and Computational Methods

#### 519.62+539.21 Numerical modeling of alloys nanostructure rearrangement by means of molecular dynamics methods

**Krasnov I. K. (Bauman Moscow State Technical University), Mozzhorina T. Y. (Bauman Moscow State Technical University), Balanin A. S. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2017-4-316

The article presents a mathematical model of the alloys nanoparticles structure rearrangement dynamics after the instantaneous thermal influence (heating or cooling). The model is based on using the method of molecular dynamics of multicomponent alloys with the Lennard-Jones and Morse interatomic potentials as well as the initial conditions of momentary expansion or compression of the alloy nanoparticle regular crystalline structure. We computationally investigate the regularities of rearranging the initially regular atomic structure of a nanoparticle over time. It is shown that depending on the number of atoms in a nanoparticle various finite settled forms of the alloys nanoparticle are possible, both amorphous and new crystalline structures different from the alloy original crystalline nanostructure. We provide numerical results for the titanium nanoparticles and the titanium-nickel alloy (nitinol).

Krasnov I.K., Mozzhorina T.Yu., Balanin A.N. Numerical modeling of alloys nanostructure rearrangement by means of molecular dynamics methods. Маthematical Modeling and Computational Methods, 2017, №4 (16), pp. 3-16

#### 510:53.072:621.1.016.4 Modeling of stochastic filtration processes in lattice systems

**Arutyunyan R. V. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2017-4-1730

The purpose of the paper was to formulate and study the system of kinetic equations modeling the process of diffusion filtration based on a stochastic approach. Within the research we proved the theorem of existence and uniqueness of the solution with respect to the case of continuous density, obtained the solutions in uniformly convergent and asymptotic series and examined its behavior at infinity. Moreover, we considered the specific cases of density of the Delta-function type and uniform distribution. As a result, the finite-difference scheme for solving the corresponding Cauchy problem on finite time intervals is built and justified. The results of computer simulation are also given.

Arutyunyan R.V. Modeling of stochastic filtration processes in lattice systems. Mathematical Modeling and Computational Methods, 2017, №4 (16), pp. 17-30

#### 539.3 Mathematical modeling of the rotating cylindrical shell torsional vibrations factored in the internal viscous fluid

**Khudoynazarov K. K. (Ташкентский государственный технический университет), Burkutboyev S. M. (Ташкентский государственный технический университет)**

doi: 10.18698/2309-3684-2017-4-3147

The article introduces a mathematical model of the rotating cylindrical shell torsional vibrations with the viscous fluid flowing inside. We have developed an algorithm for defining the stress-strain state of the considered system’s points. As an example we examine the problem of torsional vibrations of the drill column rotating with constant angular velocity. The article estimates the influence of the internal viscous fluid flow and the centrifugal inertia force on the stressed-strain state of the system.

Khudoynazarov Kh.Kh., Burkutboyev Sh.M. Mathematical modeling of the rotating cylindrical shell torsional vibrations factored in the internal viscous fluid. Mathematical Modeling and Computational Methods, 2017, №4 (16), pp. 31-47

#### 539.3 Numerical solution of inverse three-dimensional problems of recovering the loads acting upon composite structural elements

**Dimitrienko Y. I. (Bauman Moscow State Technical University), Yurin Y. V. (Bauman Moscow State Technical University), Yegoleva E. S. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2017-4-4859

The article proposes a numerical method for solving the inverse three-dimensional problems of recovering the fields of loads acting upon composite structural elements based on the results of the experimental diagnostics of structural displacements on a certain surface. The problems of this type arise when creating the systems of the built-in diagnostics of structural movements and intelligent composite structures. The restored field of loads acting upon the parts of the outer surface of the composite structure is used to calculate the stress-strain state and forecast the structural life. The proposed method uses an alternating algorithm for solving the inverse problems of restoring loads in the problem of elasticity theory, in combination with the finite element method for solving the direct problems in the theory of elasticity. We consider an example of solving the inverse problem of restoring loads acting on the structural elements made from layered fibrous composite materials.

Dimitrienko Yu.I., Yurin Yu.V., Egoleva E.S. Numerical solution of inverse three-dimensional problems of recovering the loads acting upon composite structural elements. Mathematical Modeling and Computational Methods, 2017, №4 (16), pp. 48-59

#### 533.6.011.5:004.622:004.855.5 Simulation of the pressure distribution in the disturbed region near the sphere streamlined by the inviscid flotation by means of the machine learning methods

**Kotenev V. P. (Bauman Moscow State Technical University), Puchkov A. S. (Bauman Moscow State Technical University), Sapozhnikov D. A. (Bauman Moscow State Technical University), Tonkih E. G. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2017-4-6072

The article introduces a dependency for the pressure distribution in the disturbed region near the sphere streamlined by the flow of the supersonic inviscid gas, obtained when modifying the Shepard’s Method. We use known ratios for the pressure on the body and the shockwave as well as data from the numerical experiments. We have compared the results with the data not used in the learning process of the dependency coefficients. This comparison proves high confidence of the model obtained.

Kotenev V.P., Puchkov A.S., Sapozhnikov D.A., Tonkikh E.G. Simulation of the pressure distribution in the disturbed region near the sphere streamlined by the inviscid flotation by means of the machine learning methods. Mathematical Modeling and Computational Methods, 2017, №4 (16), pp. 60-72

#### 531.6.011.32:532.582.4:517.958 Special features of vortex diagram in simulation of subsonic detached flow around the semi-infinite equivalent body

**Timofeev V. N. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2017-4-7391

The paper introduces some special features of mathematical simulation of subsonic detached flow around the bodies, the flow being localized in the vicinity of the ground shear. The formation of vortex diagram for the semi-infinite equivalent body is examined. The formulae for determining the vector functions of the vortex segments velocity are reduced to a form allowing one to easily pass to the limit as the points of the origin or ends of these segments are moved off to infinity. Furthermore, the study shows the relationships for finding the velocity function vectors of semi-infinite vortex segments and U-shaped vortex lines, the relationships being adapted for computer calculations. Findings of mathematical simulation of the flow around cylindrical bodies with the head part of the ogival form are given.

Timofeev V.N. Special features of vortex diagram in simulation of subsonic detached flow around the semi-infinite equivalent body. Mathematical Modeling and Computational Methods, 2017, №4 (16), pp. 73-91.

#### 531.36:521.1 Mathematical model of a two-tether system consisting of a space station and a dynamically symmetric asteroid

**Rodnikov A. V. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2017-4-92101

The purpose of the research was to build a mathematical model of an natural-artificial space tether system consisting of a space station and an asteroid, close in its dynamic characteristics to a dynamically symmetric solid body, connected by two tethers. The article describes some criteria of existence and stability for equilibrium configurations of the system deduced within the assumptions of the constructed model. Moreover, it classifies the types of the station motions near the asteroid surface, and gives the conditions ensuring the motions with taut tethers.

Rodnikov A.V. Mathematical model of a two-tether system consisting of a space station and a dynamically symmetric asteroid. Mathematical Modeling and Computational Methods, 2017, №4 (16), pp. 92-101.