METHODS OF OPTIMIZATION OF PARAMETRIC SYSTEMS

The paper considers methods of parametric optimization of a dynamical system, which is described by a parametric system of differential equations. The gradient of the functional in the form of Boltz is found, which is the basis of methods such as gradient descent. Another method is based on the application of the sensitivity function.

MODELING OF FINITE INHOMOGENEITIES BY DISCRET SINGULARITIES

This work focuses on development of a mathematical apparatus that allows to perform an approximate description of inhomogeneities of finite sizes in a continuous bodies by arranging the sources given on sets of smaller dimensions. The structure and properties of source densities determine the adequacy of the model. The theory of differential forms and generalized functions underlies this study. The boundary value problems with nonsmooth coefficients are formulated. The solutions of such problems is sought in the form of weakly convergent series and as an alternative - an equivalent recurrent set of boundary value problems with jumps. A feature of this approach is the ability to consistently improve the adequacy of the description of inhomogeneity. This is important because it allows to qualitatively assess the impact of real characteristic properties on the accuracy of the model description. Reducing the dimensions of inhomogeneities allows the use of efficient methods such as the Green's function and boundary integral equations to obtain a semi-analytic solution for direct and inverse problems. The work is based on a number of partial problems that demonstrate the proposed approach in modeling of inhomogeneities. The problems of modeling of the set of finite defects in an oscillating elastic beam, the set of inhomogeneities of an arbitrary shape in an oscillating plate, fragile cracks in a two-dimensional elastic body under static loading are considered.

MODELING THE DYNAMICS OF AN INFECTIOUS DISEASE TAKING INTO ACCOUNT SPATIAL-DIFFUSE PERTURBATIONS, CONCENTRATED INFLUENCES AND ENVIRONMENT CURVATURE

While the study of the interaction patterns of the immune system and the viruses detected in the body wide variety of models is used. Well-known infectious disease model by Marchuk which describes the most common mechanisms of immune defense, was obtained under the assumption that the environment of the "organism" is homogeneous and unlimited, in which all the active factors of the process are instantly mixed. The approach proposed by the authors to take into account the influence of spatially distributed diffusion "redistributions" on the nature of the infectious disease provides an opportunity to detect the reducing effect the model level of maximum antigen concentration at the infection epicenter due to their diffusion "erosion" in the disease development. In particular, in cases where the viral particles concentration at the initial time or the intensity of a concentrated source of viruses in any part of the body of infection exceeds a certain critical level of the immunological barrier such an effect of diffusion "redistribution" in a short time reduces supercritical concentrations of viral particles to values, in particular, already below the critical level and their further neutralization may be ensured by the existing level of own antibodies concentration or requires a more economical procedure of injection with a lower donor antibodies concentration. In this article the infectious disease mathematical model is generalized to take into account the curvature of the bounded environment in the conditions of spatial diffusion perturbations, convection and the presence of various concentrated influences. The corresponding singularly perturbed model problem with delay is reduced to a sequence of "solvable" problems without delay. The influence of "curvature" of a limited environment on the development of an infectious disease in the conditions of diffusion perturbations, convection and concentrated influences is illustrated.

HYDRAULIC MODELS IN THE PROBLEMS OF THERMAL POWER PLANT AUXILIARY ENERGY EFFICIENCY IMPROVEMENT

The work is devoted to the study of thermal power plants auxiliary energy efficiency. The main mechanisms in the auxiliary systems are centrifugal mechanisms that work in complex hydraulic networks with variable productivity. The main ways to adjust the parameters of the centrifugal mechanisms are to change the speed of rotor rotation, change the guide vane angle and throttle. The operation mode of a complex hydraulic network which includes a group of centrifugal mechanisms with a mixed connection scheme is analyzed. The system of equations which characterize the hydraulic system has been obtained on the basis of Kirchhoff's laws. The centrifugal mechanisms' operating characteristics are given by approximation dependences obtained with the method of least squares and similarity laws. To analyze efficiency of different methods of centrifugal mechanisms parameters regulation, optimal control problems were set and solved. The constraints for the problems are a system of equations that describe the hydraulic system operation and technical constraints that depend on the control method. Through solving the problems, values of the optimal parameters and weighted average efficiency of the group mechanisms were obtained. Studies have shown that the most effective way to regulate the centrifugal mechanisms parameters is to use an individual frequency drive, the least effective is to use only changing angle of centrifugal mechanism's guide vane. Utilization of group control is highly efficient and not inferior to individual frequency drive. However, this statement is correct under condition of the operating characteristics agreement with the centrifugal mechanisms’ operating modes similarity.

MODELING OF THERMAL INSTALLATIONS BASED ON THERMODYNAMIC APPROACHES

The most widespread approaches to the study of thermal systems involve the iterative implementation of the following steps: thermodynamics, heat and mass transfer, hydrodynamics, economics and ecology. Such methodology cannot combine economic, environmental and thermodynamic aspects from the beginning of the analysis. It does not provide information concerning not only external, but also internal, caused by thermodynamic inefficiencies of system components, impact factors on economic and ecological characteristics. Modeling methods based on the combined application of the First and Second Laws of Thermodynamics (methods of entropy and exergetic analysis), and their combination with economic and environmental assessment make it possible to identify the location, magnitude, causes, costs and environmental impact of thermodynamic inefficiencies in an energy conversion system. The paper proposes the improvement of methods for modeling thermal systems on the base of exergy analysis. It has been shown that combining exergetic, economic and ecological assessment can significantly simplify tasks of finding parameters and structure of the studied system. Examples of implementation of such studies have been presented.

MODELING OF A VENTILATED CAVITY BEHIND A STREAMLINED BODY

The results of physical and numerical modeling of a ventilated air cavity behind a streamlined body are presented. The results of laboratory experiments to determine the amount of gas flowing from the ventilated cavity are presented. It is formed behind the cavitator depending on a number of geometric and dynamic parameters. Numerical simulation of non-stationary 3D two-phase flow was performed on the basis of open source software OpenFOAM. The influence of gas blowing parameters on the formation of an air cavity, size, shape and stability has been investigated. Good qualitative agreement with experimental data was obtained. It is shown that the thickness of the ventilated cavity is determined by the diameter of the cavitator regardless of the diameter of the blow hole, and the increase in velocity or gas flow rate has a positive effect on the length and stability of the formed cavity.

THE DIFFUSION-DRIFT PROCESS WITH ACCOUNT HEATING AND RECOMBINATION IN THE p-i-n DIODES ACTIVE REGION MATHEMATICAL MODELING BY THE PERTURBATION THEORY METHODS

With prolonged transmission of an electric current through the semiconductor devices, in a particular p-i-n diodes, an electron-hole plasma of their active region is heated. This paper presents the theoretical studies results of the plasma heating effect by the Joule heat release in the p-i-n diode volume and the charge carriers recombination energy release on the plasma concentration distribution in the p-i-n diodes active region. The mathematical model is proposed for predicting the electron-hole plasma stationary concentration distribution and the temperature field in the i-region of the bulk p-i-n diodes in the form of a nonlinear boundary value problem in a given area for the equations system, which consist of the charge carrier current continuity equations, the Poisson and the thermal conductivity. It is shown that the differential equations of the model contain a small parameter in such a way that the Poisson equation is singularly perturbed and the heat conduction equation is regularly perturbed. An approximate solution of the problem posed is obtained in the form of the corresponding asymptotic series in powers of the small parameter. The asymptotic serieses, which describes the behavior of the plasma concentration and potential in the investigated region, containing near-boundary corrections to ensure the fulfillment of the boundary conditions. The terms of these series are found as a result of solving a sequence of boundary value problems, obtained as a result of splitting the original problem, for systems of linear differential equations. The boundary value problem for a nonlinear heat equation is reduced to a sequence of problems for the corresponding linear inhomogeneous equations. The process of refining solutions is iterative. The stabilization of the process is ensured by the existence of negative feedback in the system (as the temperature rises, the mobility of charge carriers decreases).

MODELING OF GAS-DYNAMIC PROCESSES IN THE ELEMENTS OF IMPULSE EJECTOR

The paper presents the numerical results on gas-dynamic processes in various elements of the impulse ejector, including pre-chamber, supersonic nozzle and mixing chamber, to determine optimal geometric parameters providing the given flow rate characteristics. At an extra-high pressure of the ejecting gas (>100 bar) it is impossible to create a nozzle design with continuously changing cross-sectional area and limited nozzle length. So, it is necessary to place a pre-chamber between the gas generator and the ejector nozzle for throttling full gas pressure. In order to optimize the pre-chamber parameters in the ejector with discrete holes of the gas generator and the operating pressure in the range of 400÷1000 bar, a series of calculations were performed to determine the pre-chamber parameters, ensuring stable operation of the supersonic annular nozzle at the high pressure of 35÷45 bar and the flow rate of 0.5÷0.6 kg/s. 3D numerical simulation of the gas flow into the pre-chamber through the gas generator holes shows the degree of the flow pattern non-uniformity in the pre-chamber at the ejector nozzle inlet is quite low. This justifies the numerical simulation of gas flow in the ejector in axisymmetric formulation and allows restricting the number of the gas generator holes without inducing significant non-uniformity in the azimuthal direction.

GAUSS APPROXIMATION FOR NUMBER DISTRIBUTION IN OF A PASCAL’S TRIANGLE

We received normal distribution parameters that approximates the distribution of numbers in the n-th row of Pascal's triangle. We calculated the values for normalized moments of even orders and shown their asymptotic tendency towards values corresponding to a normal distribution. We have received highly accurate approximations for central elements of even rows of Pascal's triangle, which allows for calculation of binomial, as well as trinomial (or, in general cases, multinomial) coefficients. A hypothesis is proposed, according to which it is possible that physical and physics-chemical processes function according to Pascal's distribution, but due to how slight its deviation is from a normal distribution, it is difficult to notice. It is also possible that as technology and experimental methodology improves, this difference will become noticeable where it is traditionally considered that a normal distribution is taking place.

DIMPLE GENERATOR OF VORTEX STRUCTURES

The paper presents the results of experimental studies of the space-time characteristics of the velocity and pressure field inside a hemispherical dimple on a flat surface. The features of the formation and development of vortex structures generated inside the dimple, as well as their interaction with the streamlined surface of the dimple and the boundary layer were established. Integral, spectral and correlation characteristics of the field of velocity, dynamic and wall pressure fluctuations were obtained. The velocities and directions of transfer of large-scale vortex structures and small-scale vortices inside the dimple were determined. The frequencies of rotations and ejections of large-scale vortices, the frequencies of oscillations of the vortex flow inside the dimple and self-oscillations of the vortex structures of the shear layer, their subharmonics and harmonics of higher orders were established.