and Computational Methods

#### 521.2:521.3:521.61 Approximate methods for studying the second harmonic generation of ultrashort laser pulses in nonlinear photonic crystals

**Ruziev Z. J. (Ташкентский государственный технический университет), Sobirov O. I. (Ташкентский государственный технический университет), Koraboev K. A. (Ташкентский государственный технический университет), Sapaev U. K. (Ташкентский государственный технический университет)**

doi: 10.18698/2309-3684-2022-1-314

Second harmonic generation of ultrashort laser pulses in nonlinear photonic crystals is investigated by numerical methods based on the approximation of slowly varying amplitudes and a unidirectional approximation, applicable to simplify the wave equation with nonlinear polarization in a dispersive medium. Under the same experimental conditions, the results of these approximations are compared. Comparative analysis shows that up to 10 fs of the main pulse duration, both approximate methods describe this process of frequency conversion in almost the same way, but below 10 fs, there is a discrepancy between their results. Mainly, the formation of the temporal profile of the second harmonic pulse and its efficiency are compared. A method for obtaining time profiles of the second harmonic pulse using a unidirectional approximation where the incident field is used entirely in both the spectral and time domains of the calculation is also shown. The effect of dispersion up to the third order of smallness is taken into account, during the use of the approximation of slowly varying amplitudes.

Рузиев З.Дж., Собиров О.И., Корабоев К.А., Сапаев У.К. Численное моделирование генерации второй гармоники ультракоротких лазерных импульсов в нелинейных фотонных кристаллах. Математическое моделирование и численные методы, 2022, № 1, с. 3–14.

#### 539.36 Modeling microstructural model of the plasticity deformation theory for transversally isotropic composites

**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: 10.18698/2309-3684-2022-1-1541

Within the framework of the deformation theory of plasticity under active loading, a model of constitutive relations for elastic-plastic composites belonging to the class of transversally isotropic materials is proposed. The theory of spectral expansions of stress and strain tensors and the spectral representation of nonlinear tensor functions for transversely isotropic media are used to develop a nonlinear constitutive relations. Specific models of plasticity functions are proposed, depending 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. Problems on the periodicity cell are formulated using the method of asymptotic averaging of periodic media. The numerical solution of problems on the periodicity cell is carried out using the finite element method within the framework of software developed at the Scientific and Educational Center "Supercomputer Engineering Modeling and Development of Software Systems" of Bauman Moscow State Technical University. An example of numerical calculation of the constants of a composite model using the proposed method for a unidirectionally reinforced composite based on carbon fibers and an aluminum alloy matrix is given. Examples of verification of the proposed model for different loading trajectories of the composite in a 6-dimensional stress space are given. 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 transversely isotropic composites

Димитриенко Ю.И., Сборщиков С.В., Димитриенко А.Ю., Юрин Ю.В. Микроструктурная модель деформационной теории пластичности трансверсально-изотропных композитов. Математическое моделирование и численные методы, 2022, № 1, с. 15–41.

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

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

doi: 10.18698/2309-3684-2022-1-4262

When designing products made of composite materials intended for use in difficult conditions of inhomogeneous deformations and temperature, it is important to take into account viscoelastic, including spectral and dynamic, properties of the binder and fillers. The article considers dynamic characteristics (complex modulus, complex malleability,their real and imaginary parts, loss angle tangent) and spectral characteristics of relaxation and creep and their dependence on each other. The characteristics mentioned above were found for all known types of creep kernel and relaxation kernel. To find the spectral characteristics, one of the numerical methods of inverting the Laplace transform was used — the method of quadrature formulas with equal coefficients. Algorithms and computer programs for the implementation of this method have been compiled. The obtained graphs are quite accurate (the maximum error of calculations in the average does not exceed 5%), despite the fact that the error is very noticeable in the initial time segments.

Валишин А.А., Тиняев М.А. Моделирование динамических и спектральных вязкоупругих характеристик материалов на основе численного обращения преобразования Лапласа. Математическое моделирование и численные методы, 2022, № 1, с. 42–62.

#### 539.26 Analysis of empirical models of deformation curves of elastoplastic materials (review). Part 1

**Belov P. A. (Institute of Applied Mechanics of RAS), Golovina N. Y. (Industrial University of Tyumen)**

doi: 10.18698/2309-3684-2022-1-6396

The article presents the result the review of works devoted to the research the properties of elastoplastic materials. The article consists of two parts. In the first part, universal single, two- and three-parametric laws describing the nonlinear dependence between the stress and deformation up to the destruction. The review includes: power laws, parabolic laws, exponential laws, harmonic law. A comparison the considered empirical curves with a sample experimental points is carried out by the standard procedure for minimizing the total quadratic deviation and using the method the gradient descent to determine the minimum function of many variables. To assess the predictive force for models on the compliance with the experiment, a representative sample used from 158 experimental points in the deformation curve of the Russian titanium alloy WT6. The analysis showed that the empirical laws of deformation containing less than four formal parameters cannot describe the universal deformation curve with the stress specified at the ends and the tangent module. Analysis of the advantages and disadvantages of existing empirical laws of deformation, made it possible to formulate certain requirements for their wording.

Головина Н.Я., Белов П.А. Анализ эмпирических моделей кривых деформирования упругопластических материалов (обзор). Часть 1. Математическое моделирование и численные методы, 2022, № 1, с. 63–96

#### 539.376 Simulation of creep in thin-walled shellsunder variable loads

**Butina T. A. (Bauman Moscow State Technical University), Dubrovin V. M. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2022-1-97108

Under prolonged loading during operation, structures are subject to the phenomenon of creep, which can affect its performance. This influence depends on the load level, loading duration, operating conditions, design features, and type of material. All of these factors are taken into account in testing to obtain creep curves for a specific material and various environmental conditions corresponding to the operating conditions of the structure. The paper considers the problem of calculating the creep deformations of thin-walled cylindrical shells under the combined action of internal pressure and axial force. A model of the theory of flow with hardening under variable loading is considered. A numerical example of calculating the creep deformations of a cylindrical shell for an aluminum alloy is given

Бутина Т.А., Дубровин В.М. Моделирования ползучести тонкостенных оболочек при переменных нагружениях. Математическое моделирование и численные методы, 2022, № 1, с. 97–108.

#### 004.942 Modelling of industrial environment with the help of discrete numerical algorithms

**Belov V. F. (МГУ им. Н.П. Огарева/АУ «Технопарк–Мордовия»), Gavryushin S. S. (Bauman Moscow State Technical University), Markova Y. N. (АУ «Технопарк–Мордовия»), Zankin A. I. (МГУ им. Н.П. Огарева)**

doi: 10.18698/2309-3684-2022-1-109128

Modelling and analysis methods for economic characteristics variation in the innovation process have become a common technique, via employing diffusion equations for a medium with given parameters. The analysis results in this case significantly depend on the measurement accuracy of the industrial environment parameters, which is hard to achieve in practice. It seems, therefore, reasonable to make a transition from the diffusion paradigm to the innovation implementation paradigm, i.e., sequential modelling of the innovation states with variables and characteristics that correspond to the practical measurement and control techniques. Applying the described approach, the economic state dynamics of the innovation development work, manufacturing and implementation can be described by systems of ordinary differential equations, where the initial conditions and coefficients depend on the parameters of the industry’s internal andexternal environments. Two discrete mathematical models developed in this work enable control of the industrial environment parameters, via application of practical measurement methods. The first discrete model is in the form of a functional (mapping), which enables conversion of the actual internal industrial environment parameters in the beginning of the innovation scaling into the coefficients of the differential equations and initial conditions that reflect the results of manufacturing process preparation. The initial data is available from the EPR data base of the industry. The second discrete model is realized as a cellular automaton. An autonomous model of the external industrial environment uses the data that can be measured by the well-developed marketing methods. The results of the computational experiments support the hypothesis of transition from the diffusion model paradigm to the paradigm of the sequential modelling of the innovation economic states.

Белов В.Ф., Гаврюшин С.С., Маркова Ю.Н., Занкин А.И. Моделирование среды предприятия с использованием дискретных вычислительных алгоритмов.Математическое моделирование и численные методы, 2022, № 1, с. 109–128

#### 519.85 Method of finding non-dominant solutions in decomposition problems

**Kiselev V. V. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2022-1-129140

The article discusses the method of finding optimal solutions in the presence of a model of a complex technical system in the optimal design problem. The method is based on the use of nondominable, lambda optimal solutions and is a generalization of the method of Krasnoshchekov P.S., Morozov V.V., Fedorov V.V. [1]. The method allows in many cases (for lambda monotone objective functions) to reduce the number of calculations and reduce the dimension of the original problem. A numerical method for constructing lambda optimal solutions has been developed. A numerical example is given in which it is shown that the number of lambda optimal solutions consists of a single point, and the set of Pareto–optimal solutions is a curve on which it is necessary to build an ε–network to find the optimal solution.

Киселев В.В. Метод нахождения недоминируемых решений в задачах декомпозиции моделей сложных систем. Математическое моделирование и численные методы, 2022, № 1, с. 129–140.