and Computational Methods

#### 539.3 Finite element modeling of elastic properties of textile polymer composites at high temperatures

**Dimitrienko Y. I. (Bauman Moscow State Technical University), Yurin Y. V. (Bauman Moscow State Technical University), Sborschikov S. V. (Bauman Moscow State Technical University), Bogdanov I. O. (Bauman Moscow State Technical University), Yakhnovskiy A. D. (Bauman Moscow State Technical University), Baymurzin R. R. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2020-1-327

The problem of multilevel model development for calculating of an elastic property of polymer composite materials with a complex reinforcement structure at high temperatures is considered. It is assumed that thermal destruction processes take place in the matrix and fibers at high temperatures. In order to take into account the change in the elastic properties of the composite depending on the temperature and heating time, a 3-level structural model of the composite is proposed. At the lower level mono-fibers and a matrix consisting of 4 phases, the ratio between which changes when heated are considered. At this level, the analytical relations proposed earlier in the works of Yu.I. Dimitrienko. At the next level of the model, a unidirectional composite is considered, consisting of bundles of monofilaments and a matrix. To calculate elastic properties at this level, the method of asymptotic averaging is used, and a finite element algorithm for solving local problems of the theory of thermoelasticity arising in this method. At the 3rd structural level of the model, composites with complex reinforcement structures, in particular, fabric composites, are considered. The method of asymptotic averaging is also used to calculate the elastic properties of the composite at this level. For the numerical calculation of the elastic characteristics of polymer composites at high temperatures, specialized software has been developed that operates under the control of the SMCM software package created at the Scientific and Educational Center for Supercomputer Engineering Modeling and Development of Software Systems of the Bauman Moscow State Technical University. The article provides examples of the application of the developed multilevel model and software for textile composites based on an epoxy matrix and glass fibers. The values of all components of the tensor of the elastic moduli of the composite are calculated, which vary depending on the heating program of the composite. The microstress fields in the composite are obtained. A comparison is made of the fields of microstresses and effective elastic constants at normal temperatures, with similar values obtained using the ANSYS software package, which has been modified to enable the calculation of effective elastic constants in accordance with the proposed model. A very good agreement was obtained between the calculation results, both of the effective constants and of the microstresses fields, which allows us to speak of the high accuracy of the developed software.

Димитриенко Ю.И., Юрин Ю.В., Сборщиков С.В., Богданов И.О., Яхновский А.Д., Баймурзин Р.Р. Конечно-элементное моделирование упругих свойств тканевых полимерных композитов при высоких температурах. Математическое моделирование и численные методы. 2020. № 1. с. 3–27

#### 532.5.032 Influence of Rayleigh, Prandtl numbers and boundary conditions on convective fluid flows in horizontal layers

**Fedyushkin A. I. (Ishlinsky Institute for Problems in Mechanics)**

doi: 10.18698/2309-3684-2020-1-2844

The paper considers two problems of thermal convection of an incompressible liquid in a horizontally extended layer: with lateral heat supply and with heating of the horizontal layer from below-the Rayleigh-Benard problem. An influence of boundary conditions and Prandtl numbers on the convective flow structure and temperature distribution is considered. The solutions of these problems are obtained using numerical modeling. The simulation is based on the numerical solution of a system of non-stationary 2D Navier-Stokes equations for an incompressible fluid, as well as for the Rayleigh-Benard problem for the case of a two-layers gas-liquid system. Navier-Stokes equations are solved using two numerical methods: the finite difference method and the control volume method. To verify the models, the results of calculations obtained by different numerical methods were compared with each other and compared with experimental data. The paper presents the results of numerical simulation of convective flows and heat and mass transfer in horizontal liquid layers under different defining dimensionless parameters and boundary conditions. The non-linear features of convective flows in horizontal liquid layers are shown, in particular, the occurrence of a counter-flow inside the layer-a liquid flow with a direction opposite to the main convective flow. The influence of boundary conditions and Rayleigh and Prandtl numbers on the existence of a countercurrent is considered. A simulation of the convective flow of a liquid in a horizontal layer when heated from the side at small Prandtl numbers, as well as at a Prandtl number equal to zero, is performed. The simulation results showed that for laminar convection (the Rayleigh number is greater than 10^5), the flow with Prandtl numbers equal to or less than 10^-2 is qualitatively different from the flow of a liquid with a zero Prandtl number. Therefore, the approximation of the zero value of the Prandtl number may not always be correct. Nonlinear peculiarities of convective flows in horizontal liquid layers are presented. It is shown that in long horizontal layers laterally heated only by thermal laminar convection (without the presence of impurities and concentration convection), it is possible to create a stable vertical density stratification of the fluid and, as a result, lead to the appearance of layered structures.

Федюшкин А.И. Влияние чисел Рэлея, Прандтля и граничных условий на конвективные течения жидкости в горизонтальных слоях. Математическое моделирование и численные методы. 2020. № 1. с. 28–44.

#### 539.3 On the method of gas turbine blade resource estimation at asymmetric cyclic loading

**Zavoichinskaya E. B. (Lomonosov Moscow State University)**

doi: 10.18698/2309-3684-2020-1-4563

One of the modern problems of the resource and safety operation estimation in turbines and compressors of gas and steam turbine construction is to study the relations of the stochastic process of micro- and macrocracks nucleation and growing in blades at operational high cycle and gigacyclic fatigue. It has been experimentally found that micro- and macrocracks occur perpendicular to the axis of the blade in the stress concentration zone root section from the bending forms of forced resonance vibrations. In general, the axial stress may be presented as the sum of the constant component and the discrete spectrum with a set of amplitudes and phase deviations between the components. Here is consider the blade asymmetric single-frequency axial loading, for which a constant stress component is determined by the known relations for an elastic cantilever beam as the maximum value of the sum of tensile stress from centrifugal forces and bending from aerodynamic forces at the point of their application, and the stress amplitude is found on a well-known solution of the bending problem for various deflections of rigidly fixed to the disk elastic beam without ties. The proposed method of the blade resource estimating at the loading is based on the theory of material scale-structural fatigue, according to which the stochastic fatigue process is considered at six scale-structural levels corresponding to different stages of the material evolution by various physical mechanisms. A hierarchical system of constitutive relations for the failure probability at the micro-, meso- and macrolevels is defined. It is chosen the stress amplitude as a variable, and material functions are determined according to standard fatigue strength tests taking into account the results of brittle cracks evolution studу and depending on the asymmetry parameter. Here is carried out the identification of basic constants for high cycle and gigacycle fatigue of no endurance materials. For each level of defects, fatigue curves are determined. The results for VD17 aluminum alloy and ZhS6K nickel alloy blades are discussed. By comparing the theoretical calculation with experimental data, the method is confirmed.

Завойчинская Э.Б. О методе оценки ресурса лопаточного аппарата газотур- бинного двигателя при асимметричном циклическом нагружении. Математиче- ское моделирование и численные методы. 2020. № 1. с. 45–63

#### 539.38:621.01:004.7 Numerical simulation the post-buckling nonlinear deformation of axisymmetric membranes

**Podkopaev S. A. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2020-1-6487

The theoretical foundations of nonlinear straining of thin-walled axisymmetric shells are considered. The operational characteristics of the membranes in various switching devices, valves and pressure sensors are presented. The types of non-linear behavior of post-buckling behavior of axisymmetric membranes are considered. A mathematical model is presented to describe nonlinear straining of axisymmetric membranes, a discrete continuation by parameter method, and the “changing the subspace of control parameters” technique. Using the hinged spherical shell as an example, a study of post-buckling behavior is performed. A rational mathematical model has been selected to describe nonlinear straining of thin-walled axisymmetric shells. A numerical algorithm for studying the processes of nonlinear straining of multi-parameter systems has been developed and implemented as an author program.

Подкопаев С.А. Численное моделирование закритического нелинейного деформирования осесимметричных мембран. Математическое моделирование и численные методы, 2020, № 1, с. 64–87.

#### 621.833.51 Моделирование динамики планетарного перемешивающего устройства с неравномерным вращательным движением рабочего органа

**Prikhodko A. A. (Kuban State Technological University), Kopteva A. A. (K.G. Razumovsky Moscow State University of technologies and management)**

doi: 10.18698/2309-3684-2020-1-88102

Article presents the results of mathematical modeling of the dynamics of a stirred tank with irregular motion of the impeller. The design of the apparatus with a stirrer is considered as a mechanical system with one degree of freedom, which is based on proposed by the authors planetary gear with elliptical wheels. The solution of the differential equation of motion of the stirred tank is proposed to be carried out by two methods: the energy-mass method and using the third-order Hermite interpolation function. There is presented a kinematic model of the mechanism, which describes the relationship between the geometric parameters of the links and allows you to bring the forces, masses and moments to the initial link. As an example, an analysis of the stirred tank according to the given initial data is carried out, there are calculated the moment of resistance on the impeller, the driving moment and the moment of inertia of the flywheel, which are of great importance at the design stage of the machine. The laws of motion of the input link are determined and constructed without taking into account and taking into account the installation of the flywheel. The convergence of the results obtained by various methods is confirmed.

Приходько А.А., Коптева А.А. Моделирование динамики планетарного перемешивающего устройства с неравномерным вращательным движением рабочего органа. Математическое моделирование и численные методы. 2020. № 1. с. 88–102.

#### 519.6:629.7.02 Software environment dgsSynchCAD for modeling and computer-aided heuristic optimization of throttle synchronizing hydrosystems

**Bushuev A. Y. (Bauman Moscow State Technical University), Ivanov M. Y., Korotaev D. V. (LLC Scientific-production association «INKOR»), Resh G. F.**

doi: 10.18698/2309-3684-2020-1-103117

A software environment was developed for computer-aided design of optimal throttle hydraulic synchronization systems of actuators of various functional purposes operating under conditions of external alternating-sign force effects. The criterion of an optimization procedure was the minimization of a mismatch time of relative movement of actuators during operation. A compute core of an object-oriented code was constructed on the basis of a dynamic mathematical model of a synchronization system consisting of four power cylinders. A model problem was solved with the help of the created software environment. This model problem demonstrated the efficiency of the proposed multidimensional optimization process. The methodology was based on the use of the well-known heuristic method (binary coded genetic algorithm) and the subsequent improvement (in the sense of a given objective functional) of the obtained solution by a method on the basis of the Hooke-Jeeves algorithm. Recommendations on the practical application of the software and mathematical support for achieving the best convergence to the extreme value of a vector of controlled parameters were formulated.

Бушуев А.Ю., Иванов М.Ю., Коротаев Д.В., Реш Г.Ф. Программная среда dgsSynchCAD для моделирования и автоматизированной эвристической оптимизации дроссельных гидросистем синхронизации. Математическое моделирование и численные методы. 2020. № 1. с 103–117.

#### 621.833.51 «Смешанные» вероятностные модели боя при переменных эффективных скорострельностях боевых единиц сторон

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

doi: 10.18698/2309-3684-2020-1-118128

On the basis of the theory of continuous Markov processes, "mixed" probabilistic models of bilateral fighting operations with exponential dependences of the effective rate of fire of the combat units of the parties on the time of battle have been developed. A numerical algorithm has been developed to calculate the main indicators of the battle of numerous groups. It is made a comparison of the results of battle simulation using a "mixed" deterministic model with exponential dependences of effective rates of fire on the time of battle, as well as with "mixed" probabilistic models with constant effective rates of fire. The scope of these types of models applicability is established.

Чуев В.Ю., Дубограй И.В. «Смешанные» вероятностные модели боя при переменных эффективных скорострельностях боевых единиц сторон. Математическое моделирование и численные методы, 2020, № 1, с. 118-128.