The article considers the issue of studying the deviation of geometric parameters of a curved surface and a facet surface approximating it. The approximating surface was constructed by triangulation. To obtain adequate results when approximating a surface, it is necessary that the geometric parameters of the approximated surface differ minimally from the parameters of the faceted surface. One of the most important geometric parameters of a surface is its normal at a given point. The deviation of the normals of the given and approximating surfaces significantly affects the calculation error.

Keywords: curved surface, faceted surface, normal to the surface, linear frame, tinted image, plane compartment

A numerical simulation was used to investigate heat transfer in plate-fin radiators with round fillet profiles of various depths, including 0.55 mm, 1.1 mm, and 1.5 mm. The issue of flowing air around a radiator with a mass flow rate of 10-3 to 4·10-3 кг/с and a temperature of 293 K was solved. The radiator was heated using a heater, whose temperature was set from 323 to 353 K. Changes in heat flow, pressure drop, and energy efficiency indicator were shown depending on the air mass flow, according to the calculation results. The research findings indicate that the radiator featuring round fillet profiles and a depth of 1.65 mm exhibits the highest heat flow and energy efficiency indicators, as well as the lowest pressure drop.

Keywords: radiator, cooling system, numerical modeling, computational fluid dynamics, heat transfer, heat flow, pressure drop, energy efficiency, calculation, electronics

The paper describes the process of modeling 3D structures of synthetic gas burners, with the analysis of the simulation results obtained for several types of models and the selection of optimal modes for them. The main use of the burner is planned for boiler equipment for household needs in private as well as apartment buildings, in the absence of central heating. In addition, the article provides a justification for the thermal parameters at the inlet and outlet of the burner.

Keywords: synthesis gas, combustion, design parameters, combustion chamber, nozzle, modeling

The analysis of N.N. Pavlovsky's method for calculating the parameters of the riverbed in the branches is given. The reasons for the limitation of this method have been established. The Lagrange transform is used to find the extremum of the objective function for equations of uneven fluid motion, which makes obtaining an equivalent resistance modulus highly visual and easy to use. The results of the experiment are presented.

Keywords: calculation method, multi-node sections of rivers, resistance modulus, N.N.Pavlovsky method

Using numerical simulation, particle deposition in a porous heat exchanger was studied; the mechanism of sediment formation was sinvestigated, and the influence of sediment on the heat flow was assessed. The influence of porous heat exchanger length, dust particle size, and air flow velocity on particle deposition efficiency, particle breakthrough, and heat flux value was studied. Research results have shown that with increasing length of the heat exchanger, the efficiency of dust particle deposition increases, which leads to the formation of sediment. In turn, the sediment reduces the value of the heat flow from the porous heat exchanger. Depending on the length of the heat exchanger and air velocity, the reduction in heat flow is 3.43-7.27%.

Keywords: porous media, heat exchanger, particle deposition, deposition efficiency, particle breakthrough, sediment, heat flow, numerical modeling, computational fluid dynamics, calculation

The purpose of the work is to develop a methodology for calculating the stress-strain state of three-layer structures made of composite material under the action of dynamic loads in robotic systems: semi-natural simulation stands, intended mainly for testing the flight characteristics. The stand consists of movable channels that ensure movement of the tested product in three degrees of freedom. The algorithm for creating a three-layer stand made of composite material is as follows. We assign the characteristics of the filler to the model of stand, create surfaces on the model and assign to the created surfaces the characteristics of a multilayer composite material with the orientation of the base of the layers along the lines of the trajectories of maximum stresses. Calculations of a three-layer stand with eight-layer load-bearing layers, with different orientations of the layers, were carried out to determine the structure of a multilayer composite material of maximum strength and rigidity based on theories of the destruction of multilayer materials stating that the destruction of one layer leads to the destruction of the multilayer material as a whole. The stress-strain state of a three-layer of the stand for semi-natural modeling of maximum rigidity and strength was obtained. The developed methodology is applicable to a wide class of problems of calculating the robotic systems made of three-layer composite material.

Keywords: material characteristics, multilayer composite material, modeling methods, three-layer structures, calculation, analysis

Tests to determine the seismic stability of elements of pipeline systems, in particular pipeline fittings, are included in the list of those necessary to confirm the operability of the element. In this study, a stand was developed for testing to determine the seismic stability of butterfly check valves, which are a type of pipeline fittings. A finite element analysis was carried out, data on maximum deformations, stresses and displacements of structural elements were obtained, and the response of the installation table to seismic impact was calculated. The resulting displacement values of the table structure do not exceed the limit values specified by the technical requirements and do not lead to distortion of the structure or destruction of joints, which makes it possible to move from design to manufacturing and commissioning of the stand.

Keywords: vibration tests, seismic resistance test bench, pipeline, pipeline valves, swing back gate, mathematical modeling, finite element method, strength calculation, simulation modeling, stress-strain state

The effectiveness of the modified Fourier method associated with the use of orthogonal splines is shown when solving the problem of dynamics of a circular membrane with an elliptical hole. A posteriori estimates of the accuracy of the obtained approximate solutions complement the previously proven theoretical convergence of the algorithm and characterize the high accuracy of solutions to the problem of membrane dynamics with a curved boundary. The differences between the approximate solutions of the problem presented in the form of finite series decrease with an increase in the number of grid nodes used in calculations.

Keywords: Fourier method, orthogonal splines, finite series, dynamics of membrane, a posteriori estimation

In this paper, a physics-informed neural network containing natural gradient descent is proposed to solve the boundary value problem of the Poisson equation. Machine learning methods used in solving partial differential equations are an alternative to the finite element method. Traditional numerical methods for solving differential equations are not capable of solving arbitrary problems of mathematical physics with equivalent efficiency, unlike machine learning methods. The loss function of the neural network is responsible for the accuracy of solving initial and boundary value problems of partial differential equations. The more efficiently the loss function is minimized, the more accurate the resulting solution is. The most traditional optimization algorithm is adaptive moment estimation, which is still used in deep learning today. However, this approach does not guarantee achieving a global minimum of the loss function. We propose to use natural gradient descent with the Dirichlet distribution which increase the accuracy of solving the Poisson equation.

Keywords: natural gradient descent, Poisson equation, Fisher matrix, finite element method, neural networks

The aim of this study is mathematical modelling of the southern part of the Siberian craton using the finite element method. The task of the study is to carry out mathematical modelling of the stress-strain state of the visco-elastic-plastic medium of the Yakutsk-Vilyui large eruptive province on the basis of the boundary value problem. Hypothesis of the study: the possibility of using the results of numerical study to determine the zones of mineral dislocations. In the southern part of the craton, on the territory of the Republic of Sakha-Yakutia, there are the richest oil and gas fields, the largest of which is the Ust-Vilyuyskoye field located in the southern part of the craton. Research method: numerical experiment carried out by the method of mathematical modelling. Results achieved: finite element studies were carried out, the possibility of using numerical methods was determined, the stress-strain state of the plate was analysed, and the locations of anomalies of dislocations of craton rocks were determined to identify potential oil and gas bearing fields.

Keywords: craton, mathematical modeling, stress-strain state, geophysics, geotectonics, stretching, igneous province, material models, Hardening Soil model, finite element method, mineral dislocation

Prism spectral devices have a spectrum-variable dispersion and therefore a nonlinear relationship between the wavelength and the position of spectral lines on the focal surface of a spectral monochromator device, which makes it difficult to calibrate such devices in terms of wavelengths and dispersion.Most often, the well-known Hartmat formula is used for graduation. However, the accuracy of its calculation is satisfactory only in a limited spectral range, and therefore the calculation is carried out on overlapping areas no wider than 200 nm with averaging in overlapping zones. Averaging gives a calibration curve (and, accordingly, a dispersion curve) with gaps at the joints. In this paper, we consider the possibility of using a single calibration for the entire spectrum region, providing smooth, continuous calibration curves. The best result is obtained by using the Hartmann formula for the entire spectral range, after which a set of deviations of the calibration points is determined. This set is interpolated by a polynomial of an arbitrarily chosen order n using the least squares method.

Keywords: monochromator, spectrum, calibration, dispersion, Hartmann formula, interpolation, polynomial

This work examines the applicability of a specific force field model – ANI-2x – to the polymer polyphenylene sulfide. The order of the actions taken is given and, as a result, the radial distribution function of sulfur atoms in systems with different temperatures is compared. It was shown that qualitatively the ANI-2x field correctly describes the situation during the transition through the glass transition temperature.

Keywords: poluphenylene sulfide, classic molecular dynamics, machine learning, force field, potential, polymer, computer simulation, model

The article considers an approach to solving the problem of optimizing the speed of aggregating queries to a continuous range of rows of a PostgreSQL database table. A program module based on PostgreSQL Extensions is created, which provides construction of a segment tree for a table and queries to it. Increased query speed by more than 80 times for a table of 100 million records compared to existing solutions.

Keywords: PostgreSQL, segment tree, query, aggregation, optimization, PosgreSQL Extensions, asymptotics, index, build, get, insert

This article analyzes data processing problems for training a neural network. The first stage of model training - feature extraction - is discussed in detail. The article discusses the method of mel-frequency cepstral coefficients. The spectrum of the voice signal was plotted. By multiplying the vectors of the signal spectrum and the window function, we found the signal energy that falls into each of the analysis windows. Next, we calculated the mel-frequency cepstral coefficients. The use of a chalk scale helps in audio analysis tasks and is used in training neural networks when working with speech. The use of mel-cepstral coefficients significantly improved the quality of recognition due to the fact that it made it possible to see the most informative coefficients. These coefficients have already been used as input to the neural network. The method with mel-frequency cepstral coefficients made it possible to reduce the input data for training, increase productivity, and improve recognition clarity.

Keywords: machine learning, data preprocessing, audio analysis, mel-cepstral coefficients, feature extraction, voice signal spectrum, Fourier transform, Hann window, discrete cosine transform, short Fourier transform

This article examines the previously studied linear in factors and non-linear in parameters modular regression model containing unary module operations. Through the use of binary, ternary, ..., l-ary module operations, a generalization of modular regression was proposed for the first time. A special case of generalization is considered - regression with a multiary operation modulus. The problem of accurately estimating such a model using least absolute deviations is reduced to a mixed integer 0-1 linear programming problem. Using data on farm productivity built into the Gretl econometric package, classical linear regression and modular regression with a multivariate operation were built. The quality of approximation of the proposed modular regression turned out to be higher than the quality of the linear model.

Keywords: regression analysis, modular regression, least absolute deviations, multiary operation modulus, mixed integer 0-1 linear programming problem

Using numerical modeling, we performed studies of the influence of the angle of inclination of the plates of the regenerative heat exchanger element on the heating time and pressure drop. The studies were conducted for models of heat exchange elements with lengths of 6 and 20 mm. Depending on the length of the element, the angle of inclination of the plates was: 10°, 20°, 30°, 40° (at L=6 mm) and 3°, 6°, 9°, 12° (at L=20 mm). At the boundary of the calculation area, the air flow velocity and temperature were established, namely 1 and 3 m/s, and 303 and 973 K. The research results demonstrated that increasing the angle of inclination of the plates helped reduce the heating time of the regenerator by 38.56-49.1%, depending on the length of the heat exchange element, the speed and temperature of the air flow.

Keywords: heat recovery, honeycomb heat exchanger, numerical modeling, calculation, heating time, pressure drop, heat exchanger geometry, angle of plate, air flow velocity, air flow temperature

Based on recent developments in the field of parallel computing, in particular at the SYCL abstraction level, the use of optimal parallel computing tools for building applications in the field of computational and applied mathematics is being considered. Examples are given of both simple computational algorithms and computations using mathematical libraries for computational linear algebra.

Keywords: parallel code,heterogeneous enviroment, intel data parallel c++, intel oneapi, sycl, onemkl, fpga accelerator, gpu accelerator

The paper proposes mathematical models that make it possible to describe the electrical conductivity of a nanocomposite based on carbon nanotubes, taking into account the waviness effect and the dispersion index. The model takes into account the contribution of various parameters, such as the concentration of nanotubes, the length, diameter and orientation of the tubes, as well as the electrical properties of the nanocomposite matrix. Using the proposed model, numerical experiments were carried out to evaluate the effects of waviness and dispersion index on the electrical conductivity of the nanocomposite. Comparisons of model data with experimental data are presented, confirming the adequacy and accuracy of the model. The results obtained can be used to optimize the process of creating nanocomposites based on carbon nanoturbines, as well as to increase the efficiency of their use in various fields, including electronics and energy.

Keywords: mathematical modeling, software package, nanocomposites, electrical conductivity, carbon nanotubes, computational experiment

The article presents the development of a software tool for modeling the influence of physical processes occurring in a single-mode optical fiber as a result of bending deformation. Model of bending deformation of single-mode optical fiber is considered. Classical and refined models of deformation and their influence on the optical fiber params are given. The initial data required to implement the software tool is discussed in detail. The development of the modeling program was gradually considered. The specifics of the implemented program obtained during the computational experiment are indicated.

Keywords: mathematical modeling, optical fiber, bending deformation, modeling of deformed fiber behavior, computational experiment, software

Skin cancer is the most common cancer pathology in the human body and one of the leading causes of death in the world. Artificial intelligence technologies can equal and even surpass the visual classification capabilities of a dermatologist. Thus, it is relevant to develop high-precision intelligent systems for auxiliary diagnostics in the field of dermatology to detect skin cancer in the early stages. The work proposes an ensemble intelligent system for analyzing heterogeneous dermatological data based on multimodal neural networks with various convolutional architectures. The accuracy of the weighted average ensemble model based on multimodal systems using convolutional architectures AlexNet, SeNet_154, Inception_v4, Densenet_161, ResNeXt_50 and ResNeXt_101 for 10 diagnostically significant categories was 87.38%.

Keywords: machine learning, artificial intelligence, convolutional neural networks, multimodal neural networks, ensemble neural networks, digital data processing, heterogeneous data, skin cancer, melanoma

Methods of computer formation of the equations of motion of multibody systems with a tree structure and algorithms for their reduction to the normal form of ordinary differential equations are considered. The equations of motion are written using Hamilton's formalism for an extended set of state variables of a mechanical system. The equations are presented in a compact visual form. Recursive formulas for determining all coefficients of equations are written out. Algorithms for reducing these equations to Hamilton equations in generalized coordinates and generalized momenta are presented. An algorithm for solving the obtained equations of motion for multibody systems using the LTDL-elimination is presented. Formulas are written that allow one to calculate the amount of arithmetic operations required to bring the equations of motion to normal form using the considered algorithms. On the basis of these formulas, a comparative analysis of the efficiency of algorithms for rigid bodies systems of various structures and with various types of connections between bodies is carried out. The results of the analysis are presented in the form of diagrams. The diagrams highlight areas in which the advantage of one or another method is manifested, depending on the type of mechanical system.

Keywords: multibody systems, equations of motion, dynamics, canonical momenta, mathematical modeling, computational efficiency

Using numerical simulation, we carried out studies on the effect of the length of a porous heat exchanger on the deposition of dust particles. The heat exchanger models with lengths of 5, 10, 20 and 30 mm were the subject of the studies. At the boundaries of the computational domain, we set the air velocity at 0.1, 1, and 5 m/s and the diameter of dust particles from 10-7 to 10-4 m. Research results have shown that with increasing length of the porous heat exchanger, the efficiency of dust particle deposition increases. This can lead to a decrease in the thermal and hydraulic characteristics of the heat exchanger.

Keywords: porous media, heat exchanger, numerical simulation, calculation, deposition of dust particles, heat exchanger length, air flow velocity, particle diameter, air cooling, microelectronics

The problem of flexible vibrations of a rectangular orthotorque plate clamped along the contour is considered. The general solution of the problem, which satisfies the vibartion equation identically, is constructed on the basis of the superposition method in the form of two Fourier series. Clamped boundary conditions lead to a homogeneous infinite system of linear algebraic equations with respect to unknown coefficients in the general solution. The uniqueness of a bounded non-trivial solution of an infinite system for the natural frequency is proved, the asymptotics of the unknowns are found, and an effective solution algorithm is constructed. Examples of the numerical implementation of the developed algorithm for calculating the natural frequencies and natural modes of the plate vibrations are given.

Keywords: plate, vibrations, natural frequencies, planar forces, superposition method, infinite system of linear equations, asymptotics

The construction and substantiation of a finite element model of the capillary tube running-in process, which was obtained by drawing on a fixed mandrel, are considered. The rolling rolls are cylindrical and absolutely rigid. The condition for performing the work is a comparison of the resulting gap between the pipe and the mandrel and the facet formed during deformation. The solid-state model of the process is described by an adaptive grid. Under these conditions, an informative model was obtained, which was used in parametric analysis.

Keywords: drawing with a mandrel, capillary tube, extraction, rolling, model construction, compression selection, pipe size tolerance

The paper deals with the construction of an algorithm for nonparametric estimation of the transition probability distribution density for the Markov model of the dynamics of the number of university employees. The problem of reconstruction of transition probability distribution density on a retrospective sample of small volume is solved. The result of the solution is the transition probability matrix, the elements of which are random variables with distribution laws obtained from retrospective data. A distinctive feature of the algorithm is the simultaneous consideration of regional and functional constraints on the values of transition probabilities. The convergence of the algorithm has been experimentally confirmed.

Keywords: probability density, nonparametric estimation, Markov model, simulation modeling, university staff movement, publication activity