and Computational Methods

#### 532.51 Modeling of thermal convection processes under unidirectional crystallization of alloys with liquid bridges motion

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

doi: 10.18698/2309-3684-2018-4-324

The paper considers the problem of thermal convection in the melt zone during unidirectional crystallization of a metal axisymmetric sample with a free surface boundary (liquid bridge) under microgravity. The mathematical problem includes a system of Navier-Stokes equations in the Boussinesq approximation with an equation for the mass transfer of impurity particles in a liquid, as well as equations for the motion of the liquid free surface.

A numerical algorithm for solving the problem based on the vortex and current function method, linearization of the problem, and finite-difference approximation using the variable direction method to solve the difference system of linear equations is developed. The physical parameters of thermal convection processes in the melt zone are calculated. It is shown that taking into account the motion of the free boundary near the crystallizing liquid phase leads to a change in the distribution of impurities near the curing surface, which in turn causes a change in the characteristics of the cured material.

Димитриенко Ю.И., Леонтьева С.В. Моделирование термоконвективных процессов при однонаправленной кристаллизации сплавов с учетом движения свободных границ. Математическое моделирование и численные методы. 2018. № 4. с. 3–24.

#### 669.162.23 Mathematical modeling of vibrating combustion mode in a distributed combustion chamber of a blast furnace air heater

**Basok B. I. (Institute of Engineering Thermophysics of Ukraine National Academy of Sciences), Gotsulenko V. V. (Institute of Engineering Thermophysics of Ukraine National Academy of Sciences)**

doi: 10.18698/2309-3684-2018-4-2540

This work attempts the construction of a quasi-three-dimensional mathematical model which describes the vibrating combustion mode in a vertical combustion chamber of a regenerative air heater in a blast furnace, considering it as a distributed dynamic system.

Басок Б.И., Гоцуленко В.В. Математическое моделирование режима вибра- ционного горения в распределенной камере горения воздухонагревателя в доменной печи. Математическое моделирование и численные методы, 2018, № 4, с. 25–40.

#### 517:519.6 Study of fast multipole boundary element method efficiency for solving Poisson equation

**Lazarev A. A. (Baranov Central Institute of Aviation Motor Development/Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2018-4-4156

The study introduces the algorithms for traditional and fast multipole boundary element methods for solving two-dimensional Poisson equation with non-potential volume sources. The influence of the method parameters on the speed and accuracy of the solu-tion is numerically studied; recommendations are given on the joint application of the methods.

Лазарев А.А. Исследование эффективности быстрого мультипольного метода граничных элементов для решения уравнения Пуассона. Математическое модели-рование и численные методы, 2018, № 4, с. 41–56.

#### 533.6 Modeling a neural network to solve the problem of classifying air frame elements

**Bulgakov V. N. (Bauman Moscow State Technical University), Raclav R. A. (Bauman Moscow State Technical University), Sapozhnikov D. A. (Bauman Moscow State Technical University), Chernyshev I. V. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2018-4-5771

The paper introduces a neural network implemented to classify air frame surface ele-ments. Within the research, we generated a sample containing the surface parameters of classification objects. In order to avoid errors associated with different measurement scales, the criteria were scaled. According to synthetic data, the neural network was trained, and the proposed model was verified. The optimal configuration of the neural network was determined experimentally. As a criterion of optimality, we used the proportion of correct answers from the test and training samples, and carried out calibration and modification of individual model parameters. The classification results of the test sample by the optimal network were summarized in the error matrix. The most significant result was achieved when distinguishing the class of ellipsoids. Separate blocks of the matrix show that the neural network accurately distinguishes the classes of ellipsoids and hyperboloids. The study proposes some ideas for further modification of the algorithm in order to increase the proportion of correct answers when distinguishing the class of paraboloids.

Булгаков В.Н., Рацлав Р.А., Сапожников Д.А., Чернышев И.В. Моделирование нейронной сети для решения задачи классификации элементов корпуса летательного аппарата. Математическое моделирование и численные методы, 2018, № 4, с. 57–71.

#### 539.3 Modeling the effective characteristics of transversely isotropic incompressible composites with finite strains

**Dimitrienko Y. I. (Bauman Moscow State Technical University), Gubareva E. A. (Bauman Moscow State Technical University), Karimov S. B. (Bauman Moscow State Technical University), Kolzhanova D. Y. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2018-4-7292

The paper considers a model of effective constitutive relations for a transversal-isotropic incompressible composite with finite strains. The model belongs to the so-called class of universal models that connect several pairs of energy stress and strain tensors simultaneously. A method is proposed for separating coupled problems of micro- and macroscopic deforming composites with finite strains that arise when the method of asymptotic homogenization (AH) of periodic structures is used. The method is based on the application of the effective constitutive relation model as an approximation dependence of the results of numerical simulation of the composite deformation curves obtained using the exact AH method. To find the elastic constants of the transversely isotropic composite model the method of minimizing the deviation of the approximation of deformation diagrams from the AH diagrams is used for a series of problems of standard deforming at finite strains. To solve minimization problems, the Nelder—Mead method is used. The results of numerical simulation by the proposed method for nonlinear elastic layered composites are presented, which showed good approximation accuracy, achieved due to application of the proposed method for the separation of coupled problems of micro- and macroscopic deforming.

Димитриенко Ю.И., Губарева Е.А., Каримов С.Б., Кольжанова Д.Ю. Модели-рование эффективных характеристик трансверсально-изотропных несжимаемых композитов с конечными деформациями. Математическое моделирование и чис-ленные методы, 2018, № 4, с. 72–92.

#### 533.16 Modeling the chemical composition of a gas in a nonequilibrium air boundary layer on a wall with a finite catalytic activity

**Gorskiy V. V. (Bauman Moscow State Technical University), Adamenko R. A.**

doi: 10.18698/2309-3684-2018-4-93106

The study of aerothermal destruction of silicon carbide is constantly given increased at-tention. This is primarily due to its widespread use as a protection against oxidation of carbon materials. The rate of aerothermochemical destruction of materials largely de-pends on the specific heat flux and on the degree of recombination of oxygen atoms on the wall. In this paper, we consider a method for solving the complex problem of high-temperature non-equilibrium air stream flowing of a material coated with a silicon car-bide film under conditions of finite catalyticity of the surface of this film.

Горский В.В., Адаменко Р.А. Моделирование химического состава газа в неравновесном воздушном пограничном слое на стенке, обладающей конечной ка-талитической активностью. Математическое моделирование и численные методы, 2018, № 4, с. 93–106.

#### 551.513 Modeling the spatial and temporal global distribution of artificial stratospheric aerosol for climate stabilization

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

doi: 10.18698/2309-3684-2018-4-107119

The purpose of the study was to estimate the possibility of climate stabilization at the present level by controlled sulfate aerosols emissions into the stratosphere, which reflect part of the incoming solar radiation, i.e. geoengineering. The study is based on a three-dimensional hydrodynamic model of the global climate, including an ocean model with

a real configuration of depths and continents, as well as a model of sea ice evolution, and a model of general atmospheric circulation. We studied the possibility of obtaining the predetermined spatial and temporal global distribution of the stratospheric aerosol under conditions when there is a limited number of aerosol sources. Wind transport and aerosol deposition were taken into account.

Пархоменко В.П. Моделирование пространственного и временного глобально-го распределения стратосферного аэрозоля искусственного происхождения для стабилизации климата. Математическое моделирование и численные методы, 2018, № 4, с. 107–119.