and Computational Methods

#### 539.26 Analysis of empirical models of deformation curves of elastoplastic materials (review). Part 3

**Golovina N. Y. (Industrial University of Tyumen)**

doi: 10.18698/2309-3684-2023-1-331

This article is the third part of a review of works devoted to the study of the properties of elastic-plastic materials. The first and second parts were devoted to the analysis of universal empirical laws of deformation, which model the material properties over the entire range of deformation, up to fracture. It was concluded that in order to create a model of the material response to stress growth, the deformation law must be at least four-parametric. The empirical Ramberg-Osgood law was found to be the most qualitative, at least for the titanium alloy VT6 considered. However, despite its accuracy, it does not reflect the material properties in the zone of large plastic strains, including in the vicinity of the point of ultimate strength. This paper presents an analysis of multilink models describing the relationship between strain and stress by different laws in the elastic zone and in the plastic zone. The review includes two-link models by Nadai, Mirambell-Real, Rasmussen, Abdella, formulated for materials whose strain curve has no positive curvature section. Also considered in the review are the three-link models of Quach; Hertele; Belov-Golovina, which allow modeling of deformation curves with a positive curvature region. The evaluation of the quality of empirical laws and their correspondence to the sample of experimental points was carried out by minimizing the standard quadratic deviation and using the method of gradient descent to determine the minimum of the function of many variables. The material for the comparative analysis of empirical models is titanium alloy VT6; for the Hertele and Belov-Golovina models — steel St3sp. It is shown that the models built on the basis of multi-line splines determine the properties of elastic-plastic materials more accurately than the models built on the basis of universal laws.

Головина Н.Я. Анализ эмпирических моделей кривых деформирования упруго-пластических материалов (обзор). Часть 3. Математическое моделирование и численные методы, 2023, No 1, с. 3–31.

#### 519.6 Mathematical modeling of the non-stationary convection-diffusion problem on the optimal choice of heat sources location

**Khayitkulov B. K. (National University of Uzbekistan)**

doi: 10.18698/2309-3684-2023-1-3242

This work is devoted to the numerical solution of the non-stationary problem of optimal placement of heat sources of minimum power. The statement of the problem requires the simultaneous fulfillment of two conditions. The first condition is to ensure that the temperature is within the limits of minimum and maximum temperatures due to the optimal placement of heat sources with a minimum power in the parallelepiped. The second condition is that the total power of the heat sources used for heating is minimal. This problem was studied under stationary conditions in the works of other scientists. However, the problem was not considered in the non-stationary case. Since it is difficult to find a continuous solution to the boundary value problem, we are looking for a numerical solution to the problem. It is difficult to find an integral operator with a continuous kernel (Green's function). The numerical value of the Green's function is found in the form of a matrix. A new algorithm for the numerical solution of a non-stationary optimal control problem for the placement of heat sources with a minimum power in processes described by parabolic partial differential equations is proposed. A new technique for numerical solution is proposed. A mathematical and numerical model of the processes described by the convection-diffusion equation given for the first boundary value problem is constructed. The boundary value problem is studied for the three-dimensional case. An implicit finite difference scheme was used to solve the problem numerically. According to this scheme, a system of difference equations was created. The formed system of difference equations is reduced to a linear programming problem. The problem of linear programming is solved using the M-method. For each time value, a linear programming problem is solved. A new approach to the numerical solution of problems is proposed. A general block diagram of the algorithm for solving the non-stationary problem of optimal control of the placement of heat sources with a minimum power is given. An algorithm and software for the numerical solution of the problem have been developed. A brief description of the software is given. On specific examples, it is shown that the numerical solution of the boundary value problem is within the specified limits, the sum of optimally placed heat sources with a minimum power gives a minimum to the functional. The results of the computational experiment are visualized

Хайиткулов Б.Х. Математическое моделирование нестационарной задачи конвекции–диффузии об оптимальном выборе местоположения источников тепла. Математическое моделирование и численные методы, 2023, No 1, с. 32–42.

#### 539.3 Finite element modeling of temperature fields in thin-walled multilayer anisotropic shells

**Dimitrienko Y. I. (Bauman Moscow State Technical University), Yurin Y. V. (Bauman Moscow State Technical University), Koryakov M. N. (Bauman Moscow State Technical University), Maremshaova A. A. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2023-1-4363

The problem of developing a model for calculating temperature fields in thin-walled multilayer curvilinear-anisotropic thin shells of arbitrary geometric shape, including composite ones, is considered. As a rule, to solve this problem, a specific coordinate notation of the equations of the theory of heat conduction is used, which creates certain difficulties for calculating complex composite shells. In this paper, it is proposed to use an invariant record of the variational formulation of problems in the theory of heat conduction, followed by the application of the finite element algorithm procedure. As a result, a matrix differential equation is derived for determining the temperature field at the nodes of a finite element mesh. A software module has been developed for the finite element solution of the problem of non-stationary thermal conductivity of shells. The module functions as part of the SMCM software package, created at the Scientific and Educational Center for Supercomputer Engineering Modeling and Development of Software Systems, Bauman Moscow State Technical University (REC SIMPLEX). An example of solving the problem of calculating a non-stationary temperature field in a cylindrical shell with longitudinal-transverse reinforcement is given. Comparison of numerical simulation with similar calculations in the ANSYS software was carried out, which showed the high accuracy of the proposed method: the relative deviation of the results does not exceed 0,5%

Димитриенко Ю.И., Юрин Ю.В., Коряков М.Н., Маремшаова А.В. Конечно-элементное моделирование температурных полей в тонкостенных многослойных оболочечных элементах конструкций. Математическое моделирование и численные методы, 2023, No 1, с. 43–63

#### 624.04 A new method for calculating the torsional stiffness of a naturally twisted bar

**Temis Y. M. (Центральный институт авиационного моторостроения им. П.И. Баранова), Ziyatdinov I. Z. (Центральный институт авиационного моторостроения им. П.И. Баранова)**

doi: 10.18698/2309-3684-2023-1-6480

At the initial stages of designing compressor blades, screws, cutting tools, it is advisable to use a finite element model based on a model of a naturally twisted beam. This model takes into account the influence of the angle of natural twist on the rigidity of the part. The torsional stiffness of a bar significantly affects the stiffness parameters of the finite element model. It is shown that the torsional stiffness correction obtained on the basis of the relations of the technical theory of naturally twisted beams makes it possible to obtain results at small angles of natural twist that are in good agreement with the three-dimensional calculation of a twisted FEM beam. At large specific angles of initial twist, the technical theory gives overestimated values of the torsional stiffness. The article proposes a modification of the relations of the technical theory to determine the torsional rigidity, taking into account large angles of initial twist.

Темис Ю.М., Зиятдинов И.З. Новый метод вычисления жесткости на кручение в модели естественно-закрученного стержня. Математическое моделирование и численные методы, 2023, No 1, с. 64–80

#### 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.

#### 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.

#### 330.43, 519.23 Dynamics of macroeconomic and bilateral trade's indicators of the BRICS countries and the USA

**Malinetsky G. G. (Keldysh Institute of Applied Mathematics of the Russian Academy of Scienсes), Makhov S. A. (Keldysh Institute of Applied Mathematics of the Russian Academy of Scienсes)**

doi: 10.18698/2309-3684-2023-1-112123

The study's aim is to predict main trends and create scenarios for the economic development of BRICS countries (Brazil, India, China, Russia, South Africa) and the USA. Regression autonomous macro models were built, as well as a model of trade between them. The autonomous submodels use Population, Fixed Capital, Gross Domestic Product and Gross Capital Formation as key indicators. Autoregressive equations describe the dynamics of these variables. The resulting system of equations allowed us to describe the historical dynamics of demographic and macroeconomic indicators from 1990 to 2015 and to do a forecast until 2030. In the trade submodel bilateral trade flows link with gross domestic products of the economies. The relationship is described by the power dependence of the export flow on the gross domestic product of both trading partners. Unlike gravity-type models, the regression equation parameters are assumed to be constant for each pair of trading partners over the entire predicted time interval. The calculations showed that the models satisfactorily describe the dynamics of monotonically changing indicators and therefore can be used as a simple tool for forecasting the national and regional economy.

Малинецкий Г.Г., Махов С.А. Динамика макроэкономических показателей и взаимной торговли стран БРИКС и США. Математическое моделирование и численные методы, 2023, No 1, с. 112–123.