and Computational Methods

#### 539.36 Finite element modeling of natural vibrations of shell structures

**Dimitrienko Y. I. (Bauman Moscow State Technical University), Yurin Y. V. (Bauman Moscow State Technical University), Bogdanov I. O. (Bauman Moscow State Technical University), Maremshaova A. A. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2023-3-317

A numerical algorithm for solving the problem of natural vibrations for thin-walled shell structures based on the finite element method is proposed. A software module has been developed as part of the SMCM software package, which implements the proposed numerical algorithm. A test problem was solved for natural vibrations of a cylindrical shell structural element. A comparative analysis of eigenfrequencies and eigenmodes was carried out with similar results obtained using a two-dimensional shell solution in the ANSYS software package, as well as with the results of solving a three-dimensional problem for natural vibrations in the ANSYS software package.

Димитриенко Ю.И., Юрин Ю.В., Богданов И.О., Маремшаова А.А. Конечно-элементное моделирование собственных колебаний оболочечных конструкций. Математическое моделирование и численные методы, 2023, № 3, с. 3–17.

#### 621.74.043 Mathematical modeling of the process of mixing of liquid metal in the mold of a continuous steel casting plant

**Odinokov V. I. (Komsomolsk-na-Amure State University), Evstigneev A. I. (Komsomolsk-na-Amure State University), Dmitriyev E. A. (Komsomolsk-na-Amure State University), Karpenko V. A. (Komsomolsk-na-Amure State University)**

doi: 10.18698/2309-3684-2023-3-1841

Experimental studies of the flow of liquid metal in the UNRS crystallizer are a long, complex and time-consuming process. Therefore, mathematical modeling by numerical methods is increasingly used for this purpose. A new technology for pouring liquid metal into a crystallizer is proposed. The original patented design of the device, consisting of a direct-flow and rotating gluhodon glasses, is given. The main results of studies of the melt flow in the crystallizer volume are presented. The objects of research were the hydrodynamic and thermal fluxes of liquid metal of a new process of casting steel into a rectangular section mold of the UNRS, and the result was a spatial mathematical model describing the fluxes and temperatures of liquid metal in the mold. To model the processes occurring during the flow of metal in the crystallizer, a specially created software package was used. The theoretical calculations are based on the fundamental equations of hydrodynamics, the equations of mathematical physics (the equation of thermal conductivity taking into account mass transfer) and a proven numerical method. The studied area was divided into elements of finite dimensions, for each element the resulting system of equations was written in a difference form. The result of the solution is the fields of velocities and temperatures of the metal current in the volume of the mold. According to the developed numerical schemes and algorithms, a calculation program has been compiled. An example of calculating the casting of steel into a mold of rectangular cross-section, flow diagrams of liquid metal along various sections of the mold is given. Vector flows of liquid metal in different sections of the mold at different numbers of revolutions of the jacket with vertical ribs are clearly presented. Areas of different turbulence have been identified. The mixing mode at n = 30 rpm is considered optimal. At n = 50 rpm, liquid metal is ejected into the slag bath.

Одиноков В.И., Евстигнеев А.И., Дмитриев Э.А., Карпенко В.А. Математическое моделирование процесса перемешивания жидкого металла в кристаллизаторе установке непрерывной разливки стали. Математическое моделирование и численные методы, 2023, № 3, с. 18–41.

#### 523.6+533.6 Modeling of the Tunguska phenomenon in 1908 within the framework of two possible hypotheses

**Andrushchenko V. A. (Institute for Computer Aided Design of the Russian Academy of Sciences), Syzranova N. G. (Institute for Computer Aided Design of the Russian Academy of Sciences)**

doi: 10.18698/2309-3684-2023-3-4261

Within the framework of the actual problem of comet-asteroid danger, the physical processes that cause the destruction and fragmentation of meteoric bodies in the Earth's atmosphere, in this case the Tunguska bolide, are numerically studied. The number of possible versions and hypotheses related to the Tunguska phenomenon is extremely large and continues to increase, therefore, an analysis and generalization of all known facts inherent in this non-standard catastrophic event is necessary, and only then proceed to the nomination of certain hypotheses explaining it. Based on the developed physical and mathematical model that determines the movement of space objects of natural origin in the atmosphere and their interaction with it, we have proposed two hypotheses explaining the processes occurring during the fall of the Tunguska body in 1908. The first hypothesis is related to the fragmentation of the body, which is a stone meteoroid into a large number of fragments, which collapsed in the dense layers of the atmosphere under the action of thermal stresses to the size of fine dust. The difficulties in identifying small particles that fell out as a result of the Tunguska event are mainly explained by the following circumstance: the timing of the initial search for traces of the fall of the body was removed from the moment of the event by as much as twenty years, during which a very significant number of other geophysical processes could occur in this area. The second hypothesis is related to phenomena that occur at small angles of entry of a body into the Earth's atmosphere. In this case, there is a change in the ballistics of its flight in the atmosphere, consisting in a transition from the fall mode to the ascent mode. This effect leads to the realization of the following possible scenarios of the event: the return of the body back to outer space at its residual velocity greater than the second cosmic one; the transition of the body to the orbit of the Earth satellite at a residual velocity greater than the first cosmic one; at lower values of the residual velocity of the body, its return after some time to the fall mode and reaching the earth's surface at a considerable distance from the intended crash site. The proposed hypotheses explain, for example, the absence of material traces, including craters, during the search for the remains of the Tunguska bolide in the vicinity of the forest collapse.

Андрущенко В.А., Сызранова Н.Г. Моделирование Тунгусского явления 1908 года в рамках двух возможных гипотез. Математическое моделирование и численные методы, 2023, № 3, с. 42–61.

#### 519.6:533.6 Numerical simulation of the flow structure near the descent vehicle and located in its wake parachute atsupersonic motion

**Babakov A. V. (Institute for Computer Aided Design of the Russian Academy of Sciences)**

doi: 10.18698/2309-3684-2023-3-6279

The article presents the results of a numerical study of the spatial non-stationary flow structure arising during supersonic motion in the atmosphere of the descent vehicle and parachute located in its vortex wake. The cases of the vehicle at angle of attack and various location of the parachute in relation to the vehicle are considered. For different distances between vehicle and parachute the patterns of the spatial non-stationary vortex structure of the flow occurring between them and in the near wake of the parachute are given. The significant influence of the distance between the vehicle and the parachute on the flow structure and force characteristics effect of the flow on the parachute is shown. Data on the influence of attack angle of the vehicle on the aerodynamic characteristics of parachute are presented. Numerical simulations are performed using two conservative numerical methods based on the approximation of conservation laws written in integral form for a finite volume. Calculations are based on the parallel algorithms implemented on modern supercomputer systems.

Бабаков В.А. Численное моделирование структуры потока около спускаемого аппарата и расположенного в его следе парашюта при сверхзвуковом движении. Математическое моделирование и численные методы, 2023, № 3, с. 62–79.

#### 519.6:532.5 The Mathematical modeling of a spots chain with different intervals in a stratified

**Gushchin V. A. (Institute for Computer Aided Design of the Russian Academy of Sciences), Smirnova I. A. (Institute for Computer Aided Design of the Russian Academy of Sciences)**

doi: 10.18698/2309-3684-2023-3-8091

A comparison is made of the spots chain dynamics of mixed liquid in a stratified environment, with different initial intervals between spots. For mathematical modeling, the SMIF method (splitting method for studying incompressible fluid flows) is used. The finitedifference scheme of the method has the second order of approximation in terms of spatial variables, minimal scheme dissipation and dispersion, and has the property of monotonicity. The results of comparing the dynamics of the stream function depending on the initial interval between spots are presented for the same values of the Reynolds and Froude numbers.

Гущин В.А., Смирнова И.А. Математическое моделирование цепочки пятен с различными интервалами в стратифицированной жидкости. Математическое моделирование и численные методы, 2023, № 3, с. 80–91.

#### 532.59 Numerical study of the effect of an internal wave on the surface of a two-layer flow flowing around a point source

**Nosov V. N. (Vernadsky Institute of Geochemistry and Analytical Chemistry, RAS), Savin A. S. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2023-3-92104

The flow around a point source localized in the lower layer of a two-layer liquid with a free surface is considered. Expressions are obtained for the perturbation of the free surfaceof the liquid associated with the manifestation of an internal wave. The source operating in the liquid is represented as a superposition of point pulse sources. This approach made it possible to find the perturbation of the flow surface as a superposition of perturbations caused by point pulse sources. The approximation used is quite justified in cases of modeling real sources of disturbances located at considerable depths, since such sources cause small disturbances of the sea surface. It is established that the internal waves appearing on the flow surface form a wedge-shaped structure. The angle of the wedge solution of the internal waves coming to the surface decreases with increasing flow velocity. The dependence of the angle of the wave wedge solution on the Froude number determined by the flow velocity and thickness of the upper liquid layer is found. The considered problem is of theoretical and practical interest, since more complex models of real disturbances of the surface of the marine environment during the flow of various inhomogeneities can be constructed as superpositions of model elementary disturbances from point sources.

Носов В.Н., Савин А.С. Численное исследование воздействия внутренней волны на поверхность двухслойного потока, обтекающего точечный источник. Математическое моделирование и численные методы, 2023, № 3, с. 92–104.

#### 519.6 Numerical solution of equations of mixed type in unlimited region on a plane

**Galanin M. P. (Bauman Moscow State Technical University/Keldysh Institute of Applied Mathematics of the Russian Academy of Scienсes), Ukhova A. R. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2023-3-105124

The purpose of the work is to build and implement an algorithm for finding a numerical solution to a problem for mixed-type equations in an unlimited region. In this case problems are considered in which the process under study is described in some limited area by the thermal conductivity equation or wave equation, and outside it by the Laplace equation. The necessary additional conditions at zero, at infinity and the conditions for conjunction at the border of the inner region are set. There is described an algorithm for finding a numerical solution to a problem with a wave equation in a limited region in one-dimensional and two-dimensional cases, problems with a thermal conductivity equation or a wave equation in a two-dimensional case. Difference schemes are built by the integro–interpolation method. The task is solved in a limited area. Nonlocal boundary conditions are set on its border so the solution of task in limited area coincides with projection of problem in unlimited area. In this case, an artificial boundary is introduced for the solution in the part of the region in which the process is described by the Laplace equation. An iterative algorithm and an algorithm with a non-local boundary condition are built. The results of calculations for examples in various fields are presented.

Галанин М.П., Ухова А.Р. Численное решение уравнений смешанного типа в неограниченной области на плоскости. Математическое моделирование и численные методы, 2023, № 3, с. 105–124.

#### 519.8 Modeling the conflict of interacting systems, taking into account the effect of obtaining information on the mutual state

**Chuev V. U. (Bauman Moscow State Technical University), Dubogray I. V. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2023-3-125133

With the help of probabilistic methods, a model of the conflict of two interacting systems consisting of numerous structural units is proposed, taking into account the effect of delaying information about the mutual state: about the structure, number and parameters of each other's structural units. A study was made of the influence of insufficient information at a particular point in time on the outcome of the process of conflict development. It is shown that the availability of information about the state of the structural units of the opposite side can significantly increase the probability of successful development of the conflict, and with an increase in the number of units of structural units, the difference in the probability of successful development of the conflict scenario increases significantly

Чуев В.Ю., Дубограй И.В. Моделирование конфликта взаимодействующих систем с учетом эффекта получения информации о взаимном состоянии. Математическое моделирование и численные методы, 2023, № 3, с. 125–133