Rubric: "1.2.2. Mathematical modeling, numerical methods and software packages (technical sciences)"
Bushuev A. Y. (Bauman Moscow State Technical University), Reznikov A. O. (Bauman Moscow State Technical University)
doi: 10.18698/2309-3684-2021-3-6273
A model of genetic algorithm with binary coding with independent Schaeffer selection is constructed, which allows one to search for a global optimum by several criteria without their scalarization. The calculations take into account the range of all possible motions of actuators under uncertain external influences in some predetermined range. An algorithm has been developed that allows storing intermediate results to eliminate the problem of a large number of repeated calculations in the course of the evolutionary algorithm, which reduced the computation time. The effectiveness of the optimization algorithm is demonstrated on the example of solving a model problem.
Бушуев А.Ю., Резников А.О. Применение генетического алгоритма в задаче моделирования и оптимизации пневмогидравлической системы синхронизации исполнительных органов. Математическое моделирование и численные методы, 2021, № 3, с. 62–73.
521.19 The perturbation hollow spheres modelling for the gravity assists in the Solar system
Borovin G. K. (Keldysh Institute of Applied Mathematics of the Russian Academy of Scienсes), Golubev Y. F. (Keldysh Institute of Applied Mathematics of the Russian Academy of Scienсes), Grushevskii A. V. (Keldysh Institute of Applied Mathematics of the Russian Academy of Scienсes), Tuchin2 A. G. (Keldysh Institute of Applied Mathematics of the Russian Academy of Scienсes)
doi: 10.18698/2309-3684-2023-4-6473
One of the types of gravitational scattering in the Solar System within the framework of the circular restricted three-body problem (CR3BP) are the gravity assist maneuvers of "particles of insignificant mass" (spacecraft, asteroids, comets, etc.). For their description, a physical analogy with the scattering of beams of charged alpha-particles in the Coulomb field is useful. However, unlike the scattering of charged particles, there are external restrictions on the ability to perform gravity assists associated with the limited size of the spheres of influence of the planet. At the same time, internal limitations on the possibility of performing gravity assists are known from the literature on CR3BP, estimated by the effective radii of planets (including gravitational capture by a planet falling into it). They depend on the asymptotic velocity of the particle relative to the planet. For obvious reasons, their influence makes it impossible to effectively use gravity assist maneuvers. The paper presents generalized estimates of the sizes of near-planetary regions (flat "perturbation rings" or "perturbation hollow spheres" rotating synchronously with a small body in the three-dimensional case), falling into which is a necessary condition for the implementation of gravity assists. A detailed analysis shows that Neptune and Saturn have characteristic of perturbation hollow spheres of the largest size in the Solar System, and Jupiter occupies only the fourth place in this list
Боровин Г.К., Голубев Ю.Ф., Грушевский А.В., Тучин А.Г. Моделирование пертурбационных оболочек для гравитационных маневров в Солнечной системе.Математическое моделирование и численные методы, 2023, № 4, с. 64–73.
519.6 Numerical solution of equations of mixed type in unlimited region on a plane
Galanin M. P. (Bauman Moscow State Technical University/Keldysh Institute of Applied Mathematics of the Russian Academy of Scienсes), Ukhova A. R. (Bauman Moscow State Technical University)
doi: 10.18698/2309-3684-2023-3-105124
The purpose of the work is to build and implement an algorithm for finding a numerical solution to a problem for mixed-type equations in an unlimited region. In this case problems are considered in which the process under study is described in some limited area by the thermal conductivity equation or wave equation, and outside it by the Laplace equation. The necessary additional conditions at zero, at infinity and the conditions for conjunction at the border of the inner region are set. There is described an algorithm for finding a numerical solution to a problem with a wave equation in a limited region in one-dimensional and two-dimensional cases, problems with a thermal conductivity equation or a wave equation in a two-dimensional case. Difference schemes are built by the integro–interpolation method. The task is solved in a limited area. Nonlocal boundary conditions are set on its border so the solution of task in limited area coincides with projection of problem in unlimited area. In this case, an artificial boundary is introduced for the solution in the part of the region in which the process is described by the Laplace equation. An iterative algorithm and an algorithm with a non-local boundary condition are built. The results of calculations for examples in various fields are presented.
Галанин М.П., Ухова А.Р. Численное решение уравнений смешанного типа в неограниченной области на плоскости. Математическое моделирование и численные методы, 2023, № 3, с. 105–124.
519.6 Agent model of two competing populations taking into account their structurality
Belotelov N. V. (Bauman Moscow State Technical University/Institution of Russian Academy of Sciences Dorodnicyn Computing Centre of RAS), Brovko A. V. (Bauman Moscow State Technical University)
doi: 10.18698/2309-3684-2022-3-7183
The article describes an agent simulation model of two populations competing for one resource. In the model, it is assumed that an individual dies if its mass-energy becomes non-positive. It is assumed that individuals of each of the populations under consideration can form flocks, this allows populations to increase their competitiveness. In the model, this is formalized through the ability to organize networks connecting individuals of the same species. At the same time, individuals can form only a certain number of connections with neighbors. The concept of "valence" is introduced in the model to describe this. It is assumed that within each network there is an instantaneous redistribution of the resource available to all members of the network by each member of the pack. In addition to the model, the article describes the structure of the program with which simulation experiments were carried out. As a result of the simulation experiments, the following was obtained. If the resource is highly productive, then in the process of competitive interaction, the population wins, the agents of which have a large "valence". And in the case of a low-productive resource, individuals of a population with a lower "valence" win in competitive interaction. This is due to the fact that more complex structures require more energy to maintain the flock.
Белотелов Н.В., Бровко А.В. Агентная модель двух конкурирующих популяций с учетом их структурности. Математическое моделирование и численные методы, 2022, № 3, с. 71–83.
Khayitkulov B. K. (National University of Uzbekistan)
doi: 10.18698/2309-3684-2023-1-3242
This work is devoted to the numerical solution of the non-stationary problem of optimal placement of heat sources of minimum power. The statement of the problem requires the simultaneous fulfillment of two conditions. The first condition is to ensure that the temperature is within the limits of minimum and maximum temperatures due to the optimal placement of heat sources with a minimum power in the parallelepiped. The second condition is that the total power of the heat sources used for heating is minimal. This problem was studied under stationary conditions in the works of other scientists. However, the problem was not considered in the non-stationary case. Since it is difficult to find a continuous solution to the boundary value problem, we are looking for a numerical solution to the problem. It is difficult to find an integral operator with a continuous kernel (Green's function). The numerical value of the Green's function is found in the form of a matrix. A new algorithm for the numerical solution of a non-stationary optimal control problem for the placement of heat sources with a minimum power in processes described by parabolic partial differential equations is proposed. A new technique for numerical solution is proposed. A mathematical and numerical model of the processes described by the convection-diffusion equation given for the first boundary value problem is constructed. The boundary value problem is studied for the three-dimensional case. An implicit finite difference scheme was used to solve the problem numerically. According to this scheme, a system of difference equations was created. The formed system of difference equations is reduced to a linear programming problem. The problem of linear programming is solved using the M-method. For each time value, a linear programming problem is solved. A new approach to the numerical solution of problems is proposed. A general block diagram of the algorithm for solving the non-stationary problem of optimal control of the placement of heat sources with a minimum power is given. An algorithm and software for the numerical solution of the problem have been developed. A brief description of the software is given. On specific examples, it is shown that the numerical solution of the boundary value problem is within the specified limits, the sum of optimally placed heat sources with a minimum power gives a minimum to the functional. The results of the computational experiment are visualized
Хайиткулов Б.Х. Математическое моделирование нестационарной задачи конвекции–диффузии об оптимальном выборе местоположения источников тепла. Математическое моделирование и численные методы, 2023, No 1, с. 32–42.
Valishin A. A. (Bauman Moscow State Technical University), Zaprivoda A. V. (Bauman Moscow State Technical University), Klonov A. S. (Bauman Moscow State Technical University)
doi: 10.18698/2309-3684-2024-1-93109
With the development of forecasting methods, the exclusion of random effects from the initial information and the studied processes becomes essential. These effects are associated not only with the impossibility of taking into account all factors, but also with the fact that some of them are often not taken into account at all. It is important not to forget about random measurement errors. In the predicted values, due to these effects, a kind of random offset or "noise" is created. Filtering (exclusion) of noise should, of course, increase the reliability and justifiability of forecasts. This article discusses the principles of real-time data filtering. The problem statement is given, as well as the main evaluation criteria that must be met to obtain a satisfactory result. In addition, the principle of operation of the two most common types of filters – absolutely optimal and conditionally optimal - is analyzed, their advantages and disadvantages are described. The application of Kalman and Pugachev filters to a model with two sensors is considered. Some conclusions and recommendations are presented on in which cases it is better to use one or another filter.
Валишин А.А., Запривода А.В., Клонов А.С. Математическое моделирование и сравнительный анализ численных методов решения задачи непрерывнодискретной фильтрации случайных процессов в реальном времени. Математическое моделирование и численные методы, 2024, № 1, с. 93–109.
Andrianov I. K. (Комсомольский-на-Амуре государственный технический университет), Chepurnova E. K. (Комсомольский-на-Амуре государственный технический университет)
doi: 10.18698/2309-3684-2024-2-316
The study considers the problem of optimizing the crack detection system of gas turbine blades. The shell of the capsule of the damage detection system, which is under the influence of internal pressure, is considered as an object of research. The task of the study was devoted to the mathematical modeling of optimal pressure in capsules of the damage detection system. As part of solving the research problem, a mathematical formulation of the problem of optimizing the nonlinear pressure function was carried out in the presence of restrictions on variable parameters: wall thickness and outer diameter of the cylindrical capsule shell. The construction of the optimization objective function was carried out on the basis of the equilibrium condition of the shell element in the area of crack opening of the turbine blade, the limit state criterion using the Tresk-Saint-Venant strength theory. The research methodology was based on the approximate decomposition of the function into a Taylor series, the Lagrange multiplier method, and the Kuhn-Tucker theorem. When solving the problem of conditional optimization, the cases of violation of the regularity conditions of the limiting functions are analyzed. According to the calculation results, the minimum value of the required pressure for the destruction of the capsule shell in case of opening of the crack banks of the turbine blade is achieved at the maximum value of the outer diameter of the shell and the minimum thickness of its wall. According to the test calculation data, the area of acceptable solutions to the optimization problem is graphically presented, and the lines of the level of the target function of pressure optimization are shown. The constructed mathematical model and calculation algorithm will automate the process of calculating the required pressure in the capsules of the turbine blade crack detection system and obtain an estimate of the minimum pressure value in the presence of restrictions on the absolute and relative values of the capsule shell wall thickness, the outer diameter of the capsule.
Андрианов И.К., Чепурнова Е.К. Математическая модель условной оптимизации давления в системе обнаружения трещин лопаток газовых турбин. Математическое моделирование и численные методы, 2024, № 2, с. 3–16.
Chuev V. U. (Bauman Moscow State Technical University), Dubogray I. V. (Bauman Moscow State Technical University)
doi: 10.18698/2309-3684-2022-2-102113
Using the method of dynamics of averages, a "mixed" model of numerous groupings confrontation has been developed with linear dependencies on the time of effective speeds of striking by units of the parties. An algorithm is constructed that allows to investigate the course of the process and calculate its main indicators. It is established that the use of confrontation models with constant effective speed of strikes in many cases leads to significant errors in the calculation of the main indicators of the process. The influence of a preemptive strike by one of the opposing sides on the course of the progress and the final outcome are investigated.
Чуев В.Ю., Дубограй И.В. «Смешанная» модель противоборства многочисленных группировок при линейных зависимостях от времени эффективных скоростей воздействий единицами сторон. Математическое моделирование и численные методы, 2022, № 2, с. 104–115
004.855.5 Neural network methods for solving the problem of credit scoring
Kadiev A. D. (Bauman Moscow State Technical University), Chibisova A. V. (Bauman Moscow State Technical University)
doi: 10.18698/2309-3684-2022-4-8192
The mathematical derivation of the presented neural network model is demonstrated. Reduction of the classification problem to an optimization problem. Produced recon-naissance data analysis, as well as their preprocessing for further use in training classification algorithms. The architectures of neural networks were designed depending on the activation function, the number of hidden layers of the neural network and the number of neurons in the hidden layers. More than ten neural networks were trained to solve the task of credit scoring. The calculation of the learning time of neural networks was made. The solution of the problem using classical machine learning algorithms is presented. It could be seen that the standard deviation of accuracy and ROC AUC for neural networks is greater than that of a random forest. This is due to the fact that we choose the initial weights randomly and calculate the gradients not on the entire sample, but on small parts, which adds some learning error. But these deviations were not only for the worse. In the best situations, according to both metrics, neural networks showed the worst result by a couple of percent. The analysis of results is made. Comparative analysis shows that neural networks have better classification quality than classical machine learning algorithms, and also that neural networks have less training time than classical machine learning algorithms. Graphs and tables displaying the results obtained are presented.
Кадиев А.Д., Чибисова А.В. Нейросетевые методы решения задачи кредитного скоринга. Математическое моделирование и численные методы, 2022, № 4, с. 81–92.