and Computational Methods

#### 539.3 Modeling the bending of beams made of rubber-like materials

**Firsanov V. V. (Moscow Aviation Institute (National Research University))**

doi: 10.18698/2309-3684-2021-4-316

Since the classical hypotheses of Bernoulli for beams and Kirchhoff for thin plates contradict the additional condition of incompressibility for rubber-like (incompressible) materials (invariability of the volume during deformation), a calculation model for a bending beam is proposed, which does not lead to a serious complication of the problem in comparison with the classical solution. The invariability of the volume is manifested under the action of a power load; in the case of a temperature load, the deformation of the volume change is not zero. The absence of volumetric deformations for rubber-like materials is a consequence of Hooke's law for such materials. Summing the linear deformations expressed in terms of stresses and taking Poisson's ratio 0.5, we obtain the equality of the indicated sum to zero. Many rubber-like materials are incompressible and low-modulus, which means their weak resistance to tension and shear, but the resistance of the material to change in volume tends to infinity, therefore the physical relations of the generalized Hooke's law are transformed into the so-called "neo- Hooke " equations of the relationship between stresses and strains. Of the two independent physical characteristics (modules) for incompressible materials, only one module remains, which characterizes the resistance of the medium to change in shape. In physical relations for an incompressible material, the product of an infinitely large volumetric modulus by the deformation of a change in volume equal to zero is an uncertainty that is replaced by some force function that has the dimension of stresses and is an additional unknown. At the same time, the system of governing equations of the mechanics of incompressible media is supplemented by the equation of invariability of volume. The scheme for solving the problem in displacements for traditional structural materials turns into a mixed scheme for rubber-like materials, since for them not only displacements but also the mentioned force function act as the main unknown sought function.

Фирсанов В.В. Моделирование изгиба балок из резиноподобных материалов. Математическое моделирование и численные методы, 2021, № 4, с. 3–16.

#### 539.36 Modeling microstructural model of the plasticity deformation theory for quasi-isotropic composite materials

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

doi: 10.18698/2309-3684-2021-4-1744

A model of constitutive relations for elastic-plastic composites with cubic symmetry of properties is proposed. This class includes a significant number of composite materials: dispersed-reinforced composites, which have an ordered rather than a chaotic reinforcement system, as well as some types of spatially reinforced composites. To build a model of nonlinear constitutive relations, a tensor-symmetry approach was used, based on the spectral expansions of stress and strain tensors, as well as the spectral representation of nonlinear tensor relations between these tensors. The deformation theory of plasticity is considered, for which the tensor-symmetric approach is used, and specific models are proposed for plasticity functions that depend on the spectral invariants of the strain tensor. To determine the model constants, a method is proposed in which these constants are calculated based on the approximation of deformation curves obtained by direct numerical solution of three-dimensional problems on the periodicity cell of elastic-plastic composites. These problems arise in the method of asymptotic averaging of periodic media. To solve problems on a periodicity cell, a finite element method and special software was used that implements solutions to problems on periodicity cells, developed at the Scientific and Educational Center for Supercomputer Engineering Modeling and Development of Software Packages of Bauman Moscow State Technical University. An example of calculating the constants of a composite model using the proposed method for a dispersed-reinforced composite based on a metal matrix is considered. Also, the verification of the proposed model for various ways of multiaxial loading of the composite was carried out with direct numerical simulation. It is shown that the proposed microstructural model and the algorithm for determining its constants provide a sufficiently high accuracy in predicting the elastic-plastic deformation of the composite.

Димитриенко Ю.И., Сборщиков С.В., Димитриенко А.Ю., Юрин Ю.В. Микроструктурная модель деформационной теории пластичности квази-изотропных композиционных материалов. Математическое моделирование и численные методы, 2021, № 4, с. 17–44.

#### 539.3 Mathematical modeling of fatigue failure during high-frequency bending vibrations of titanium alloy specimens

**Stratula B. A. (Institute for Computer Aided Design of the Russian Academy of Sciences)**

doi: 10.18698/2309-3684-2021-4-4557

A unified numerical method for different fatigue fracture modes from low-cycle to very-high-cycle fatigue is described on the basis of a multi-mode two-criterion model of cyclic damage. This method allows for a through calculation of the evolution of crack-like fatigue fracture zones in a material, as well as an assessment of the durability of specimens from crack nucleation to macrofracture. Fatigue fracture calculations of titanium alloy specimens under prolonged cyclic loading under three-point bending scheme with development of "quasi-cracks" in modes from multi-cycle to super-multi-cycle fatigue have been carried out. Numerical and experimental results are compared to each other.

Стратула Б.А. Математическое моделирование усталостного разрушения при высокочастотных изгибных колебаниях образцов из титановых сплавов. Математическое моделирование и численные методы, 2021, № 4, с. 45–57.

#### 536.2 Finite-difference approximation of mixed derivatives in mathematical physics equations

**Gorskiy V. V. (Bauman Moscow State Technical University), Resh V. G. (Moscow State Textile University named after A. N. Kosygin)**

doi: 10.18698/2309-3684-2021-4-5879

A qualitative numerical solution of the equations of mathematical physics is intimately connected with ensuring a high accuracy of approximation of all differential operators included in these equations. The solution of this problem for the first and second derivatives in the equations of mathematical physics, which are used to describe a wide range of scientific and technical problems has been described in numerous literary publications. At the same time, mixed derivatives are not so often present in the equations of mathematical physics and, therefore, issues related to the quality of finite-difference approximation of these derivatives are not given enough attention in literary publications. One of the main reasons for the appearance of mixed derivatives in the equations of mathematical physics is the use of an affine transformation of the coordinate system, which provides the transition to domain of a substantially simpler form. The solution of this problem is the subject of the present paper. The problem is solved by the example of approximation of mixed derivatives on rectangular domain of definition of the required function with constant steps in each direction. A detailed derivation of the finite-difference relations used for the finite-difference approximation of mixed derivatives in all typical nodes of the function domain is given, which makes it possible to develop the proposed technique on domains of different types.

Горский В.В., Реш В.Г. Конечно-разностная аппроксимация смешанных производных в уравнениях математической физики. Математическое моделирование и численные методы, 2021, № 4, с. 58–79

#### 531.6.011.32:532.582.4 Method for organizing an iterative process in modeling subsonic separated flow around elongated bodies using an equivalent surface and cubic splines

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

doi: 10.18698/2309-3684-2021-4-80102

In the aspect of improving the methodology of mathematical modeling of subsonic detachable flow around elongated bodies with partial implementation of the concept of viscosity-inviolable interaction, the issues of organizing the iterative process of constructing the surface of an equivalent body are considered. A flow diagram with a semi-infinite equivalent surface was used. Numerical modeling was carried out according to the algorithms of the technique using the method of discrete vortices. For greater versatility, when constructing a calculated grid on the surface of an equivalent body, approximation with smoothing cubic splines was used. Data on the influence of the shape of the tail section of the equivalent surface on the distribution of speed and pressure during axisymmetric flow around bodies with a bottom section are presented.

Тимофеев В.Н. Методика организации итерационного процесса при моделировании дозвукового отрывного обтекания удлиненных тел с применением эквивалентной поверхности и кубических сплайнов. Математическое моделирование и численные методы, 2021, № 4, с. 80–102.

#### 519.2 Simulation of locally homogeneous radar images using different statistical criteria

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

doi: 10.18698/2309-3684-2021-4-103120

The article deals with the problem of classifying pixels of the radar image (RI). A locally homogeneous radar image model was used, in which the readings of each small area (local area) were considered to belong to only one class. The classification results of several real radar images by local areas are compared using the statistical criteria for the maximum a posteriori probability, Kolmogorov and Cramer-Mises-Smirnov. At the same time, in the case when the listed criteria made it difficult to classify a local area — when it hit the interface of the underlying surfaces, it was considered to be assigned to a special, boundary class, and its readings were processed using the grid method for separating mixtures of probability distributions. For each criterion, the classification accuracy was evaluated as the proportion of correctly classified pixels within the selected homogeneous areas. It has been established that in the case of significant interclass differences, the best classification accuracy is ensured by the use of the least powerful Kolmogorov criterion among nonparametric criteria. Also, using a real image as an example, it is shown that when the differences in the characteristics of objects of the same class are comparable to interclass differences, the highest classification accuracy is achieved when using the maximum a posteriori probability criterion. Such cases are typical for a wide class of classification problems, including those not related to image processing.

Достовалова А.М. Моделирование локально-однородных радиолокационных изображений при использовании различных статистических критериев. Математическое моделирование и численные методы, 2021, № 4, с. 103–120.

#### 519.87 Modeling of the effects of double standards and soft power in competition by cellular

**Bobrov V. A. (MPGU), Brodsky Y. I. (ФИЦ ИУ РАН)**

doi: 10.18698/2309-3684-2021-4-12134

We consider a discrete analog of the classical A. Lotka – V. Volterra competition model in the environment of cellular automata. In the classical model, we know that the type of its evolution over time primarily depends on the coefficients of double standards affiliation to certain ranges of their possible values. The paper shows that the same situation holds for the discrete model either. We can see there is a soft power effect for the classical model. The classic competition model turns into a cooperative positional differential game, the limitations of which are the original system of competition equations by A. Lotka – V. Volterra. The controls are the coefficients of double standards when considering it concerning social systems. The effect of soft power is that the parties tend to compare the competitive pressure on them by the rival population with the one within the domestic population and may take the less stress of the opponent for his favorable attitude towards them, and more extensive — for the hostile manifestation. Whereas comparing the external competitive pressure with the internal pressure in this game does not give us any information — everything depends exclusively on the coefficients of double standards, which are controls and therefore are unavailable to the opponents in this game. Simulation experiments with the discrete analog of the competition model implemented in the cellular automata environment show that the effect of soft power also takes place in this case

Бобров В.А., Бродский Ю.И. Моделирование клеточными автоматами эффектов двойных стандартов и мягкой силы при конкуренции. Математическое моделирование и численные методы, 2021, № 4, с. 121–134.