and Computational Methods

#### 539.3 Modeling of thermal stresses in composite shells based on asymptotic theory. Part 2. Calculation of cylindrical shells

**Dimitrienko Y. I. (Bauman Moscow State Technical University), Gubareva E. A. (Bauman Moscow State Technical University), Pichugina A. Y. (Bauman Moscow State Technical University), Bel’kova K. V. (Bauman Moscow State Technical University), Borin D. M. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2022-4-330

The general asymptotic theory of thin multilayer shells developed by the authors earlier in Part 1 of this study is applied to cylindrical anisotropic thermoelastic shells. It is shown that for cylindrical shells the general theory is substantially simplified: general two-dimensional averaged thermoelasticity equations for multilayer shells are obtained. These equations are similar to the classical equations of cylindrical shells in the Kirchhoff-Love theory, but they are obtained in a completely different way: on the basis of only an asymptotic analysis of the general three-dimensional equations of the theory of thermoelasticity. No hypotheses regarding the distribution of displacements or stresses over the thickness are used in this theory, which makes it logically consistent. In addition, the developed theory makes it possible to obtain explicit analytical expressions for all 6 components of the stress tensor in cylindrical anisotropic shells. Explicit expressions are obtained for all tensor constants included in these stress formulas. An example of calculating thermal stresses in a cylindrical composite shell with axisymmetric bending due to the combined action of external pressure and one-sided non-stationary heating is given. An example of a layered-fiber 4-layer shell with different angles of helical winding of reinforcing fibers is considered. It is shown that the developed one allows one to study in detail such complex effects as the formation of significant transverse thermal stresses during heating, which significantly exceed the level of interlayer shear stresses, which are traditionally considered the most dangerous for layered composites.

Димитриенко Ю.И., Губарева Е.А., Пичугина А.Е., Белькова К.В., Борин Д.М. Моделирование термонапряжений в композитных оболочках на основе асимптотической теории. Часть 2. Расчет цилиндрических оболочек. Математическое моделирование и численные методы, 2022, № 3, с. 3–30

#### 551.5; 519.6 Organization of numerical experiments on a joint global model of the atmosphere and ocean

**Parkhomenko V. P. (ФИЦ ИУ РАН/Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2022-4-3147

A three-dimensional hydrodynamic model of the global climate is presented, including a block of the atmosphere general circulation, a block of the ocean in geostrophic approximation with a frictional term in the horizontal momentum equations with a real distribution of depths and continents, a block of the sea ice evolution. Calculations of possible climate change up to 2100 are given on the basis of several CO2 growth scenarios. A significant decrease in the meridional flow of water in the Atlantic has been established during the implementation of a strong CO2 growth scenario. Numerical experiments have been carried out to identify potential hysteresis associated with attenuation, up to blocking (under certain conditions) Atlantic meridional thermohaline circulation.

Пархоменко В.П. Организация численных экспериментов на совместной глобальной модели атмосферы и океана. Математическое моделирование и численные методы, 2022, № 4, с. 31–4

#### 519.2 Comparative analysis of modeling methods and time series forecasting based on the theory of fractal Brownian motion

**Oblakova T. V. (Bauman Moscow State Technical University), Alekseev D. S. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2022-4-4862

The work summarizes the results obtained in the course of the implementation of Bachelor's final qualifying work and is devoted to the methods of simulating and applying the fractional Brownian motion in the problems of time series analysis. Software modules have been implemented to generate trajectories of fractal Brownian motion using the methods of stochastic representation, Cholesky decomposition and Davis-Hart. Algorithms vere compared in terms of their complexity and the quality of the resulting trajectories. The Hurst exponent was estimated by the Minkowski and R/S analysis methods. An approximation of time series by fractal Brownian motion using a power function is proposed and implemented for the subsequent application of a linear prediction algorithm based on the normal correlation theorem. It has been established that with the help of the presented approximation it is possible to achieve a satisfactory forecast of the exchange rate for several values ahead.

Облакова Т.В., Алексеев Д.С. Сравнительный анализ методов моделирования и прогнозирования временных рядов на основе теории фрактального броуновского движения. Математическое моделирование и численные методы, 2022, № 4, с. 48–62

#### 519.654 Modeling of cyclic processes (of time series type experimental date) by solution of piecewise linear differential equations with constant coefficients

**Smirnov V. Y. (LTD Azforus/MONIKI), Kuznetsova A. V. (IBCP RAS)**

doi: 10.18698/2309-3684-2022-4-6380

This paper presents a method for modeling of real macroworld cyclic processes by the solutions of two (or more) linear differential equations with constant coefficients. It is shown that possible to send such model from any initial condition (point in phase space) to a prescribed final point in phase space for a prescribed amount of steps. As a result, conditions for a constructing closed cycle in phase space of any dimension were found. Switching from one system of equations to another is obtained then integral curve reaches some threshold — frontier in phase sheet (space). The analysis of convergence rate of the solution for such closed cycles is made. The integral curve behavior significant dependence of initial condition was shown.The obtained model is connected with experimental data in the form of time series and approximates them with solution of linear differential equations on condition of minimizing the RMS deviation. Proposed model also can be applied to a task of some index (technical, economical) reaching the prescribed meaning to a prescribed moment.

Смирнов В.Ю., Кузнецова А.В. О моделировании циклических процессов решениями кусочно-линейных разностных уравнений с постоянными коэффициентами по экспериментальным данным в виде временных рядов. Математическое моделирование и численные методы, 2022, № 4, с. 63–80.

#### 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.

#### 519.87 Structural theory of complex systems. Geometric theory and humanitarian aspects of modeling

**Brodsky Y. I. (ФИЦ ИУ РАН)**

doi: 10.18698/2309-3684-2022-4-93113

We propose a formal definition of the complex system computer model, as a species of structure in the sense of N. Bourbaki — the M (Model) species of structure. The class of mathematical objects defined by the M species of structure has the following two properties: a complex created by combining mathematical objects of the M species of structure, according to the certain rules, is itself a mathematical object of the same M species of structure. The computation organization process is same for all the mathematical objects of the M species of structure and therefore can be implemented by a single universal program for the simulation calculations organization. The presence of these two properties of the M species of structure representatives allows us to build an end-to-end technology for the description, synthesis and software implementation of the complex systems models — Model Synthesis and Model-Oriented Programming. By studying the morphisms of the M species of structure base sets of the model constructed with the model synthesis help, and the invariants limiting such morphisms, we obtain a formal mathematical language for the study of complex open (changing their composition) systems. By conducting a traditional humanitarian discourse, one can always correlate it with the corresponding object of the M species of structure — translating the higher-level language of humanitarian concepts into mathematical language. The conclusions obtained using this language are, for example, that sustainable development is the modus vivendi of a complex open system and that in complex open systems, unlike closed physical systems, the conservation of laws plays a leading role (the system sacrifices power to maintain its axioms and structure), but not the conservation laws (which of course take a place).

Бродский Ю.И. Структурная теория сложных систем. Геометрическая теория и гуманитарные ас-пекты моделирования. Математическое моделирование и численные методы, 2022, № 4, с. 93–113.

#### 519.2 Numerical research of persistent time series based on the ARFIMA model

**Oblakova T. V. (Bauman Moscow State Technical University), Kasupovich E. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2022-4-114125

The work is devoted to methods for detecting long-term memory in financial time series. Using the method of analysis with the help of the original program code, a number of values of the real financial index S & P500 were studied, estimates of the Hurst index were obtained, and persistence was demonstrated. To solve the problem of predicting the future values of a series, the ARFIMA model is proposed, which is a generalization of the standard ARIMA model and involves the use of a fractional differentiation opera-tor. A two-stage algorithm for constructing a forecast for a series of logarithmic profits is presented and implemented. It is shown that the use of the ARFIMA model improves the quality of the forecast in comparison with ARIMA for all standard metrics.

Облакова Т.В., Касупович Э. Численное исследование персистентных временных рядов на основе модели ARFIMA. Математическое моделирование и численные методы, 2022, № 4, с. 114–125.