. A mathematical formulation of the dynamic problem of elastic structure is developed using the variational principle of virtual displacements, which takes into account the work done by the inertia forces, aerodynamic and hydrodynamic pressures. Calculation of complex eigenvalues of the coupled system of two equations is performed using an algorithm based on implicitly restarted Arnoldi method. The stability criterion is based on an analysis of the complex eigenvalues of system of two equations obtained for increasing flow or gas velocity. The reliability of the obtained numerical solution has been estimated by comparing it with the available theoretical data. A few numerical examples were considered to demonstrate the existence of different types of instability depending on the velocities of fluid or gas flow, combinations of kinematic boundary conditions prescribed at the edges of the plate, and the fluid layer height. It has been found that a violation of the smoothness of the obtained relationships and diagrams of stability is caused by a change in the flutter mode, or change of the type of loss of stability.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2016;20(3):552-566

552-566

### Two-dimensional convection of an incompressible viscous fluid with the heat exchange on the free border

#### Abstract

The exact stationary solution of the boundary-value problem that describes the convective motion of an incompressible viscous fluid in the two-dimensional layer with the square heating of a free surface in Stokes’s approach is found. The linearization of the Oberbeck-Boussinesq equations allows one to describe the flow of fluid in extreme points of pressure and temperature. The condition under which the counter-current flows (two counter flows) in the fluid can be observed, is introduced. If the stagnant point in the fluid exists, six non-closed whirlwinds can be observed.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2016;20(3):567-577

567-577