Rubric: "1.2.2. Mathematical modeling, numerical methods and software packages (technical sciences)"



521.2:521.3:521.61 Approximate methods for studying the second harmonic generation of ultrashort laser pulses in nonlinear photonic crystals

Ruziev Z. J. (Ташкентский государственный технический университет), Sobirov O. I. (Ташкентский государственный технический университет), Koraboev K. A. (Ташкентский государственный технический университет), Sapaev U. K. (Ташкентский государственный технический университет)


doi: 10.18698/2309-3684-2022-1-314


Second harmonic generation of ultrashort laser pulses in nonlinear photonic crystals is investigated by numerical methods based on the approximation of slowly varying amplitudes and a unidirectional approximation, applicable to simplify the wave equation with nonlinear polarization in a dispersive medium. Under the same experimental conditions, the results of these approximations are compared. Comparative analysis shows that up to 10 fs of the main pulse duration, both approximate methods describe this process of frequency conversion in almost the same way, but below 10 fs, there is a discrepancy between their results. Mainly, the formation of the temporal profile of the second harmonic pulse and its efficiency are compared. A method for obtaining time profiles of the second harmonic pulse using a unidirectional approximation where the incident field is used entirely in both the spectral and time domains of the calculation is also shown. The effect of dispersion up to the third order of smallness is taken into account, during the use of the approximation of slowly varying amplitudes.


Рузиев З.Дж., Собиров О.И., Корабоев К.А., Сапаев У.К. Численное моделирование генерации второй гармоники ультракоротких лазерных импульсов в нелинейных фотонных кристаллах. Математическое моделирование и численные методы, 2022, № 1, с. 3–14.



519.63:536.4 Numerical modelling of porcesses of formation, growth, and decomposition of agglomerates in porous medium under different modes of heating

Donskoy I. G. (Melentiev Energy Systems Institute SB RAS)


doi: 10.18698/2309-3684-2021-3-2441


The paper considers a numerical model of flow in a porous medium containing particles of a melting component (polymer). When heated, these particles swell, deform and fill the pore spaces, as a result of which the permeability is significantly reduced. The relationship between porosity and permeability is described by a simple Kozeny-Karman formula. Then, near the lower (inlet) boundary, a region with low permeability (i.e.agglomerate) is formed, the growth of which is determined by the conditions at the side wall and inlet boundaries. As a result of calculations, typical scenarios of porous medium blocking at different heating temperatures were obtained. It is shown that when heated through the wall, the polymer may decompose, so the porous medium partially restores its permeability. When heated by the inlet gas, agglomerate is much more stable, since it blocks the heating source.


Донской И.Г. Численное моделирование процессов образования, роста и разложения агломератов в пористой среде при разных режимах нагрева. Математическое моделирование и численные методы, 2021, № 3, с. 24–41.



519.6 Agent-based model of cultural interactions on non-metrizable Hausdorff spaces

Belotelov N. V. (Bauman Moscow State Technical University/Institution of Russian Academy of Sciences Dorodnicyn Computing Centre of RAS), Pavlov S. A. (Bauman Moscow State Technical University)


doi: 10.18698/2309-3684-2021-3-105119


The need to develop formalized computer-oriented approaches to conducting interdisciplinary research of intercultural interactions is an urgent task. The article describes an approach to the development of agent models of intercultural interactions based on the use of non-metrizable Hausdorff spaces using genetic algorithms to introduce dynamic changes in the structure of cultural agents under consideration. The article considers a prototype of an agent model in which the state of agents is described in Hausdorff spaces. Using the choice of reference points for each agent, the Uryson function is built, which allows you to enter the preferences of agents. Further, using the technology of gentic algorithms, it is possible to obtain the clock dynamics of changes in the entire system of agents. The article describes some simulation experiments. Possible prospects for the development of this approach are discussed.


Белотелов Н.В, Павлов С.А. Агентная модель культурных взаимодействий на неметризуемых хаусдорфовых пространствах. Математическое моделирование и численные методы, 2021, № 3, с. 105–119.



519.8 A "mixed" model of the confrontation of numerous groups with linear dependencies on the time of the effective speed of strikes by units of the parties

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


doi: 10.18698/2309-3684-2022-2-102113


Using the method of dynamics of averages, a "mixed" model of numerous groupings confrontation has been developed with linear dependencies on the time of effective speeds of striking by units of the parties. An algorithm is constructed that allows to investigate the course of the process and calculate its main indicators. It is established that the use of confrontation models with constant effective speed of strikes in many cases leads to significant errors in the calculation of the main indicators of the process. The influence of a preemptive strike by one of the opposing sides on the course of the progress and the final outcome are investigated.


Чуев В.Ю., Дубограй И.В. «Смешанная» модель противоборства многочисленных группировок при линейных зависимостях от времени эффективных скоростей воздействий единицами сторон. Математическое моделирование и численные методы, 2022, № 2, с. 104–115



004.855.5 Neural network methods for solving the problem of credit scoring

Kadiev A. D. (Bauman Moscow State Technical University), Chibisova A. V. (Bauman Moscow State Technical University)


doi: 10.18698/2309-3684-2022-4-8192


The mathematical derivation of the presented neural network model is demonstrated. Reduction of the classification problem to an optimization problem. Produced recon-naissance data analysis, as well as their preprocessing for further use in training classification algorithms. The architectures of neural networks were designed depending on the activation function, the number of hidden layers of the neural network and the number of neurons in the hidden layers. More than ten neural networks were trained to solve the task of credit scoring. The calculation of the learning time of neural networks was made. The solution of the problem using classical machine learning algorithms is presented. It could be seen that the standard deviation of accuracy and ROC AUC for neural networks is greater than that of a random forest. This is due to the fact that we choose the initial weights randomly and calculate the gradients not on the entire sample, but on small parts, which adds some learning error. But these deviations were not only for the worse. In the best situations, according to both metrics, neural networks showed the worst result by a couple of percent. The analysis of results is made. Comparative analysis shows that neural networks have better classification quality than classical machine learning algorithms, and also that neural networks have less training time than classical machine learning algorithms. Graphs and tables displaying the results obtained are presented.


Кадиев А.Д., Чибисова А.В. Нейросетевые методы решения задачи кредитного скоринга. Математическое моделирование и численные методы, 2022, № 4, с. 81–92.



519.6:621.791.05+544.015.4 Investigation of the effect of wettability and concentration of modifying nanoparticles on the weld structure in laser welding of an aluminum alloy

Isaev V. I. (Khristianovich Institute of Theoretical and Applied Mechanics SB RAS), Cherepanov A. N. (Khristianovich Institute of Theoretical and Applied Mechanics SB RAS), Shapeev V. P. (Khristianovich Institute of Theoretical and Applied Mechanics SB RAS/Новосибирский государственный университет)


doi: 10.18698/2309-3684-2023-1-8191


We proposed a self-consistent thermokinetic model of binary alloy crystallization in the weld joint modified by nanodimensional inoculators added to the welding pool during the laser weling process. This combined model of laser welding of uniform metals describes thermophysical prcesses of macroscopic properties of the weld joint and its structure depending on welding regimes and properties of nanomodifying powders (capillary angle, powder concentration). It is based on a thermophysical model of the action of the laser radiation on metal during the laser welding of metallic plates. The simulation is supplemented with a nonequilibrium model of heerogeneous nucleation and the growth of crystalline phase on nanomodifying particles added to the welding pool during the cooling and solidification of alloy in the weld joint. Numerical simulations based on the collocations and least squared methods were performed for butt weling of plates made of a binary aluminum alloy. We calculated the temperature distribution in the plates, the shape of the cross-section of the welding pool, and quantitative properties of its crystal structure. We investigated the effect of the capillary angle and nanomodifying powder concentration on the crystal grain size


Исаев В.И., Черепанов А.Н., Шапеев В.П. Исследование влияния смачиваемости и концентрации модифицирующих наночастиц на структуру шва при лазерной сварке алюминиевого сплава. Математическое моделирование и численные методы, 2023, No 1, с. 81–91.



519.8 A Diffusion Model of Cluster Evolution in a Heat-Resistant Nickel Alloy Metal Melt

Tyagunov A. G. (Ural Federal University), Zeyde K. M. (Universidad Politècnica Salesiana/University of Genoa), Milder O. B. (Ural Federal University), Tarasov D. A. (Ural Federal University)


doi: 10.18698/2309-3684-2023-2-332


In this work, a mathematical model of the thermo-temporal evolution of a cluster in the melt of a heat-resistant nickel alloy ZhS6U is constructed. An initial-boundary value problem with a moving boundary is formulated, for the solution of which numerical modeling is used by the particle trajectory method, and a number of classical physical theories are used to describe evolutionary processes. To check the accuracy of the model, a physical experiment is involved in constructing polytherms and isotherms of the electrical resistance of the alloy under consideration. It has been confirmed that the Brownian diffusion model and Drude's theory of conductivity are applicable to describe both the temporal and temperature evolution of a cluster. The approach to modeling based on "hard balls" also justified itself. According to the simulation results, in the time range from 1690 to 1752 K, the number of particles in the cluster varies from 5000 to 2000, the average dynamic viscosity of the cluster varies from 3 to 2 * 1010 Pa * s, however, it is assumed that the central part is much denser than periphery. The cluster radius varies from 24 to 18 A, and the radius of the free zone around the cluster varies from 56 to 43 A. The directions of further development of the model are determined.


Тягунов А.Г., Зейде К.М., Мильдер О.Б., Тарасов Д.А. Диффузионная модель эволюции кластера в металлическом расплаве жаропрочного никелевого сплава. Математическое моделирование и численные методы, 2023, № 2, с. 3–32.



519.6 Agent model of two competing populations taking into account their structurality

Belotelov N. V. (Bauman Moscow State Technical University/Institution of Russian Academy of Sciences Dorodnicyn Computing Centre of RAS), Brovko A. V. (Bauman Moscow State Technical University)


doi: 10.18698/2309-3684-2022-3-7183


The article describes an agent simulation model of two populations competing for one resource. In the model, it is assumed that an individual dies if its mass-energy becomes non-positive. It is assumed that individuals of each of the populations under consideration can form flocks, this allows populations to increase their competitiveness. In the model, this is formalized through the ability to organize networks connecting individuals of the same species. At the same time, individuals can form only a certain number of connections with neighbors. The concept of "valence" is introduced in the model to describe this. It is assumed that within each network there is an instantaneous redistribution of the resource available to all members of the network by each member of the pack. In addition to the model, the article describes the structure of the program with which simulation experiments were carried out. As a result of the simulation experiments, the following was obtained. If the resource is highly productive, then in the process of competitive interaction, the population wins, the agents of which have a large "valence". And in the case of a low-productive resource, individuals of a population with a lower "valence" win in competitive interaction. This is due to the fact that more complex structures require more energy to maintain the flock.


Белотелов Н.В., Бровко А.В. Агентная модель двух конкурирующих популяций с учетом их структурности. Математическое моделирование и численные методы, 2022, № 3, с. 71–83.



621.822.2, 519.63 Numerical study of the influence of lubricant viscosity grade on thrust bearing operation

Sokolov N. V. (HMS Compressors Business Unit (HMS Group)/Казанский национальный исследовательский технологический университет), Khadiev M. B. (Казанский национальный исследовательский технологический университет), Fedotov P. E. (Kazan (Volga region) Federal University,/LLC «AST Volga region»), Fedotov E. M. (LLC «AST Volga region»)


doi: 10.18698/2309-3684-2023-1-92111


The study of the influence of the viscosity grade of the supplied oil ISO VG32 and ISO VG46 in a wide range of rotor speeds and operating clearances on the local and integral characteristics of a thrust plain bearing with fixed pads of the compressor is presented. The studies were carried out using the Sm2Px3Txτ calculation program based on the results of numerical experiments of the bearing. The program is built by numerical implementation of a nonstationary periodic thermoelastohydrodynamic (PTEHD) mathematical model of the thrust bearing operation. The research results indicate a significant influence of the oil viscosity grade on the main characteristics and temperature conditions of the thrust bearing. When changing from ISO VG46 to the lighter oil ISO VG32, there is a noticeable reduction in bearing pad temperatures and power loss. However, the level of this change is determined by the specified operating clearance between the rotating thrust collar and the bearing pads. The influence of oil viscosity grade and the profile of the working surface on the temperature regime of the pad is analyzed. The value and location of the maximum temperature of the thrust bearing pad is determined, as well as the possibility of applying the standard point 75/75 from API-670 in practice.


Соколов Н.В., Хадиев М.Б., Федотов П.Е., Федотов Е.М. Численное исследование влияния класса вязкости смазки на работу упорного подшипника скольжения. Математическое моделирование и численные методы, 2023, No 1, с. 92–111.



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