and Computational Methods

doi: 10.18698/2309-3684-2024-2-8599

In this paper, optimization of the control of the flight of a small spacecraft (spacecraft) on ion engines to the orbit of Venus is considered, taking into account the attraction of the Earth and the time of departure from the geostationary orbit. When solving the problem, the following assumptions were made: the orbits of the planets are circular, lying in the same plane. A detailed consideration of the influence of Venus when approaching the orbit of the planet was not considered. The problem is solved using the Pontryagin maximum principle by numerical targeting method. The spacecraft motion simulation was divided into 3 stages: acceleration of the spacecraft to a speed that allows overcoming the Earth's attraction with the help of short-term operation of the jet engine, optimization of control near the Earth at a distance of the spacecraft to the Earth of no more than 950 000 km and for the main interorbital flight between planets. The algorithm for solving the problem is implemented in the C++ programming language. Optimal control of the angle of action of the thrust vector is obtained. The analysis of the obtained results showed that, while minimizing the time to reach the orbit of Venus, in addition to significantly influencing the efficiency criterion of the longest interorbital section of the flight, the moment of the start (departure from Earth orbit) is fundamentally important.

Мозжорина Т.Ю., Закуражная А.А. Моделирование влияния времени схода с орбиты Земли на оптимальное управление перелетом малоразмерного КА на Венеру. Математическое моделирование и численные методы, 2024, № 2, с. 88–99.

doi: 10.18698/2309-3684-2021-3-7487

In this paper, the optimization of the transfer of a low–mass satellite from the Earth's orbit to the Mars orbit under a solar sail is considered. Optimization of the control of the pitch angle of the solar sail is carried out using the Pontryagin maximum principle while minimizing the flight time. In contrast to previous works on this topic, the solution of the boundary value problem, to the solution of which the maximum principle is reduced, was obtained by the false position method. The calculation program is written in the C++ programming language. Despite the computational difficulties arising when using the false position method, it was possible to achieve good convergence of the Newton method underlying the algorithm. The analysis of the accuracy of the results obtained is carried out and the possibility of using the false position method in solving such problems is shown. A comparison is made with the data of previously published works. Despite some assumptions used in the development of the calculation algorithm, the work has its value in terms of assessing the possibility of using the false position method, which gives the most accurate numerical optimization results.

Мозжорина Т.Ю., Рахманкулов Д.А. Моделирование и оптимизация управлением спутника малой массы при перелете с орбиты Земли на орбиту Марса под солнечным парусом. Математическое моделирование и численные методы, 2021, № 3, с. 74–87.

doi: 10.18698/2309-3684-2017-4-316

The article presents a mathematical model of the alloys nanoparticles structure rearrangement dynamics after the instantaneous thermal influence (heating or cooling). The model is based on using the method of molecular dynamics of multicomponent alloys with the Lennard-Jones and Morse interatomic potentials as well as the initial conditions of momentary expansion or compression of the alloy nanoparticle regular crystalline structure. We computationally investigate the regularities of rearranging the initially regular atomic structure of a nanoparticle over time. It is shown that depending on the number of atoms in a nanoparticle various finite settled forms of the alloys nanoparticle are possible, both amorphous and new crystalline structures different from the alloy original crystalline nanostructure. We provide numerical results for the titanium nanoparticles and the titanium-nickel alloy (nitinol).

Krasnov I.K., Mozzhorina T.Yu., Balanin A.N. Numerical modeling of alloys nanostructure rearrangement by means of molecular dynamics methods. Маthematical Modeling and Computational Methods, 2017, №4 (16), pp. 3-16

doi: 10.18698/2309-3684-2017-2-94106

We conducted a numerical experiment in applying the shooting method to solve problems of optimal control with switching. We used the problem of lunar soft-landing to test an algorithm that ensures convergence of Newton's method in problems of this type. We analysed the accuracy of our computations.

Mozzhorina T. Yu. Numerical solution to problems of optimal control with switching by means of the shooting method. Маthematical Modeling and Coтputational Methods, 2017, №2 (14), pp. 94-106

doi: 10.18698/2309-3684-2017-3-7182

The article considers the application of the direct statistical simulation method to the problems of gas dynamics in a rarefied region. An analytical method for assignment and taking into account complex boundary conditions associated with the geometry of the body located in the computational domain is proposed. An algorithm for the rational description of a body streamlined by a gas is developed.

Krasnov I.K.,Mozzhorina T.Yu., Dzhus D.V. Numerical statistical simulation of the process of rarefied gas flow over an aircraft .Маthematical Modeling and Computational Methods, 2017, №3 (15), pp. 71–82

doi: 10.18698/2309-3684-2021-2-5467

In this paper, optimization of the transfer of a low–mass satellite from Earth orbit to Mars orbit using ion thrusters is considered. The ion engine allows you to minimize fuel consumption and accelerate the spacecraft to fairly high speeds far from the planets of the solar system. The heliocentric section of the flight is subject to consideration. The task is to minimize the flight time. The following assumptions are made in the work: the orbits of the Earth and Mars are circular and lying in the same plane. The angle between the tangential velocity of the spacecraft in the heliocentric system and the direction of thrust action is selected as a control. When compiling the optimization algorithm, the Pontryagin maximum principle was used, which leads the optimization problem of a functional to a boundary value problem for a system of ordinary differential equations. The solution to the boundary value problem was found by one of the numerical methods — the false position method, which gives the most accurate results. The analysis of the results obtained is carried out and a comparison with the data obtained earlier in similar calculations by foreign authors by another numerical solution method is carried out. The conclusion is made about the efficiency of the false position method when solving such problems.

Мозжорина Т.Ю., Чуванова Л.О. Моделирование и оптимизация перелета спутников малой массы с Земной орбиты на орбиту Марса с помощью ионных двигателей. Математическое моделирование и численные методы, 2021, № 2, с. 54–67.

doi: 10.18698/2309-3684-2014-2-327

The paper considers two-dimensional boundary value problem of propagation of plane electromagnetic wave through a periodic stratified medium with one-dimensional photonic crystal structure. The structure contains a finite number of slabs. Each periodicity cell consists of two layers with different real values of constant dielectric permittivity and different thicknesses. We show that under certain additional condition, which connects the angle of incidence of the plane wave, thicknesses of the layers, frequencies and dielectric permittivity of the layers, we can solve the problem completely and explicitly, the solution leading to simple expressions for both the field reflected from the structure, and the field which has passed through it. Herewith in case of H-polarized field, unlike E-polarization, properties of this medium depend on the ratio of thickness of the layers multiplied by their dielectric permittivity (with E-polarization they depend on thickness ratio only). As a result, depending on the field frequency, photonic crystal can behave as perfectly reflecting structure, while with the same ratio of thicknesses of the layers in case of E-polarization, it becomes a wave guiding structure, and vice-versa. We have compared numerical computations with those for cases of E-polarization.

Apeltsin V., Mozzhorina T. Properties of one-dimensional photonic crystal as a reflective or wave guiding structure when excited by H-polarization. Маthematical Modeling and Coтputational Methods, 2014, №2 (2), pp. 3-27

doi: 10.18698/2309-3684-2014-3-7488

We examined effects of typical for different climatic zones atmospheric conditions on flight program optimization for a subsonic long-haul passenger aircraft. Simulation of flight and power plant performance was based on current traditional approaches used in solving problems of this kind. The acceleration-climb flight segment has been optimized by minimizing fuel consumption at this flight segment. The cruising flight segment has been optimized considering operating limitations accepted for civil aviation. The in-built model of bypass turbojet engine was used for simulating the flight. This model allows calculating power plant performances under any flight conditions. The flight of subsonic aircraft has been examined in one vertical plane. Calculations have been performed for 6 standard air temperature variations with altitude (depending on climatic zone). Atmospheric pressure variation near Earth surface was considered and effects of atmospheric conditions on flight program optimization were estimated.

Mozzhorina T., Gubareva E. Simulating atmospheric conditions influence on flight program optimization for a subsonic passenger aircraft. Маthematical Modeling and Coтputational Methods, 2014, №3 (3), pp. 74-88

doi: 10.18698/2309-3684-2022-2-88101

In this paper, optimization of the flight of a low-mass satellite from Earth orbit to the orbit of Venus using ion engines is considered. The first flight to the planet took place in 1961 by the Soviet automatic interplanetary station "Venus-1", which passed 100,000 kilometers from Venus. In addition, in 1962, the American station "Mariner-2" was flown. The most recent spacecraft launched to the planet was the European Space Agency's Venus Express in 2005, which flew to Venus in 153 days. When solving the current problem, the following assumptions were made: an interorbital flight is considered without taking into account the attraction of the planets, and the orbits of the planets are considered circular and lying in the same plane. The angle between the tangential velocity of the spacecraft and the thrust direction was chosen as the control. Optimization of satellite control was carried out using the Pontryagin maximum principle. The resulting boundary value problem for a system of ordinary differential equations was solved by a numerical method — the targeting method. Newton's method was used to solve systems of nonlinear algebraic equations. The calculation program was written using the C++ programming language. As a result of the work, it was possible to minimize the flight time between orbits, thus the operability of the shooting method for solving optimization problems was shown.

Мозжорина Т.Ю., Закуражная Д.А. Моделирование и оптимизация управления полетом космического аппарата с орбиты Земли на орбиту Венеры с помощью ионных двигателей. Математическое моделирование и численные методы, 2022, № 2, с. 90–103

doi: 10.18698/2309-3684-2020-2-8194

In this paper, we consider one of the possible feedback algorithms for the vertical landing of the returned first stage of the spacecraft for its reuse in the future. It is proposed to use for thrust correction not correction engines, but the main engine of the power plant of the spacecraft, which can be throttled to 60% of the maximum thrust value. A numerical experiment using the Monte Carlo method is performed to evaluate the performance of this algorithm. A soft landing is a landing with a speed of zero or no more than a few meters per second. The last section of the vertical landing is subject to investigation. The optimal program control in this problem statement in terms of minimum fuel consumption is free fall, then turning the engine on at full power until the moment of landing. It is assumed that such parameters as the speed and mass of the spacecraft's return module at an altitude of 2000 m, the specific impulse, as well as the air density and the coefficient of aerodynamic drag can be randomly deviated from the calculated values. It is assumed that these random variables are distributed according to the normal law, are independent, and their deviations from the calculated values do not exceed 1% for the engine pulse and 5% for all other variables. The landing speed is a random value for which the distribution parameters are calculated. The results of the calculation are analyzed.

Мозжорина Т.Ю., Осипов В.В. Моделирование и синтез оптимального управления вертикальной посадкой возвращаемых космических модулей. Математическое моделирование и численные методы. 2020. № 2. с. 81–94.