and Computational Methods

#### 519.245:004.942 Numerical stochastic simulation of electron transport in irradiated ferroelectrics

**Pavelchuk A. V. (ФГБОУ ВО "Амурский государственный университет"), Maslovskaya A. G. (ФГБОУ ВО "Амурский государственный университет")**

doi: 10.18698/2309-3684-2018-2-320

The article presents 3D-simulation system designed to analyze the characteristics of electron injection processes at diagnostics and modification of ferroelectrics with the scanning electron microscope techniques. The discrete stochastic model of electron trajectories was based on physical model of single interactions. The computational scheme was constructed with use of Monte Carlo simulation. The software application was suggested in order to perform the estimations of characteristics of electron injection effects in typical ferroelectrics. The characteristics specification of electron beam interaction area with irradiated target as well as the analytical expression for electron energy losses distribution were also descried with use of computing experiments data. The relationship of principal simulation parameters were indicated to control the field effects of the injected charges in ferroelectrics under electron irradiation.

Павельчук А.В., Масловская А.Г. Численное стохастическое моделирование транспорта электронов в облученных сегнетоэлектрических материалах. Математическое моделирование и численные методы, 2018, № 2, с. 3–20.

#### 51-72:519.688 Simulation of the characteristics of a nanocomposite material with spherical inclusions using the genetic algorithm

**Romanchuk S. P. (Yuri Gagarin State Technical University of Saratov), Korchagin S. A. (Yuri Gagarin State Technical University of Saratov), Terin D. V. (Yuri Gagarin State Technical University of Saratov)**

doi: 10.18698/2309-3684-2018-2-2131

The approach of solving the problem of selecting the composition of a nanocomposite material with the required characteristics is considered. As an object of research is a nanocomposite material that consists of a homogeneous matrix and inclusions of a spheroidal shape. To solve this problem, a software package is implemented, the selection algorithm is developed based on the genetic algorithm. The efficiency of the developed method and the dependence of the rate of convergence of the method on various parameters are analyzed.

Романчук С.П., Корчагин С.А., Терин Д.В. Моделирование характеристик нанокомпозитного материала со сферическими включениями применяя генетический алгоритм. Математическое моделирование и численные методы, 2018, № 2, с. 21–31

#### 539.376 Simulation of the bearing capacity of a cylindrical shell reinforced by a power set under conditions of material creep

**Dubrovin V. M. (Bauman Moscow State Technical University), Semyonov K. S. (Bauman Moscow State Technical University/RSC Energia)**

doi: 10.18698/2309-3684-2018-2-3246

Cylindrical shell, which is under the influence of considerable loads for a long time, can lose the ability to withstand the level of these loads, as its carrying capacity decreases. This is due to the fact that the shell material is subject to the creep phenomenon. As studies [1-3] show, creep is noticeably manifested even at normal temperature and stresses, much lower than the yield point of the shell material. Experimental and theoretical work on the stability of shells show [4-5] that the main reason for reducing the critical load for real shells in comparison with ideal shells is the initial design imperfections. Therefore, it is to be expected that additional deflections that arise as a result of creep deformation have a significant effect on the critical load (bearing capacity) of the shell. A method is proposed for calculating the load-bearing capacity of a cylindrical shell reinforced by a longitudinal (stringers) and a final (frame) power set under the action of axial and transverse loads, as well as internal excess pressure. As an example, a shell is considered, the material of which is an aluminum-magnesium alloy AMg6-M and AMg6-H. The dependence of the bearing capacity on the operating time is obtained.

Дубровин В.М., Семенов К.С. Моделирование несущей способности подкрепленной силовым набором цилиндрической оболочки в условиях ползучести материала. Математическое моделирование и численные методы, 2018, № 2, с. 32–46.

#### 539.374 Numerical simulation of nonisothermal elastoplastic deformation processes of structural materials

**Temis Y. M. (Центральный институт авиационного моторостроения им. П.И. Баранова), Hudyakova A. D. (Центральный институт авиационного моторостроения им. П.И. Баранова/Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2018-2-4769

The algorithm of the loading surface equation account during integration of constitutive equations system of invariant nonisothermal plasticity theory, based on returning of a representing point to a loading surface equation on each step of calculation was offered. Efficiency of proposed algorithm in combination with different linearization schemas for number of proportional and nonproportional thermomechanical deformation trajectories was explored. Results of simulation of the processes of tubular nickel-based alloy IN738LC specimens testing under proportional (tension-compression, tension-compression with torsion) and nonproportional («circle» and «diamond» with out-of-phase axis strain and temperature changing) cyclic thermomechanical deformation trajectories in operating-temperature range from 450 to 950°C.

Темис Ю.М., Худякова А.Д. Численное моделирование процессов неизотермического упругопластического деформирования конструкционных материалов. Математическое моделирование и численные методы, 2018, № 2, с. 47–69.

#### 532.51 Mathematical simulation of non-isothermal steady flow of non-Newtonian fluid by finite element method

**Dimitrienko Y. I. (Bauman Moscow State Technical University), Li S. -. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2018-2-7095

The finite element method is used to simulate the nonisothermal flow of non-Newtonian viscous fluids in complex geometries. The Carreau-Yasuda model of a non-Newtonian fluid is considered, in which the dependence of the viscosity coefficient on the second invariant of the strain rate tensor has a power form. A variational formulation of the problem of the motion of a non-Newtonian fluid for a plane case is obtained. The iteration algorithm of Newton-Raphson is used to solve the Navier-Stokes equations system, and the Picard iteration algorithm is used to solve the energy equation. The problem of the movement of a polymer mass in a mold of complex variable cross section in the presence of an uneven temperature field is considered. With the help of finite element modeling, a numerical analysis of the effect of various parameters on the movement of a liquid and the heat transfer of a polymer material at different values of external pressure was carried out. It is shown that the nature of the motion of a non-Newtonian fluid essentially depends on the rheological properties of the fluid and the characteristics of the geometric shape, which must be taken into account in technological processes of plastics processing.

Димитриенко Ю.И., Шугуан Ли Конечно-элементное моделирование неизотермического стационарного течения неньютоновской жидкости в сложных областях. Математическое моделирование и численные методы, 2018, № 2, с. 70–95.

#### 533.16 Procedure for numerical solution of equations of laminar-turbulent boundary layer on an axially symmetrical blunt body in a jet of combustion gases of liquid-propellant engine

**Gorskiy V. V. (Bauman Moscow State Technical University), Kovalsky M. G.**

doi: 10.18698/2309-3684-2018-2-96108

Currently, carbon-based materials are widely used as an ablative thermal protection for high-heat constructional elements in air and space en-gineering. In its turn, forecasting a change in shape of external surfaces of the specified elements with the course of time, which is determined by scorching of thermal protection, cannot be separated from the use of calcula-tion and theoretical procedures, which describe course of different phys-ico-chemical and mechanical processes associated with course of the considered event. Thereby, it is mandatory to approve such procedures through the re-sults of experi-mental studies, conducted in jets inside aerodynamic tunnels.

Among primary evidences of ablation of carbon-based materials is erosion (mass loss), which is usually observed in high-pressure gas flows. Meanwhile, it is required to carry out studies on large-scale models in the course of development tests, and that had de-termined the wide use of under-expanded jets of combustion gases of liquid-propellant rocket engines (LPE) for simulation of erosion of thermal protection.

Among the basic problems, encountering during solution of this kind of tasks, is calcu-lation of laminar-turbulent heat exchange in the conditions of gradient flow past blunt-ed point of the tested model by a diverging gas jet. This article is dedicated to the solu-tion of this problem and includes the modified version of semi-empirical model of ap-parent turbulent viscosity, approved by the results of experimental studies. This article shows that the use of this method makes it possible to specify significantly thermal con-di-tions of the model in comparison with the use of effective length method, which is used universally in practical work.

Горский В.В., Ковальский М.Г. Методика численного решения уравнений ламинарно-турбулентного пограничного слоя на осесимметричном затупленном теле в струе продуктов сгорания ЖРД. Математическое моделирование и численные методы, 2018, № 2, с. 96–108.

#### 533.6.011.5 The restoration of the pressure distribution in the perturbed region near the sphere in supersonic gas flow with an arbitrary effective adiabatic index

**Kotenev V. P. (Bauman Moscow State Technical University), Puchkov A. S. (Bauman Moscow State Technical University), Sapozhnikov D. A. (Bauman Moscow State Technical University), Tonkih E. G. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2018-2-109121

The generalization of the dependency which was proposed earlier for determining of the pressure in perturbed area streamlined by the supersonic flow of the inviscid perfect gas was provided. The modification allows to consider effects which occur when the sphere is streamlined by high temperature gas with adiabatic index which do not equal 1.4. According to article [2-3], made adjustment to the function which describes the behavior of the shock wave which depends on the adiabatic index. The Shepard function’s coefficient also consider of adiabatic index. First and second order members of Shepard function which describe a pressure in area does not change. Comparison of application versions with the available data calculations for the high temperature gas and approximations based on perfect gas shows high accuracy of the proposed approach.

Котенев В.П., Пучков А.С., Сапожников Д.А., Тонких Е.Г. Восстановление распределения давления в возмущенной области около сферы, обтекаемой сверхзвуковым потоком газа с произвольным эффективным показателем адиабаты. Математическое моделирование и численные методы, 2018, № 2, с. 109–121

#### 519.8 Stochastic models of bilateral hostilities of numerous groups while linear dependence of military units’ effective firing rate on the duration of the battle

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

doi: 10.18698/2309-3684-2018-2-122132

The article presents developed on the basis of the theory of Markov’s processes the models of bilateral hostilities with the linear dependence of effective firing rate of military units on the time of the battle. Developed the algorithm allows to calculate the main indicators of the battle of numerous groups. Fulfilled the comparison with the results of a battle simulation, received on the basis of probabilistic models of the battle with constant effective firing rate and the deterministic model of a combat with a linear dependency of effective firing rate on the time of the battle. The range of the last models applicability presented.

Чуев В.Ю., Дубограй И.В. Стохастические модели двухсторонних боевых тдействий многочисленных группировок при линейных зависимостях эффективнх скорострельностей боевых единиц сторон от времени боя. Математическое моделирование и численные методы, 2018, № 2, с. 122–132.