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The article is devoted to the effect of thermal feedback, which occurs during the operation of integrated circuits and electronic systems with their use. Thermal feedback is due to the fact that the power consumed by the functioning of the microchip heats it and, due to the significant dependence of its electrical parameters on temperature, interactive interaction arises between its electrical and thermal processes. The effect of thermal feedback leads to a change in both electrical parameters and temperature levels in microcircuits. Positive thermal feedback is an undesirable phenomenon, because it causes the output of the electrical parameters of the microcircuits beyond the permissible values, the reduction in reliability and, in some cases, burn out. Negative thermal feedback is manifested in stabilizing the electrical and thermal regimes at lower temperature levels. Therefore, when designing microcircuits and electronic systems with their application, it is necessary to achieve the implementation of negative feedback. In this paper, we propose a method for modeling of thermal modes in electronic systems, taking into account the effect of thermal feedback. The method is based on introducing into the thermal model of the electronic system new model circuit elements that are nonlinearly dependent on temperature, the number of which is equal to the number of microcircuits in the electronic system. This approach makes it possible to apply matrix-topological equations of thermal processes to the thermal model with new circuit elements introduced into it and incorporate them into existing thermal design software packages. An example of modeling a thermal process in a real electronic system is presented, taking into account the effect of thermal feedback on the example of a microcircuit installed on a printed circuit board. It is shown that in order to adequately model the electrical and thermal processes of microcircuits and electronic systems, it is necessary to take into account the effects of thermal feedback in order to avoid design errors and create competitive electronic systems.

In real problems of exploration seismology we deal with a heterogeneity of the nature of elastic waves interaction with the surface of a fracture by the propagation through it. The fracture is a complex heterogeneous structure. In some locations the surfaces of fractures are placed some distance apart and are separated by filling fluid or emptiness, in some places we can observe the gluing of surfaces, when under the action of pressure forces the fracture surfaces are closely adjoined to each other. In addition, fractures can be classified by the nature of saturation: fluid or gas. Obviously, for such a large variety in the structure of fractures, one cannot use only one model that satisfies all cases.

This article is concerned with description of developed mathematical fracture models which can be used for numerical solution of exploration seismology problems using the grid-characteristic method on unstructured triangular (in 2D-case) and tetrahedral (in 3D-case) meshes. The basis of the developed models is the concept of an infinitely thin fracture, whose aperture does not influence the wave processes in the fracture area. These fractures are represented by bound areas and contact boundaries with different conditions on contact and boundary surfaces. Such an approach significantly reduces the consumption of computer resources since there is no need to define the mesh inside the fracture. On the other side, it allows the fractures to be given discretely in the integration domain, therefore, one can observe qualitatively new effects, such as formation of diffractive waves and multiphase wave front due to multiple reflections between the surfaces of neighbor fractures, which cannot be observed by using effective fracture models actively used in computational seismology.

The computational modeling of seismic waves propagation through layers of mesofractures was produced using developed fracture models. The results were compared with the results of physical modeling in problems in the same statements.

The article deals with the problem of a distributed system operation reliability. The system core is an open integration platform that provides interaction of varied software for modeling gas transportation. Some of them provide an access through thin clients on the cloud technology “software as a service”. Mathematical models of operation, transmission and computing are to ensure the operation of an automated dispatching system for oil and gas transportation. The paper presents a system solution based on the theory of Markov random processes and considers the stable operation stage. The stationary operation mode of the Markov chain with continuous time and discrete states is described by a system of Chapman–Kolmogorov equations with respect to the average numbers (mathematical expectations) of the objects in certain states. The objects of research are both system elements that are present in a large number – thin clients and computing modules, and individual ones – a server, a network manager (message broker). Together, they are interacting Markov random processes. The interaction is determined by the fact that the transition probabilities in one group of elements depend on the average numbers of other elements groups.

The authors propose a multi-criteria dispersion model of risk assessment for such systems (both in the broad and narrow sense, in accordance with the IEC standard). The risk is the standard deviation of estimated object parameter from its average value. The dispersion risk model makes possible to define optimality criteria and whole system functioning risks. In particular, for a thin client, the following is calculated: the loss profit risk, the total risk of losses due to non-productive element states, and the total risk of all system states losses.

Finally the paper proposes compromise schemes for solving the multi-criteria problem of choosing the optimal operation strategy based on the selected set of compromise criteria.

The paper considers the process of flow formation in an outflow of a binary gas mixture through a thin slit into vacuum. An approach to modeling the flows of rarefied gas mixtures in the transient regime is proposed based on the direct solution of the Boltzmann kinetic equation, in which the conservative projection method is used to calculate the collision integrals. Calculation formulas are provided; the calculation procedure is described in detail in relation to the flow of a binary gas mixture. The Lennard–Jones potential is used as an interaction potential of molecules. A software modeling environment has been developed that makes it possible to study the flows of gas mixtures in a transitional regime on systems of cluster architecture. Due to the use of code parallelization technologies, an acceleration of calculations by 50–100 times was obtained. Numerical simulation of a two-dimensional outflow of a binary argon-neon gas mixture from a vessel into vacuum through a thin slit is carried out for various values of the Knudsen number. The graphs of the dependence of gas mixture components output flow on time in the process of establishing the flow are obtained. Non-stationary regions of strong separation of gas mixture components, in which the molecular densities ratio reaches 10 or more, were discovered. The discovered effect can have applications in the problem of gas mixtures separation.

The work is devoted to the study of the influence of nonlinear processes in the boundary layer on the general nature of gas flows in microchannels of technical systems. Such a study is actually concerned with nanotechnology problems. One of the important problems in this area is the analysis of gas flows in microchannels in the case of transient and supersonic flows. The results of this analysis are important for the gas-dynamic spraying techique and for the synthesis of new nanomaterials. Due to the complexity of the implementation of full-scale experiments on micro- and nanoscale, they are most often replaced by computer simulations. The efficiency of computer simulations is achieved by both the use of new multiscale models and the combination of mesh and particle methods. In this work, we use the molecular dynamics method. It is applied to study the establishment of a gas microflow in a metal channel. Nitrogen was chosen as the gaseous medium. The metal walls of the microchannels consisted of nickel atoms. In numerical experiments, the accommodation coefficients were calculated at the boundary between the gas flow and the metal wall. The study of the microsystem in the boundary layer made it possible to form a multicomponent macroscopic model of the boundary conditions. This model was integrated into the macroscopic description of the flow based on a system of quasi-gas-dynamic equations. On the basis of such a transformed gas-dynamic model, calculations of microflow in real microsystem were carried out. The results were compared with the classical calculation of the flow, which does not take into account nonlinear processes in the boundary layer. The comparison showed the need to use the developed model of boundary conditions and its integration with the classical gas-dynamic approach.

This paper provides an overview of studies on nonlinear supratransmission and related phenomena. This effect consists in the transfer of energy at frequencies not supported by the systems under consideration. The supratransmission does not depend on the integrability of the system, it is resistant to damping and various classes of boundary conditions. In addition, a nonlinear discrete medium, under certain general conditions imposed on the structure, can create instability due to external periodic influence. This instability is the generative process underlying the nonlinear supratransmission. This is possible when the system supports nonlinear modes of various nature, in particular, discrete breathers. Then the energy penetrates into the system as soon as the amplitude of the external harmonic excitation exceeds the maximum amplitude of the static breather of the same frequency.

The effect of nonlinear supratransmission is an important property of many discrete structures. A necessary condition for its existence is the discreteness and nonlinearity of the medium. Its manifestation in systems of various nature speaks of its fundamentality and significance. This review considers the main works that touch upon the issue of nonlinear supratransmission in various systems, mainly model ones.

Many teams of authors are studying this effect. First of all, these are models described by discrete equations, including sin-Gordon and the discrete Schr¨odinger equation. At the same time, the effect is not exclusively model and manifests itself in full-scale experiments in electrical circuits, in nonlinear chains of oscillators, as well as in metastable modular metastructures. There is a gradual complication of models, which leads to a deeper understanding of the phenomenon of supratransmission, and the transition to disordered structures and those with elements of chaos structures allows us to talk about a more subtle manifestation of this effect. Numerical asymptotic approaches make it possible to study nonlinear supratransmission in complex nonintegrable systems. The complication of all kinds of oscillators, both physical and electrical, is relevant for various real devices based on such systems, in particular, in the field of nano-objects and energy transport in them through the considered effect. Such systems include molecular and crystalline clusters and nanodevices. In the conclusion of the paper, the main trends in the research of nonlinear supratransmission are given.

An approximate mathematical model of blood flow in an axisymmetric blood vessel is studied. Such a vessel is understood as an infinitely long circular cylinder, the walls of which consist of elastic rings. Blood is considered as an incompressible fluid flowing in this cylinder. Increased pressure causes radially symmetrical stretching of the elastic rings. Following J. Lamb, the rings are located close to each other so that liquid does not flow between them. To mentally realize this, it is enough to assume that the rings are covered with an impenetrable film that does not have elastic properties. Only rings have elasticity. The considered model of blood flow in a blood vessel consists of three equations: the continuity equation, the law of conservation of momentum and the equation of state. An approximate procedure for reducing the equations under consideration to the Korteweg – de Vries (KdV) equation is considered, which was not fully considered by J. Lamb, only to establish the dependence of the coefficients of the KdV equation on the physical parameters of the considered model of incompressible fluid flow in an axisymmetric vessel. From the KdV equation, by a standard transition to traveling waves, ODEs of the third, second and first orders are obtained, respectively. Depending on the different cases of arrangement of the three stationary solutions of the first-order ODE, a cnoidal wave and a soliton are standardly obtained. The main attention is paid to an unbounded periodic solution, which we call a degenerate cnoidal wave. Mathematically, cnoidal waves are described by elliptic integrals with parameters defining amplitudes and periods. Soliton and degenerate cnoidal wave are described by elementary functions. The hemodynamic meaning of these types of decisions is indicated. Due to the fact that the sets of solutions to first-, second- and third-order ODEs do not coincide, it has been established that the Cauchy problem for second- and third-order ODEs can be specified at all points, and for first-order ODEs only at points of growth or decrease. The Cauchy problem for a first-order ODE cannot be specified at extremum points due to the violation of the Lipschitz condition. The degeneration of the cnoidal wave into a degenerate cnoidal wave, which can lead to rupture of the vessel walls, is numerically illustrated. The table below describes two modes of approach of a cnoidal wave to a degenerate cnoidal wave.

The approach to optimization of integral estimation of bio-systems state is presented. The approach is included the procedures of decreasing of variability of integral estimation based on statistical modeling of experimental data set and optimization the quantity of a state characteristics on a base of their relative contribution to the integral estimation using parallel calculation.

By numerical simulation methods the interaction processes of oscillating soliton (breather) with a 180-degree Neel domain wall in the framework of a (2 + 1)-dimensional supersymmetric O(3) nonlinear sigma model is studied. The purpose of this paper is to investigate nonlinear evolution and stability of a system of interacting localized dynamic and topological solutions. To construct the interaction models, were used a stationary breather and domain wall solutions, where obtained in the framework of the two-dimensional sine-Gordon equation by adding specially selected perturbations to the A3-field vector in the isotopic space of the Bloch sphere. In the absence of an external magnetic field, nonlinear sigma models have formal Lorentz invariance, which allows constructing, in particular, moving solutions and analyses the experimental data of the nonlinear dynamics of an interacting solitons system. In this paper, based on the obtained moving localized solutions, models for incident and head-on collisions of breathers with a domain wall are constructed, where, depending on the dynamic parameters of the system, are observed the collisions and reflections of solitons from each other, a long-range interactions and also the decay of an oscillating soliton into linear perturbation waves. In contrast to the breather solution that has the dynamics of the internal degree of freedom, the energy integral of a topologically stable soliton in the all experiments the preserved with high accuracy. For each type of interaction, the range of values of the velocity of the colliding dynamic and topological solitons is determined as a function of the rotation frequency of the A3-field vector in the isotopic space. Numerical models are constructed on the basis of methods of the theory of finite difference schemes, using the properties of stereographic projection, taking into account the group-theoretical features of constructions of the O(N) class of nonlinear sigma models of field theory. On the perimeter of the two-dimensional modeling area, specially developed boundary conditions are established that absorb linear perturbation waves radiated by interacting soliton fields. Thus, the simulation of the interaction processes of localized solutions in an infinite two-dimensional phase space is carried out. A software module has been developed that allows to carry out a complex analysis of the evolution of interacting solutions of nonlinear sigma models of field theory, taking into account it’s group properties in a two-dimensional pseudo-Euclidean space. The analysis of isospin dynamics, as well the energy density and energy integral of a system of interacting dynamic and topological solitons is carried out.

The paper contains numerical simulation of nonisothermal nonlinear flow in a porous medium. Twodimensional unsteady problem of heavy oil, water and steam flow is considered. Oil phase consists of two pseudocomponents: light and heavy fractions, which like the water component, can vaporize. Oil exhibits viscoplastic rheology, its filtration does not obey Darcy's classical linear law. Simulation considers not only the dependence of fluids density and viscosity on temperature, but also improvement of oil rheological properties with temperature increasing.

To solve this problem numerically we use streamline method with splitting by physical processes, which consists in separating the convective heat transfer directed along filtration from thermal conductivity and gravitation. The article proposes a new approach to streamline methods application, which allows correctly simulate nonlinear flow problems with temperature-dependent rheology. The core of this algorithm is to consider the integration process as a set of quasi-equilibrium states that are results of solving system on a global grid. Between these states system solved on a streamline grid. Usage of the streamline method allows not only to accelerate calculations, but also to obtain a physically reliable solution, since integration takes place on a grid that coincides with the fluid flow direction.

In addition to the streamline method, the paper presents an algorithm for nonsmooth coefficients accounting, which arise during simulation of viscoplastic oil flow. Applying this algorithm allows keeping sufficiently large time steps and does not change the physical structure of the solution.

Obtained results are compared with known analytical solutions, as well as with the results of commercial package simulation. The analysis of convergence tests on the number of streamlines, as well as on different streamlines grids, justifies the applicability of the proposed algorithm. In addition, the reduction of calculation time in comparison with traditional methods demonstrates practical significance of the approach.