and Computational Methods

#### 539.3 Modeling the viscoelastic characteristics of materials based on the numerical inversion of the Laplace transform

**Valishin A. A. (Bauman Moscow State Technical University), Tinyaev M. A. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2020-3-321

When designing products made of composite materials intended for operation in difficult conditions of inhomogeneous deformations and temperatures, it is important to take into account the viscoelastic properties of the binder and fillers. The article analyzes the relationship between relaxation and creep characteristics. All known creep and re-laxation kernels in the literature are considered. The problem of transformation of creep characteristics into relaxation characteristics and vice versa is discussed in de-tail. There is a simple relationship between the creep and relaxation functions in the Laplace image space. However, when returning to the space of the originals, in many cases there are great difficulties in reversing the Laplace transform. Two numerical methods for inverting the Laplace transform are considered: the use of the Fourier series in sine and the method of quadrature formulas. Algorithms and computer programs for realization of these methods are made. It is shown that the operating time of a computer program implementing the Fourier method by sine is almost 2 times less than the operating time of a computer program implementing the quadrature formula method. However, the first method is inferior to the latter method in accuracy of calculations: the relaxation functions and relaxation rates, it is advisable to find the first method, since the computational error is almost indiscernible, and the functions of creep and creep speed, the second way, because for most functions, the result obtained by the second method is much more accurate than the result obtained by the first method. The Gavrillac-Negami creep function and the Gavrillac-Negami creep rate function could not be constructed due to a complex recursive formula for the series coefficients, but using both methods, these functions can still be obtained and compared with each other.

Валишин А.А., Тиняев М.А. Моделирование вязкоупругих характеристик материалов на основе численного обращения преобразования Лапласа. Математическое моделирование и численные методы, 2020, № 3, с. 3–21.

#### 539.3 Modeling of effective relaxation and creep kernels of viscoelastic composites by asymptotic averaging method

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

doi: 10.18698/2309-3684-2020-3-2246

The problem of calculating the integral characteristics of the viscoelasticity of composite materials is considered, based on information on similar characteristics of the composite components and its microstructure. An algorithm is proposed for predicting the effective relaxation and creep kernels of composites with an arbitrary reinforcement microstructure. The algorithm is based on the Fourier transform application and the inverse Fourier transform, as well as the method of asymptotic averaging for composites under steady-state polyharmonic vibrations. The algorithm uses exponential relaxation and creep kernels for the initial components of the composite. The basis of the computational procedure of the proposed algorithm is the finite element solution of local viscoelasticity problems over the composite periodicity cell. The result of the algorithm application is the determination of the exponential relaxation and creep kernels parameters for composite materials, which makes it possible to obtain a problem solution in a completely closed form. As an example, a numerical simulation of the viscoelastic-tic characteristics of unidirectionally reinforced carbon /epoxy composites has been carried out. It is shown that the developed algorithm allows one to obtain effective relaxation and creep kernels of the composite with high accuracy, without oscillations, which, as a rule, ac-company the methods of inverting Fourier transforms.

Димитриенко Ю.И., Юрин Ю.В., Сборщиков С.В., Яхновский А.Д., Баймурзин Р.Р. Моделирование эффективных ядер релаксации и ползучести вязко-упругих композитов методом асимптотического осреднения. Математическое моделирование и численные методы, 2020, № 3, с. 22–46.

#### 539.3 Mathematical modelling of conjugate natural convection in vapor and liquid during ventless storage of cryogenic fuel components

**Gordonov A. O. (АО ГНЦ "Центр Келдыша")**

doi: 10.18698/2309-3684-2020-3-4767

Method for numerical modelling of natural convection heat and mass transfer in liquid and vapor phases during ventless storage of cryogenic fuel components in tanks is presented in this paper. The proposed mathematical model is based on low Much number approach for vapor phase and Boussinesq approach for liquid phase. Method for numerical modelling of inhomogeneous initial conditions in vapor phase is given. Presented model and method is validated using experimental data for ventless storage of liquid hydrogen and nitrogen.

Городнов А.О. Математическое моделирование сопряженной естественной конвекции в паре и жидкости при бездренажном хранении криогенных компонентов топлива. Математическое моделирование и численные методы, 2020, № 3, с. 47–67.

#### 629.762 Optimization of the method of burning surface area calculation for solid propellant charges of comparatively simple spatial forms

**Plyusnin A. V. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2020-3-6884

It may be expedient to integrate the preliminary design of solid propellant motors (SPM), especially the motors aimed for launching, into the overall design, when lifting vehicles are created with SPM included in their structure. This implies that the internal ballistic calculations have to be provided by utilizing the engineering mathematical models, the only complex point of which is the evaluation of the burning surface of the charge under consideration. A method for the computation of this relation is presented in this work, which optimizes one of the previously proposed methods as in efficiency of calculations so in the precision of the results.

Плюснин А.В. Оптимизация метода расчета площади поверхности горения для твердотопливных зарядов сравнительно несложных пространственных форм. Математическое моделирование и численные методы, 2020, № 3, с. 68–84.

#### 519.6 Conservative difference schemes for the optimal selection of the location of heat sources in the rod

**Khayitkulov B. K. (National University of Uzbekistan)**

doi: 10.18698/2309-3684-2020-8598

In this paper, we have developed a method and an algorithm for solving the problem of the optimal selection of the density of heat sources on the rod in such a way that the temperature inside the considered area is within the specified limits is developed. In this case, the heat sources must provide the specified temperature regime of the minimum total power and temperature in the specified temperature corridor. Conservative finite-dimensional approximations of the original problem are constructed in the form of a linear programming problem. A method for constructing conservative difference schemes for solving the heat conduction equation with variable coefficients, a brief description of the developed software application for constructing computational grids and solving equations is presented. A new method for the numerical solution of non-stationary problems of optimal selection of heat sources in a rod is proposed and substantiated. A software application has been created for carrying out numerical experiments to solve the problem. The description of the based algorithm and the results of numerical experiments are given.

Хайиткулов Б.Х. Консервативные разностные схемы по оптимальному выбору местоположения источников тепла в стержне. Математическое моделирование и численные методы, 2020, № 3, с. 85–98.

#### 532.582.4:517.958 Simulation of subsonic detachable flow of axisymmetric bodies with bottom slice at non-zero angles of attack

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

doi: 10.18698/2309-3684-2020-3-99116

According to the current scheme with the equivalent half-bottom surface, the subsonic streamlining of the axisymmetric bodies with the separation line located in the bottom cut area is examined. Recommendations have been made to improve the calculation method in the case of non-zero angles of attack. The estimated ratios of the technique are adapted to use the method of discrete vortexes. Detailed data on the distribution of speed and pressure on the surface of the cylindrical body with a bottom cut and the head part of the liveable shape at non-zero angles of attack are presented.

Тимофеев В.Н. Моделирование дозвукового отрывного обтекания осесимметричных тел с донным срезом при ненулевых углах атаки. Математическое моделирование и численные методы, 2020, № 3, с. 99–116.

#### 519.2 Application of a finite mixture model in radar image processing

**Dostavalova A. M. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2020-3-117130

The problem of target detection on a radar image (RI) is considered. When solving it, it is proposed to use a finite mixture as a model describing the structure of the processed image. It is assumed that each of its components corresponds to one of the classes of objects present on the radar image, which are similar in their reflective properties. This hypothesis allows one to detect targets by solving the problem of finite mixture separating. It is possible to use for this existing methods of mixture decomposition — the EM-algorithm. However, the structure of the image, in which target classes are local inhomogeneities, consisting of a small number of samples, imposes restrictions on the possibility of using these algorithms in their pure form and leads to the need to create their adaptations that take into account this feature of the input data. The article presents the results of applying the adapted EM-algorithm using the generated radar image with pixels obeying the normal law as an input data. The efficiency of the created algorithm is assessed in comparison with the results of applying the classical version of the EM-algorithm for this model. The data obtained made it possible to reveal the peculiarities of the method due to both the created mechanics of image processing and the properties of the procedure for separating mixtures — the EM-algorithm, which must be taken into account in the further use of this method of image processing.

Достовалова А.М. Применение модели смеси вероятностных распределений в обработке радиолокационных изображений. Математическое моделирование и численные методы, 2020, № 3, с. 117–130.