and Computational Methods

#### 534.131.2 Modeling of dynamic stability of thin-walled structures partially filled with liquid under hydrostatic action

**Park S. (Moscow Aviation Institute (National Research University)), Grigor’ev V. G. (Moscow Aviation Institute (National Research University))**

doi: 10.18698/2309-3684-2022-3-317

In this paper, we consider the problem of stability of a thin-walled shell structure with two hemispherical bottoms of the same thickness, partially filled with liquid, which is immersed in an external liquid medium and is under hydrostatic pressure. The dynamic characteristics of such a structure containing a limited volume of liquid under internal pressure and hydrostatic pressure are obtained. The developed program for calculating the dynamic characteristics of axisymmetric shell structures containing liquid is based on the finite element method. The finite elements have an annular shape when rotated around the axis of symmetry. The program is implemented in Excel spreadsheet using Visual Basic for Applications (VBA). It allows to calculate the natural frequencies of thin-walled elastic structures interacting with an arbitrary number of liquids, considering the influence of the static stress-strain state caused by hydrostatic and internal pressure and other external forces that do not violate the axial symmetry of the problem. At a fixed value of the internal pressure, the calculation of the lowest natural frequencies of vibrations with different numbers of waves along the circumference is performed. By successive refinement, the critical thickness of the shell is determined, at which at least one of the natural frequencies reaches zero. The internal pressure p varies from 0 to 1 atm. in increments of 0,1 atm. and the calculations are repeated to obtain each critical value. At each pressure value, curves are plotted on the graph "number of waves — natural frequency". On the coordinate plane "pressure — shell thickness" the boundary of the instability region is constructed.

Пак Сонги, Григорьев В.Г. Моделирование динамической устойчивости тонкостенных конструкций, частично заполненных жидкостью, при гидростатическом воздействии. Математическое моделирование и численные методы, 2022, № 3, с. 3–17.

#### 519.62 Application of the one–step Galerkin method for solving a system of ordinary differential equations with initial conditions

**Russikikh S. V. (Moscow Aviation Institute (National Research University)), Shklyarchuk F. N. (Moscow Aviation Institute (National Research University))**

doi: 10.18698/2309-3684-2022-3-1832

A nonlinear oscillatory system described by ordinary differential equations with variable coefficients is considered. It is assumed that in the time interval under consideration, the solution of the system is sufficiently smooth - without discontinuities, collisions and bifurcations. From an inhomogeneous system of equations, terms that depend linearly on coordinates, velocities and accelerations and terms that depend non-linearly on these variables are explicitly distinguished. A new approach is proposed for numerical solution by the step method of the initial problem described by such a system of ordinary differential equations of the second order. At the integration step, unknown functions are represented as a sum of functions satisfying the initial conditions: a linear Euler solution and several given correcting functions in the form of polynomials of the second and higher degrees with unknown coefficients. The differential equations at the step are satisfied approximately in the sense of a weak solution by the Galerkin method on a system of corrective functions. Algebraic equations with nonlinear terms are obtained, which are solved by iteration, starting in the first approximation with a linear solution. The resulting solution at the end of this step is used as the initial conditions for the next step. As an example, we consider one homogeneous second-order differential equation without the first derivative with strong cubic nonlinearity in coordinate (at maximum amplitude, the nonlinear force is twice the linear force). This equation has an exact periodic solution in the form of an integral of the energy of a conservative system, which is used to estimate the accuracy of numerical solutions obtained by Galerkin, Runge-Kutta and Adams methods of the second order, as well as by Radau5 and BDF methods at various time intervals (up to 8000 periods of free oscillations of the system) using various constant integration steps (from 0.0025 fractions of a period). At the same time, in the Galerkin method, four identical correction functions were used at each step in the form of polynomials from the second to the fifth degree. It is shown that for large time intervals of calculations, the Galerkin method has a higher accuracy compared to other numerical methods considered. Therefore, it can be used for the numerical solution of nonlinear problems in which it is required to solve them over long time intervals; for example, when calculating steady-state limit cycles of nonlinear oscillations and chaotic nonlinear oscillations with strange attractors.

Русских С.В., Шклярчук Ф.Н. Применение одношагового метода Галеркина для решения системы обыкновенных дифференциальных уравнений с начальными условиями. Математическое моделирование и численные методы, 2022, № 3, с. 18–32.

#### 533.6.011.55 Numerical simulation high-speed flow around a cylindrical–conical body and a double cone

**Kharchenko N. A. (Central Aerohydrodynamic Institute (TsAGI)/MEPhI/Moscow Aviation Institute (National Research University)), Nosenko N. A. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2022-3346

The paper presents a classical validation problem of high-speed modeling. This problem is about interaction of a shock wave with a boundary layer in a laminar air flow around a cylindrical–conical body and a double cone. The main computational complexity of this problem is the detailed resolution of the near-wall region in order to further reproduce the experimental distributions of the surface characteristics of pressure and heat flux. Depending on the conditions of the undisturbed flow of the researched flow mode, the problem can have a recirculation zone, which is a vortex flow. This flow has a significant effect on the structure of the near-wall flow.

Харченко Н.А., Носенко Н.А. Численное моделирование обтекания высокоскоростным потоком цилиндрически–конического тела и двойного конуса. Математическое моделирование и численные методы, 2022, № 3, с. 33–46.

#### 539.36 Microstructural model anisotropic flow theory for elastic-plastic layered composites

**Dimitrienko Y. I. (Bauman Moscow State Technical University), Черкасова М. С. (Bauman Moscow State Technical University), Dimitrienko A. Y. (Lomonosov Moscow State University)**

doi: 10.18698/2309-3684-2022-3-4770

A microstructural model of layered elastic-plastic composites based on the anisotropic flow theory is proposed. The model represents the effective constitutive relations of the transversally isotropic theory of plastic flow, in which the model constants are determined not experimentally, but on the basis of approximations of the deformation curves of composites obtained by direct numerical solution of problems on the periodicity cell for basic loading trajectories, which arise in the method of asymptotic averaging. The problem of identifying the constants of this composite model is formulated; for the numerical solution of this problem, methods of optimizing the error functional are used. The results of numerical simulation by the proposed method for layered elastic-plastic composites are presented, which showed good accuracy of approximation of numerical strain diagrams.

Димитриенко Ю.И., Черкасова М.С., Димитриенко А.Ю. Микроструктурная модель анизотропной теории течения для упруго-пластических слоистых композитов. Математическое моделирование и численные методы, 2022, № 3, с. 47–70.

#### 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.

#### 519.866 Mathematical modeling of an advertising campaign

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

doi: 10.18698/2309-3684-2022-3-8497

This article proposes a method for optimizing the dynamic budget allocation policy for an advertising campaign placed through an advertising tool built into the search engine. This method takes into account the unique features of social media marketing, provides an optimal budget allocation policy over time for one advertising campaign and minimizes the duration of the campaign, taking into account the specific budget and the desired level of coverage of each marketing segment. The model includes a general "efficiency function" that determines the relationship between the cost of an advertising bid at a given time and the number of new users shown at that time. This goal is achieved by implementing an algorithm for optimal solution of the problem of dynamic distribution of the advertising budget under certain boundary conditions, as well as by analyzing data on advertising campaign for June 2018. In the course of the study, an algorithm for optimal solution of the problem of dynamic distribution of the advertising budget under appropriate boundary conditions was implemented, examples of specific cases of the efficiency function were given and some models of real advertising campaigns of the enterprise were analyzed. Then, the data registered by the advertising agency of a particular enterprise was analyzed in relation to an advertising campaign registeredusing a built-in search engine tool for calculating bids and audience coverage for 30 days.

Чибисова А.В., Шинаков Д.С. Математическое моделирование рекламной кампании. Математическое моделирование и численные методы, 2022, № 3, с. 84–97.

#### 519.87 Structural theory of complex systems. Model synthesis

**Brodsky Y. I. (ФИЦ ИУ РАН)**

doi: 10.18698/2309-3684-2022-3-98123

The purpose of this work is to organize from a unified viewpoint the results of the author's work in the field of the structural theory of complex systems modeling and the practice of their implementing of the last two decades. We propose a formal definition of the complex system computer model, as a species of structure in the sense of N. Bourbaki — the М (System) species of structure, based on the humanitarian analysis of the complex systems key properties, recognized by a number of authoritative researchers and practicians in this field, and the assumption of the possibility of constructing a mathematical computer model of a complex system, — the closure hypothesis. The class of mathematical objects defined by the М species of structure has the following two properties: a complex created by combining a finite number of mathematical objects of the М species of structure, according to the certain rules, is itself a mathematical object of the same М species of structure. The computation organization process is same for all the mathematical objects of the М 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 М 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 М 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 М species of structure — translating the higher-level language of humanitarian concepts into mathematical language. The proposed theory has a practical application in the field of development, description and implementation of complex software systems. A new programming paradigm is proposed — Model-Oriented Programming, which is a complete implementation of CAD methods in programming. When developing a software system, it is possible to stay within the framework of declarative programming, avoiding imperative, which greatly simplifies both its development and implementation, and subsequent debugging.

Бродский Ю.И. Структурная теория сложных систем. Модельный синтез. Математическое моделирование и численные методы, 2022, № 3, с. 98–123.