and Computational Methods

#### 539.3 Modeling of effective elastic–plastic properties of composites under cyclic loading

**Dimitrienko Y. I. (Bauman Moscow State Technical University), Sborschikov S. V. (Bauman Moscow State Technical University), Yurin Y. V. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2020-4-326

A method is proposed for calculating the effective elastic–plastic properties of composites under cyclic loading. The technique is based on the application of the method of asymptotic averaging of periodic structures for the case of materials with elastic-plastic properties under cyclic loading. A model of the deformation theory of plasticity by A.A. Il’yushin – V.V. Moskvitin under cyclic loading using the Masing model for changing the plasticity function under cyclic deformation. Local problems of the theory of plasticity for the periodicity cell of a composite material, as well as averaged problems of the theory of anisotropic plasticity under cyclic loading are formulated. A software module has been developed for the finite element solution of local problems on the periodicity cell. The software of the SMCM complex developed at the Scientific and Educational Center "Supercomputer Engineering Modeling and Development of Software Systems" of the Bauman Moscow State Technical University was used. The SMCM complex is designed for finite element modeling of the properties of composite materials. Numerical calculations of the elastic-plastic properties of dispersed-reinforced composites based on an aluminum alloy and SiC ceramic particles have been carried out. Calculations have shown that the developed technique can be used to predict cyclic deformation diagrams of elastic-plastic composites in a wide range of loading conditions, as well as to design new composite materials with specified properties.

Димитриенко Ю.И., Сборщиков С.В., Юрин Ю.В. Моделирование эффектив-ных упруго–пластических свойств композитов при циклическом нагружении. Ма-тематическое моделирование и численные методы, 2020, № 4, с. 3–26.

#### 531.6.011.32:532.582.4 Modeling of subsonic separation flow around bodies by the method of discrete vortices based on the concept of an equivalent surface with cubic splines

**Timofeev V. N. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2020-4-2743

For mathematical modeling of subsonic separation flow around axisymmetric bodies with a bottom section, a technique with a partial implementation of the concept of visco–inviscid interaction was used. A flow scheme with an equivalence semi-infinite surface was used. Numerical simulations were carried out according to the algorithms of the method using the method of discrete vortices and approximation by smoothing cubic splines. Data on the influence of the shape of the tail section of an equivalent surface on the velocity and pressure distribution during axisymmetric flow around bodies with a bottom section are presented. The proposed recommendations make it possible to apply this technique more universally.

Тимофеев В.Н. Моделирование дозвукового отрывного обтекания тел методом дискретных вихрей на основе концепции эквивалентной поверхности с кубическими сплайнами. Математическое моделирование и численные методы, 2020, № 4, с. 27–43.

#### 629.762 Some examples of numerical simulation of unsteady gas dynamical phenomena for the purpose of lifting vehicles design

**Plyusnin A. V. (Bauman Moscow State Technical University/JSC MIC NPO Mashinostroyenia)**

doi: 10.18698/2309-3684-2020-4-4460

The problem of unsteady gas flow between two volumes has been considered for the spatial and for the engineering formulations. The comparison of the two solutions has been presented which allows to assess the accuracy of the engineering approach usually utilized in practical calculations. The problem has been also considered of the determination of the unsteady force acting on the bottom of the canister at the stage of the lifting vehicle exit in the process of the gas dynamical ejection. The numerical simulation of the spatial problems has been completed in accordance with the classical Godunov’s method including the moving grid technique.

Плюснин А.В. Некоторые примеры численного моделирования нестационар-ных газодинамических явлений в обеспечение проектирования летательных аппа-ратов. Математическое моделирование и численные методы, 2020, № 4, с. 44–60.

#### 532.5:551.465 Numerical study of the amplitude–frequency characteristics of the ice cover disturbed by an immersed pulsating source

**Savin A. S. (Bauman Moscow State Technical University), Sidnyaev N. I. (Bauman Moscow State Technical University), Теделури М. А. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2020-4-6172

In connection with the implementation of programs for the development of vast Arctic spaces, adopted in several countries, the attention of many researchers is attracted by the properties of the ice sheets of the seas and land bodies of water. At the same time, the following trend can be noted. If earlier theoretical works related to mathematical modeling of the ice sheet dynamics were mainly devoted to the propagation of free waves, then in recent years the work aimed at studying the processes of wave generation on the ice sheet under the influence of various sources of disturbances has clearly prevailed. To date, analytical solutions have been obtained for a number of problems concerning the generation of waves on the ice sheet by model sources of disturbances that are identical to some point hydrodynamic features, for example, point sources or dipoles. In this case, the ice was considered as a thin elastic plate floating on the surface of the water. Even in such an idealized setting, it was possible to reveal far from obvious properties of the ice cover. Modeling of sources of fluid perturbations by point hydrodynamic features was previously used in classical hydrodynamics to calculate perturbations occurring on the surface of a fluid. This approach has also shown its effectiveness in the problems of ice cover perturbations. A significant advantage of the method of modeling the sources of fluid disturbances using various systems of point hydrodynamic features can be attributed to the absence of the need to set boundary conditions in the area of localization of the sources of disturbances. Continuously distributed sources of disturbances can be approximated with varying accuracy in the form of a superposition of point hydrodynamic features, which makes it possible to model many processes occurring in the aquatic environment, for example, the flow around the bottom irregularities, the release of matter, the displacement of the bottom sections, etc. Thus, model sources of perturbations with point localization are of interest both from the point of view of modeling more complex sources, and from the point of view of obtaining the simplest estimates of practical significance. In this paper, we con-sider the spatial problem of perturbation of the ice cover by a point source localized in the thickness of an infinitely deep liquid, and having an intensity that varies according to the harmonic law. A numerical study of the amplitude-frequency characteristics of the ice cover of different thickness under the influence of such a source is carried out. The main attention is paid to the disturbances of the ice cover that occur directly above the source. The frequencies of the source intensity fluctuations to which the ice cover responds to the greatest extent are determined. The dependences of such frequencies on the thickness of the ice cover are obtained.

Савин А.С., Сидняев Н.И., Теделури М. М. Численное исследование амплитудно–частотной характеристики ледяного покрова, возмущаемого погруженным пульсирующим источником. Математическое моделирование и численные методы, 2020, № 4, с. 61–72.

#### 533.6:532.526 Simulation of a three-dimensional flow of a perfect gas in a laminar boundary layer on the lateral surface of a circular blunt cone of small elongation

**Щебнева А. А. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2020-4-7383

The paper presents an analysis of heat transfer on the surface of a blunt cone of low elongation, streamlined at an angle of attack, which is based on systematic numerical solutions of the equations of a three-dimensional laminar boundary layer. The main attention in the work is paid to the study of the influence exerted on the intensity of heat exchange by the Mach and Reynolds numbers, as well as by the temperature of the cone surface. It is pointed out that in recent years there have been no literary publications devoted to the solution of three-dimensional equations of the boundary layer, which is associated with the emergence of the possibility of solving problems of this kind within the framework of the Navier–Stokes equations. At the same time, the validity of carrying out studies of heat transfer within the framework of three-dimensional equations of the boundary layer for the case of flow around bodies of a simple geometric shape is noted.

Щебнева А.А. Моделирование трехмерного течения совершенного газа в ла-минарном пограничном слое на боковой поверхности кругового затупленного ко-нуса малого удлинения. Математическое моделирование и численные методы, 2020, № 4, с. 73–83.

#### 539.3 Thermal stress modeling in composite shells based on asymptotic theory. Part 1. General shell theory

**Dimitrienko Y. I. (Bauman Moscow State Technical University), Gubareva E. A. (Bauman Moscow State Technical University), Pichugina A. Y. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2020-4-84110

An asymptotic theory of thermoelasticity of multilayer composite shells is proposed, the derivation of the basic equations of which is based on the asymptotic expansion in terms of a small geometric parameter of three-dimensional thermoelasticity equations. This method was previously developed by the authors for thin composite plates, and in this article it is applied to thin-walled shells of an arbitrary frame. According to the developed method, the original three-dimensional problem of thermoelasticity decomposes into a recurrent successor of one-dimensional local problems of thermoelasticity and an averaged two-dimensional problem of thin shells. For local problems of thermoelasticity, analytical solutions are obtained, which make it possible to close the averaged formulation of the problem of the theory of shells with respect to 5 unknown functions: longitudinal displacements, deflection, and two shear forces. It is shown that the averaged problem for multilayer shells coincides with the classical system of equations for Kirchhoff–Love shells, however, it is more substantiated, since the asymptotic theory does not contain any assumptions regarding the pattern of the distribution of permutations and stresses over thickness. In addition, the asymptotic theory makes it possible to calculate all the stresses in the shell, without solving any additional problems, but only by differentiating the averaged displacements.

Димитриенко Ю.И., Губарева Е.А., Пичугина А.Е. Моделирование термона-пряжений в композитных оболочках на основе асимптотической теории. Часть 1. Общая теория оболочек. Математическое моделирование и численные методы, 2020, № 4, с. 84–110.

#### 519.6 Comparison of methods for calculating values special functions of mathematical physics

**Апельцин В. Ф. (Bauman Moscow State Technical University), Krasnov I. K. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2020-4-111119

Comparison of two approaches to calculate values of Chebishev polynomials via recurrent procedures is realized. At that, first approach is based upon recursion upwards with respect to the index starting from the least index value. Second approach is based upon recursion downwards starting from evident asymptotical expressions of the functions with high values of index.

Апельцин В.Ф., Краснов И.К. Сравнение методов вычисления значений специальных функций математической физики. Математическое моделирование и численные методы, 2020, № 4, с. 111–119