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Evgeniya Tsarkova, Alexandr Belyaev and Elena Andreeva
Research of the mathematical model of the immune system
The paper considers a mathematical model describing the processes occurring in the human immune system. The algorithm of numerical solution of the problem by gradient descent method is developed. The dependence of the optimal control problem solution on the model parameters is investigated.

Vadim Salmin and Alexey Chetverikov
Algorithm for narrowing of the region of the final trajectory parameters deflection during the flight to geostationary orbit with low thrust engines
The article deals with the problem of control during the flight to the geostationary orbit of interorbital transport vehicles with low-thrust electric propulsion systems. The analysis of the influence on the final trajectory parameters of disturbing factors of various kinds, including the presence of a systematic error in the implementation of electric propulsion thrust. An algorithm for narrowing the area of deviation of the final trajectory parameters during the flight to geostationary orbit using low-thrust engines has been developed, which includes an algorithm for forming nominal programs for controlling the thrust vector and an algorithm for refining the amount of thrust created by an electric propulsion system, based on measurements of the actual period of circulation.

Vadim Salmin, Konstantin Peresypkin, Alexey Chetverikov and Ivan Tkachenko
Determination initial approximation in solving the problem of numerical optimization of a large-sized space structure using linear extrapolation of optimal solutions
The parameters of a large-sized space structure are determined using numerical optimization using the gradient method. The complexity of the boundary of design constraints makes the optimal solution poorly conditioned relative to the starting point of the iterative process. The starting point is selected based on the optimal solutions found with an incomplete set of design variables. The starting point is found by linear extrapolation of these optimal solutions. In the problem under consideration, this approach made it possible to obtain a better locally optimal solution than was possible with the help of an intuitive choice of the starting point.

Aurthur Vimalachandran Thomas Jayachandran, Andrey Tkachenko, Hewa Hussein Omar and Krishnakumar Aruljothi
Performance computing of an open cycle micro gas turbine powerplant using data aided modeling and simulation.
In this paper the study of the thermodynamic performance for a 6kW micro gas turbine power plant has been optimized by considering Air- fuel ratio at various altitudes along with fuel inlet temperature. The parametric analysis was performed to improve the overall efficiency of the power plant to obtain the required power needed by a multi-copter. The research deals with parametric optimization for determining realistic and practical combinations for improving the specific fuel consumption and overall efficiency of the Micro gas turbine power plant. The use of recuperators has not been beneficial for the micro gas turbine power system due to the increase in the total volume and weight of the engine. The fuel inventory should be decreased significantly to justify the addition of a recuperator system. The addition of heat exchangers improves the Specific fuel consumption that results in environmental friendly micro gas turbine engines. The result of this work helps to develop and design the micro gas turbine with high efficiency and at the optimized output. A model for a 6 kW micro gas turbine power plant was optimized for the application of primary power units for multi-copter or drones at low altitude conditions. To validate the model, JET CAT PHT3 shaft turbine design parameters were used and the results are compared with the experimental tests on this engine.

Aurthur Vimalachandran Thomas Jayachandran, Andrey Tkachenko, Krishnakumar Aruljothi and Hewa Hussein Omar
Modeling and parametric optimization for a solar-powered closed-cycle micro gas turbine for space applications.
The research describes the theory design and the modeling of a closed cycle micro gas turbine power plant (CMGTP). A computer simulation model in ASTRA is used to model specifically, for this cycle. The research is a feasibility study to examine the technology for Brayton-cycle based solar thermal power plants and to optimize the model in comparison with solar array photovoltaic cells to increase the power density by increasing the power generated per unit solar-collector area. A mathematical model for CMGTP was designed to power satellites in space environments. The goal is to provide electricity from a parabolic mirror acting as a heat source that is collected by pointing at the sun. The Closed Brayton Cycle power was simulated to produce an output of 6 kW at the generator terminal. The heat source generated from the sun was considered to be around 750 K to 1250 K. The vacuum at the space radiation environment was considered to be 200 K acting as a coolant sink. The CMGTP consists of a centrifugal compressor, a radial in-flow turbine on a single shaft powering a two- pole permanent magnet alternator.

Vadim Krysko
Nonlinear dynamics rectangular in plan nanoshells
Mathematical model of non-linear vibrations of rectangular in plan nano-shells on the basis of the first-order approximation of the Kirchhoff-Love theory under the action of a transverse sign-variable are proposed. The geometric non-linearity is introduced by means of the von Kármán shell theory and the shells are assumed to be shallow, elastic and isotropic. The governing equations are yielded by Hamilton–Ostrogradsky's variational principle, taking into account the modified couple stress theory. The continuous problem is reduced to the Cauchy problem by the finite-difference method of the second order accuracy. The Cauchy problem is solved via the fourth-order Runge–Kutta method with respect to time. We analyzed the convergence of these solutions depending on the step of integration over time and spatial coordinate. In particular, we found that taking nano-effects into account increases area of harmonic vibrations and leads to the appearance of as chaotic so hyperchaotic vibrations. Furthermore, we have detected the transition of vibrations from harmonic to chaotic according to Feigenbauman's scenario. We analyzed the character vibration using signs of spectrum the Lyapunov exponents. We found that hyperchaotic vibrations are characterized by two positive Lyapunov exponents and chaotic vibration by one positive Lyapunov exponent. For nano-shells, this phenomenon was discovered for the first time. The spectrum of Lyapunov exponents was determined by the methods of Wolf, Kanz, Rosenstein and Sawada to obtain reliable results. Computational examples of the theoretical investigations are given.

Hewa Hussein Omar, Venedikt Kuz’michev, Andrey Tkachenko and Aurthur Vimalachandran Thomas Jayachandran
Development of a mathematical model of the compact heat exchanger used for optimizing thermodynamic parameters of the aviation gas turbine engine
Nowadays, the task of increasing the economic and technical efficiency of the gas turbine engine and decrement of the specific weight and specific fuel consumption is one of the main targets in the development of the aviation gas turbine engine. A promising approach for decrement of specific fuel consumption and obtaining high thermal efficiency of the gas turbine engine (30% and above) is based on the concept of the recuperative cycle of the gas turbine engine. The article presents the computer-aided calculation for analysing the heat exchanger surface and developing novel mathematical models of heat exchanger in terms of weight goodness, flow passage goodness, and optimum weight and flow passage. 18 different heat transfer surfaces for heat exchanger cores have been evaluated and the results are presented. The same hydraulic diameter assumed for all heat transfer surfaces and only their thermo-hydraulic performances evaluated during the calculation design. To assess the reliability of the obtained models, the results of the design calculations of the developed models compared with the data of other authors and with the data of the existing gas turbine engine with heat exchanger. The obtained models focus on mathematical design calculation for optimizing the main thermodynamic parameters of a gas turbine engine coupled with a heat exchanger at the stage of conceptual design of aviation gas turbine engine with heat recovery of exhaust gas.

Valentina Burmistrova, Alexander Butov, Boris Kostishko and Marina Yavtushenko
The problem of optimal control of the intensities of processes with change-point
This article we consider the formulation of the problem of optimal control of processes with change-point. At the same time, for one important quotient, the problem of finding the optimal parameter value characterizing the intensity of the change-point is formed and solved. As a criterion, we consider a functional that takes into account the distribution of the moment of the first crossing fixed boundary by a process with a change-point with a controlled parameter.

Vladimir Volotsuev
A digital model of the density of the earth's residual atmosphere for ballistic analysis in the design of low-orbit spacecraft

Larisa Stepanova
A photoelastic and finite element study of the stress field in the vicinity of two interacting cracks: stress intensity factors, T-stresses and higher order terms
Multi-parameter stress field near the Mode I crack tip in an isotropic linear elastic material is presented. The coefficients of the Williams series expansion of the stress field in the vicinity of the Mode I crack in the isotropic linear elastic material are obtained by the digital photoelasticity technique and by finite element analysis. The main objective of this study is to determine first fifteen coefficients with good accuracy and to compare the numerical results with the analytical solution for the infinite plate with the central crack.

Larisa Stepanova and Oksana Belova
Determination of the higher-order coefficients of the Williams asymptotic expansion for notched semidiscs using the photoelasticity method and finite element analysis
The coefficients of the Williams series expansion in notched semidisks with different angles of the notch are obtained. The study is based on the experimental procedure of the photoelasticity method and the finite element analysis. The large series of experiments for semidiscs was performed. Digital photoelasticity method is used to analyze experimentally the stress and displacement fields in the vicinity of the crack tip in isotropic linear elastic cracked specimens under Mixed Mode loading. The distribution of the isochromatic fridge patterns is employed for obtaining the stress field near the crack tip by the use of the complete Williams asymptotic expansion for various classes of the experimental specimens.

Hewa Hussein Omar, Venedikt Kuz’michev and Andrey Tkachenko
Optimization of the parameters of the working process of aviation turbofan engines with heat recovery
Improving the fuel efficiency of aircraft engines is the main global trend of modern propulsion engineering. One of the promising ways to further improve fuel efficiency is the use of complex thermodynamic cycles with heat recovery of exhaust gases and intermediate cooling in the process of air compression. An important task in the design of engines with heat recovery is to select the parameters of the working process that ensure maximum efficiency of the power plant - aircraft system. In the article presents the problem of optimization and selection of rational parameters of the working process of turbofan engines with heat recovery. Based on the developed method of multi-criteria optimization by means of numerical simulations conducted and presents the results of optimization of parameters of working process of turbofan engines with heat recovery system in passenger aircraft of type Airbus A310 on criteria such as the total mass of the power plant and fuel required for the flight, and the specific fuel consumption of the aircraft on a ton per kilometer. The developed mathematical model for calculating the mass of a compact heat exchanger, designed to solve optimization problems at the stage of conceptual design of the engine, is presented. The developed methods and models are implemented in the ASTRA program.

Elena Shchepakina
Order reduction for critical travelling waves problems
The paper deals with the order reduction for critical travelling waves problems. The specificity of such travelling waves is that they separate waves with qualitatively different behaviors. We show how the application of the geometric theory of singular perturbations allows us to reduce the problem of finding the travelling wave of the original PDE system to the analysis the projection of the system onto its slow invariant manifold. We illustrate this approach to the problem of finding the point-to-periodic travelling wave for the reaction-diffusion model.