and Computational Methods

#### 537.611+530.146 Mathematical modeling of breathers of two-dimensional O(3) nonlinear sigma model

**Shokirov F. S. (S.U. Umarov Physical-Technical Institute of Academy of Sciences of the Republic of Tajikistan)**

doi: 10.18698/2309-3684-2016-4-316

The study examined the formation and evolution of stationary and moving breathers of a two-dimensional O(3) nonlinear sigma model. We detected analytical form of trial functions of two-dimensional sine-Gordon equations, which over time evolve into periodic (breather) solutions. According to the solutions found, by adding the rotation to an A3-field vector in isotopic space S^2 we obtained the solutions for the O(3) nonlinear sigma model. Furthermore, we conducted the numerical study of the solutions dynamics and showed their stability in a stationary and a moving state for quite a long time, although in the presence of a weak radiation.

Shokirov F. Mathematical modeling of breathers of two-dimensional O(3) nonlinear sigma model. Маthematical Modeling and Coтputational Methods, 2016, №4 (12), pp. 3-16

#### 536.2(075) Applying the control−volume method to extended period simulations in pipe network hydraulics

**Volkov V. Y. (JSC OKB GIROPRESS,), Golibrodo L. A. (JSC OKB GIROPRESS,), Zorina I. G. (Bauman Moscow State Technical University), Kudryavtsev O. V. (JSC OKB GIROPRESS,), Krutikov A. A. (JSC OKB GIROPRESS,), Skibin A. A. (JSC OKB GIROPRESS,)**

doi: 10.18698/2309-3684-2016-4-3446

For modeling piping systems we made a transition from the mass balance equations, based on 1m and 2m Kirchhoff laws, to the mathematical description of a hydraulic network using the continuity equation discretization. For this purpose we applied a controlvolume method. This paper introduces an extension of the developed control-volume method for extended period simulations in hydraulic networks. This extension is developed for slow time-varying conditions in the hydraulic networks and is not intended to calculate rapidly occurring local phenomena such as waterhammer. The control-volume method was successfully applied to test tasks.

Volkov V., Golibrodo L., Zorina I., Kudryavtsev O., Krutikov A., Skibin A. Applying the control−volume method to extended period simulations in pipe network hydraulics. Маthematical Modeling and Coтputational Methods, 2016, №4 (12), pp. 34-46

#### 534.142:536.24+621.63 Simulation of pressure gas flow self-oscillation excited by heat supply

**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: 10.18698/2309-3684-2016-4-1733

We obtained a mathematical model for determining the parameters of longitudinal selfoscillations, excited in the pressure gas flow at local flow heat supply. In our research we established that under certain conditions the gas heat supply alters the flow hydraulic characteristics, creating the "negative" resistance effect. In this case, the self-oscillation excitation is possible even with the monotonically decreasing supercharger pressure characteristic.

Basok B., Gotsulenko V. Simulation of pressure gas flow self-oscillation excited by heat supply. Маthematical Modeling and Coтputational Methods, 2016, №4 (12), pp. 17-33

#### 539.3 Multiscale modeling of thin multilayer composite plates with solitary defects

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

doi: 10.18698/2309-3684-2016-4-4766

We developed a multiscale model of deformation of thin multilayer composite plates with solitary defects. The model is based on the asymptotic analysis of general threedimensional equations of deformable solid mechanics. The general solution of threedimensional equations is reduced to the solution of two classes of problems: problems for thin plates without defects and local three-dimensional problems in the vicinity of the defect with the condition of damping solution at the distance from the defect. A solution of local problems is used for averaged problems of the multilayer plates theory, which enables us to find an explicit solution for all six components of the stress tensor in the field without the defect, based on the solution of the averaged two-dimensional problem of the plate theory. In the defect area the general solution is a superposition of the two solutions: the one obtained on the basis of the plates theory and local three-dimensional mechanics problems. The paper gives an example of a numerical finite element solution of the local mechanics problem for the three-layer composite plate with a solitary defect in the middle layer. Moreover, findings of the research show that the defect impact is localized in its immediate vicinity and the maximum transverse stress concentration is achieved in the vicinity of the defect peak.

Dimitrienko Y., Yurin Y. Multiscale modeling of thin multilayer composite plates with solitary defects. Маthematical Modeling and Coтputational Methods, 2016, №4 (12), pp. 47-66

#### 531.6.011.32:532.582.4:517.958 Construction of a semi-infinite equivalent body in mathematical modeling of subsonic separated axisymmetric flow

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

doi: 10.18698/2309-3684-2016-4-6783

The purpose of the work was to do mathematical modeling of axisymmetric body separation flow at subsonic velocities of incident flow. In our research we used the concept of viscous-inviscid interaction. We found velocities and pressures on the surface of the body under study according to the results of calculating of some equivalent body inviscid flow. The wake turbulence effect was simulated by the tailed section of the equivalent body. We examined the semi-infinite tailed sections of the equivalent body instead of the tailed sections of finite length. Moreover, we studied flow separation conditions in the base region. For the numerical simulation we used the discrete vortex method. The base pressure was found by Horner formula. We carried out mathematical modeling of the flow around cylindrical bodies with the head part of the ogival form.

Timofeev V. Construction of a semi-infinite equivalent body in mathematical modeling of subsonic separated axisymmetric flow. Маthematical Modeling and Coтputational Methods, 2016, №4 (12), pp. 67-83

#### 519.6 Use of hybrid algorithms in extremum eigenproblems of Lagrangian dynamical systems

**Sulimov V. D. (Bauman Moscow State Technical University), Shkapov P. M. (Bauman Moscow State Technical University), Goncharov D. A. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2016-4-84102

The study examines extremum problems for eigen spectra components of Lagrangian dynamical systems. Mathematical models of the systems studied are described by the matrices depending on the parameters. The eigenproblems defined for such systems, in general, are characterized by a spectrum, which can contain multiple eigenvalues. Subtests in extremum problems are assumed to be continuous, Lipschitzian, multiextremum and maybe not everywhere differentiable functions. The search for global solutions is conducted using new hybrid algorithms that combine a stochastic algorithm for scanning the variables space and deterministic local search methods. The study gives numerical examples of solving the problems of global nondifferentiable minimization of the maximum systems eigenvalues.

Sulimov V., Shkapov P., Goncharov D. Use of hybrid algorithms in extremum eigenproblems of Lagrangian dynamical systems. Маthematical Modeling and Coтputational Methods, 2016, №4 (12), pp. 84-102

#### 626.422.23 Quality criteria and algorithm for selecting reduced finite element models for technical design monitoring

**Meschikhin I. A. (Bruk Institute of Electronic Controlling Machines), Gavryushin S. S. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2016-4-103121

The article centers on developing the models of complex engineering structures for monitoring their operational status. In order to organize effective monitoring of the structures state, we often need to restore the state parameters values according to the measuremen tresults and use monitoring object models. The study suggests using a model built by calculation and reduced to small size. Among the state parameters there may be either load values and their combinations on which direct restrictions are imposed, or limit states defined by their values. The work proposes both criteria for assessing the quality of the reduced model, and an efficient algorithm for searching them.

Meschikhin I., Gavryushin S. Quality criteria and algorithm for selecting reduced finite element models for technical design monitoring. Маthematical Modeling and Coтputational Methods, 2016, №4 (12), pp. 103-121