Вестник Южно-Уральского государственного университета. Серия: Математическое моделирование и программирование @vestnik-susu-mmp
Статьи журнала - Вестник Южно-Уральского государственного университета. Серия: Математическое моделирование и программирование
Все статьи: 806
Neural net decoders for linear block codes
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The work is devoted to neural network decoders of linear block codes. Analytical methods for calculating synaptic weights based on a generator and parity-check matrices are considered. It is shown that to build a neural net decoder based on a parity-check matrix was sufficiently four layers feedforward neural net. The activation functions and weight matrices for each layer are determined, as well as the number of weights for the neural net decoder. An example of error correction with uses of the BCH neural net decoder is considered. As a special case of a neural network decoder built on the basis of a parity-check matrix, a model for decoding Hamming codes has been proposed. This is the two-layer feedforward neural net for with a neuron number equal to the length of the codeword and a number of weight coefficients equal to the square of the codeword length. The graphs of the number of a synaptic weight of neural net decoders based on the generator and parity-check matrices, on the number of bits and the number of corrected errors, are shown.
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Статья научная
In this paper, we present two new P-type and D-type iterative learning control (ILC) update laws for linear stochastic systems with random data dropout modeled with a Bernoulli random variable. We prove that the P-type and D-type ILC update laws converge to the desired input in the almost sure sense. We show that the convergence conditions of the inputs corresponding to the P-type and D-type ILC update laws for networked control systems are the same. We present the performance comparison of the P-type and D-type ILC update laws. In this comparison, we conclude that the P-type ILC update law is more effective than the D-type ILC update law for networked control systems.
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New approximate method for solving the Stokes problem in a domain with corner singularity
Статья научная
In this paper we introduce the notion of an Rv-generalized solution to the Stokes problem with singularity in a two-dimensional non-convex polygonal domain with one reentrant corner on its boundary in special weight sets. We construct a new approximate solution of the problem produced by weighted finite element method. An iterative process for solving the resulting system of linear algebraic equations with a block preconditioning of its matrix is proposed on the basis of the incomplete Uzawa algorithm and the generalized minimal residual method. Results of numerical experiments have shown that the convergence rate of the approximate Rv-generalized solution to an exact one is independent of the size of the reentrant corner on the boundary of the domain and equals to the first degree of the grid size h in the norm of the weight space W12,v(Ω) for the velocity field components in contrast to the approximate solution produced by classical finite element or finite difference schemes convergence to a generalized one no faster than at an O(hα) rate in the norm of the space W12(Ω) for the velocity field components, where α
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New features of parallel implementation of N-body problems on GPU
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This paper focuses on the parallel implementation of a direct N-body method (particle-particle algorithm) and the application of multiple GPUs for galactic dynamics simulations. Application of a hybrid OpenMP-CUDA technology is considered for models with a number of particles N:105-107. By means of N-body simulations of gravitationally unstable stellar galactic we have investigated the algorithms parallelization efficiency for various Nvidia Tesla graphics processors (K20, K40, K80). Particular attention was paid to the parallel performance of simulations and accuracy of the numerical solution by comparing single and double floating-point precisions (SP and DP). We showed that the double-precision simulations are slower by a factor of 1,7 than the single-precision runs performed on Nvidia Tesla K-Series processors. We also claim that application of the single-precision operations leads to incorrect result in the evolution of the non-axisymmetric gravitating N-body systems. In particular, it leads to significant quantitative and even qualitative distortions in the galactic disk evolution. For instance, after 104 integration time steps for the single-precision numbers the total energy, momentum, and angular momentum of a system with N=220 conserve with accuracy of 10-3, 10-2 and 10-3 respectively, in comparison to the double-precision simulations these values are 10-5, 10-15 and 10-13, respectively. Our estimations evidence in favour of usage of the second-order accuracy schemes with double-precision numbers since it is more efficient than in the fourth-order schemes with single-precision numbers.
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Статья научная
In this paper, we prove some new results on operational second order differential equations of elliptic type with general Robin boundary conditions in a non-commutative framework. The study is performed when the second member belongs to a Sobolev space. Existence, uniqueness and optimal regularity of the classical solution are proved using interpolation theory and results on the class of operators with bounded imaginary powers. We also give an example to which our theory applies. This paper improves naturally the ones studied in the commutative case by M. Cheggag, A. Favini, R. Labbas, S. Maingot and A. Medeghri: in fact, introducing some operational commutator, we generalize the representation formula of the solution given in the commutative case and prove that this representation has the desired regularity.
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Non-Uniqueness of Solutions to Boundary Value Problems with Wentzell Condition
Статья научная
Recently, in the mathematical literature, theWentzell boundary condition is considered from two points of view. In the first case, let us call it classical one, this condition is an equation containing a linear combination of the values of the function and its derivatives on the boundary of the domain. Moreover, the function itself also satisfies the equation with an elliptic operator defined in the domain. In the second case, which we call neoclassical one, the Wentzell condition is an equation with the Laplace–Beltrami operator defined on the boundary of the domain understood as a smooth compact Riemannian manifold without boundary, and the external action is represented by the normal derivative of a function defined in the domain. The paper shows the non-uniqueness of solutions to boundary value problems with the Wentzell condition in the neoclassical sense both for the equation with the Laplacian and for the equation with the Bi-Laplacian given in the domain.
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Статья научная
Degenerate differential equations, as part of the differential-algebraic equations, the last few decades cause increasing interest among researchers, both because of the attractiveness of the considered theoretical questions, and by virtue of their applications. Currently, advanced methods developed in this area are used for system modelling and analysis of electrical and electronic circuits, chemical reaction simulations, optimization theory and automatic control, and many other areas. In this paper, the theory of normal forms of differential equations, originated in the works of Poincare and recently developed in the works of Arnold and his school, adapted to the simplest case of a degenerate differential equations. For this purpose we are using technique of Jordan chains, which was widely used in various problems of bifurcation theory. We study the normal forms of degenerate differential equations in the case of the existence of the maximal Jordan chain. Two and three dimensional spaces are studied in detail. Normal forms are the simplest representatives of the degenerate differential equations, which are equivalent to more complex ones. Therefore, normal forms should be considered as a model type of degenerate differential equations.
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Статья научная
The process of oil reservoir development in the elastic-water-drive mode is considered. It is assumed that the displacement of oil by the edge water occurs completely and a clear boundary between two liquids is formed in the reservoir, which moves according to a previously unknown law. Within the framework of a one-dimensional model of the elastic-water-drive development regime, the task is set to identify the main hydrodynamic parameters of the reservoir, i.e. the pressure at the interface between liquids, the pressure distribution in the reservoir and the position of the interface between liquids, only on the basis of information obtained from the gallery of production wells. The problem set belongs to the class of boundary inverse problems. By applying the methods of front straightening and difference approximation, the problem is reduced to solving a system of difference equations. A special representation is proposed to solve the system of difference equations, having previously written it down as a variational problem with local regularization. As a result, an explicit formula is obtained for determining the approximate value of the pressure at the interface of liquids and recurrent formulas for determining the distribution of pressure and the position of the interface of liquids in the reservoir at each time layer. Based on the proposed computational algorithm, numerical experiments were carried out for a model oil reservoir.
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Numerical Study of Swirling Jet Streams Based on Modern Turbulence Models
Статья научная
This article examines a swirling jet based on a two-fluid turbulence model. This task, despite its simplicity, is quite a difficult task for many turbulence models. Because anisotropic turbulence is observed in swirling flows. Therefore, many modern RANS models are not able to describe such flows even qualitatively. The two-fluid model used in this work has been developed recently. Pioneering work shows that the basis for constructing this model is the possibility of representing a turbulent flow as a heterogeneous mixture of two liquids. Approach was proposed by Spaulding. The idea of the approach is to represent turbulence as the interpenetrating motion of two fluids, with the pulsating nature of the turbulent flow being caused by the relative moverment of them. For each fluid, it's own equation of motion is written, which leads to a closed system of equations. These studies also show that the developed two-fluid model is able to adequately describe complex anisotropic turbulence. To numerically implement the equations of a turbulent axisymmetric swirling jet, a uniform staggered calculation grid and a control volume method were used, and velocity correction was carried out using a simple method. The numerical results obtained are compared with experimental data from the ERCOFTAC database. It is shown that the results of the two-fluid model, despite the use of a rather rough computational grid, are in satisfactory agreement with experimental data.
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Numerical algorithm and computational experiments for one linear stochastic Hoff model
Статья научная
Investigated is a model of deformation in a structure composed of I-beams with random external effect; it is based on stochastic Hoff equations with an initial-final condition. The article describes an algorithm for a numerical solution of the initial-final problem for stochastic Hoff equations; the algorithm is based on the Galerkin method. Provided is a numerical investigation algorithm providing for numerical solutions for both degenerate and non-degenerate equations. The main theoretical results that enabled this numerical investigation are the methods of the theory of degenerate groups of operators and of the theory of the Sobolev type equations. The algorithms are represented by schemes enabling building flowcharts of programs for computational experiments. Results of computational experiments. In addition, numerical investigation of the stochastic model involves further obtaining and processing the results of experiments at various values of a random variable, including those related to rare events.
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Numerical analysis of fractional order integral dynamical models with piecewise continuous kernels
Статья научная
Volterra integral equations find their application in many areas, including mathematical physics, control theory, mechanics, electrical engineering, and in various industries. In particular, dynamic Volterra models with discontinuous kernels are effectively used in power engineering to determine the operating modes of energy storage devices, as well as to solve the problem of load balancing. This article proposes the numerical scheme for solution of the fractional order linear Volterra integral equations of the first kind with piecewise continuous kernels. The developed approach is based on a polynomial collocation method and effectively approximate such a weakly singular integrals. The efficiency of proposed numerical scheme is illustrated by two examples.
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Numerical investigation of the Boussinesq - Love mathematical models on geometrical graphs
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The article is devoted to the numerical investigation of the Boussinesq - Love mathematical models on geometrical graphs representing constructions made of thin elastic rods. The first paragraph describes the developed algorithm for numerical solution of the Boussinesq - Love equation with initial conditions and boundary conditions in the vertices. The block diagram of the algorithm is given and described. The result of computation experiment is given in the second paragraph.
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The process of unsteady flow of incompressible viscoelastic fluid in a cylindrical tube of constant cross-section is considered. To describe the rheological properties of a viscoelastic fluid, the Kelvin-Voigt model is used and the mathematical model of this process is presented as an integro-differential partial differential equation. Within the framework of this model, the problem is to determine the pressure drop along the length of the pipe, which ensures the passage of a given flow rate of viscoelastic fluid through the pipe. This problem belongs to the class of inverse problems related to the recovery of the right parts of integro-differential equations. By replacing variables, the integro-differential equation is transformed into a third-order partial differential equation. First, a discrete analog of the problem is constructed using finite-difference approximations. To solve the resulting difference problem, we propose a special representation that allows splitting the problems into two mutually independent second-order difference problems. As a result, an explicit formula is obtained for determining the approximate value of the pressure drop along the length of the pipeline for each discrete value of the time variable. Based on the proposed computational algorithm, numerical experiments were performed for model problems.
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Numerical modelling of the dynamics of the galactic halos in the colliding galaxies
Статья научная
Based on parallel three-dimensional simulation of N-body and gas self-consistent dynamics, we study the behavior of hot coronal gas in the colliding galaxies with "live'' dark matter halos. We model a few scenarios of the galactic collisions including "bull-eye'' and non-central ones, and use different values of the initial velocities of the colliding galaxies. Taking into account the self-gravity, we demonstrate that the collision of gaseous and stellar components does not lead to the formation of a gaseous "protogalaxy'' observed in some numerical simulations. Also, we show that about sixty percent of hot halo gas is expelled into intergalactic space during the collision. Numerical simulations show that a considerable amount of gas (up to 70% for a bull-eye collisions) exchanges between two colliding galaxies.
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Numerical research of the Barenblatt - Zheltov - Kochina stochastic model
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At present, investigations of Sobolev-type models are actively developing. In the solution of applied problems the results allowing to get their numerical solutions are very significant. In the article the algorithm for numerical solving of the initial boundary value problem is developed. The problem describes the pressure distribution of the homogeneous fluid in the horizontal layer in the circle. The layer is opened by a vertical well of a small radius. In our research we suppose that random disturbing loads have an influence on the fluid. The problem was solved under two assumptions. Firstly, we suppose that an unstable fluid flow is axially symmetric, and secondly, that in initial moment the pressure in the layer is constant. After the process of the discretization we modify the original model to the Cauchy problem for the system of ordinary differential equations. For the numerical solution we use algorithms based on explicit one-step formulas of the Runge - Kutta type with the seventh-order accuracy and with the selection of the integration step. We also use the scheme of the eighth-order accuracy to evaluate the calculation accuracy on each steps of time. According to the results of this control, we choose the time-step. A lot of numerical experiments have shown high numerical efficiency of the algorithm that we use to solve the investigated initial-boundary problem.
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Numerical research of the mathematical model for traffic flow
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The problems of distribution of transport flows are currently relevant in connection with the increase in vehicles. In the 50s of the last century, the first macroscopic (hydrodynamic) models appeared, where the transport flow resembles the flow "motivated" compressible liquid. The scientific approach based on the Navier - Stokes system. The main idea of the scholars is consideration the hydrodynamic models on the grounds of interrelation between the transport flow and incompressible fluid. For modelling traffic flows we examine Oskolkov equation on the geometric graph, where the edge has two positive values corresponding to it "length" and "width". Certainly, in the context of mathematical model the values lk and bk are dimensionless, but for clarity it is convenient to imagine that lk is measured in linear metric units, for example, kilometers or miles, and bk is equal to the number of traffic lanes on the roadway in one direction. In terms of the Oskolkov model, we obtained a non-classical multipoint initial-final value condition. We will study such a model using the idea and methods of the Sobolean equation theory. These notes describe a numerical experiment based on the Galerkin method for the Oskolkov equation with a multipoint initial-final condition on the graph.
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Статья научная
The existing mathematical models of pressure swing adsorption (PSA) apply various assumptions regarding the mass and heat transfer mechanisms in the "gas mixture-adsorbent'' system. An increase in the number of assumptions leads to a simplification of the model, a decrease in the calculation time of one iteration in the model and, at the same time, a decrease in its accuracy. The simplification of the model is especially important in PSA processes, since the calculation of the model is carried out before the cyclic steady state and takes tens and even hundreds of cycles (iterations). Ensuring high accuracy of the PSA model and its minimum complexity is a contradictory requirement; therefore it is important to reasonably consider only those transfer mechanisms that are dominant in the model. The paper proposes a mathematical model of the PSA process, which takes into account the thermal effects of sorption, external and internal diffusion mechanisms of adsorptive transfer. A numerical research was carried out to determine the dominant transfer mechanism, and recommendations were proposed for using the preferred PSA model in terms of its accuracy and calculation time (for the processes of air oxygen enrichment and synthesis gas separation). It was found that to calculate PSA oxygen units with a capacity of less than 4 l/min at NTP, it is advisable to use an isothermal model, which saves at least 24,3% of the calculation time with a loss of accuracy of no more than 0,084 vol%. To calculate PSA hydrogen units, the use of an isothermal model is impractical even at the lowest productivity of 50 l/min at NTP. When the diameter of the adsorbent particles is less than 2 mm, it is advisable to use an external diffusion model, which saves at least 54,2% of the calculation time for oxygen units and at least 47,1% of the calculation time for hydrogen units with a slight loss of accuracy. At a gas flow velocity of more than 0,05 m/s, the model can ignore the diffusion in the gas. The research results can be used to calculate various PSA processes for separation of gas mixtures: rPSA, ultra rPSA, VSA, VPSA, and related processes.
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Numerical study of the SUSUPLUME air pollution model
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In this paper, we propose a SUSUPLUME air pollution as a modern application of the classical Gaussian plume model. The presented model takes into account meteorological conditions and parameters of the pollution sources. The classical model is supplemented by the equations of motion of the center of mass of a single emission. A numerical study has shown that in stationary weather conditions the presented model qualitatively coincides with other known models. The results of calculating the concentrations of pollutants do not contradict the obtained values based on the official methodology for calculating the maximum concentrations of pollutants approved for usege in the territory of the Russian Federation. The SUSUPLUME model contains a number of identifiable parameters and it can be adapted to real conditions. The computational model consists of two blocks: a block for recording measurement information and a block for calculating the concentrations of pollutants. The measurement information registration unit has a low labor intensity (over a million registrations per second). The pollutant concentrations calculation block is laborious (400 points of calculations per second). Concentrations are calculated independently, it allows to use parallelization of the computational process in the future.
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