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The article presents a method for linearization the effective function of material instantaneous deformation in order to generalize the torsional vibration equation to the case of nonlinearly deformable rheologically active shafts. It is considered layered and structurally heterogeneous, on average isotropic shafts made of nonlinearly viscoelastic components. The technique consists in determining the approximate shear modulus by minimizing the root-mean-square deviation in approximation of the effective diagram of instantaneous deformation.

The method allows to estimate analytically values of natural frequencies of layered and structurally heterogeneous nonlinearly viscoelastic shaft. This makes it possible to significantly reduce resources in vibration analysis, as well as to track changes in values of natural frequencies with changing geometric, physico-mechanical and structural parameters of shafts, which is especially important at the initial stages of modeling and design. In addition, the paper shows that only a pronounced nonlinearity of the effective state equation has an effect on the natural frequencies, and in some cases the nonlinearity in determining the natural frequencies can be neglected.

As equations of state of the composite material components, the article considers the equations of nonlinear heredity with instantaneous deformation functions in the form of the Prandtl’s bilinear diagrams. To homogenize the state equations of layered shafts, it is applied the Voigt’s hypothesis on the homogeneity of deformations and the Reuss’ hypothesis on the homogeneity of stresses in the volume of a composite body. Using these assumptions, effective secant and tangential shear moduli, proportionality limits, as well as creep and relaxation kernels of longitudinal, axial and transversely layered shafts are obtained. In addition, it is obtained the indicated effective characteristics of a structurally heterogeneous, on average isotropic shaft using the homogenization method previously proposed by the authors, based on the determination of the material deformation parameters by the rule of a mixture for the Voigt’s and the Reuss’ state equations.

The work is dedicated to the use of the software package Turbulence Problem Solver (TPS) for numerical simulation of a wide range of laser problems. The capabilities of the package are demonstrated by the example of numerical simulation of the interaction of femtosecond laser pulses with thin metal bonds. The software package TPS developed by the authors is intended for numerical solution of hyperbolic systems of differential equations on multiprocessor computing systems with distributed memory. The package is a modern and expandable software product. The architecture of the package gives the researcher the opportunity to model different physical processes in a uniform way, using different numerical methods and program blocks containing specific initial conditions, boundary conditions and source terms for each problem. The package provides the the opportunity to expand the functionality of the package by adding new classes of problems, computational methods, initial and boundary conditions, as well as equations of state of matter. The numerical methods implemented in the software package were tested on test problems in one-dimensional, two-dimensional and three-dimensional geometry, which included Riemann's problems on the decay of an arbitrary discontinuity with different configurations of the exact solution.

Thin films on substrates are an important class of targets for nanomodification of surfaces in plasmonics or sensor applications. Many articles are devoted to this subject. Most of them, however, focus on the dynamics of the film itself, paying little attention to the substrate, considering it simply as an object that absorbs the first compression wave and does not affect the surface structures that arise as a result of irradiation. The paper describes in detail a computational experiment on the numerical simulation of the interaction of a single ultrashort laser pulse with a gold film deposited on a thick glass substrate. The uniform rectangular grid and the first-order Godunov numerical method were used. The presented results of calculations allowed to confirm the theory of the shock-wave mechanism of holes formation in the metal under femtosecond laser action for the case of a thin gold film with a thickness of about 50 nm on a thick glass substrate.

The article covers several mathematical problems relating to elastic stability of thin shells in view of inconsistencies that have been recently identified between the experimental data and the predictions based on the shallow- shell theory. It is highlighted that the contradictions were caused by new algorithms that enabled updating the values of the so called “low critical stresses” calculated in the 20th century and adopted as a buckling criterion for thin shallow shells by technical standards. The new calculations often find the low critical stress close to zero. Therefore, the low critical stress cannot be used as a safety factor for the buckling analysis of the thinwalled structure, and the equations of the shallow-shell theory need to be replaced with other differential equations. The new theory also requires a buckling criterion ensuring the match between calculations and experimental data.

The article demonstrates that the contradiction with the new experiments can be resolved within the dynamic nonlinear three-dimensional theory of elasticity. The stress when bifurcation of dynamic modes occurs shall be taken as a buckling criterion. The nonlinear form of original equations causes solitary (solitonic) waves that match non-smooth displacements (patterns, dents) of the shells. It is essential that the solitons make an impact at all stages of loading and significantly increase closer to bifurcation. The solitonic solutions are illustrated based on the thin cylindrical momentless shell when its three-dimensional volume is simulated with twodimensional surface of the set thickness. It is noted that the pattern-generating waves can be detected (and their amplitudes can by identified) with acoustic or electromagnetic devices.

Thus, it is technically possible to reduce the risk of failure of the thin shells by monitoring the shape of the surface with acoustic devices. The article concludes with a setting of the mathematical problems requiring the solution for the reliable numerical assessment of the buckling criterion for thin elastic shells.

A classical for nonlinear dynamics model, Brusselator, is considered, being augmented by addition of a third variable, which plays the role of a fast-diffusing inhibitor. The model is investigated in one-dimensional case in the parametric domain, where two types of diffusive instabilities of system’s homogeneous stationary state are manifested: wave instability, which leads to spontaneous formation of autowaves, and Turing instability, which leads to spontaneous formation of stationary dissipative structures, or Turing structures. It is shown that, due to the subcritical nature of Turing bifurcation, the interaction of two instabilities in this system results in spontaneous formation of stationary dissipative structures already before the passage of Turing bifurcation. In response to different perturbations of spatially uniform stationary state, different stable regimes are manifested in the vicinity of the double bifurcation point in the parametric region under study: both pure regimes, which consist of either stationary or autowave dissipative structures; and mixed regimes, in which different modes dominate in different areas of the computational space. In the considered region of the parametric space, the system is multistable and exhibits high sensitivity to initial noise conditions, which leads to blurring of the boundaries between qualitatively different regimes in the parametric region. At that, even in the area of dominance of mixed modes with prevalence of Turing structures, the establishment of a pure autowave regime has significant probability. In the case of stable mixed regimes, a sufficiently strong local perturbation in the area of the computational space, where autowave mode is manifested, can initiate local formation of new stationary dissipative structures. Local perturbation of the stationary homogeneous state in the parametric region under investidation leads to a qualitatively similar map of established modes, the zone of dominance of pure autowave regimes being expanded with the increase of local perturbation amplitude. In two-dimensional case, mixed regimes turn out to be only transient — upon the appearance of localized Turing structures under the influence of wave regime, they eventually occupy all available space.

A hierarchical method of mathematical and computer modeling of interval-stochastic thermal processes in complex electronic systems for various purposes is developed. The developed concept of hierarchical structuring reflects both the constructive hierarchy of a complex electronic system and the hierarchy of mathematical models of heat exchange processes. Thermal processes that take into account various physical phenomena in complex electronic systems are described by systems of stochastic, unsteady, and nonlinear partial differential equations and, therefore, their computer simulation encounters considerable computational difficulties even with the use of supercomputers. The hierarchical method avoids these difficulties. The hierarchical structure of the electronic system design, in general, is characterized by five levels: Level 1 — the active elements of the ES (microcircuits, electro-radio-elements); Level 2 — electronic module; Level 3 — a panel that combines a variety of electronic modules; Level 4 — a block of panels; Level 5 — stand installed in a stationary or mobile room. The hierarchy of models and modeling of stochastic thermal processes is constructed in the reverse order of the hierarchical structure of the electronic system design, while the modeling of interval-stochastic thermal processes is carried out by obtaining equations for statistical measures. The hierarchical method developed in the article allows to take into account the principal features of thermal processes, such as the stochastic nature of thermal, electrical and design factors in the production, assembly and installation of electronic systems, stochastic scatter of operating conditions and the environment, non-linear temperature dependencies of heat exchange factors, unsteady nature of thermal processes. The equations obtained in the article for statistical measures of stochastic thermal processes are a system of 14 non-stationary nonlinear differential equations of the first order in ordinary derivatives, whose solution is easily implemented on modern computers by existing numerical methods. The results of applying the method for computer simulation of stochastic thermal processes in electron systems are considered. The hierarchical method is applied in practice for the thermal design of real electronic systems and the creation of modern competitive devices.

The review considers a wide range of phenomena and problems associated with the explosion. Detailed numerical studies revealed an interesting physical effect — the formation of discrete vortex structures directly behind the front of a shock wave propagating in dense layers of a heterogeneous atmosphere. The necessity of further investigation of such phenomena and the determination of the degree of their connection with the possible development of gas-dynamic instability is shown. The brief analysis of numerous works on the thermal explosion of meteoroids during their high-speed movement in the Earth’s atmosphere is given. Much attention is paid to the development of a numerical algorithm for calculating the simultaneous explosion of several fragments of meteoroids and the features of the development of such a gas-dynamic flow are analyzed. The work shows that earlier developed algorithms for calculating explosions can be successfully used to study explosive volcanic eruptions. The paper presents and discusses the results of such studies for both continental and underwater volcanoes with certain restrictions on the conditions of volcanic activity.

The mathematical analysis is performed and the results of analytical studies of a number of important physical phenomena characteristic of explosions of high specific energy in the ionosphere are presented. It is shown that the preliminary laboratory physical modeling of the main processes that determine these phenomena is of fundamental importance for the development of sufficiently complete and adequate theoretical and numerical models of such complex phenomena as powerful plasma disturbances in the ionosphere. Laser plasma is the closest object for such a simulation. The results of the corresponding theoretical and experimental studies are presented and their scientific and practical significance is shown. The brief review of recent years on the use of laser radiation for laboratory physical modeling of the effects of a nuclear explosion on asteroid materials is given.

As a result of the analysis performed in the review, it was possible to separate and preliminarily formulate some interesting and scientifically significant questions that must be investigated on the basis of the ideas already obtained. These are finely dispersed chemically active systems formed during the release of volcanoes; small-scale vortex structures; generation of spontaneous magnetic fields due to the development of instabilities and their role in the transformation of plasma energy during its expansion in the ionosphere. It is also important to study a possible laboratory physical simulation of the thermal explosion of bodies under the influence of highspeed plasma flow, which has only theoretical interpretations.

We consider a model of spontaneous formation of a computational structure in the human brain for solving a given class of tasks in the process of performing a series of similar tasks. The model is based on a special definition of a numerical measure of the complexity of the solution algorithm. This measure has an informational property: the complexity of a computational structure consisting of two independent structures is equal to the sum of the complexities of these structures. Then the probability of spontaneous occurrence of the structure depends exponentially on the complexity of the structure. The exponential coefficient requires experimental determination for each type of problem. It may depend on the form of presentation of the source data and the procedure for issuing the result. This estimation method was applied to the results of a series of experiments that determined the strategy for solving a series of similar problems with a growing number of initial data. These experiments were described in previously published papers. Two main strategies were considered: sequential execution of the computational algorithm, or the use of parallel computing in those tasks where it is effective. These strategies differ in how calculations are performed. Using an estimate of the complexity of schemes, you can use the empirical probability of one of the strategies to calculate the probability of the other. The calculations performed showed a good match between the calculated and empirical probabilities. This confirms the hypothesis about the spontaneous formation of structures that solve the problem during the initial training of a person. The paper contains a brief description of experiments, detailed computational schemes and a strict definition of the complexity measure of computational structures and the conclusion of the dependence of the probability of structure formation on its complexity.

The work is devoted to the structural analysis of complex technical systems. Mechanical structures are considered, the properties of which affect the behavior of products during assembly, repair and operation. The main source of data on parts and mechanical connections between them is a hypergraph. This model formalizes the multidimensional basing relation. The hypergraph correctly describes the connectivity and mutual coordination of parts, which is achieved during the assembly of the product. When developing complex products in CAD systems, an engineer often makes serious design mistakes: overbasing of parts and non-sequential assembly operations. Effective ways of identifying these structural defects have been proposed. It is shown that the property of independent assembly can be represented as a closure operator whose domain is the boolean of the set of product parts. The images of this operator are connected and coordinated subsets of parts that can be assembled independently. A lattice model is described, which is the state space of the product during assembly, disassembly and decomposition into assembly units. The lattice model serves as a source of various structural information about the project. Numerical estimates of the cardinality of the set of admissible alternatives in the problems of choosing an assembly sequence and decomposition into assembly units are proposed. For many technical operations (for example, control, testing, etc.), it is necessary to mount all the operand parts in one assembly unit. A simple formalization of the technical conditions requiring the inclusion (exclusion) of parts in the assembly unit (from the assembly unit) has been developed. A theorem that gives an mathematical description of product decomposition into assembly units in exact lattice terms is given. A method for numerical evaluation of the robustness of the mechanical structure of a complex technical system is proposed.

When a supersonic air flow interacts with a transverse secondary jet injected into this flow through an orifice on a flat wall, a special flow structure is formed. This flow takes place during fuel injection into combustion chambers of supersonic aircraft engines; therefore, in recent years, various approaches to intensifying gas mixing in this type of flow have been proposed and studied in several countries. The approach proposed in this work implies using spark discharges for pulsed heating of the gas and generating the instabilities in the shear layer at the boundary of the secondary jet. Using simulation in the software package FlowVision 3.13, the characteristics of this flow were obtained in the absence and presence of pulsed-periodic local heat release on the wall on the windward side of the injector opening. A comparison was made of local characteristics at different periodicities of pulsed heating (corresponding to the values of the Strouhal number 0.25 and 0.31). It is shown that pulsed heating can stimulate the formation of perturbations in the shear layer at the jet boundary. For the case of the absence of heating and for two modes of pulsed heating, the values of an integral criterion for mixing efficiency were calculated. It is shown that pulsed heating can lead both to a decrease in the average mixing efficiency and to its increase (up to 9% in the considered heating mode). The calculation method used (unsteady Reynolds-averaged Navier – Stokes equations with a modified $k-\varepsilon$ turbulence model) was validated by considering a typical case of the secondary transverse jet interaction with a supersonic flow, which was studied by several independent research groups and well documented in the literature. The grid convergence was shown for the simulation of this typical case in FlowVision. A quantitative comparison was made of the results obtained from FlowVision calculations with experimental data and calculations in other programs. The results of this study can be useful for specialists dealing with the problems of gas mixing and combustion in a supersonic flow, as well as the development of engines for supersonic aviation.

In this work, we use analytical and numerical methods to consider the problem of the structure and dynamics of coupled localized nonlinear waves in the sine-Gordon model with three identical attractive extended “impurities”, which are modeled by spatial inhomogeneity of the periodic potential. Two possible types of coupled nonlinear localized waves are found: breather and soliton. The influence of system parameters and initial conditions on the structure, amplitude, and frequency of localized waves was analyzed. Associated oscillations of localized waves of the breather type as in the case of point impurities, are the sum of three harmonic oscillations: in-phase, in-phase-antiphase and antiphase type. Frequency analysis of impurity-localized waves that were obtained during a numerical experiment was performed using discrete Fourier transform. To analyze localized breather-type waves, the numerical finite difference method was used. To carry out a qualitative analysis of the obtained numerical results, the problem was solved analytically for the case of small amplitudes of oscillations localized on impurities. It is shown that, for certain impurity parameters (depth and width), it is possible to obtain localized solitontype waves. The ranges of values of the system parameters in which localized waves of a certain type exist, as well as the region of transition from breather to soliton types of oscillations, have been found. The values of the depth and width of the impurity at which a transition from the breather to the soliton type of localized oscillations is observed were determined. Various scenarios of soliton-type oscillations with negative and positive amplitude values for all three impurities, as well as mixed cases, were obtained and considered. It is shown that in the case when the distance between impurities much less than one, there is no transition region where which the nascent breather, after losing energy through radiation, transforms into a soliton. It is shown that the considered model can be used, for example, to describe the dynamics of magnetization waves in multilayer magnets.