and Computational Methods

#### 519.634 2D model and the numerical method of countercurrent flow in a rotating viscous heat-conducting gas

**Aksenov A. G. (Institute for Computer Aided Design of the Russian Academy of Sciences)**

doi: 10.18698/2309-3684-2023-4-314

A countercurrent vortex of a gas centrifuge is simulated. The mathematical model of the motion of a viscous heat-conducting gas includes an equation for density, velocities and specific energy in cylindrical geometry. After the introduction of the grid, the partial derivatives over the space are replaced by finite differences, and the problem is reduced to a system of ordinary differential equations (ODES). This technique is called the Lines Method. Since the flow is supersonic, and the design area includes thin boundary layers, the ODE system is stiff due to the presence of different-time scales and a decay. In the language of mathematics, this means a significant difference between the eigenvalues of the Jacobi matrix and the negative real parts. Therefore, to solve the problem, it is useful to use the implicit Geer method for the ODE system without splitting the problem into physical processes and directions. An effective method for solving the Jacobin matrix inversion is the use of the cyclic reduction method in the matrix variant. As an example, the countercurrent flow arising due to the temperature gradient is demonstrated.

Аксенов А.Г. Двумерная модель и метод расчета противотока во вращающемся вязком теплопроводном газе. Математическое моделирование и численные методы, 2023, № 4, с. 3–14.

#### 519.63 Mathematical Modeling of Detonation Initiation in the Channel with the Profiled End Using Parallel Computations

**Lopato A. I. (Institute for Computer Aided Design of the Russian Academy of Sciences)**

doi: 10.18698/2309-3684-2023-4-1526

The work is devoted to numerical studies of detonation initiation in a gas mixture in a rectangular channel with a profiled end. Detonation is initiated as a result of the interaction of shock waves which are formed as a result of the reflection of an incident shock wave of relatively low intensity from the end of the channel. The mathematical model includes the system of gas dynamics equations supplemented by Arrhenius kinetics for a model hydrogen-oxygen mixture with tabular kinetic parameters corresponding to the operating range of pressures and temperatures of the mixture. Numerical calculations are carried out using the finite volume method. The construction of computational grids consisting of triangular cells is carried out using the free software SALOME. The numerical algorithm is parallelized by the computational domain decomposition method using the METIS library. The exchange of grid functions between computational cores is carried out using the functions of the MPI library. The problem of acceleration of the parallel algorithm realized in the code is considered in comparison with the case of the linear dependence of the number of computational cores. A number of calculations were carried out using a different number of triangular cells and a comparison of patterns of detonation initiation was carried out. The performed calculations show that the detonation initiation time is approximately the same in all computations. The main difference in detonation patterns is associated with gas flow and Physical and chemical reactions in the mixture.

Лопато А.И. Математическое моделирование инициирования детонации в канале с профилированным торцом с использованием параллельных вычислений. Математическое моделирование и численные методы, 2023, № 4, с. 15–26

#### 533.6.011.5:533.6.011.72:519.6 Numerical simulation of hysteresis around a flat nozzle

**Maksimov F. A. (Institute for Computer Aided Design of the Russian Academy of Sciences)**

doi: 10.18698/2309-3684-2023-4-2746

The results of numerical simulation of two-dimensional plane laminar flows near two inclined plates forming a constricting nozzle along the velocity vector of an incoming supersonic perfect gas flow are presented. A multi-block computational technology is applied with the use of local curvilinear grids adapted to the surface of bodies, which have finite areas of overlap with a global rectangular grid for the entire computational domain. Viscous boundary layers are resolved on local grids using the Navier-Stokes equations, and the effects of aerodynamic interference of accompanying shock-wave structures are described in terms of the Euler equations. In areas of grid overlap, function interpolation is applied up to the boundaries of the transition from one grid to another. With a successive increase or decrease in the Mach number of the oncoming supersonic flow, a qualitative rearrangement of the flow structure near the nozzle is detected - either a detached shock wave and a subsonic flow zone in front of the nozzle, or oblique shocks near inclined plates are formed. A hysteresis is revealed, which is expressed in the fact that in a certain range of Mach numbers, the flow structure and the aerodynamic load on the nozzle depend not only on the value, but also on the prehistory of the change in the Mach number. The possibility of changing the flow structure by introducing a density inhomogeneity into the oncoming flow is shown.

Максимов Ф.А. Численное моделирование гистерезиса при обтекании плоского сопла. Математическое моделирование и численные методы, 2023, № 4,с. 27–46.

#### 539.36 Modeling of microstructure changes and elastic properties of alloys in the process of spot welding

**Dimitrienko Y. I. (Bauman Moscow State Technical University), Salnikova A. A. (Bauman Moscow State Technical University), Oreshnikova E. A. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2023-4-4763

A mathematical model of phase transformations in steel alloys during resistance spot welding is proposed, taking into account all stages of the process: from heating and partial melting of the metal, which cause irreversible physical and chemical transformations of the steel microstructure, to the cooling stage, during which solidification and “return” formation of alloy phases occurs . The model describes changes in the 3D microstructure of a steel alloy during heating and subsequent cooling with the formation of ferritic and austenitic structures. An algorithm for calculating model constants using a special procedure for solving the inverse problem is proposed, as well as an algorithm for numerically solving the problem of predicting changes in the elastic properties of steel during the welding process, which includes finite element 3D modeling using the SMCM software package, developed at the Department of Computational Mathematics and Mathematics physics" of Bauman Moscow State Technical University. An example of numerical simulation using the proposed model and algorithm for a steel alloy is given.

Димитриенко Ю.И., Сальникова А.А., Орешникова Е.А. Моделирование изменения микроструктуры и упругих свойств сплавов в процессе контактной точечной сварки. Математическое моделирование и численные методы, 2023, № 4, с. 47–63

#### 521.19 The perturbation hollow spheres modelling for the gravity assists in the Solar system

**Borovin G. K. (Keldysh Institute of Applied Mathematics of the Russian Academy of Scienсes), Golubev Y. F. (Keldysh Institute of Applied Mathematics of the Russian Academy of Scienсes), Grushevskii A. V. (Keldysh Institute of Applied Mathematics of the Russian Academy of Scienсes), Tuchin2 A. G. (Keldysh Institute of Applied Mathematics of the Russian Academy of Scienсes)**

doi: 10.18698/2309-3684-2023-4-6473

One of the types of gravitational scattering in the Solar System within the framework of the circular restricted three-body problem (CR3BP) are the gravity assist maneuvers of "particles of insignificant mass" (spacecraft, asteroids, comets, etc.). For their description, a physical analogy with the scattering of beams of charged alpha-particles in the Coulomb field is useful. However, unlike the scattering of charged particles, there are external restrictions on the ability to perform gravity assists associated with the limited size of the spheres of influence of the planet. At the same time, internal limitations on the possibility of performing gravity assists are known from the literature on CR3BP, estimated by the effective radii of planets (including gravitational capture by a planet falling into it). They depend on the asymptotic velocity of the particle relative to the planet. For obvious reasons, their influence makes it impossible to effectively use gravity assist maneuvers. The paper presents generalized estimates of the sizes of near-planetary regions (flat "perturbation rings" or "perturbation hollow spheres" rotating synchronously with a small body in the three-dimensional case), falling into which is a necessary condition for the implementation of gravity assists. A detailed analysis shows that Neptune and Saturn have characteristic of perturbation hollow spheres of the largest size in the Solar System, and Jupiter occupies only the fourth place in this list

Боровин Г.К., Голубев Ю.Ф., Грушевский А.В., Тучин А.Г. Моделирование пертурбационных оболочек для гравитационных маневров в Солнечной системе.Математическое моделирование и численные методы, 2023, № 4, с. 64–73.

#### 519.63 Modification of aluminum by laser shock wave detected in atomistic modeling

**Perov E. A. (JIHT), Zhakhovsky V. V. (Dukhov Automatics Research Institute/JIHT), Алимович N. A. (L.D. Landau Institute for Theoretical Physics RAS), Shepelev V. V. (Institute for Computer Aided Design of the Russian Academy of Sciences), Fortova S. V. (Institute for Computer Aided Design of the Russian Academy of Sciences), Doludenko A. N. (JIHT)**

doi: 10.18698/2309-3684-2023-4-7492

Plastic deformations are the basis of such industrial technology as laser thermal hardening or laser pinning (LSP, laser shock peening). In this paper, the possibility of hardening the surface layer of an aluminum sample irradiated by a single femtosecond laser pulse is investigated by the method of classical molecular dynamics. Several initialstates of the film are simulated; three orientations of the crystal lattice are investigated — [1, 0, 0] (the first orientation of the crystal lattice), [1, 1, 0] (the second orientation of the crystal lattice), [1, 1, 1] (the third orientation of the crystal lattice). A numerical study of the effect of various values of the invested energy in the range from 120,98 J/m2 to 2540,01 J/m2 of a laser pulse on the depth of plastic deformations affecting the hardening of the material under study was carried out. The energy values were selected in such a way that the plastic front of the UV (shock wave) stopped before it reached the right boundary of the simulated film. If this condition is not observed, then the dependence cannot be considered correctly constructed, since the stretching wave reflected from the right boundary of the sample will slow down the plastic shock front, acting as an unloading wave. With the help of this dependence, the threshold value of the invested energy is determined, when exceeded, aluminum begins to deform plastically.

Перов E.А., Жаховский В.В., Иногамов Н.А., Шепелев В.В., Фортова С.В., Долуденко А.Н.. Молекулярно-динамическое моделирование модификации алюминия лазерной ударной волной. Математическое моделирование и численные методы, 2023, № 4, с. 74-92

#### 519.2 Polynomial chaos and regression based on KolmogorovGabor polynomials: comparative modeling

**Oblakova T. V. (Bauman Moscow State Technical University), Pham Q. (-)**

doi: 10.18698/2309-3684-2023-4-93108

The application of the generalized expansion of polynomial chaos (PC) and models based on Kolmogorov-Gabor polynomials in regression problems is considered. When choosing PC expansion, the Wiener-Askey scheme was used, which sets the correspondence between the feature distribution law and the orthogonal polynomial basis. To calculate the expansion coefficients, non-intrusive methods were used: least squares, elastic network, as well as Ivakhnenko's inductive evolutionary algorithm. Kolmogorov-Gabor polynomials are used as a reference function of a polynomial neural network. Model errors and performance were calculated on a test set. Models were compared on a linear transport problem under uncertainty: the diffusion coefficient and drift were modeled by uniformly distributed random variables. It is shown that with a small interval of variation in the values of random variables, both models give good efficiency, but the PC model demonstrates a smaller spread of errors and is faster in time. For the de-cay equation with random coefficients distributed according to the Gaussian law, the influence of the correlation of these coefficients on the rate of convergence is studied. It is shown that with dependent coefficients, the best performance is observed in higher-order PC models. On the basis of comparative modeling, it has been established that the use of PC is unambiguously preferable in the following cases: a small dimension of the space of input features, a known law of distribution of input data, and correlated features. It is also shown that the use of PC with a large dimension of the space of input features is inefficient due to the rapid increase in the number of terms in the expansion, leading to a sharp increase in the time to process the task. In this case, the regression model based on the Kolmogorov-Gabor polynomials in combination with the GMDH turned out to be clearly preferable.

Облакова Т.В., Фам Куок Вьет. Сравнительное моделирование на основе многочленов Колмогорова-Габора в задачах полиномиального хаоса и регрессии. Математическое моделирование и численные методы, 2023, № 4, с. 93–108.

#### 519.866 Mathematical modeling of a strategic advertising campaign

**Chibisova A. V. (Bauman Moscow State Technical University), Shinakov D. S. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2023-4-109121

In this article, it is proposed to create a profit model for an ordinary advertiser who will advertise on a video platform. Video platforms provide video content services to meet the needs of users in entertainment, as well as advertising space to meet the needs of advertisers in making a profit. When watching videos, consumers can gain perceived usefulness by satisfying their need for entertainment and curiosity. For example, people watch TV shopping channels to buy or learn about products; the net usefulness of advertising for such users is posi-tive. Nevertheless, the inclusion of advertising may negatively affect some consumers, thereby reducing their usefulness. For example, users may get bored with video advertis-ing due to disinterest. Consequently, the relationship between the duration of video and advertising is a key factor affecting the perceived usefulness of consumers. In this article, we investigate the relationship between the duration of the commercial and the advertiser's profit in the short video advertising market.

Чибисова А.В., Шинаков Д.С. Математическое моделирование стратегической рекламной кампании. Математическое моделирование и численные методы, 2023, № 4, с. 109-121.